Mock Version: 2.9 Mock Version: 2.9 Mock Version: 2.9 ENTER ['do_with_status'](['bash', '--login', '-c', '/usr/bin/rpmbuild -bs --target aarch64 --nodeps /builddir/build/SPECS/gap-pkg-digraphs.spec'], chrootPath='/var/lib/mock/f35-build-27301512-3529047/root'env={'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;\\007"', 'PS1': ' \\s-\\v\\$ ', 'LANG': 'C.UTF-8'}shell=Falselogger=timeout=201600uid=1000gid=425user='mockbuild'nspawn_args=[]unshare_net=TrueprintOutput=False) Executing command: ['bash', '--login', '-c', '/usr/bin/rpmbuild -bs --target aarch64 --nodeps /builddir/build/SPECS/gap-pkg-digraphs.spec'] with env {'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;\\007"', 'PS1': ' \\s-\\v\\$ ', 'LANG': 'C.UTF-8'} and shell False Building target platforms: aarch64 Building for target aarch64 setting SOURCE_DATE_EPOCH=1621036800 Wrote: /builddir/build/SRPMS/gap-pkg-digraphs-1.4.1-1.fc35.src.rpm Child return code was: 0 ENTER ['do_with_status'](['bash', '--login', '-c', '/usr/bin/rpmbuild -bb --target aarch64 --nodeps /builddir/build/SPECS/gap-pkg-digraphs.spec'], chrootPath='/var/lib/mock/f35-build-27301512-3529047/root'env={'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;\\007"', 'PS1': ' \\s-\\v\\$ ', 'LANG': 'C.UTF-8'}shell=Falselogger=timeout=201600uid=1000gid=425user='mockbuild'nspawn_args=[]unshare_net=TrueprintOutput=False) Executing command: ['bash', '--login', '-c', '/usr/bin/rpmbuild -bb --target aarch64 --nodeps /builddir/build/SPECS/gap-pkg-digraphs.spec'] with env {'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;\\007"', 'PS1': ' \\s-\\v\\$ ', 'LANG': 'C.UTF-8'} and shell False Building target platforms: aarch64 Building for target aarch64 setting SOURCE_DATE_EPOCH=1621036800 Executing(%prep): /bin/sh -e /var/tmp/rpm-tmp.9dEPlD + umask 022 + cd /builddir/build/BUILD + cd /builddir/build/BUILD + rm -rf digraphs-1.4.1 + /usr/bin/gzip -dc /builddir/build/SOURCES/digraphs-1.4.1.tar.gz + /usr/bin/tar -xof - + STATUS=0 + '[' 0 -ne 0 ']' + cd digraphs-1.4.1 + /usr/bin/chmod -Rf a+rX,u+w,g-w,o-w . + rm -fr extern/edge-addition-planarity-suite-Version_3.0.1.0 + RPM_EC=0 ++ jobs -p + exit 0 Executing(%build): /bin/sh -e /var/tmp/rpm-tmp.OwUwZi + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.1 + export LC_ALL=C.UTF-8 + LC_ALL=C.UTF-8 + CFLAGS='-O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection' + export CFLAGS + CXXFLAGS='-O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection' + export CXXFLAGS + FFLAGS='-O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I/usr/lib64/gfortran/modules' + export FFLAGS + FCFLAGS='-O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I/usr/lib64/gfortran/modules' + export FCFLAGS + LDFLAGS='-Wl,-z,relro -Wl,--as-needed -Wl,-z,now -specs=/usr/lib/rpm/redhat/redhat-hardened-ld ' + export LDFLAGS + LT_SYS_LIBRARY_PATH=/usr/lib64: + export LT_SYS_LIBRARY_PATH + CC=gcc + export CC + CXX=g++ + export CXX + '[' '-flto=auto -ffat-lto-objectsx' '!=' x ']' ++ find . -type f -name configure -print + for file in $(find . -type f -name configure -print) + /usr/bin/sed -r --in-place=.backup 's/^char \(\*f\) \(\) = /__attribute__ ((used)) char (*f) () = /g' ./configure + diff -u ./configure.backup ./configure + mv ./configure.backup ./configure + /usr/bin/sed -r --in-place=.backup 's/^char \(\*f\) \(\);/__attribute__ ((used)) char (*f) ();/g' ./configure + diff -u ./configure.backup ./configure + mv ./configure.backup ./configure + /usr/bin/sed -r --in-place=.backup 's/^char \$2 \(\);/__attribute__ ((used)) char \$2 ();/g' ./configure + diff -u ./configure.backup ./configure --- ./configure.backup 2021-05-14 19:23:40.000000000 +0000 +++ ./configure 2021-05-15 21:17:40.423564393 +0000 @@ -1797,7 +1797,7 @@ #ifdef __cplusplus extern "C" #endif -char $2 (); +__attribute__ ((used)) char $2 (); /* The GNU C library defines this for functions which it implements to always fail with ENOSYS. Some functions are actually named something starting with __ and the normal name is an alias. */ + /usr/bin/sed --in-place=.backup '1{$!N;$!N};$!N;s/int x = 1;\nint y = 0;\nint z;\nint nan;/volatile int x = 1; volatile int y = 0; volatile int z, nan;/;P;D' ./configure + diff -u ./configure.backup ./configure + mv ./configure.backup ./configure + /usr/bin/sed --in-place=.backup 's#^lt_cv_sys_global_symbol_to_cdecl=.*#lt_cv_sys_global_symbol_to_cdecl="sed -n -e '\''s/^T .* \\(.*\\)$/extern int \\1();/p'\'' -e '\''s/^$symcode* .* \\(.*\\)$/extern char \\1;/p'\''"#' ./configure + diff -u ./configure.backup ./configure --- ./configure.backup 2021-05-15 21:17:40.423564393 +0000 +++ ./configure 2021-05-15 21:17:40.463563896 +0000 @@ -7700,7 +7700,7 @@ # Transform an extracted symbol line into a proper C declaration. # Some systems (esp. on ia64) link data and code symbols differently, # so use this general approach. -lt_cv_sys_global_symbol_to_cdecl="sed -n"\ +lt_cv_sys_global_symbol_to_cdecl="sed -n -e 's/^T .* \(.*\)$/extern int \1();/p' -e 's/^$symcode* .* \(.*\)$/extern char \1;/p'" $lt_cdecl_hook\ " -e 's/^T .* \(.*\)$/extern int \1();/p'"\ " -e 's/^$symcode$symcode* .* \(.*\)$/extern char \1;/p'" @@ -20123,7 +20123,7 @@ compiler='`$ECHO "$compiler" | $SED "$delay_single_quote_subst"`' GCC='`$ECHO "$GCC" | $SED "$delay_single_quote_subst"`' lt_cv_sys_global_symbol_pipe='`$ECHO "$lt_cv_sys_global_symbol_pipe" | $SED "$delay_single_quote_subst"`' -lt_cv_sys_global_symbol_to_cdecl='`$ECHO "$lt_cv_sys_global_symbol_to_cdecl" | $SED "$delay_single_quote_subst"`' +lt_cv_sys_global_symbol_to_cdecl="sed -n -e 's/^T .* \(.*\)$/extern int \1();/p' -e 's/^$symcode* .* \(.*\)$/extern char \1;/p'" lt_cv_sys_global_symbol_to_import='`$ECHO "$lt_cv_sys_global_symbol_to_import" | $SED "$delay_single_quote_subst"`' lt_cv_sys_global_symbol_to_c_name_address='`$ECHO "$lt_cv_sys_global_symbol_to_c_name_address" | $SED "$delay_single_quote_subst"`' lt_cv_sys_global_symbol_to_c_name_address_lib_prefix='`$ECHO "$lt_cv_sys_global_symbol_to_c_name_address_lib_prefix" | $SED "$delay_single_quote_subst"`' + '[' 1 = 1 ']' +++ dirname ./configure ++ find . -name config.guess -o -name config.sub + for i in $(find $(dirname ./configure) -name config.guess -o -name config.sub) ++ basename ./cnf/config.guess + '[' -f /usr/lib/rpm/redhat/config.guess ']' + /usr/bin/rm -f ./cnf/config.guess ++ basename ./cnf/config.guess + /usr/bin/cp -fv /usr/lib/rpm/redhat/config.guess ./cnf/config.guess '/usr/lib/rpm/redhat/config.guess' -> './cnf/config.guess' + for i in $(find $(dirname ./configure) -name config.guess -o -name config.sub) ++ basename ./cnf/config.sub + '[' -f /usr/lib/rpm/redhat/config.sub ']' + /usr/bin/rm -f ./cnf/config.sub ++ basename ./cnf/config.sub + /usr/bin/cp -fv /usr/lib/rpm/redhat/config.sub ./cnf/config.sub '/usr/lib/rpm/redhat/config.sub' -> './cnf/config.sub' + '[' 1 = 1 ']' + '[' x '!=' 'x-Wl,-z,now -specs=/usr/lib/rpm/redhat/redhat-hardened-ld' ']' ++ find . -name ltmain.sh + for i in $(find . -name ltmain.sh) + /usr/bin/sed -i.backup -e 's~compiler_flags=$~compiler_flags="-Wl,-z,now -specs=/usr/lib/rpm/redhat/redhat-hardened-ld"~' ./cnf/ltmain.sh + ./configure --build=aarch64-redhat-linux-gnu --host=aarch64-redhat-linux-gnu --program-prefix= --disable-dependency-tracking --prefix=/usr --exec-prefix=/usr --bindir=/usr/bin --sbindir=/usr/sbin --sysconfdir=/etc --datadir=/usr/share --includedir=/usr/include --libdir=/usr/lib64 --libexecdir=/usr/libexec --localstatedir=/var --sharedstatedir=/var/lib --mandir=/usr/share/man --infodir=/usr/share/info --with-gaproot=/usr/lib/gap --disable-silent-rules --with-external-planarity --without-intrinsics checking build system type... aarch64-redhat-linux-gnu checking host system type... aarch64-redhat-linux-gnu checking for a BSD-compatible install... /usr/bin/install -c checking whether build environment is sane... yes checking for a race-free mkdir -p... /usr/bin/mkdir -p checking for gawk... gawk checking whether make sets $(MAKE)... yes checking whether make supports nested variables... yes checking whether make supports nested variables... (cached) yes checking whether make supports the include directive... yes (GNU style) checking for aarch64-redhat-linux-gnu-gcc... gcc checking whether the C compiler works... yes checking for C compiler default output file name... a.out checking for suffix of executables... checking whether we are cross compiling... no checking for suffix of object files... o checking whether the compiler supports GNU C... yes checking whether gcc accepts -g... yes checking for gcc option to enable C11 features... none needed checking whether gcc understands -c and -o together... yes checking dependency style of gcc... none checking for aarch64-redhat-linux-gnu-ar... no checking for aarch64-redhat-linux-gnu-lib... no checking for aarch64-redhat-linux-gnu-link... no checking for ar... ar checking the archiver (ar) interface... ar checking whether the compiler supports GNU C++... yes checking whether g++ accepts -g... yes checking for g++ option to enable C++11 features... none needed checking dependency style of g++... none checking how to print strings... printf checking for a sed that does not truncate output... /usr/bin/sed checking for grep that handles long lines and -e... /usr/bin/grep checking for egrep... /usr/bin/grep -E checking for fgrep... /usr/bin/grep -F checking for ld used by gcc... /usr/bin/ld checking if the linker (/usr/bin/ld) is GNU ld... yes checking for BSD- or MS-compatible name lister (nm)... /usr/bin/nm -B checking the name lister (/usr/bin/nm -B) interface... BSD nm checking whether ln -s works... yes checking the maximum length of command line arguments... 1572864 checking how to convert aarch64-redhat-linux-gnu file names to aarch64-redhat-linux-gnu format... func_convert_file_noop checking how to convert aarch64-redhat-linux-gnu file names to toolchain format... func_convert_file_noop checking for /usr/bin/ld option to reload object files... -r checking for aarch64-redhat-linux-gnu-objdump... no checking for objdump... objdump checking how to recognize dependent libraries... pass_all checking for aarch64-redhat-linux-gnu-dlltool... no checking for dlltool... no checking how to associate runtime and link libraries... printf %s\n checking for aarch64-redhat-linux-gnu-ar... ar checking for archiver @FILE support... @ checking for aarch64-redhat-linux-gnu-strip... no checking for strip... strip checking for aarch64-redhat-linux-gnu-ranlib... no checking for ranlib... ranlib checking command to parse /usr/bin/nm -B output from gcc object... ./configure: line 7706: -e 's/^T .* \(.*\)$/extern int \1();/p' -e 's/^[ABCDGIRSTW][ABCDGIRSTW]* .* \(.*\)$/extern char \1;/p': No such file or directory ok checking for sysroot... no checking for a working dd... /usr/bin/dd checking how to truncate binary pipes... /usr/bin/dd bs=4096 count=1 checking for aarch64-redhat-linux-gnu-mt... no checking for mt... no checking if : is a manifest tool... no checking for stdio.h... yes checking for stdlib.h... yes checking for string.h... yes checking for inttypes.h... yes checking for stdint.h... yes checking for strings.h... yes checking for sys/stat.h... yes checking for sys/types.h... yes checking for unistd.h... yes checking for dlfcn.h... yes checking for objdir... .libs checking if gcc supports -fno-rtti -fno-exceptions... no checking for gcc option to produce PIC... -fPIC -DPIC checking if gcc PIC flag -fPIC -DPIC works... yes checking if gcc static flag -static works... no checking if gcc supports -c -o file.o... yes checking if gcc supports -c -o file.o... (cached) yes checking whether the gcc linker (/usr/bin/ld) supports shared libraries... yes checking whether -lc should be explicitly linked in... no checking dynamic linker characteristics... GNU/Linux ld.so checking how to hardcode library paths into programs... immediate checking for shl_load... no checking for shl_load in -ldld... no checking for dlopen... no checking for dlopen in -ldl... yes checking whether a program can dlopen itself... yes checking whether a statically linked program can dlopen itself... yes checking whether stripping libraries is possible... yes checking if libtool supports shared libraries... yes checking whether to build shared libraries... yes checking whether to build static libraries... no checking how to run the C++ preprocessor... g++ -E checking for ld used by g++... /usr/bin/ld checking if the linker (/usr/bin/ld) is GNU ld... yes checking whether the g++ linker (/usr/bin/ld) supports shared libraries... yes checking for g++ option to produce PIC... -fPIC -DPIC checking if g++ PIC flag -fPIC -DPIC works... yes checking if g++ static flag -static works... no checking if g++ supports -c -o file.o... yes checking if g++ supports -c -o file.o... (cached) yes checking whether the g++ linker (/usr/bin/ld) supports shared libraries... yes checking dynamic linker characteristics... (cached) GNU/Linux ld.so checking how to hardcode library paths into programs... immediate checking for GAP root directory... /usr/lib/gap checking for GAP architecture... checking whether to enable debug mode... no checking whether to enable stats mode... no checking whether to build with code coverage support... no checking whether enable compiler warnings... no checking whether to use external planarity... yes checking for gp_InitGraph in -lplanarity... yes checking for planarity/graph.h... yes checking whether to use external bliss... no configure: compiler intrinsics will not be used even if available checking that generated files are newer than configure... done configure: creating ./config.status config.status: creating Makefile config.status: creating src/_pkgconfig.h config.status: executing depfiles commands config.status: executing src/digraphs-config.h commands config.status: creating src/digraphs-config.h - prefix DIGRAPHS for src/_pkgconfig.h defines config.status: executing libtool commands + /usr/bin/make -O -j5 V=1 VERBOSE=1 make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-uintseqhash.lo `test -f 'extern/bliss-0.73/uintseqhash.cc' || echo './'`extern/bliss-0.73/uintseqhash.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/uintseqhash.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-uintseqhash.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-defs.lo `test -f 'extern/bliss-0.73/defs.cc' || echo './'`extern/bliss-0.73/defs.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/defs.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-defs.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-timer.lo `test -f 'extern/bliss-0.73/timer.cc' || echo './'`extern/bliss-0.73/timer.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/timer.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-timer.o extern/bliss-0.73/timer.cc: In member function ‘bliss_digraphs::Timer::reset()’: extern/bliss-0.73/timer.cc:39:24: warning: ‘clkticks.tms_utime’ is used uninitialized [-Wuninitialized] 39 | ((double) clkticks.tms_utime + (double) clkticks.tms_stime) / | ~~~~~~~~~^~~~~~~~~ extern/bliss-0.73/timer.cc:35:14: note: ‘clkticks’ declared here 35 | struct tms clkticks; | ^~~~~~~~ extern/bliss-0.73/timer.cc:39:54: warning: ‘clkticks.tms_stime’ is used uninitialized [-Wuninitialized] 39 | ((double) clkticks.tms_utime + (double) clkticks.tms_stime) / | ~~~~~~~~~^~~~~~~~~ extern/bliss-0.73/timer.cc:35:14: note: ‘clkticks’ declared here 35 | struct tms clkticks; | ^~~~~~~~ extern/bliss-0.73/timer.cc: In member function ‘bliss_digraphs::Timer::get_duration()’: extern/bliss-0.73/timer.cc:50:24: warning: ‘clkticks.tms_utime’ is used uninitialized [-Wuninitialized] 50 | ((double) clkticks.tms_utime + (double) clkticks.tms_stime) / | ~~~~~~~~~^~~~~~~~~ extern/bliss-0.73/timer.cc:46:14: note: ‘clkticks’ declared here 46 | struct tms clkticks; | ^~~~~~~~ extern/bliss-0.73/timer.cc:50:54: warning: ‘clkticks.tms_stime’ is used uninitialized [-Wuninitialized] 50 | ((double) clkticks.tms_utime + (double) clkticks.tms_stime) / | ~~~~~~~~~^~~~~~~~~ extern/bliss-0.73/timer.cc:46:14: note: ‘clkticks’ declared here 46 | struct tms clkticks; | ^~~~~~~~ make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-heap.lo `test -f 'extern/bliss-0.73/heap.cc' || echo './'`extern/bliss-0.73/heap.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/heap.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-heap.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-orbit.lo `test -f 'extern/bliss-0.73/orbit.cc' || echo './'`extern/bliss-0.73/orbit.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/orbit.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-orbit.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-utils.lo `test -f 'extern/bliss-0.73/utils.cc' || echo './'`extern/bliss-0.73/utils.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/utils.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-utils.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-bitarray.lo `test -f 'src/bitarray.c' || echo './'`src/bitarray.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/bitarray.c -fPIC -DPIC -o src/.libs/digraphs_la-bitarray.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-conditions.lo `test -f 'src/conditions.c' || echo './'`src/conditions.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/conditions.c -fPIC -DPIC -o src/.libs/digraphs_la-conditions.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-bliss_C.lo `test -f 'extern/bliss-0.73/bliss_C.cc' || echo './'`extern/bliss-0.73/bliss_C.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/bliss_C.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-bliss_C.o extern/bliss-0.73/bliss_C.cc: In function ‘void fill_size(BlissStats*, const bliss_digraphs::Stats&)’: extern/bliss-0.73/bliss_C.cc:156:21: warning: comparison of integer expressions of different signedness: ‘int’ and ‘std::vector::size_type’ {aka ‘long unsigned int’} [-Wsign-compare] 156 | for(int i = 0; i < sizes.size(); ++i) { | ~~^~~~~~~~~~~~~~ make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-partition.lo `test -f 'extern/bliss-0.73/partition.cc' || echo './'`extern/bliss-0.73/partition.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/partition.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-partition.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-homos-graphs.lo `test -f 'src/homos-graphs.c' || echo './'`src/homos-graphs.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/homos-graphs.c -fPIC -DPIC -o src/.libs/digraphs_la-homos-graphs.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-cliques.lo `test -f 'src/cliques.c' || echo './'`src/cliques.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/cliques.c -fPIC -DPIC -o src/.libs/digraphs_la-cliques.o src/cliques.c: In function ‘init_graph_from_digraph_obj’: src/cliques.c:175:14: warning: unused variable ‘out’ [-Wunused-variable] 175 | Obj out = FuncOutNeighbours(0L, digraph_obj); | ^~~ make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-digraphs.lo `test -f 'src/digraphs.c' || echo './'`src/digraphs.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/digraphs.c -fPIC -DPIC -o src/.libs/digraphs_la-digraphs.o src/digraphs.c: In function ‘FLOYD_WARSHALL.constprop.0’: src/digraphs.c:1216:13: warning: ‘adj’ may be used uninitialized in this function [-Wmaybe-uninitialized] 1216 | free(adj); | ^~~~~~~~~ make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-perms.lo `test -f 'src/perms.c' || echo './'`src/perms.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/perms.c -fPIC -DPIC -o src/.libs/digraphs_la-perms.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-planar.lo `test -f 'src/planar.c' || echo './'`src/planar.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/planar.c -fPIC -DPIC -o src/.libs/digraphs_la-planar.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-schreier-sims.lo `test -f 'src/schreier-sims.c' || echo './'`src/schreier-sims.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/schreier-sims.c -fPIC -DPIC -o src/.libs/digraphs_la-schreier-sims.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CC --mode=compile gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o src/digraphs_la-homos.lo `test -f 'src/homos.c' || echo './'`src/homos.c libtool: compile: gcc -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c src/homos.c -fPIC -DPIC -o src/.libs/digraphs_la-homos.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=compile g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS="/usr/lib/gap" -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c -o extern/bliss-0.73/digraphs_la-graph.lo `test -f 'extern/bliss-0.73/graph.cc' || echo './'`extern/bliss-0.73/graph.cc libtool: compile: g++ -DHAVE_CONFIG_H -I. -I./src -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -I/usr/lib/gap/src -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -c extern/bliss-0.73/graph.cc -fPIC -DPIC -o extern/bliss-0.73/.libs/digraphs_la-graph.o make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /bin/sh ./libtool --tag=CXX --mode=link g++ -I./extern/ -O2 -flto=auto -ffat-lto-objects -fexceptions -g -grecord-gcc-switches -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -specs=/usr/lib/rpm/redhat/redhat-hardened-cc1 -fstack-protector-strong -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -mbranch-protection=standard -fasynchronous-unwind-tables -fstack-clash-protection -Wl,-z,relro -Wl,--as-needed -module -avoid-version -Wl,-z,relro -Wl,--as-needed -Wl,-z,now -specs=/usr/lib/rpm/redhat/redhat-hardened-ld -o digraphs.la -rpath /usr/lib64 src/digraphs_la-digraphs.lo src/digraphs_la-bitarray.lo src/digraphs_la-conditions.lo src/digraphs_la-homos.lo src/digraphs_la-cliques.lo src/digraphs_la-homos-graphs.lo src/digraphs_la-perms.lo src/digraphs_la-planar.lo src/digraphs_la-schreier-sims.lo extern/bliss-0.73/digraphs_la-defs.lo extern/bliss-0.73/digraphs_la-graph.lo extern/bliss-0.73/digraphs_la-partition.lo extern/bliss-0.73/digraphs_la-orbit.lo extern/bliss-0.73/digraphs_la-uintseqhash.lo extern/bliss-0.73/digraphs_la-heap.lo extern/bliss-0.73/digraphs_la-timer.lo extern/bliss-0.73/digraphs_la-utils.lo extern/bliss-0.73/digraphs_la-bliss_C.lo -lplanarity libtool: link: g++ -fPIC -DPIC -shared -nostdlib /usr/lib/gcc/aarch64-redhat-linux/11/../../../../lib64/crti.o /usr/lib/gcc/aarch64-redhat-linux/11/crtbeginS.o src/.libs/digraphs_la-digraphs.o src/.libs/digraphs_la-bitarray.o src/.libs/digraphs_la-conditions.o src/.libs/digraphs_la-homos.o src/.libs/digraphs_la-cliques.o src/.libs/digraphs_la-homos-graphs.o src/.libs/digraphs_la-perms.o src/.libs/digraphs_la-planar.o src/.libs/digraphs_la-schreier-sims.o extern/bliss-0.73/.libs/digraphs_la-defs.o extern/bliss-0.73/.libs/digraphs_la-graph.o extern/bliss-0.73/.libs/digraphs_la-partition.o extern/bliss-0.73/.libs/digraphs_la-orbit.o extern/bliss-0.73/.libs/digraphs_la-uintseqhash.o extern/bliss-0.73/.libs/digraphs_la-heap.o extern/bliss-0.73/.libs/digraphs_la-timer.o extern/bliss-0.73/.libs/digraphs_la-utils.o extern/bliss-0.73/.libs/digraphs_la-bliss_C.o -lplanarity -L/usr/lib/gcc/aarch64-redhat-linux/11 -L/usr/lib/gcc/aarch64-redhat-linux/11/../../../../lib64 -L/lib/../lib64 -L/usr/lib/../lib64 -L/usr/lib/gcc/aarch64-redhat-linux/11/../../.. -lstdc++ -lm -lc -lgcc_s /usr/lib/gcc/aarch64-redhat-linux/11/crtendS.o /usr/lib/gcc/aarch64-redhat-linux/11/../../../../lib64/crtn.o -Wl,-z,now -specs=/usr/lib/rpm/redhat/redhat-hardened-ld -O2 -flto=auto -g -grecord-gcc-switches -fstack-protector-strong -mbranch-protection=standard -Wl,-z -Wl,relro -Wl,--as-needed -Wl,-z -Wl,relro -Wl,--as-needed -Wl,-z -Wl,now -Wl,-soname -Wl,digraphs.so -o .libs/digraphs.so libtool: link: ( cd ".libs" && rm -f "digraphs.la" && ln -s "../digraphs.la" "digraphs.la" ) make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' make[1]: Entering directory '/builddir/build/BUILD/digraphs-1.4.1' /usr/bin/mkdir -p ./bin/aarch64-redhat-linux-gnu-default64-kv7 ./bin/lib cp .libs/digraphs.so /builddir/build/BUILD/digraphs-1.4.1/bin/aarch64-redhat-linux-gnu-default64-kv7/ SUCCESS! make[1]: Leaving directory '/builddir/build/BUILD/digraphs-1.4.1' + mkdir ../pkg + ln -s ../digraphs-1.4.1 ../pkg + gap -l '/builddir/build/BUILD/digraphs-1.4.1/..;/usr/lib/gap' ┌───────┐ GAP 4.11.1 of 2021-03-02 │ GAP │ https://www.gap-system.org └───────┘ Architecture: aarch64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.1, PrimGrp 3.4.1, SmallGrp 1.4.2, TransGrp 3.1 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> gap> gap> gap> gap> gap> gap> gap> gap> "Digraphs" gap> gap> true gap> gap> > > > > > > > > > > > > > > > > [ "attr.xml", "cliques.xml", "constructors.xml", "digraph.xml", "display.xml", "examples.xml", "grahom.xml", "grape.xml", "io.xml", "isomorph.xml", "labels.xml", "oper.xml", "orbits.xml", "planar.xml", "prop.xml", "utils.xml", "../PackageInfo.g" ] gap> gap> > > > #I Composing XML document . . . #I Parsing XML document . . . #I Checking XML structure . . . #I Text version (also produces labels for hyperlinks): #I First run, collecting cross references, index, toc, bib and so on . . . #I Table of contents complete. #I Producing the index . . . #I Reading bibliography data files . . . #I BibTeX format: digraphs #I checking and translating to BibXMLext . . . #I Writing bibliography . . . #I Second run through document . . . #I Producing simplified search strings and labels for hyperlinks . . . #I Constructing LaTeX version and calling pdflatex: #I Writing LaTeX file, 4 x pdflatex with bibtex and makeindex, #I Writing manual.six file ... #I Finally the HTML version . . . #I First run, collecting cross references, index, toc, bib and so on . . . #I Table of contents complete. #I Producing the index . . . #I Writing bibliography . . . #I Second run through document . . . #I - also HTML version for MathJax . . . #I First run, collecting cross references, index, toc, bib and so on . . . #I Table of contents complete. #I Producing the index . . . #I Writing bibliography . . . #I Second run through document . . . rec( Name := "digraphs", attributes := rec( ), bibdata := "digraphs", bibentries := [ , , , , , , , , ], bibkeys := [ "BM04", "BM06", "B06", "B12", "CK86", "Gab00", "JK07", "MP14", "vLS81" ], biblabels := [ "BM04", "BM06", "Boy06", "Boy12", "CK86", "Gab00", "JK07", "MP14", "vLS81" ], bibpath := dir("/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/d\ oc/"), bibstrings := [ [ 0, 0 ] ], bibtext := "\n

\n

\n[BM04] Boyer, J. M.\ and Myrvold, W. J.,\n On the Cutting Edge: Simplified\ O(n) Planarity by Edge\n Addition,\n Journa\ l of Graph Algorithms and Applications,\n 8 \ (3)\n (2004),\n 241--273.\n

\n\n\n

\n

\n[BM06] \ Boyer, J. M. and Myrvold, W. J.,\n On the Cutting Edge: Simplified O(n) Planarity by Edge\n Addition,\ \n in Graph Algorithms and Applications 5,\n {WORLD} {SCIENTIFIC}\n (2\ 006),\n 241--273.\n

\n\n\n

\n

\n[\ Boy06] Boyer, J. M. (Healy, P. and Nikolov, N. S., Eds.),\n A New Metho\ d for Efficiently Generating Planar Graph Visibility\n Representations,\n in Graph Drawing,\n Springer Berlin Heidelberg,\n Berlin, Heid\ elberg\n (2006),\n \ 508--511.\n

\n\n\n

\n

\n[Boy12] B\ oyer, J. M.,\n Subgraph Homeomorphism via the Edge Add\ ition Planarity\n Algorithm,\n Journal of Gr\ aph Algorithms and Applications,\n 16 (2)\n (2012),\n 381--410.\n

\n\n\n

\n

\n[CK86] Calderbank, R. and Kantor, W. M.,\n The geometry \ of two-weight codes,\n Bull. London Math. Soc.,\n 18 (2)\n (1986),\n 97--122.\n\n\n\n

\n

\ \n[Gab00] Gabow, H. N.,\ \n Path-based depth-first search for strong and biconnecte\ d\n components,\n Information Processin\ g Letters,\n 74 (34<\ /span>)\n (2000),\n 107 - \ 114.\n

\n\n\n

\n

\n[JK07] Juntt\ ila, T. and Kaski, P. (Applegate, D., Brodal, G. S\ ., Panario, D. and Sedgewick, R., Eds.),\n Engineer\ ing an efficient canonical labeling tool for large and\n sparse grap\ hs,\n in Proceedings of the Ninth Workshop on Al\ gorithm Engineering and\n Experiments and the Fourth Workshop on Analytic Alg\ orithms and\n Combinatorics,\n SI\ AM\n (2007),\n 135-\ -149.\n

\n\n\n

\n

\n[MP14] McKa\ y, B. D. and Piperno, A.,\n Practical graph isomorphis\ m, II,\n Journal of Symbolic Computation,\ \n 60 (0)\n (2014),\n 94 - 112.\n

\n\ \n\n

\n

\n[<\ span class='BibKeyLink'>vLS81] van Lint, J. H. and Schrijver, \ A.,\n Construction of strongly regular graphs, two-wei\ ght codes and\n partial geometries by finite fields,\n Combinatorica,\n 1 (1)\n (1981),\n 63--73.\n

\n\n", bookname := "digraphs", chapsectlinks := rec( 1 := "\n", 2 := "
2 Installing Digraphs\n
 2.1 For those in a hurry\n\n
\n
  2.1-1 Config\ uration options\n
\n
\n\n\n
 <\ a href=\"chap2_mj.html#X857CBE5484CF703A\">2.4 Rebuild\ ing the documentation\n\n
\n\ \n\
\n", 3 := "\n
 3.3 \ New digraphs from old\n\n
\n
 &nb\ sp;3.3-1 DigraphImmutableCo\ py
\n
  \ 3.3-2 DigraphImmutableCopyI\ fImmutable
\n
&nbs\ p; 3.3-3 InducedSubdig\ raph
\n
 &nbs\ p;3.3-4 ReducedDigraph<\ /span>\n
  \ 3.3-5 MaximalSymmetricSubdigraph
\n
  
3.3-6 MaximalAntiSymmetricSubdi\ graph
\n
 &nb\ sp;3.3-7 UndirectedSpanning\ Forest
\n
 &n\ bsp;3.3-8 QuotientDigraph
\n
  
3.3-9 DigraphReverse\ \n
  3.3-10 DigraphDual
\n
  3.3-11 DigraphSymmetricClosure
\n
  3.3-12 DigraphTransitiveClosure
\n
  3.3-13 DigraphTransitiveReduction
\n
  3.3-14 DigraphAddVertex\n
  3.3-15 DigraphAddVertices
\n
  3.3-16 DigraphAddEdge
\n
  3.3-17 DigraphAddEdgeOrbit
\n\
  3.3-18 DigraphAddEdges
\n
<\ span class=\"nocss\">  
3.3-19 DigraphRemoveVertex\n
  
3.3-20 DigraphRemoveVertices\n
  
3.3-21 DigraphRemoveEdge\n
  3.\ 3-22 DigraphRemoveEdgeOrbit
\n
  3.\ 3-23 DigraphRemoveEdges
\n
  3.3-2\ 4 DigraphRemoveLoops
\n
  3.3-25 Di\ graphRemoveAllMultipleEdges
\n
  3.\ 3-26 DigraphReverseEdges
\n
  3.3-2\ 7 DigraphDisjointUnion
\n
  3.3-28 \ DigraphEdgeUnion
\n
  3.3-29 Digra\ phJoin
\n
 &n\ bsp;3.3-30 DigraphCartesian\ Product
\n
 &\ nbsp;3.3-31 DigraphDirectPr\ oduct
\n
 &nb\ sp;3.3-32 DigraphCartesianP\ roductProjections
\n
  3.3-33 Digra\ phDirectProductProjections
\n
  3.3\ -34 LineDigraph
\n
  3.3-35 LineUnd\ irectedDigraph
\n
\   3.3-36 DoubleDi\ graph
\n
 &nb\ sp;3.3-37 BipartiteDoubleDi\ graph
\n
 &nb\ sp;3.3-38 DigraphAddAllLoop\ s
\n
  <\ /span>3.3-39 DistanceDigraph\n
  <\ a href=\"chap3_mj.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure
\n<\ span class=\"ContSS\">
  3.3-41 DigraphMycielskian
\n
\n\n\n", 4 := "\n", 5 := "\n\n\n\n\n
\n", 6 := "
6 Properties of digraphs\n
 6.1 Edge propertie\ s\n\n\n\n\n\n", 7 := "
7 Homomorphisms\n\n\n\n
\n", 8 := "\n", 9 := "\n", A := "
A \n Grape to Digraphs Command Map\n \n\n\n\n\n\n\n\n\n" ), chapsectlinkstext := rec( 1 := "\n", 2 := "
2 Installing Digraphs\n
 2.1 For those in a hurry\n\n
\n
  2.1-1 Config\ uration options\n
\n
\n\n\n
 <\ a href=\"chap2_mj.html#X857CBE5484CF703A\">2.4 Rebuild\ ing the documentation\n\n
\n\ \n\
\n", 3 := "\n
 3.3 \ New digraphs from old\n\n
\n
 &nb\ sp;3.3-1 DigraphImmutableCo\ py
\n
  \ 3.3-2 DigraphImmutableCopyI\ fImmutable
\n
&nbs\ p; 3.3-3 InducedSubdig\ raph
\n
 &nbs\ p;3.3-4 ReducedDigraph<\ /span>\n
  \ 3.3-5 MaximalSymmetricSubdigraph
\n
  
3.3-6 MaximalAntiSymmetricSubdi\ graph
\n
 &nb\ sp;3.3-7 UndirectedSpanning\ Forest
\n
 &n\ bsp;3.3-8 QuotientDigraph
\n
  
3.3-9 DigraphReverse\ \n
  3.3-10 DigraphDual
\n
  3.3-11 DigraphSymmetricClosure
\n
  3.3-12 DigraphTransitiveClosure
\n
  3.3-13 DigraphTransitiveReduction
\n
  3.3-14 DigraphAddVertex\n
  3.3-15 DigraphAddVertices
\n
  3.3-16 DigraphAddEdge
\n
  3.3-17 DigraphAddEdgeOrbit
\n\
  3.3-18 DigraphAddEdges
\n
<\ span class=\"nocss\">  
3.3-19 DigraphRemoveVertex\n
  
3.3-20 DigraphRemoveVertices\n
  
3.3-21 DigraphRemoveEdge\n
  3.\ 3-22 DigraphRemoveEdgeOrbit
\n
  3.\ 3-23 DigraphRemoveEdges
\n
  3.3-2\ 4 DigraphRemoveLoops
\n
  3.3-25 Di\ graphRemoveAllMultipleEdges
\n
  3.\ 3-26 DigraphReverseEdges
\n
  3.3-2\ 7 DigraphDisjointUnion
\n
  3.3-28 \ DigraphEdgeUnion
\n
  3.3-29 Digra\ phJoin
\n
 &n\ bsp;3.3-30 DigraphCartesian\ Product
\n
 &\ nbsp;3.3-31 DigraphDirectPr\ oduct
\n
 &nb\ sp;3.3-32 DigraphCartesianP\ roductProjections
\n
  3.3-33 Digra\ phDirectProductProjections
\n
  3.3\ -34 LineDigraph
\n
  3.3-35 LineUnd\ irectedDigraph
\n
\   3.3-36 DoubleDi\ graph
\n
 &nb\ sp;3.3-37 BipartiteDoubleDi\ graph
\n
 &nb\ sp;3.3-38 DigraphAddAllLoop\ s
\n
  <\ /span>3.3-39 DistanceDigraph\n
  <\ a href=\"chap3_mj.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure
\n<\ span class=\"ContSS\">
  3.3-41 DigraphMycielskian
\n
\n\n\n", 4 := "\n", 5 := "\n\n\n\n\n
\n", 6 := "
6 Properties of digraphs\n
 6.1 Edge propertie\ s\n\n\n\n\n\n", 7 := "
7 Homomorphisms\n\n\n\n
\n", 8 := "\n", 9 := "\n", A := "
A \n Grape to Digraphs Command Map\n \n\n\n\n\n\n\n\n\n" ), chaptitle := rec( 1 := "\n The Digraphs package\n ", 2 := "Installing Digraphs<\ /strong>", 3 := "Creating digraphs", 4 := "Operators", 5 := "Attributes and operations", 6 := "Properties of digraphs", 7 := "Homomorphisms", 8 := "Finding cliques and independent sets", 9 := "Visualising and IO", A := "\n Grape to Digraphs Command Map\n " ), content := [ rec( content := "Book SYSTEM \"gapdoc.dtd\"\n [ Digraphs\">\n https://gap-packages.github.io/grapeGrape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suiteedge-addition-planarity-suite\">\n https://p\ allini.di.uniroma1.it/nauty\">\n https://github.com/gap-packages/NautyTracesInterfaceNautyTracesInterface\"\ >\n \n https://gap-packag\ es.github.io/io/\nIO\">\ \n \n https://gap-packa\ ges.github.io/orb/\n Orb\">\n https://github.com/gap-packages/datastructuresdatastructures\">\n \n\n\ \n\n\n\n\n\n\n\ \n ]", count := [ 0, 0, 0, 1 ], name := "XMLDOCTYPE", root := ~ ), rec( attributes := rec( Name := "digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ) ], count := [ 0, 0, 0, 1 ], name := "Title", next := 2013, root := ~, start := 1988, stop := 2012 ), rec( attributes := rec( ), content := [ rec( content := "Version 1.4.1", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Version", next := 2052, root := ~, start := 2016, stop := 2051 ), rec( attributes := rec( ), content := [ rec( content := "Jan De Beule\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "jdebeule@cage.ugent.be", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2118, root := ~, start := 2081, stop := 2117 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://homepages.vub.ac.b\ e/~jdbeule", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2179, root := ~, start := 2123, stop := 2178 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2191, root := ~, start := 2056, stop := 2190 ), rec( attributes := rec( ), content := [ rec( content := "Julius Jonušas\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "julius.jonusas@tuwien.ac.\ at", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2269, root := ~, start := 2227, stop := 2268 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://julius.jonusas.wor\ k", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2321, root := ~, start := 2274, stop := 2320 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2333, root := ~, start := 2195, stop := 2332 ), rec( attributes := rec( ), content := [ rec( content := "James D. Mitchell\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "jdm3@st-and.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2399, root := ~, start := 2367, stop := 2398 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://jdbm.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2440, root := ~, start := 2404, stop := 2439 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2452, root := ~, start := 2337, stop := 2451 ), rec( attributes := rec( ), content := [ rec( content := "Michael C. Torpey\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "mct25@st-andrews.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2523, root := ~, start := 2486, stop := 2522 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://mtorpey.github.io\ ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2574, root := ~, start := 2528, stop := 2573 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2586, root := ~, start := 2456, stop := 2585 ), rec( attributes := rec( ), content := [ rec( content := "Maria Tsalakou\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "mt200@st-andrews.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2654, root := ~, start := 2617, stop := 2653 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://mariatsalakou.git\ hub.io", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2711, root := ~, start := 2659, stop := 2710 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2723, root := ~, start := 2590, stop := 2722 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. Wilson\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "gap@wilf-wilson.net", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2788, root := ~, start := 2754, stop := 2787 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://wilf.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2829, root := ~, start := 2793, stop := 2828 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2841, root := ~, start := 2727, stop := 2840 ), rec( attributes := rec( ), content := [ rec( content := "Stuart Burrell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2876, root := ~, start := 2845, stop := 2875 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Cirpons", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2911, root := ~, start := 2880, stop := 2910 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elliott", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2944, root := ~, start := 2915, stop := 2943 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2973, root := ~, start := 2948, stop := 2972 ), rec( attributes := rec( ), content := [ rec( content := "Christopher Jefferson", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3015, root := ~, start := 2977, stop := 3014 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pfeiffer", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3051, root := ~, start := 3019, stop := 3050 ), rec( attributes := rec( ), content := [ rec( content := "Christopher Russell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3091, root := ~, start := 3055, stop := 3090 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smith", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3122, root := ~, start := 3095, stop := 3121 ), rec( attributes := rec( ), content := [ rec( content := "Murray White", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3155, root := ~, start := 3126, stop := 3154 ), rec( attributes := rec( ), content := [ rec( content := "© 2014-21 by\n Jan De Beule, \ Julius Jonušas, James D. Mitchell, Michael\n Torpey, Wilf A. Wilson et al.\ ", count := [ 0, 0, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "P", next := 3305, root := ~, start := 3301, stop := 3304 ), rec( content := "\n\n ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is free software; you can redis\ tribute it and/or modify it\n under the terms of the ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "GNU General Public License" ), content := [ rec( content := "\n https://www.fsf.org\ /licenses/gpl.html", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "URL", next := 3495, root := ~, start := 3408, stop := 3494 ), rec( content := " as published by the Free Softwa\ re\n Foundation; either version 3 of the License, or (at your option) any l\ ater\n version.\n ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 1 ], name := "Copyright", next := 3636, root := ~, start := 3159, stop := 3635 ), rec( attributes := rec( ), content := [ rec( content := "\n The ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package is a ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " package containing methods for \ graphs,\n digraphs, and multidigraphs.\n ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], name := "Abstract", next := 3774, root := ~, start := 3640, stop := 3773 ), rec( attributes := rec( ), content := [ rec( content := "\n We would like to thank Chri\ stopher Jefferson for his help in including \n ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ) , content := [ rec( content := "http://www.tcs.tk\ k.fi/Software/bliss/", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 0, 0, 3, 1 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 0, 0, 3, 1 ], name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " in ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n\n This package's methods \ for computing digraph homomorphisms are based on work\n by Max Neunhöffer,\ and independently Artur Schäfer.\n ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], name := "Acknowledgements", next := 4069, root := ~, start := 3778, stop := 4068 ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "TitlePage", next := 4083, root := ~, start := 1974, stop := 4082 ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "TableOfContents", next := 4104, root := ~, start := 4086, stop := 4103 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "The Digraphs package" ), content := [ rec( content := "\n ", count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The ", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\n ", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 4205, root := ~, start := 4156, stop := 4204 ), rec( content := "\n ", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package overview", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "Index", next := 4265, root := ~, start := 4208, stop := 4264 ), rec( content := "\n\n", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Introduction", count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 4307, root := ~, start := 4276, stop := 4306 ), rec( content := "\n\n This is the manual\ for version 1.4.1 of the ", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package.\n This packag\ e was developed at the University of St Andrews by:\n ", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Jan De Be\ ule,", count := [ 1, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 5 ], name := "Item", next := 4483, root := ~, start := 4457, stop := 4482 ), rec( attributes := rec( ), content := [ rec( content := "Julius Jo\ nušas,", count := [ 1, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 7 ], name := "Item", next := 4521, root := ~, start := 4488, stop := 4520 ), rec( attributes := rec( ), content := [ rec( content := "James D. \ Mitchell,", count := [ 1, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 9 ], name := "Item", next := 4557, root := ~, start := 4526, stop := 4556 ), rec( attributes := rec( ), content := [ rec( content := "Michael C\ . Torpey,", count := [ 1, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 11 ], name := "Item", next := 4593, root := ~, start := 4562, stop := 4592 ), rec( attributes := rec( ), content := [ rec( content := "Maria Tsa\ lakou, and", count := [ 1, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 13 ], name := "Item", next := 4630, root := ~, start := 4598, stop := 4629 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. W\ ilson.", count := [ 1, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 15 ], name := "Item", next := 4663, root := ~, start := 4635, stop := 4662 ) ], count := [ 1, 1, 0, 4 ], mathmode := "Text", name := "List", next := 4673, root := ~, start := 4446, stop := 4672 ), rec( content := "\n Additional contribut\ ions were made by:\n ", count := [ 1, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Stuart Bu\ rrell,", count := [ 1, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 19 ], name := "Item", next := 4756, root := ~, start := 4728, stop := 4755 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Ci\ rpons,", count := [ 1, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 21 ], name := "Item", next := 4789, root := ~, start := 4761, stop := 4788 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elli\ ott,", count := [ 1, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 23 ], name := "Item", next := 4820, root := ~, start := 4794, stop := 4819 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn,\ ", count := [ 1, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 25 ], name := "Item", next := 4847, root := ~, start := 4825, stop := 4846 ), rec( attributes := rec( ), content := [ rec( content := "Christoph\ er Jefferson,", count := [ 1, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 27 ], name := "Item", next := 4887, root := ~, start := 4852, stop := 4886 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pf\ eiffer,", count := [ 1, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 29 ], name := "Item", next := 4921, root := ~, start := 4892, stop := 4920 ), rec( attributes := rec( ), content := [ rec( content := "Christoph\ er Russell,", count := [ 1, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 31 ], name := "Item", next := 4959, root := ~, start := 4926, stop := 4958 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smit\ h, and", count := [ 1, 1, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 33 ], name := "Item", next := 4992, root := ~, start := 4964, stop := 4991 ), rec( attributes := rec( ), content := [ rec( content := "Murray Wh\ yte.", count := [ 1, 1, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 35 ], name := "Item", next := 5023, root := ~, start := 4997, stop := 5022 ) ], count := [ 1, 1, 0, 18 ], mathmode := "Text", name := "List", next := 5033, root := ~, start := 4717, stop := 5032 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package contains a vari\ ety of methods for efficiently creating\n and storing mutable and immutable d\ igraphs and computing information about\n them. Full explanations of all t\ he functions contained in the package are\n provided below. ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "P", next := 5295, root := ~, start := 5291, stop := 5294 ), rec( content := "\n\n If the ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "https://g\ ap-packages.github.io/grape", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 130, root := ~, start := 106, stop := 129 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package is available, i\ t will be loaded automatically.\n Digraphs created with the ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package can be converte\ d to ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "https://g\ ap-packages.github.io/grape", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 130, root := ~, start := 106, stop := 129 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := "\n graphs with ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ) , content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 5477, root := ~, start := 5458, stop := 5476 ), rec( content := ", and conversely ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "https://g\ ap-packages.github.io/grape", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 130, root := ~, start := 106, stop := 129 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " graphs can be\n conver\ ted to ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " objects with ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph" ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 5576, root := ~, start := 5555, stop := 5575 ), rec( content := ". ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "https://g\ ap-packages.github.io/grape", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 130, root := ~, start := 106, stop := 129 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " is not\n required for \ ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to run. ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "P", next := 5632, root := ~, start := 5628, stop := 5631 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " tool ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Cite", next := 5671, root := ~, start := 5653, stop := 5670 ), rec( content := " is included in this pac\ kage. It\n is an open-source tool for computing automorphism groups and cano\ nical forms\n of graphs, written by Tommi Junttila and Petteri Kaski. Several\ of the methods\n in the ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package rely on ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ". If the ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/gap-packages/NautyT\ racesInterface", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := "\n package for GAP is a\ vailable then it is also possible to use ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "https://p\ allini.di.uniroma1.it/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 125, root := ~, start := 101, stop := 124 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Cite", next := 6033, root := ~, start := 6011, stop := 6032 ), rec( content := " for computing automorph\ ism groups and canonical forms\n in ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ". See Section \n ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Isomorphisms and canoni\ cal labellings" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Ref", next := 6171, root := ~, start := 6120, stop := 6170 ), rec( content := " for more details. \n \ ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "P", next := 6198, root := ~, start := 6194, stop := 6197 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "edge-addition-p\ lanarity-suite" ), content := [ rec( content := "https://github.com/graph-algo\ rithms/edge-addition-planarity-suite", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 1, 1, 0, 40 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "edge-addi\ tion-planarity-suite", count := [ 1, 1, 0, 40 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 40 ], name := "Package", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " is also included in ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := "; see\n ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM04" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Cite", next := 6294, root := ~, start := 6276, stop := 6293 ), rec( content := ", ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "B06" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Cite", next := 6313, root := ~, start := 6296, stop := 6312 ), rec( content := ", ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Cite", next := 6333, root := ~, start := 6315, stop := 6332 ), rec( content := ", and ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "B12" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Cite", next := 6358, root := ~, start := 6339, stop := 6357 ), rec( content := " . The ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "edge-addition-p\ lanarity-suite" ), content := [ rec( content := "https://github.com/graph-algo\ rithms/edge-addition-planarity-suite", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 1, 1, 0, 40 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "edge-addi\ tion-planarity-suite", count := [ 1, 1, 0, 40 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 40 ], name := "Package", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " is an open-source\n im\ plementation of the edge addition planar graph embedding algorithm and\n rela\ ted algorithms by John M. Boyer. See Section ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Planarity" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Ref", next := 6565, root := ~, start := 6542, stop := 6564 ), rec( content := " for\n more details.\n \ ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 41 ], mathmode := "Text", name := "P", next := 6592, root := ~, start := 6588, stop := 6591 ), rec( content := "\n\n From version 1.0.0\ of this package, digraphs can be either mutable or immutable.\n Mutable digr\ aphs can be changed in-place by many of the methods in the\n package, which a\ voids unnecessary copying. Immutable digraphs cannot be\n changed in-place, b\ ut their advantage is that the value of an attribute of\n an immutable digrap\ h is only ever computed once. Mutable digraphs can be\n converted into immuta\ ble digraphs in-place using ", count := [ 1, 1, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "MakeImmutable" ), content := 0, count := [ 1, 1, 0, 41 ], mathmode := "Text", name := "Ref", next := 7073, root := ~, start := 7027, stop := 7072 ), rec( content := ". One of the motivation\ s for introducing mutable\n digraphs in version 1.0.0 was that in practice th\ e authors often wanted\n to create a digraph and immediately modify it (remov\ ing certain edges,\n loops, and so on). Before version 1.0.0, this involved c\ opying the digraph\n several times, with each copy being discarded almost imm\ ediately.\n From version 1.0.0, this unnecessary copying can be eliminated by\ \n first creating a mutable digraph, then changing it in-place, and finally\ \n converting the mutable digraph to an immutable one in-place\n (if desirab\ le).\n\n ", count := [ 1, 1, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Definitions" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Definitio\ ns", count := [ 1, 1, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 2 ], mathmode := "Text", name := "Heading", next := 7704, root := ~, start := 7674, stop := 7703 ), rec( content := "\n\n For the \ purposes of this package and its documentation, the following\n definitions\ apply: ", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "P", next := 7807, root := ~, start := 7803, stop := 7806 ), rec( content := "\n\n A ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7829, root := ~, start := 7815, stop := 7828 ), rec( content := " ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E=(E^0,E^\ 1,r,s)", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7852, root := ~, start := 7830, stop := 7851 ), rec( content := ", also known as \ a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed\\ n graph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7897, root := ~, start := 7870, stop := 7896 ), rec( content := ", consists of a \ set of vertices ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E^0", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7939, root := ~, start := 7929, stop := 7938 ), rec( content := " and a set of ed\ ges\n ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E^1", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7973, root := ~, start := 7963, stop := 7972 ), rec( content := " together with f\ unctions ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s, r: E^1\ \\to E^0", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 8022, root := ~, start := 7998, stop := 8021 ), rec( content := ", called the\n \ ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "source", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 8052, root := ~, start := 8039, stop := 8051 ), rec( content := " and ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "range", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 8069, root := ~, start := 8057, stop := 8068 ), rec( content := ", respectively.\\ n\n The source and range of an edge is respectively the values of ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s, r", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 8163, root := ~, start := 8152, stop := 8162 ), rec( content := " at\n that ed\ ge.\n\n An edge is called a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "loop", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 8218, root := ~, start := 8207, stop := 8217 ), rec( content := " if its source a\ nd range are the same.\n\n A digraph is called a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "multidigr\ aph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 8303, root := ~, start := 8284, stop := 8302 ), rec( content := " if there exist \ two or more edges\n with the same source and the same range. ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "P", next := 8386, root := ~, start := 8382, stop := 8385 ), rec( content := "\n\n A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ walk", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8414, root := ~, start := 8394, stop := 8413 ), rec( content := " on a digraph is\ a sequence of alternating vertices\n and edges ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1}, v_n)", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8526, root := ~, start := 8480, stop := 8525 ), rec( content := " such that each \ edge\n ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e_i", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8561, root := ~, start := 8551, stop := 8560 ), rec( content := " has source ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_i", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8583, root := ~, start := 8573, stop := 8582 ), rec( content := " and range ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8608, root := ~, start := 8594, stop := 8607 ), rec( content := ". A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed\\ n path", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8638, root := ~, start := 8612, stop := 8637 ), rec( content := " is a directed w\ alk where no vertex (and hence no edge) is\n repeated. A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ circuit", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8737, root := ~, start := 8714, stop := 8736 ), rec( content := " is a directed w\ alk where ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_1 =\n \ v_n", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8785, root := ~, start := 8763, stop := 8784 ), rec( content := ", and a ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ cycle", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8814, root := ~, start := 8793, stop := 8813 ), rec( content := " is a directed c\ ircuit where where no\n vertex is repeated, except for ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_1 = v_n\ ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8903, root := ~, start := 8887, stop := 8902 ), rec( content := ". ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "P", next := 8910, root := ~, start := 8906, stop := 8909 ), rec( content := "\n\n The ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "length", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8933, root := ~, start := 8920, stop := 8932 ), rec( content := " of a directed w\ alk ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1},\n v_n)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9005, root := ~, start := 8953, stop := 9004 ), rec( content := " is equal to ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n-1", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9028, root := ~, start := 9018, stop := 9027 ), rec( content := ", the number of \ edges it contains. A\n directed walk (or path) ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1}, v_n)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9140, root := ~, start := 9094, stop := 9139 ), rec( content := " is\n sometim\ es called a directed walk (or path) ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "from vert\ ex ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ _1", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9216, root := ~, start := 9206, stop := 9215 ), rec( content := " to\n \ vertex ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ _n", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9243, root := ~, start := 9233, stop := 9242 ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 9247, root := ~, start := 9191, stop := 9246 ), rec( content := ". A directed wa\ lk of zero length, i.e. a sequence\n ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9312, root := ~, start := 9302, stop := 9311 ), rec( content := " for some vertex\ ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9337, root := ~, start := 9329, stop := 9336 ), rec( content := ", is called ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "trivial", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 9363, root := ~, start := 9349, stop := 9362 ), rec( content := ". A trivial\n \ directed walk is considered to be both a circuit and a cycle, as is the\n \ empty directed walk ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "()", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 9484, root := ~, start := 9475, stop := 9483 ), rec( content := ". A ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "simple ci\ rcuit", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 9509, root := ~, start := 9488, stop := 9508 ), rec( content := " is another name\ for a\n non-trivial and non-empty directed cycle.", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "P", next := 9581, root := ~, start := 9577, stop := 9580 ), rec( content := "\n\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 1 ], mathmode := "Text", name := "Subsection", next := 9598, root := ~, start := 7642, stop := 9597 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 9611, root := ~, start := 4267, stop := 9610 ), rec( content := "\n", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 9622, root := ~, start := 4115, stop := 9621 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " intro ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Installing Digraphs" ) , content := [ rec( content := "\n ", count := [ 2, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Installing ", count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ) ], count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 9721, root := ~, start := 9681, stop := 9720 ), rec( content := "\n\n ", count := [ 2, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "For those in a hurry" ), content := [ rec( content := "\n ", count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "For those in a hu\ rry", count := [ 2, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 9807, root := ~, start := 9768, stop := 9806 ), rec( content := "\n In this section we\ give a brief description of how to start using\n ", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "P", next := 9897, root := ~, start := 9893, stop := 9896 ), rec( content := "\n\n It is assumed th\ at you have a working copy of ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " with version number\n \ 4.10.0 or higher. The most up-to-date version of ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " and instructions\n o\ n how to install it can be obtained from the main ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " webpage\n ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://www.gap-s\ ystem.org", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "URL", next := 10164, root := ~, start := 10127, stop := 10163 ), rec( content := ".", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "P", next := 10169, root := ~, start := 10165, stop := 10168 ), rec( content := "\n\n The following is\ a summary of the steps that should lead to a\n successful installation of \ ", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ":\n\n ", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ensure that the ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n https://gap-package\ s.github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL" , next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 2, 1, 0, 7 ], name := \ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Package", next := 123, root := ~, start := 102 , stop := 122 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " package\ version 4.5.1 or higher is available.\n ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n https://gap-package\ s.github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL" , next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 2, 1, 0, 7 ], name := \ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Package", next := 123, root := ~, start := 102 , stop := 122 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " must be\ compiled before ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ loaded.\n ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Item", next := 10460, root := ~, start := 10299, stop := 10459 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ensure that the ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Orb" ), content := [ rec( content := "\n https://gap-packag\ es.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL" , next := 80, root := ~, start := 17, stop := 79 ) , rec( content := "\ \n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 92, root := ~, start := 1, stop := 91 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Orb", count := [ 2, 1, 0, 9 ], name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Package", next := 131, root := ~, start := 109 , stop := 130 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 92, stop := 136 ), rec( content := " package\ version 4.8.2 or higher is available.\n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Orb" ), content := [ rec( content := "\n https://gap-packag\ es.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL" , next := 80, root := ~, start := 17, stop := 79 ) , rec( content := "\ \n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 92, root := ~, start := 1, stop := 91 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Orb", count := [ 2, 1, 0, 9 ], name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Package", next := 131, root := ~, start := 109 , stop := 130 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 92, stop := 136 ), rec( content := " has bet\ ter performance when compiled, but although compilation is\n recommende\ d, it is not required to be compiled for ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to be\n\ loaded.\n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Item", next := 10740, root := ~, start := 10468, stop := 10739 ), rec( attributes := rec( ), content := [ rec( content := " ensure t\ hat the ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "datastructures" ), content := [ rec( content := "https://github\ .com/gap-packages/datastructures", count := [ 2, 1, 0, 11 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Tex\ t", name := "URL", next := 103, root := ~, start := 17, stop := 102 ) ], count := [ 2, 1, 0, 11 ], name := "Alt", next := 109, root := ~, start := 1, stop := 108 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "datastructures", count := [ 2, 1, 0, 1\ 1 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Packa\ ge", next := 162, root := ~, start := 129, stop := 161 ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Alt", next := 168, root := ~, start := 109, stop := 167 ), rec( content := " package\ version\n 0.2.5 or higher is available. \n ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Item", next := 10869, root := ~, start := 10748, stop := 10868 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "B", next := 10921, root := ~, start := 10892, stop := 10920 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package\ to be available; see Section\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 11061, root := ~, start := 11027, stop := 11060 ), rec( content := " for ful\ l details. To use these\n functions make sure\n that the ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package\ version 4.8.1 or higher is available.\n If ", count := [ 2, 1, 0, 13 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " is not \ available, then ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ used as normal with\n the exception that the functions listed in Subsec\ tion\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 11383, root := ~, start := 11349, stop := 11382 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Item", next := 11412, root := ~, start := 10877, stop := 11411 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "B", next := 11470, root := ~, start := 11441, stop := 11469 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " package\ to be available.\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO \n\ see Section\n for full d\ etails. ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n \ If you want to make use of these functions, please ensure that the\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " package\ version 0.2 or\n higher is available. If ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is not \ available,\n then ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ used as normal with the exception that \n functions whose names conta\ in ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ auty", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Q", next := 12029, root := ~, start := 12017, stop := 12028 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Item", next := 12058, root := ~, start := 11426, stop := 12057 ), rec( attributes := rec( ), content := [ rec( content := "\n \ download the package archive ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.4.1.tar.gz", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 12137, root := ~, start := 12110, stop := 12136 ), rec( content := " from\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "the Dig\ raph package webpage" ), content := [ rec( content := "https://digraphs.github\ .io/Digraphs/", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "URL", next := 12233, root := ~, start := 12151, stop := 12232 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], name := "Item", next := 12248, root := ~, start := 12066, stop := 12247 ), rec( attributes := rec( ), content := [ rec( content := "\n \ unzip and untar the file, this should create a directory called\n ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.4.1", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "F", next := 12363, root := ~, start := 12343, stop := 12362 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], name := "Item", next := 12378, root := ~, start := 12256, stop := 12377 ), rec( attributes := rec( ), content := [ rec( content := "\n \ locate the ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12422, root := ~, start := 12412, stop := 12421 ), rec( content := " directo\ ry of your ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " directo\ ry, which contains\n the directories ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ib", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12507, root := ~, start := 12497, stop := 12506 ), rec( content := ", ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ oc", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12519, root := ~, start := 12509, stop := 12518 ), rec( content := " and so \ on. Move the directory\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.4.1", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12578, root := ~, start := 12558, stop := 12577 ), rec( content := " into th\ e ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12598, root := ~, start := 12588, stop := 12597 ), rec( content := " directo\ ry.\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], name := "Item", next := 12623, root := ~, start := 12386, stop := 12622 ), rec( attributes := rec( ), content := [ rec( content := "\n \ it is necessary to compile the ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n Inside the ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg/digraphs-1.4.1", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "F", next := 12740, root := ~, start := 12716, stop := 12739 ), rec( content := " directo\ ry, type\n ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ n./configure\nmake", count := [ 2, 1, 0, 24 ], name := "PCDATA", root := ~ ) ] , count := [ 2, 1, 0, 24 ], mathmode := "Text", name := "Listing", next := 12813, root := ~, start := 12765, stop := 12812 ), rec( content := "\n \ Further information about this step can be found in Section\n ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Compili\ ng the kernel module" ), content := 0, count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "Ref", next := 12931, root := ~, start := 12890, stop := 12930 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], name := "Item", next := 12946, root := ~, start := 12631, stop := 12945 ), rec( attributes := rec( ), content := [ rec( content := "\n \ start ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " in the \ usual way (i.e. type ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "C", next := 13019, root := ~, start := 13009, stop := 13018 ), rec( content := " at the \ command line).\n ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], name := "Item", next := 13055, root := ~, start := 12954, stop := 13054 ), rec( attributes := rec( ), content := [ rec( content := "\n \ type ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ oadPackage(\"digraphs\");", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "C", next := 13114, root := ~, start := 13083, stop := 13113 ), rec( content := "\n \ ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], name := "Item", next := 13128, root := ~, start := 13063, stop := 13127 ) ], count := [ 2, 1, 0, 6 ], mathmode := "Text", name := "List", next := 13140, root := ~, start := 10285, stop := 13139 ), rec( content := "\n\n If you want to c\ heck that the package is working correctly, you\n should run some of the te\ sts described in Section\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Testing your installati\ on" ), content := 0, count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "Ref", next := 13307, root := ~, start := 13268, stop := 13306 ), rec( content := ".\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Configuration options"\ ), content := [ rec( content := "\n ", count := [ 2, 1, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Configura\ tion options", count := [ 2, 1, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 2 ], mathmode := "Text", name := "Heading", next := 13402, root := ~, start := 13362, stop := 13401 ), rec( content := "\n\nIn addition \ to the usual autoconf generated configuration flags, the following\nflags are \ provided.\n\n", count := [ 2, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "C\ onfiguration flags", count := [ 2, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 1, 4 ], name := "Caption", next := 13565, root := ~, start := 13527, stop := 13564 ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 1, 4 ], name := "HorLine", next := 13577, root := ~, start := 13567, stop := 13576 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Option", count := [ 2, 1, 1, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 5 ], name := "Ite\ m", next := 13608, root := ~, start := 13589, stop := 13607 ), rec( attributes :\ = rec( ), content := [ rec( content := "Meaning", count := [ 2, 1, 1, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 7 ], name := "Ite\ m", next := 13628, root := ~, start := 13608, stop := 13627 ) ], count := [ 2, 1, 1, 4 ], name := "Row", next := 13637, root := ~, start := 13580, stop := 13636 ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 1, 8 ], name := "HorLine", next := 13649, root := ~, start := 13639, stop := 13648 ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 1, 8 ], name := "HorLine", next := 13661, root := ~, start := 13651, stop := 13660 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--enable-code-coverage", count := [ 2, 1, 1, 9 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 9 ], mathmode := "Text", name \ := "C", next := 13707, root := ~, start := 13678, stop := 13706 ) ], count := [ 2, 1, 1, 9 ], name := "Ite\ m", next := 13714, root := ~, start := 13672, stop := 13713 ), rec( attributes :\ = rec( ), content := [ rec( content := " enable code coverage support", count\ := [ 2, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 11 ], name := "Ite\ m", next := 13756, root := ~, start := 13714, stop := 13755 ) ], count := [ 2, 1, 1, 8 ], name := "Row", next := 13764, root := ~, start := 13663, stop := 13763 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--enable-compile-warnings", count := [ 2, 1, 1, 13 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 13 ], mathmode := "Text", name\ := "C", next := 13813, root := ~, start := 13781, stop := 13812 ) ], count := [ 2, 1, 1, 13 ], name := "Ite\ m", next := 13820, root := ~, start := 13775, stop := 13819 ), rec( attributes :\ = rec( ), content := [ rec( content := " enable compiler warnings", count := [ \ 2, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 15 ], name := "Ite\ m", next := 13858, root := ~, start := 13820, stop := 13857 ) ], count := [ 2, 1, 1, 12 ], name := "Row", next := 13866, root := ~, start := 13766, stop := 13865 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--enable-debug", count := [ 2, 1, 1, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 1, 17 ], mathmode := "Text", name := "C" , next\ := 13904, root := ~, start := 13883, stop := 13903 ) ], count := [ 2, 1, 1, 17 ], name := "Ite\ m", next := 13911, root := ~, start := 13877, stop := 13910 ), rec( attributes :\ = rec( ), content := [ rec( content := " enable debug mode", count := [ 2, 1,\ 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 19 ], name := "Ite\ m", next := 13942, root := ~, start := 13911, stop := 13941 ) ], count := [ 2, 1, 1, 16 ], name := "Row", next := 13950, root := ~, start := 13868, stop := 13949 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--with-external-bliss", count := [ 2, 1, 1, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 21 ], mathmode := "Text", name\ := "C", next := 13995, root := ~, start := 13967, stop := 13994 ) ], count := [ 2, 1, 1, 21 ], name := "Ite\ m", next := 14002, root := ~, start := 13961, stop := 14001 ), rec( attributes :\ = rec( ), content := [ rec( content := " use external ", count := [ 2, 1, 1, \ 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), co\ ntent := [ rec( content := "http://www.tcs.tkk.fi/Software/bliss/", count := [\ 2, 1, 1, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 2, 1, 1, 23 ], mathmode := "Text", name := "URL", next := 81, root := ~, sta\ rt := 17, stop := 80 ) ], count := [ 2, 1, 1, 23 ], name := "Alt", next := 87, root\ := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "bliss", cou\ nt := [ 2, 1, 1, 23 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 23\ ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], coun\ t := [ 2, 1, 1, 23 ], mathmode := "Text", name := "Alt", next := 137, root := \ ~, start := 87, stop := 136 ) ], count := [ 2, 1, 1, 23 ], name := "Item", next := 1403\ 6, root := ~, start := 14002, stop := 14035 ) ], count := [ 2, 1, 1, 20 ], name := "Row", next := 14044, root := ~, start := 13952, stop := 14043 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--with-external-planarity", count := [ 2, 1, 1, 25 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 25 ], mathmode := "Text", name\ := "C", next := 14093, root := ~, start := 14061, stop := 14092 ) ], count := [ 2, 1, 1, 25 ], name := "Ite\ m", next := 14100, root := ~, start := 14055, stop := 14099 ), rec( attributes :\ = rec( ), content := [ rec( content := " use external ", count := [ 2, 1, 1, \ 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( Not := "Text" ), content := [ rec( attributes := rec( Text := "edge-additio\ n-planarity-suite" ), content := [ rec( content := "https://github.com/graph-a\ lgorithms/edge-addition-planarity-suite", count := [ 2, 1, 1, 27 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 27 ], mathmode \ := "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ] , coun\ t := [ 2, 1, 1, 27 ], name := "Alt", next := 146, root := ~, start := 1, stop \ := 145 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attribut\ es := rec( ), content := [ rec( content := "edge-addition-planarity-suite", c\ ount := [ 2, 1, 1, 27 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, \ 27 ], name := "Package", next := 218, root := ~, start := 170, stop := 217 ) ] , coun\ t := [ 2, 1, 1, 27 ], mathmode := "Text", name := "Alt", next := 224, root := \ ~, start := 146, stop := 223 ) ], count := [ 2, 1, 1, 27 ], name := "Item", next := 1415\ 8, root := ~, start := 14100, stop := 14157 ) ], count := [ 2, 1, 1, 24 ], name := "Row", next := 14166, root := ~, start := 14046, stop := 14165 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--with-gaproot", count := [ 2, 1, 1, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 1, 29 ], mathmode := "Text", name := "C" , next\ := 14204, root := ~, start := 14183, stop := 14203 ) ], count := [ 2, 1, 1, 29 ], name := "Ite\ m", next := 14211, root := ~, start := 14177, stop := 14210 ), rec( attributes :\ = rec( ), content := [ rec( content := " specify root of GAP installation", coun\ t := [ 2, 1, 1, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 31 ], name := "Ite\ m", next := 14257, root := ~, start := 14211, stop := 14256 ) ], count := [ 2, 1, 1, 28 ], name := "Row", next := 14265, root := ~, start := 14168, stop := 14264 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "--without-intrinsics", count := [ 2, 1, 1, 33 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 33 ], mathmode := "Text", name \ := "C", next := 14309, root := ~, start := 14282, stop := 14308 ) ], count := [ 2, 1, 1, 33 ], name := "Ite\ m", next := 14316, root := ~, start := 14276, stop := 14315 ), rec( attributes :\ = rec( ), content := [ rec( content := " do not use compiler intrinsics even \ if available", count := [ 2, 1, 1, 35 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 2, 1, 1, 35 ], name := "Item", next := 14378, root := ~, start := 143\ 16, stop := 14377 ) ], count := [ 2, 1, 1, 32 ], name := "Row", next := 14386, root := ~, start := 14267, stop := 14385 ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 1, 36 ], name := "HorLine", next := 14398, root := ~, start := 14388, stop := 14397 ) ], count := [ 2, 1, 1, 4 ], mathmode := "Text", name := "Table", next := 14407, root := ~, start := 13504, stop := 14406 ), rec( content := "\n", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 1, 1 ], mathmode := "Text", name := "Subsection", next := 14421, root := ~, start := 13313, stop := 14420 ), rec( content := "\n ", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 14434, root := ~, start := 9725, stop := 14433 ), rec( content := "\n\n ", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 1, 1, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Optional package dependencies"\ ), content := [ rec( content := "\n ", count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Optional package \ dependencies", count := [ 2, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 14699, root := ~, start := 14651, stop := 14698 ), rec( content := "\n\n ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "************************\ ********************************************", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n The ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package is written in " , count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " and C code and requires\ the\n ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ rec( content := "\n ht\ tps://gap-packages.github.io/io/\n", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "IO", count := [ 2, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], name := "Package", next := 123, root := ~, start := 102, stop := 122 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " package. The ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ rec( content := "\n ht\ tps://gap-packages.github.io/io/\n", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "IO", count := [ 2, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], name := "Package", next := 123, root := ~, start := 102, stop := 122 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " package is used to read\ and write transformations,\n partial permutations, and bipartitions to a f\ ile. ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "P", next := 14994, root := ~, start := 14990, stop := 14993 ), rec( content := "\n\n ", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "************************\ ********************************************", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "The Grape package" ), content := [ rec( content := "\n ", count := [ 2, 2, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "The Grape\ package", count := [ 2, 2, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 2 ], mathmode := "Text", name := "Heading", next := 15162, root := ~, start := 15126, stop := 15161 ), rec( content := "\n\n The ", count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "h\ ttps://gap-packages.github.io/grape", count := [ 2, 2, 1, 3 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 2, 2, 1, 3 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "G\ rape", count := [ 2, 2, 1, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], name := "Package", next := 130, root := ~, start := 106, stop := 129 ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package must be\ available for the following operations to be\n available:\n\n ", count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( Oper := "Graph" ), content := 0, count := [ 2, 2, 1, 5 ], mathmode := "T\ ext", name := "Ref", next := 15307, root := ~, start := 15288, stop := 15306 ) , rec( content := "\ with a digraph argument", count := [ 2, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 2, 1, 5 ], name := "Item", next := 15338, root := ~, start := 15282, stop := 15337 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Attr := "AsGraph" ), content := 0, count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "Ref", next := 15374, root := ~, start := 15353, stop := 15373\ ), rec( content := " with a digraph argument", count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 7 ], name := "Item", next := 15405, root := ~, start := 15347, stop := 15404 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Oper := "Digraph" ), content := 0, count := [ 2, 2, 1, 9 ], mathmode := \ "Text", name := "Ref", next := 15441, root := ~, start := 15420, stop := 15440\ ), rec( content := " with a ", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( content := "https://gap-pac\ kages.github.io/grape", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "\ URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 2, 2, 1, 9 ], name := "Alt\ ", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "Grape", count := [ 2, 2, 1, 9 ], name := "PCDATA", root := ~ ) ], count := [ \ 2, 2, 1, 9 ], name := "Package", next := 130, root := ~, start := 106, stop :=\ 129 ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := "\ graph argument", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 2, 2, 1, 9 ], name := "Item", next := 15478, root := ~, start := 15414, stop := 15477 ) ], count := [ 2, 2, 1, 4 ], mathmode := "Text", name := "List", next := 15492, root := ~, start := 15267, stop := 15491 ), rec( content := "\n\n If ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "h\ ttps://gap-packages.github.io/grape", count := [ 2, 2, 1, 11 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 2, 2, 1, 11 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "G\ rape", count := [ 2, 2, 1, 11 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], name := "Package", next := 130, root := ~, start := 106, stop := 129 ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " is not availabl\ e, then ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs" , count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be used as \ normal with\n the exception that the functions above will not work.\n \ ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 1 ], mathmode := "Text", name := "Subsection", next := 15649, root := ~, start := 15081, stop := 15648 ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 15662, root := ~, start := 14599, stop := 15661 ), rec( content := "\n\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Compiling the kernel module" ) , content := [ rec( content := "\n ", count := [ 2, 3, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Compiling the ker\ nel module", count := [ 2, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 15923, root := ~, start := 15877, stop := 15922 ), rec( content := "\n The ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package has a ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " kernel component in\n \ C which should be compiled. This component contains certain low-level\n f\ unctions required by ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "P", next := 16103, root := ~, start := 16099, stop := 16102 ), rec( content := "\n\n It is not possib\ le to use the ", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package without compili\ ng it.", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "P", next := 16183, root := ~, start := 16179, stop := 16182 ), rec( content := "\n\n To compile the k\ ernel component inside the ", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "pkg/digraphs-1.4.\ 1", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "F", next := 16256, root := ~, start := 16232, stop := 16255 ), rec( content := "\n directory, type\n" , count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n\n./configure\n\ make", count := [ 2, 3, 0, 6 ], name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 6 ], mathmode := "Text", name := "Listing", next := 16326, root := ~, start := 16277, stop := 16325 ), rec( content := "\n ", count := [ 2, 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "P", next := 16335, root := ~, start := 16331, stop := 16334 ), rec( content := "\n\n If you installed\ the package in another 'pkg' directory than the standard 'pkg'\n directory\ in your ", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " installation, then you \ have to do two things. Firstly\n during compilation you have to use the opt\ ion '--with-gaproot=PATH' of the\n 'configure' script where 'PATH' is a pat\ h to the main GAP root directory (if\n not given the default '../..' is ass\ umed).", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "P", next := 16713, root := ~, start := 16709, stop := 16712 ), rec( content := "\n\n If you installed\ ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " on several architecture\ s, you must execute the\n configure/make step for each of the architectures\ . You can either do this\n immediately after configuring and compiling GAP \ itself on this architecture, or\n alternatively set the environment\n va\ riable 'CONFIGNAME' to the name of the configuration you used when compiling\n\ GAP before running './configure'. Note however that your compiler choice \ and\n flags (environment variables 'CC' and 'CFLAGS') need to be chosen to \ match the\n setup of the original GAP compilation. For example you have to \ specify 32-bit\n or 64-bit mode correctly!\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 17361, root := ~, start := 15827, stop := 17360 ), rec( content := "\n\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Rebuilding the documentation" ), content := [ rec( content := "\n ", count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Rebuilding the do\ cumentation", count := [ 2, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 17624, root := ~, start := 17577, stop := 17623 ), rec( content := "\n The ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package comes complete \ with pdf, html, and text versions of\n the documentation. However, you migh\ t find it necessary, at some point, to\n rebuild the documentation. To rebu\ ild the documentation, please use the\n function ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsMakeDoc" ), content := 0, count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "Ref", next := 17901, root := ~, start := 17872, stop := 17900 ), rec( content := ".\n\n ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsMakeDoc\ " ), content := 0, count := [ 2, 4, 1, 2 ], mathmode := "Text", name := "Func", next := 17960, root := ~, start := 17923, stop := 17959 ), rec( attributes := rec( ), content := [ rec( content := "Nothing", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 17989, root := ~, start := 17963, stop := 17988 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function should be called with no argument to compile the ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := "\n do\ cumentation.\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 18119, root := ~, start := 17992, stop := 18118 ) ], count := [ 2, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 18133, root := ~, start := 17908, stop := 18132 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 18147, root := ~, start := 17526, stop := 18146 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Testing your installation" ), content := [ rec( content := "\n ", count := [ 2, 5, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Testing your inst\ allation", count := [ 2, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 18404, root := ~, start := 18360, stop := 18403 ), rec( content := "\n\n In this section \ we describe how to test that ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is working as\n inte\ nded. To test that ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is installed correctly \ use\n ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsTestInstall" ), content := 0, count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Ref", next := 18583, root := ~, start := 18550, stop := 18582 ), rec( content := " or for more extensive t\ ests use\n ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsTestStandard" ) , content := 0, count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Ref", next := 18654, root := ~, start := 18620, stop := 18653 ), rec( content := ". ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "P", next := 18660, root := ~, start := 18656, stop := 18659 ), rec( content := "\n\n If something goe\ s wrong, then please review the instructions in Section ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "For those in a hurry" ) , content := 0, count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "Ref", next := 18778, root := ~, start := 18738, stop := 18777 ), rec( content := " and ensure that ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " has been\n properly \ installed. If you continue having problems, please use the ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "issue tracker" ), content := [ rec( content := "https://github.co\ m/digraphs/Digraphs/issues", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "URL", next := 18969, root := ~, start := 18888, stop := 18968 ), rec( content := "\n to report the issu\ es you are having.\n\n ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestIns\ tall" ), content := 0, count := [ 2, 5, 1, 2 ], mathmode := "Text", name := "Func", next := 19072, root := ~, start := 19031, stop := 19071 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "K", next := 19095, root := ~, start := 19084, stop := 19094 ), rec( content := " or ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "K", next := 19111, root := ~, start := 19099, stop := 19110 ), rec( content := ".", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 19122, root := ~, start := 19075, stop := 19121 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function can be called without arguments to test your installation of\n " , count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is work\ ing correctly. These tests should take no more than a\n few seconds to comp\ lete. To test more comprehensively that\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is work\ ing correctly, use ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestStandard" ), content := 0, count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Ref", next := 19433, root := ~, start := 19399, stop := 19432 ), rec( content := ".\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 19451, root := ~, start := 19125, stop := 19450 ) ], count := [ 2, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 19465, root := ~, start := 19016, stop := 19464 ), rec( content := "\n\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestSta\ ndard" ), content := 0, count := [ 2, 5, 2, 2 ], mathmode := "Text", name := "Func", next := 19528, root := ~, start := 19486, stop := 19527 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "K", next := 19551, root := ~, start := 19540, stop := 19550 ), rec( content := " or ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "K", next := 19567, root := ~, start := 19555, stop := 19566 ), rec( content := ".", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 19578, root := ~, start := 19531, stop := 19577 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function can be called without arguments to test all of the methods\n inc\ luded in ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n T\ hese tests should take less than a minute to complete. ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "P", next := 19764, root := ~, start := 19760, stop := 19763 ), rec( content := "\n\n \ To quickly test that ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is inst\ alled correctly use ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestInstall" ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 19866, root := ~, start := 19829, stop := 19865 ), rec( content := ". For a\ more thorough test, use ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestExtreme" ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 19936, root := ~, start := 19899, stop := 19935 ), rec( content := ".\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 19954, root := ~, start := 19581, stop := 19953 ) ], count := [ 2, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 19968, root := ~, start := 19471, stop := 19967 ), rec( content := "\n\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestExt\ reme" ), content := 0, count := [ 2, 5, 3, 2 ], mathmode := "Text", name := "Func", next := 20030, root := ~, start := 19989, stop := 20029 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 20060, root := ~, start := 20033, stop := 20059 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function should be called with no argument. It executes a series of\n ver\ y demanding tests, which measure the performance of a variety of\n function\ s on large examples. These tests take a long time to\n complete, at least \ several minutes. ", count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "P", next := 20336, root := ~, start := 20332, stop := 20335 ), rec( content := "\n\n \ For these tests to complete, the digraphs library ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-lib", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "C", next := 20411, root := ~, start := 20392, stop := 20410 ), rec( content := " must\n \ be downloaded and placed in the ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "C", next := 20468, root := ~, start := 20453, stop := 20467 ), rec( content := " directo\ ry in a subfolder\n named ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-lib", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "F", next := 20523, root := ~, start := 20504, stop := 20522 ), rec( content := ".\n T\ his library can be found on the ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " website\ .\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 20598, root := ~, start := 20063, stop := 20597 ) ], count := [ 2, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 20612, root := ~, start := 19974, stop := 20611 ), rec( content := "\n\n\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 20627, root := ~, start := 18312, stop := 20626 ), rec( content := "\n\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n\n", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 20801, root := ~, start := 9641, stop := 20800 ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " installing ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Creating digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Creating digraphs", count := [ 3, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 20896, root := ~, start := 20860, stop := 20895 ), rec( content := "\n\n In this chapter we describ\ e how to create digraphs.", count := [ 3, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 0, 0, 4 ], mathmode := "Text", name := "P", next := 20955, root := ~, start := 20951, stop := 20954 ), rec( content := "\n\n ", count := [ 3, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Creating digraphs\ ", count := [ 3, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 21004, root := ~, start := 20968, stop := 21003 ), rec( content := "\n ", count := [ 3, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsDigraph", Type := "Category" ), content := 0, count := [ 3, 1, 1, 2 ], mathmode := "Text", name := "Filt", next := 21064, root := ~, start := 21024, stop := 21063 ), rec( attributes := rec( ), content := [ rec( content := "\n Eve\ ry digraph in ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " belong\ s to the category\n ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "C", next := 21163, root := ~, start := 21147, stop := 21162 ), rec( content := ". Some \ basic attributes and operations for digraphs\n are ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 21253, root := ~, start := 21224, stop := 21252 ), rec( content := ", ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 21281, root := ~, start := 21255, stop := 21280 ), rec( content := ", and\n \ ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 21318, root := ~, start := 21291, stop := 21317 ), rec( content := ".\n " , count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 21338, root := ~, start := 21067, stop := 21337 ) ], count := [ 3, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 21352, root := ~, start := 21009, stop := 21351 ), rec( content := "\n\n ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsMutableDigrap\ h", Type := "Category" ), content := 0, count := [ 3, 1, 2, 2 ], mathmode := "Text", name := "Filt", next := 21420, root := ~, start := 21373, stop := 21419 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMutableDigraph", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "C", next := 21464, root := ~, start := 21441, stop := 21463 ), rec( content := " is a sy\ nonym for ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 21506, root := ~, start := 21482, stop := 21505 ), rec( content := " and ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsMutable" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 21554, root := ~, start := 21511, stop := 21553 ), rec( content := ". A mut\ able digraph may be changed\n in-place by methods in the ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ , and is not attribute-storing\n – see ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsAttributeStoringRep" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 21733, root := ~, start := 21682, stop := 21732 ), rec( content := ".\n " , count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "P", next := 21743, root := ~, start := 21739, stop := 21742 ), rec( content := "\n\n \ A mutable digraph may be converted into an immutable attribute-storing\n di\ graph by calling ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "MakeImmutable" ), content := 0, count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "Ref", next := 21885, root := ~, start := 21843, stop := 21884 ), rec( content := " on the\\ n digraph.\n ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 21924, root := ~, start := 21423, stop := 21923 ) ], count := [ 3, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 21938, root := ~, start := 21358, stop := 21937 ), rec( content := "\n\n ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsImmutableDigr\ aph", Type := "Category" ), content := 0, count := [ 3, 1, 3, 2 ], mathmode := "Text", name := "Filt", next := 22008, root := ~, start := 21959, stop := 22007 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "C", next := 22054, root := ~, start := 22029, stop := 22053 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 22099, root := ~, start := 22075, stop := 22098 ), rec( content := ".\n D\ igraphs that lie in ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "C", next := 22151, root := ~, start := 22126, stop := 22150 ), rec( content := " are imm\ utable and\n attribute-storing. In particular, they lie in\n ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsAttributeStoringRep" ), content := 0, count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 22275, root := ~, start := 22224, stop := 22274 ), rec( content := ". ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "P", next := 22281, root := ~, start := 22277, stop := 22280 ), rec( content := "\n\n \ A mutable digraph may be converted to an immutable\n digraph that lies in t\ he category ", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "C", next := 22401, root := ~, start := 22376, stop := 22400 ), rec( content := " by call\ ing\n ", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "MakeImmutable" ), content := 0, count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "Ref", next := 22459, root := ~, start := 22417, stop := 22458 ), rec( content := " on the \ digraph.", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "P", next := 22479, root := ~, start := 22475, stop := 22478 ), rec( content := "\n\n \ The operation ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ MutableCopy" ), content := 0, count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 22531, root := ~, start := 22499, stop := 22530 ), rec( content := " can be \ used\n to construct a mutable copy of an immutable digraph. It is however n\ ot\n possible to convert an immutable digraph into a mutable digraph in-pla\ ce.\n ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 22713, root := ~, start := 22011, stop := 22712 ) ], count := [ 3, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 22727, root := ~, start := 21944, stop := 22726 ), rec( content := "\n\n ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsCayleyDigraph\ ", Type := "Category" ), content := 0, count := [ 3, 1, 4, 2 ], mathmode := "Text", name := "Filt", next := 22794, root := ~, start := 22748, stop := 22793 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCayleyDigraph", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "C", next := 22837, root := ~, start := 22815, stop := 22836 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "C", next := 22874, root := ~, start := 22858, stop := 22873 ), rec( content := ". Digrap\ hs that\n are Cayley digraphs of a group and that are constructed by the op\ eration\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "Ref", next := 22998, root := ~, start := 22971, stop := 22997 ), rec( content := " are con\ structed in this category, and are\n always immutable.\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 23081, root := ~, start := 22797, stop := 23080 ) ], count := [ 3, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 23095, root := ~, start := 22733, stop := 23094 ), rec( content := "\n\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsDigraphWithAd\ jacencyFunction", Type := "Category" ), content := 0, count := [ 3, 1, 5, 2 ], mathmode := "Text", name := "Filt", next := 23177, root := ~, start := 23116, stop := 23176 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphWithAdjacencyFunction", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "C", next := 23235, root := ~, start := 23198, stop := 23234 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "C", next := 23272, root := ~, start := 23256, stop := 23271 ), rec( content := ".\n D\ igraphs that are ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ reated", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "E", next := 23310, root := ~, start := 23296, stop := 23309 ), rec( content := " using a\ n adjacency function are constructed\n in this category.\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 23393, root := ~, start := 23180, stop := 23392 ) ], count := [ 3, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 23407, root := ~, start := 23101, stop := 23406 ), rec( content := "\n\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "DigraphByOutNei\ ghboursType" ), content := 0, count := [ 3, 1, 6, 2 ], mathmode := "Text", name := "Var", next := 23468, root := ~, start := 23428, stop := 23467 ), rec( attributes := rec( Name := "DigraphFamily" ), content := 0, count := [ 3, 1, 6, 4 ], mathmode := "Text", name := "Fam", next := 23498, root := ~, start := 23471, stop := 23497 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ type of all digraphs is ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphByOutNeighboursType", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "C", next := 23580, root := ~, start := 23547, stop := 23579 ), rec( content := ".\n T\ he family of all digraphs is ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFamily", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "C", next := 23636, root := ~, start := 23616, stop := 23635 ), rec( content := ".\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "Description", next := 23654, root := ~, start := 23501, stop := 23653 ) ], count := [ 3, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 23668, root := ~, start := 23413, stop := 23667 ), rec( content := "\n\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt,] obj[, so\ urce, range]", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 2 ], mathmode := "Text", name := "Oper", next := 23750, root := ~, start := 23689, stop := 23749 ), rec( attributes := rec( Arg := "[filt,] list, fu\ nc", Label := "for a list and function", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 4 ], mathmode := "Text", name := "Oper", next := 23843, root := ~, start := 23753, stop := 23842 ), rec( attributes := rec( Arg := "[filt,] G, list,\ act, adj", Label := "for a group, list, function, and function", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 6 ], mathmode := "Text", name := "Oper", next := 23960, root := ~, start := 23846, stop := 23959 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Returns", next := 23992, root := ~, start := 23963, stop := 23991 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 24055, root := ~, start := 24044, stop := 24054 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 24201, root := ~, start := 24190, stop := 24200 ), rec( content := " is ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 24235, root := ~, start := 24205, stop := 24234 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 24303, root := ~, start := 24292, stop := 24302 ), rec( content := " is ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 24343, root := ~, start := 24307, stop := 24342 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 24428, root := ~, start := 24417, stop := 24427 ), rec( content := " is not \ present, then ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 24486, root := ~, start := 24450, stop := 24485 ), rec( content := " is used\ by default.\n ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "for a list (i.e. an adjacency list)", coun\ t := [ 3, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 8 ], name := "Mar\ k", next := 24572, root := ~, start := 24524, stop := 24571 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 9\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := \ "Text", name := "A", next := 24607, root := ~, start := 24597, stop := 24606 ) , rec( cont\ ent := " is a list of lists of positive integers in\n the range from ", coun\ t := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 3, 1, 7, 9 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], math\ mode := "Text", name := "C", next := 24682, root := ~, start := 24674, stop :=\ 24681 ), rec( content := " to ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "Length(", count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj",\ count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ],\ count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "A", next := 24706, root\ := ~, start := 24696, stop := 24705 ), rec( content := ")", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "C", next := 24711, root := ~, start := 24\ 686, stop := 24710 ), rec( content := ", then this function\n returns t\ he digraph with vertices\n ", count := [ 3, 1, 7, 9 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "E ^ 0 = ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name :=\ "M", next := 24797, root := ~, start := 24782, stop := 24796 ), rec( attribut\ es := rec( ), content := [ rec( content := "[1 .. Length(", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := \ rec( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode\ := "Text", name := "A", next := 24823, root := ~, start := 24813, stop := 248\ 22 ), rec( content := ")]", count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "C", next := 24829, root := ~, start := 24797, stop := 24828 ), rec( conten\ t := ",\n and edges corresponding to the entries of ", count := [ 3, 1,\ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmod\ e := "Text", name := "A", next := 24891, root := ~, start := 24881, stop := 24\ 890 ), rec( content := ". ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 3, 1, 7, 10 ], mathmode := "Text", name := "P", next := 24897, root := ~, star\ t := 24893, stop := 24896 ), rec( content := "\n\n More precisely, ther\ e is an edge from vertex ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 24960, root := ~, start := 24952, stop := 24959 ), rec( content := " to ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 24972, root := ~, start := 24964, stop :=\ 24971 ), rec( content := " if and\n only if ", count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "j", count := [ 3, 1, 7, 10 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 25004, root := ~, start := 24996, stop := 25003 ), rec( conte\ nt := " is in ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := re\ c( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "A", next := 25024, root := ~, start := 25014, stop := 250\ 23 ), rec( content := "[i]", count := [ 3, 1, 7, 10 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 25031, root := ~, start := 25011, stop := 25030 ), rec( conte\ nt := "; the source of this edge is\n ", count := [ 3, 1, 7, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 25076, root := ~, start := 25068, stop := 25075 ), rec( conte\ nt := " and the range is ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "j", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 25102, root := ~, start := 25094, stop := 25101 ), rec( content := ". If ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 25115, root := ~, start := 25107, stop :=\ 25114 ), rec( content := " occurs in\n ", count := [ 3, 1, 7, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "A", next := 25147, root := \ ~, start := 25137, stop := 25146 ), rec( content := "[i]", count := [ 3, 1, 7,\ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 10 ], mathmode := "Text", name := "C", next := 25154, root := ~, start := 25\ 134, stop := 25153 ), rec( content := " with multiplicity ", count := [ 3, 1, \ 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmod\ e := "Text", name := "C", next := 25181, root := ~, start := 25173, stop := 25\ 180 ), rec( content := ", then there are ", count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "k", count := [ 3, 1, 7, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 25206, root := ~, start := 25198, stop := 25205 ), rec( conte\ nt := "\n edges from ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 25234, root := ~, start := 25226, stop := 25233 ), rec( content := " to ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 25246, root := ~, start := 25238, stop :=\ 25245 ), rec( content := ".\n ", count := [ 3, 1, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], name := "Item\ ", next := 25261, root := ~, start := 24579, stop := 25260 ), rec( attributes :\ = rec( ), content := [ rec( content := "for three lists", count := [ 3, 1, 7,\ 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 11 ], name := "Mar\ k", next := 25297, root := ~, start := 25269, stop := 25296 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 1\ 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 12 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode \ := "Text", name := "A", next := 25332, root := ~, start := 25322, stop := 2533\ 1 ), rec( content := " is a duplicate-free list, and ", count := [ 3, 1, 7, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "source", count := [ 3, 1, 7, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmod\ e := "Text", name := "A", next := 25376, root := ~, start := 25363, stop := 25\ 375 ), rec( content := " and\n ", count := [ 3, 1, 7, 12 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "range", count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "A", next := 25401, root := ~, start := 25389, stop := 25400 ), rec( conte\ nt := " are lists of equal length consisting of positive integers\n in \ the list ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[1 .. Lengt\ h(", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "obj", count := [\ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "A", next := 25506, root := ~, st\ art := 25496, stop := 25505 ), rec( content := ")]", count := [ 3, 1, 7, 12 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ] , math\ mode := "Text", name := "C", next := 25512, root := ~, start := 25480, stop :=\ 25511 ), rec( content := ", then this function\n returns a digraph wit\ h vertices\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "E ^ 0 = ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "M", next \ := 25596, root := ~, start := 25581, stop := 25595 ), rec( attributes := rec( \ ), content := [ rec( content := "[1 .. Length(", count := [ 3, 1, 7, 12 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), c\ ontent := [ rec( content := "obj", count := [ 3, 1, 7, 12 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Tex\ t", name := "A", next := 25622, root := ~, start := 25612, stop := 25621 ), re\ c( content := ")]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C" , next\ := 25628, root := ~, start := 25596, stop := 25627 ), rec( content := ", and\ \n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Length(", c\ ount := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), re\ c( attributes := rec( ), content := [ rec( content := "source", count := [ 3,\ 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3\ , 1, 7, 12 ], mathmode := "Text", name := "A", next := 25665, root := ~, start\ := 25652, stop := 25664 ), rec( content := ")", count := [ 3, 1, 7, 12 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "C", next := 25670, root := ~, start := 25642, stop :=\ 25669 ), rec( content := " edges. For each ", count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 12 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "C", next := 25696, root := ~, start := 25688, stop := 25695 ), rec( conte\ nt := " in\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "[1 .. Length(", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "sou\ rce", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "A", next := 2573\ 7, root := ~, start := 25724, stop := 25736 ), rec( content := ")]", count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 25743, root := ~, star\ t := 25708, stop := 25742 ), rec( content := " there exists an edge with sourc\ e\n vertex ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ source[i]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next :\ = 25808, root := ~, start := 25792, stop := 25807 ), rec( content := " and ran\ ge vertex ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "range[i]", \ count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 25841, root := \ ~, start := 25826, stop := 25840 ), rec( content := ".\n See ", count :\ = [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( Attr := "DigraphSource" ), content := 0, count := [ 3, 1, 7, 12\ ], mathmode := "Text", name := "Ref", next := 25882, root := ~, start := 2585\ 5, stop := 25881 ), rec( content := " and ", count := [ 3, 1, 7, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "D\ igraphRange" ), content := 0, count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "Ref", next := 25913, root := ~, start := 25887, stop := 25912 ), rec( cont\ ent := ".\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1\ , 7, 13 ], mathmode := "Text", name := "P", next := 25927, root := ~, start :=\ 25923, stop := 25926 ), rec( content := "\n\n The vertices of the digr\ aph will be labelled by the elements of\n ", count := [ 3, 1, 7, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "obj", count := [ 3, 1, 7, 13 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 13 ], mathmode := "Text", name\ := "A", next := 26019, root := ~, start := 26009, stop := 26018 ), rec( conte\ nt := ".\n ", count := [ 3, 1, 7, 13 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 7, 12 ], name := "Item", next := 26034, root := ~, start := 253\ 04, stop := 26033 ), rec( attributes := rec( ), content := [ rec( content := \ "for an integer, and two lists", count := [ 3, 1, 7, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 14 ], name := "Mark", next := 2608\ 4, root := ~, start := 26042, stop := 26083 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n if ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "obj", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 2\ 6119, root := ~, start := 26109, stop := 26118 ), rec( content := " is an inte\ ger, and ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "source", co\ unt := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 26152, root := \ ~, start := 26139, stop := 26151 ), rec( content := " and\n ", count :=\ [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "range", count := [ 3, 1, 7, 1\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, \ 15 ], mathmode := "Text", name := "A", next := 26177, root := ~, start := 2616\ 5, stop := 26176 ), rec( content := " are lists of equal length consisting of \ positive integers\n in the list ", count := [ 3, 1, 7, 15 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "[1 .. ", count := [ 3, 1, 7, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( con\ tent := "obj", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", nex\ t := 26275, root := ~, start := 26265, stop := 26274 ), rec( content := "]", c\ ount := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next := 26280, root := \ ~, start := 26256, stop := 26279 ), rec( content := ", then this function\n \ returns a digraph with vertices\n ", count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "E ^ 0 = ", count := [ 3, 1, 7, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "T\ ext", name := "M", next := 26364, root := ~, start := 26349, stop := 26363 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[1 .. ", count := [ 3, 1, 7, \ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 15 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode\ := "Text", name := "A", next := 26383, root := ~, start := 26373, stop := 263\ 82 ), rec( content := "]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "C", next := 26388, root := ~, start := 26364, stop := 26387 ), rec( conte\ nt := ", and\n ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "Length(", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", \ root := ~ ), rec( attributes := rec( ), content := [ rec( content := "source"\ , count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) \ ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 26425, r\ oot := ~, start := 26412, stop := 26424 ), rec( content := ")", count := [ 3, \ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "C", next := 26430, root := ~, start \ := 26402, stop := 26429 ), rec( content := " edges. For each ", count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "i", count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "C", next := 26456, root := ~, start := 26448, stop :=\ 26455 ), rec( content := " in\n ", count := [ 3, 1, 7, 15 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "[1 .. Length(", count := [ 3, 1, 7, 15 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "source", count := [ 3, 1, 7, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "A", next := 26497, root := ~, start := 26484, stop := 26496 ), rec( conten\ t := ")]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next :=\ 26503, root := ~, start := 26468, stop := 26502 ), rec( content := " there ex\ ists an edge with source\n vertex ", count := [ 3, 1, 7, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "source[i]", count := [ 3, 1, 7, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Tex\ t", name := "C", next := 26568, root := ~, start := 26552, stop := 26567 ), rec( cont\ ent := " and range vertex ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "range[i]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next\ := 26601, root := ~, start := 26586, stop := 26600 ), rec( content := ".\n \ See ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( Attr := "DigraphSource" ), content := 0, coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "Ref", next := 26642, root :\ = ~, start := 26615, stop := 26641 ), rec( content := " and ", count := [ 3, 1\ , 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( Attr := "DigraphRange" ), content := 0, count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "Ref", next := 26673, root := ~, start := 26647, stop \ := 26672 ), rec( content := ".\n ", count := [ 3, 1, 7, 15 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], name := "Ite\ m", next := 26688, root := ~, start := 26091, stop := 26687 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a list and a function", count := [ \ 3, 1, 7, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 16 ], name := "Mar\ k", next := 26734, root := ~, start := 26696, stop := 26733 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode\ := "Text", name := "A", next := 26770, root := ~, start := 26759, stop := 267\ 69 ), rec( content := " is a list and ", count := [ 3, 1, 7, 17 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "func", count := [ 3, 1, 7, 17 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name\ := "A", next := 26796, root := ~, start := 26785, stop := 26795 ), rec( conte\ nt := " is a function taking 2\n arguments that are elements of ", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "A", next := 26870, root := ~, start := 26859 , stop\ := 26869 ), rec( content := ", and ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "func", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "A", next := 26887, root := ~, start := 26876, stop := 26886 ), rec( content \ := " returns\n ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "true", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "K", next :=\ 26915, root := ~, start := 26904, stop := 26914 ), rec( content := " or ", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "false", count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, \ 17 ], mathmode := "Text", name := "K", next := 26931, root := ~, start := 2691\ 9, stop := 26930 ), rec( content := ", then this operation creates a digraph w\ ith\n vertices ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "[1 .. Length(", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", next := 2702\ 0, root := ~, start := 27009, stop := 27019 ), rec( content := ")]", count := \ [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 27026, root := ~, star\ t := 26993, stop := 27025 ), rec( content := " and an edge from vertex\n \ ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "i", count := [ 3\ , 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 27067, root := ~, star\ t := 27059, stop := 27066 ), rec( content := " to vertex ", count := [ 3, 1, 7\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "j", count := [ 3, 1, 7, 17 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode :\ = "Text", name := "C", next := 27086, root := ~, start := 27078, stop := 27085\ ), rec( content := " if and only if\n ", count := [ 3, 1, 7, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "func", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], cou\ nt := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", next := 27124, root :=\ ~, start := 27113, stop := 27123 ), rec( content := "(", count := [ 3, 1, 7, \ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmod\ e := "Text", name := "A", next := 27136, root := ~, start := 27125, stop := 27\ 135 ), rec( content := "[i], ", count := [ 3, 1, 7, 17 ], mathmode := "Text", \ name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( c\ ontent := "list", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", \ next := 27152, root := ~, start := 27141, stop := 27151 ), rec( content := "[j\ ])", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 27160 , root\ := ~, start := 27110, stop := 27159 ), rec( content := " returns ", count := [ \ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "true", count := [ 3, 1, 7, 17 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ] , math\ mode := "Text", name := "K", next := 27180, root := ~, start := 27169, stop :=\ 27179 ), rec( content := ".\n ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], name := "Ite\ m", next := 27195, root := ~, start := 26741, stop := 27194 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a group, a list, and two function\ s", count := [ 3, 1, 7, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 18 ], name := "Mar\ k", next := 27254, root := ~, start := 27203, stop := 27253 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The arguments will be ", coun\ t := [ 3, 1, 7, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G, list, act, adj", count := \ [ 3, 1, 7, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 19 ], mathmode := "Text", name := "A", next := 27322, root := ~, star\ t := 27298, stop := 27321 ), rec( content := ". ", count := [ 3, 1, 7, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "P", next := 2\ 7328, root := ~, start := 27324, stop := 27327 ), rec( content := "\n\n \ Let ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count := \ [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 27350, root := ~, star\ t := 27342, stop := 27349 ), rec( content := " be a group acting on the object\ s in ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "list", count \ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 27398, root := ~ , star\ t := 27387, stop := 27397 ), rec( content := " via\n the action ", coun\ t := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "act", count := [ 3, 1, 7, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20\ ], mathmode := "Text", name := "A", next := 27432, root := ~, start := 27422, stop\ := 27431 ), rec( content := ", and let ", count := [ 3, 1, 7, 20 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "adj", count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "A", next := 27452, root := ~, start := 27442, stop := 27451 ), rec( conte\ nt := " be a function taking\n two objects from ", count := [ 3, 1, 7, \ 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "list", count := [ 3, 1, 7, 20 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmod\ e := "Text", name := "A", next := 27510, root := ~, start := 27499, stop := 27\ 509 ), rec( content := " as arguments and returning ", count := [ 3, 1, 7, 20 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "true", count := [ 3, 1, 7, 20 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text" , name\ := "C", next := 27549, root := ~, start := 27538, stop := 27548 ), rec( conte\ nt := " or\n ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "false", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "C", next := \ 27573, root := ~, start := 27561, stop := 27572 ), rec( content := ". The func\ tion ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "adj", count :\ = [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 27598, root := ~, star\ t := 27588, stop := 27597 ), rec( content := " will describe the adjacency\n \ between objects from ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "list", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next\ := 27667, root := ~, start := 27656, stop := 27666 ), rec( content := ", whic\ h is invariant under the action of\n ", count := [ 3, 1, 7, 20 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "G", count := [ 3, 1, 7, 20 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "A", next := 27724, root := ~, start := 27716, stop := 27723 ), rec( conte\ nt := ". This variant of the constructor returns a digraph with\n verti\ ces the objects of ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "list", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 27\ 824, root := ~, start := 27813, stop := 27823 ), rec( content := " and directe\ d edges ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[x, y]", co\ unt := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 20 ], mathmode := "Text", name := "C", next := 27857, root := \ ~, start := 27844, stop := 27856 ), rec( content := "\n when ", count :\ = [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "f(x, y)", count := [ 3, 1, 7,\ 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 20 ], mathmode := "Text", name := "C", next := 27885, root := ~, start := 27\ 871, stop := 27884 ), rec( content := " is ", count := [ 3, 1, 7, 20 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "true", count := [ 3, 1, 7, 20 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "C", next := 27900, root := ~, start := 27889, stop := 27899 ), rec( conte\ nt := ". ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "P", next := 27906, root := ~, start := 27902, stop :=\ 27905 ), rec( content := "\n\n The action of the group ", count := [ 3\ , 1, 7, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "G", count := [ 3, 1, 7, 21 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "A", next := 27948, root := ~, start := 27940, stop :=\ 27947 ), rec( content := " on the objects in ", count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "list", count := [ 3, 1, 7, 21 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 21 ], mathmode := "Text" , name\ := "A", next := 27978, root := ~, start := 27967, stop := 27977 ), rec( conte\ nt := " is stored\n in the attribute ", count := [ 3, 1, 7, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := \ "DigraphGroup" ), content := 0, count := [ 3, 1, 7, 21 ], mathmode := "Text", name\ := "Ref", next := 28040, root := ~, start := 28014, stop := 28039 ), rec( cont\ ent := ", and is used to speed up\n operations like ", count := [ 3, 1,\ 7, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Attr := "DigraphDiameter" ), content := 0, count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "Ref", next := 28119, root := ~, start := 28090, stop \ := 28118 ), rec( content := ".\n ", count := [ 3, 1, 7, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 19 ], name := "Ite\ m", next := 28134, root := ~, start := 27261, stop := 28133 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a Grape package graph", count := [ \ 3, 1, 7, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 22 ], name := "Mar\ k", next := 28180, root := ~, start := 28142, stop := 28179 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 2\ 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 23 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmode \ := "Text", name := "A", next := 28215, root := ~, start := 28205, stop := 2821\ 4 ), rec( content := " is a ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content :=\ [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "ht\ tps://gap-packages.github.io/grape", count := [ 3, 1, 7, 23 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmode := "T\ ext", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count\ := [ 3, 1, 7, 23 ], name := "Alt", next := 86, root := ~, start := 1, stop :=\ 85 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes \ := rec( ), content := [ rec( content := "Grape", count := [ 3, 1, 7, 23 ], na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Package", ne\ xt := 130, root := ~, start := 106, stop := 129 ) ], count := [ 3, 1, 7, 23 ], math\ mode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 13\ 5 ), rec( content := " package graph (i.e. a record for which the\n fun\ ction ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "IsGraph", co\ unt := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 23 ], mathmode := "Text", name := "C", next := 28303, root := \ ~, start := 28289, stop := 28302 ), rec( content := " returns ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "true", count := [ 3, 1, 7, 23 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], math\ mode := "Text", name := "K", next := 28323, root := ~, start := 28312, stop :=\ 28322 ), rec( content := "), then this function returns\n a digraph is\ omorphic to ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 23 ], mathmode := "Text", name := "A", next := 28395, root := \ ~, start := 28385, stop := 28394 ), rec( content := ".\n ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Ite\ m", next := 28410, root := ~, start := 28187, stop := 28409 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a binary relation", count := [ 3,\ 1, 7, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 24 ], name := "Mar\ k", next := 28452, root := ~, start := 28418, stop := 28451 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 2\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 25 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode \ := "Text", name := "A", next := 28487, root := ~, start := 28477, stop := 2848\ 6 ), rec( content := " is a binary relation on the points ", count := [ 3, 1, \ 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "[1 .. n]", count := [ 3, 1, 7, 25 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "C", next := 28538, root := ~, start := 28523, stop :=\ 28537 ), rec( content := " for\n some positive integer ", count := [ 3\ , 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "n", count := [ 3, 1, 7, 25 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "M", next := 28581, root := ~, start := 28573, stop :=\ 28580 ), rec( content := ", then this function returns the digraph\n d\ efined by ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "A", next := 28651, root := \ ~, start := 28641, stop := 28650 ), rec( content := ". Specifically, this func\ tion returns a digraph\n which has ", count := [ 3, 1, 7, 25 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "n", count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "M", next := 28725, root := ~, start := 28717, stop := 28724 ), rec( conte\ nt := " vertices, and which has an edge with source ", count := [ 3, 1, 7, 25 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 25 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "C", next := 28778, root := ~, start := 28770, stop := 28777 ), rec( conte\ nt := "\n and range ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "j", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "C", next :=\ 28805, root := ~, start := 28797, stop := 28804 ), rec( content := " if and o\ nly if ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[i,j]", cou\ nt := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "C", next := 28833, root := \ ~, start := 28821, stop := 28832 ), rec( content := " is a pair in\n th\ e binary relation ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ obj", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "A", next := 2888\ 5, root := ~, start := 28875, stop := 28884 ), rec( content := ".\n ", coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], name := "Ite\ m", next := 28900, root := ~, start := 28459, stop := 28899 ) ], count := [ 3, 1, 7, 8 ], mathmode := "Text", name := "List", next := 28912, root := ~, start := 24511, stop := 28911 ), rec( content := "\n\n \ ", count := [ 3, 1, 7, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 5, 8, 10], [2, 3, 4, 2, 5, 6, 8, 9, 10], [1],\n> \ [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9], [1, 2, 7,\ 8], [3, 5]]);\n\ngap> gr :\ = Digraph([\"a\", \"b\", \"c\"], [\"a\"], [\"b\"]);\n\ngap> gr := Digraph(5, [1, 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, \ 1]);\n\ngap> Petersen := Grap\ h(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y) return Intersection(x\ , y) = []; end);;\ngap> Digraph(Petersen);\n\ngap> gr := Digraph([1 .. 10], ReturnTrue);\n", count := [ 3, 1, 7, 28 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 28 ], mathmode := "Text", name := "Example", next := 29654, root := ~, start := 28918, stop := 29653 ), rec( content := "\n\n \ The next example illustrates the uses of the fourth and fifth variants of\n \ this constructor. The resulting digraph is a strongly regular graph, and\ \n it is actually the point graph of the van Lint-Schrijver partial\n \ geometry, ", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "vLS81" ) , content := 0, count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "Cite", next := 29921, root := ~, start := 29902, stop := 29920 ), rec( content := ". The a\ lgebraic description\n is taken from the seminal paper of Calderbank and \ Kantor ", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "CK86" ), content := 0, count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "Cite", next := 30039, root := ~, start := 30013, stop := 30038 ), rec( content := ".\n", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, x -> Order(x) = 5);\nZ(\ 3^4)^64\ngap> L := Union([Zero(f)], List(Group(gamma)));\n[ 0*Z(3), Z(3)^0, Z(\ 3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ]\ngap> omega := Union(List(L, x -> L\ ist(Difference(L, [x]), y -> x - y)));\n[ Z(3)^0, Z(3), Z(3^4)^5, Z(3^4)^7, Z(\ 3^4)^8, Z(3^4)^13, Z(3^4)^15, \n Z(3^4)^16, Z(3^4)^21, Z(3^4)^23, Z(3^4)^24, \ Z(3^4)^29, Z(3^4)^31, \n Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4)^45, Z(3^4)^4\ 7, Z(3^4)^48, \n Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(3^4)^63, Z(3^4\ )^64, \n Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 ]\ngap> adj :=\ function(x, y)\n> return x - y in omega;\n> end;\nfunction( x, y ) ... end\ \ngap> digraph := Digraph(AsList(f), adj);\n\ngap> group := Group(Z(3));;\ngap> act := \\*;\n\ngap> digraph := Digraph(group, List(f), act, adj);\n\n", count := [ 3, 1, 7, 30 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 30 ], mathmode := "Text", name := "Example", next := 31009, root := ~, start := 30041, stop := 31008 ), rec( content := "\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Description", next := 31026, root := ~, start := 23995, stop := 31025 ) ], count := [ 3, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 31040, root := ~, start := 23674, stop := 31039 ), rec( content := "\n\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByAdjace\ ncyMatrix" ), content := 0, count := [ 3, 1, 8, 2 ], mathmode := "Text", name := "Oper", next := 31119, root := ~, start := 31061, stop := 31118 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 31151, root := ~, start := 31122, stop := 31150 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 31214, root := ~, start := 31203, stop := 31213 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 31360, root := ~, start := 31349, stop := 31359 ), rec( content := " is ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 31394, root := ~, start := 31364, stop := 31393 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 31462, root := ~, start := 31451, stop := 31461 ), rec( content := " is ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 31502, root := ~, start := 31466, stop := 31501 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 31587, root := ~, start := 31576, stop := 31586 ), rec( content := " is not \ present, then ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 31645, root := ~, start := 31609, stop := 31644 ), rec( content := " is used\ by default.", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "P", next := 31669, root := ~, start := 31665, stop := 31668 ), rec( content := "\n\n \ If ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "A", next := 31689, root := ~, start := 31678, stop := 31688 ), rec( content := " is the \ adjacency matrix of a digraph in the sense of\n ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "Ref", next := 31776, root := ~, start := 31747, stop := 31775 ), rec( content := ", then t\ his operation returns the digraph\n which is defined by ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "A", next := 31853, root := ~, start := 31842, stop := 31852 ), rec( content := ". ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "P", next := 31859, root := ~, start := 31855, stop := 31858 ), rec( content := "\n\n \ Alternatively, if ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 31894, root := ~, start := 31883, stop := 31893 ), rec( content := " is a sq\ uare boolean matrix, then this\n operation returns the digraph with ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 31986, root := ~, start := 31972, stop := 31985 ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 31997, root := ~, start := 31986, stop := 31996 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 32005, root := ~, start := 31997, stop := 32004 ), rec( content := "\n ve\ rtices which has the edge ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 32050, root := ~, start := 32038, stop := 32049 ), rec( content := " if and \ only if\n ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 32081, root := ~, start := 32070, stop := 32080 ), rec( attributes := rec( ), content := [ rec( content := "[\ i][j]", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 32094, root := ~, start := 32081, stop := 32093 ), rec( content := " is ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "K", next := 32109, root := ~, start := 32098, stop := 32108 ), rec( content := ". ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 6 ], mathmode := "Text", name := "P", next := 32115, root := ~, start := 32111, stop := 32114 ), rec( content := "\n\n \ ", count := [ 3, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphByAdjacencyMatrix([\n> [0, 1, 0, 2, 0],\n> [1, 1, 1, 0, 1],\n> [0\ , 3, 2, 1, 1],\n> [0, 0, 1, 0, 1],\n> [2, 0, 0, 0, 0]]);\n\ngap> D := DigraphByAdjacencyMatrix([\n> [true,\ false, true],\n> [false, false, true],\n> [false, true, false]]);\n\ngap> OutNeighbours(D);\n[ [ 1, 3 ], [ 3 ],\ [ 2 ] ]\ngap> D := DigraphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, fal\ se, true],\n> [false, false, true],\n> [false, true, false]]);\n\n", count := [ 3, 1, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 7 ], mathmode := "Text", name := "Example", next := 32711, root := ~, start := 32121, stop := 32710 ), rec( content := "\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 32728, root := ~, start := 31154, stop := 32727 ) ], count := [ 3, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 32742, root := ~, start := 31046, stop := 32741 ), rec( content := "\n\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list[, n\ ]", Name := "DigraphByEdges" ), content := 0, count := [ 3, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 32816, root := ~, start := 32763, stop := 32815 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 32848, root := ~, start := 32819, stop := 32847 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 32911, root := ~, start := 32900, stop := 32910 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 33057, root := ~, start := 33046, stop := 33056 ), rec( content := " is ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 33091, root := ~, start := 33061, stop := 33090 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 33159, root := ~, start := 33148, stop := 33158 ), rec( content := " is ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 33199, root := ~, start := 33163, stop := 33198 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 33284, root := ~, start := 33273, stop := 33283 ), rec( content := " is not \ present, then ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 33342, root := ~, start := 33306, stop := 33341 ), rec( content := " is used\ by default.", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "P", next := 33366, root := ~, start := 33362, stop := 33365 ), rec( content := "\n\n \ If ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "A", next := 33386, root := ~, start := 33375, stop := 33385 ), rec( content := " is list\ of pairs of positive integers, then this function\n returns the digraph wi\ th the minimum number of vertices ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "C", next := 33513, root := ~, start := 33505, stop := 33512 ), rec( content := " such th\ at\n its list equal ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "A", next := 33554, root := ~, start := 33543, stop := 33553 ), rec( content := ".", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "P", next := 33559, root := ~, start := 33555, stop := 33558 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 33605, root := ~, start := 33597, stop := 33604 ), rec( content := " is a po\ sitive integer with\n ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 1, 9, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := \ "Text", name := "A", next := 33648, root := ~, start := 33640, stop := 33647 ) , rec( content := "\ >= m", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 33657, root := ~, start := 33637, stop := 33656 ), rec( content := " (with " , count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 33672, root := ~, start := 33664, stop := 33671 ), rec( content := " defined\ as above),\n then this function returns the digraph with ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 33748, root := ~, start := 33740, stop := 33747 ), rec( content := " vertice\ s and list\n ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 33782, root := ~, start := 33771, stop := 33781 ), rec( content := ". ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "P", next := 33788, root := ~, start := 33784, stop := 33787 ), rec( content := "\n\n \ See ", count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "Ref", next := 33824, root := ~, start := 33798, stop := 33823 ), rec( content := ".\n\n \ ", count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6], \ [5, 2], [5, 4], [5, 6], [6, 6]]);\n\ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6\ ], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\ngap> DigraphByEdges(IsMutableDigraph, \n> [[1, 3], [2, 1], [2, 3],\ [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\n", count := [ 3, 1, 9, 7 ], name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 9, 7 ], mathmode := "Text", name := "Example", next := 34354, root := ~, start := 33831, stop := 34353 ), rec( content := "\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 34371, root := ~, start := 32851, stop := 34370 ) ], count := [ 3, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 34385, root := ~, start := 32748, stop := 34384 ), rec( content := "\n\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "G, edges[, n]", Name := "EdgeOrbitsDigra\ ph" ), content := 0, count := [ 3, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 34458, root := ~, start := 34406, stop := 34457 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable digraph.\n ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 34509, root := ~, start := 34461, stop := 34508 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34541, root := ~, start := 34533, stop := 34540 ), rec( content := " is a pe\ rmutation group, ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34578, root := ~, start := 34566, stop := 34577 ), rec( content := " is an e\ dge or list of\n edges, and ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34624, root := ~, start := 34616, stop := 34623 ), rec( content := " is a no\ n-negative integer such that ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34669, root := ~, start := 34661, stop := 34668 ), rec( content := " fixes\n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 1, 10, 3 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := \ "Text", name := "A", next := 34697, root := ~, start := 34689, stop := 34696 ) , rec( content := "\ ]", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "C", next := 34702, root := ~, start := 34680, stop := 34701 ), rec( content := " setwise\ , then\n this operation returns an immutable digraph with ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34778, root := ~, start := 34770, stop := 34777 ), rec( content := " vertice\ s and the union\n of the orbits of the edges in ", count := [ 3, 1, 10, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ edges ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34850, root := ~, start := 34836, stop := 34849 ), rec( content := " under t\ he action of the\n permutation group ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 34905, root := ~, start := 34897, stop := 34904 ), rec( content := ". An ed\ ge in this context is simply a pair of\n positive integers.\n ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "P", next := 34983, root := ~, start := 34979, stop := 34982 ), rec( content := "\n\n \ If the optional third argument ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "A", next := 35028, root := ~, start := 35020, stop := 35027 ), rec( content := " is not \ present, then the largest\n moved point of the permutation group ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "A", next := 35111, root := ~, start := 35103, stop := 35110 ), rec( content := " is used\ by default.", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 10, 5 ], mathmode := "Text", name := "P", next := 35135, root := ~, start := 35131, stop := 35134 ), rec( content := "\n\n \ ", count := [ 3, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),\n> \ [[1, 2], [4, 5]], 5);\n\ngap> OutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4,\ 5 ], [ 1, 3, 5 ], [ ] ]\ngap> RepresentativeOutNeighbours(digraph);\n[ [ 2, \ 4, 5 ], [ ] ]\n", count := [ 3, 1, 10, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 6 ], mathmode := "Text", name := "Example", next := 35493, root := ~, start := 35141, stop := 35492 ), rec( content := "\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 35510, root := ~, start := 34512, stop := 35509 ) ], count := [ 3, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 35524, root := ~, start := 34391, stop := 35523 ), rec( content := "\n\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbours" ), content := 0, count := [ 3, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 35600, root := ~, start := 35545, stop := 35599 ), rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbors" ), content := 0, count := [ 3, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 35657, root := ~, start := 35603, stop := 35656 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Returns", next := 35689, root := ~, start := 35660, stop := 35688 ), rec( attributes := rec( ), content := [ rec( content := "\n If th\ e optional first argument ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 35750, root := ~, start := 35739, stop := 35749 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph be\ ing created will\n belong to. For example, if ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 35892, root := ~, start := 35881, stop := 35891 ), rec( content := " is ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 35926, root := ~, start := 35896, stop := 35925 ), rec( content := ",\n the\ n the digraph being created will be mutable, if ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 35992, root := ~, start := 35981, stop := 35991 ), rec( content := " is ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 36030, root := ~, start := 35996, stop := 36029 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 36113, root := ~, start := 36102, stop := 36112 ), rec( content := " is not \ present, then ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 36169, root := ~, start := 36135, stop := 36168 ), rec( content := " is used\ by default.", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "P", next := 36193, root := ~, start := 36189, stop := 36192 ), rec( content := "\n\n If\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "A", next := 36211, root := ~, start := 36200, stop := 36210 ), rec( content := " is a li\ st of lists of positive integers list the range\n ", count := [ 3, 1, 11, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. Length(", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 36296, root := ~, start := 36285, stop := 36295 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 36302, root := ~, start := 36269, stop := 36301 ), rec( content := ", then t\ his function returns the\n digraph with vertices ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ^0=", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "M", next := 36370, root := ~, start := 36359, stop := 36369 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. Length(", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 36397, root := ~, start := 36386, stop := 36396 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 36403, root := ~, start := 36370, stop := 36402 ), rec( content := ", and\n \ edges corresponding to the entries of ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "A", next := 36460, root := ~, start := 36449, stop := 36459 ), rec( content := ". More p\ recisely, there is an\n edge with source vertex ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 36524, root := ~, start := 36516, stop := 36523 ), rec( content := " and ran\ ge vertex ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 36550, root := ~, start := 36542, stop := 36549 ), rec( content := " if ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 36562, root := ~, start := 36554, stop := 36561 ), rec( content := " is list\ \n ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 36587, root := ~, start := 36576, stop := 36586 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 36594, root := ~, start := 36573, stop := 36593 ), rec( content := ". ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "P", next := 36600, root := ~, start := 36596, stop := 36599 ), rec( content := "\n\n If\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 36615, root := ~, start := 36607, stop := 36614 ), rec( content := " occurs \ list ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 7 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := \ "Text", name := "A", next := 36642, root := ~, start := 36631, stop := 36641 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 36649, root := ~, start := 36628, stop := 36648 ), rec( content := " with mu\ ltiplicity ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 36676, root := ~, start := 36668, stop := 36675 ), rec( content := ",\n the\ n there are ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 36703, root := ~, start := 36695, stop := 36702 ), rec( content := " multipl\ e edges from ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 36732, root := ~, start := 36724, stop := 36731 ), rec( content := " to ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 36744, root := ~, start := 36736, stop := 36743 ), rec( content := ". ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "P", next := 36750, root := ~, start := 36746, stop := 36749 ), rec( content := "\n\n Se\ e ", count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InNeigh\ bours" ), content := 0, count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "Ref", next := 36784, root := ~, start := 36758, stop := 36783 ), rec( content := ".\n\n \ ", count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphByInNeighbours([\n> [2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10]\ ,\n> [1], [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9],\ [1, 2, 7, 8], [3, 5]]);\n\ngap>\ D := DigraphByInNeighbours([[2, 3, 2], [1], [1, 2, 3]]);\n\ngap> D := DigraphByInNeighbours(IsMutableDigra\ ph, \n> [[2, 3, 2], [1], [1, 2, 3]]);\n\n", count := [ 3, 1, 11, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 11, 9 ], mathmode := "Text", name := "Example", next := 37320, root := ~, start := 36791, stop := 37319 ), rec( content := "\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 37337, root := ~, start := 35692, stop := 37336 ) ], count := [ 3, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 37351, root := ~, start := 35530, stop := 37350 ), rec( content := "\n\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "G[, gens]", Name := "CayleyDigraph" ), content := 0, count := [ 3, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 37416, root := ~, start := 37372, stop := 37415 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble digraph.", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 37459, root := ~, start := 37419, stop := 37458 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37492, root := ~, start := 37484, stop := 37491 ), rec( content := " be any \ group and let ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37525, root := ~, start := 37514, stop := 37524 ), rec( content := " be a li\ st of elements of\n ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37563, root := ~, start := 37555, stop := 37562 ), rec( content := ". This o\ peration returns an immutable digraph that corresponds to\n the Cayley grap\ h of ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37661, root := ~, start := 37653, stop := 37660 ), rec( content := " with re\ spect\n ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37690, root := ~, start := 37679, stop := 37689 ), rec( content := ". The ve\ rtices are the elements of ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37733, root := ~, start := 37725, stop := 37732 ), rec( content := ". There \ exists an edge\n from the vertex ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 37784, root := ~, start := 37776, stop := 37783 ), rec( content := " to the \ vertex ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 37807, root := ~, start := 37799, stop := 37806 ), rec( content := " if and \ only if there exists\n a generator ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 37860, root := ~, start := 37852, stop := 37859 ), rec( content := " in ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 37875, root := ~, start := 37864, stop := 37874 ), rec( content := " such th\ at ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ * g = y", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 37902, root := ~, start := 37886, stop := 37901 ), rec( content := ". ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "P", next := 37908, root := ~, start := 37904, stop := 37907 ), rec( content := "\n\n \ ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( content := " FIXME W\ hat is the correspondence between vertices and elements? ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "A", next := 38035, root := ~, start := 38024, stop := 38034 ), rec( content := " is not \ present, then the\n generators of ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "A", next := 38087, root := ~, start := 38079, stop := 38086 ), rec( content := " are use\ d by default.", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "P", next := 38112, root := ~, start := 38108, stop := 38111 ), rec( content := "\n\n \ The digraph created by this operation belongs to the category ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 38215, root := ~, start := 38180, stop := 38214 ), rec( content := ", the gr\ oup ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "A", next := 38235, root := ~, start := 38227, stop := 38234 ), rec( content := " can be \ recovered from the\n digraph using ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "GroupOf\ CayleyDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 38314, root := ~, start := 38280, stop := 38313 ), rec( content := ", and th\ e generators\n ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "A", next := 38350, root := ~, start := 38339, stop := 38349 ), rec( content := " can be \ obtained using ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfCayleyDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 38412, root := ~, start := 38373, stop := 38411 ), rec( content := ".", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 6 ], mathmode := "Text", name := "P", next := 38417, root := ~, start := 38413, stop := 38416 ), rec( content := "\n\n \ Note that this function can only return an immutable digraph.\n\n ", count := [ 3, 1, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(8);\n\ngap> Cay\ leyDigraph(G);\n\ngap> G := Dihed\ ralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\ \n\ngap> digraph := CayleyDigraph\ (G, [()]);\n\ngap> GroupOfCayleyDi\ graph(digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]", count := [ 3, 1, 12, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 12, 7 ], mathmode := "Text", name := "Example", next := 38973, root := ~, start := 38490, stop := 38972 ), rec( content := "\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 38990, root := ~, start := 37462, stop := 38989 ) ], count := [ 3, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 39004, root := ~, start := 37357, stop := 39003 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 39018, root := ~, start := 20959, stop := 39017 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Changing represen\ tations", count := [ 3, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 39074, root := ~, start := 39031, stop := 39073 ), rec( content := "\n ", count := [ 3, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsBinaryRelatio\ n" ), content := 0, count := [ 3, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 39139, root := ~, start := 39094, stop := 39138 ), rec( attributes := rec( ), content := [ rec( content := "A binary \ relation.", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 39179, root := ~, start := 39142, stop := 39178 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "A", next := 39217, root := ~, start := 39203, stop := 39216 ), rec( content := " is a di\ graph with a positive number of vertices\n ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "M", next := 39278, root := ~, start := 39270, stop := 39277 ), rec( content := ", and no\ multiple edges, then this operation returns a binary\n relation on the poi\ nts ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..n]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 39380, root := ~, start := 39367, stop := 39379 ), rec( content := ".\n T\ he pair ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 39407, root := ~, start := 39395, stop := 39406 ), rec( content := " is in t\ he binary relation if and only if\n ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 39465, root := ~, start := 39453, stop := 39464 ), rec( content := " is an e\ dge in ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "A", next := 39494, root := ~, start := 39480, stop := 39493 ), rec( content := ".\n " , count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 1, 4 ], mathmode := "Text", name := "P", next := 39504, root := ~, start := 39500, stop := 39503 ), rec( content := "\n", count := [ 3, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 2], [1, 2], [2], [3, 4]]);\n\ngap> AsBinaryRelation(D);\nBinary Relation on 4 points\n", count := [ 3, 2, 1, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 1, 5 ], mathmode := "Text", name := "Example", next := 39686, root := ~, start := 39505, stop := 39685 ), rec( content := "\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 39703, root := ~, start := 39182, stop := 39702 ) ], count := [ 3, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 39717, root := ~, start := 39079, stop := 39716 ), rec( content := "\n\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]trans[, \ n]", Name := "AsDigraph" ), content := 0, count := [ 3, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 39787, root := ~, start := 39738, stop := 39786 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "K", next := 39824, root := ~, start := 39813, stop := 39823 ), rec( content := ".", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 39835, root := ~, start := 39790, stop := 39834 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 39898, root := ~, start := 39887, stop := 39897 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 40044, root := ~, start := 40033, stop := 40043 ), rec( content := " is ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 40078, root := ~, start := 40048, stop := 40077 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 40146, root := ~, start := 40135, stop := 40145 ), rec( content := " is ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 40186, root := ~, start := 40150, stop := 40185 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 40271, root := ~, start := 40260, stop := 40270 ), rec( content := " is not \ present, then ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 40329, root := ~, start := 40293, stop := 40328 ), rec( content := " is used\ by default.", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "P", next := 40353, root := ~, start := 40349, stop := 40352 ), rec( content := "\n\n \ If ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 40374, root := ~, start := 40362, stop := 40373 ), rec( content := " is a tr\ ansformation, and ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 40408, root := ~, start := 40400, stop := 40407 ), rec( content := " is a no\ n-negative integer\n such that the restriction of ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 40480, root := ~, start := 40468, stop := 40479 ), rec( content := " to ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := \ "Text", name := "A", next := 40501, root := ~, start := 40493, stop := 40500 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 40506, root := ~, start := 40484, stop := 40505 ), rec( content := " defines\ \n a transformation of ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := \ "Text", name := "A", next := 40556, root := ~, start := 40548, stop := 40555 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 40561, root := ~, start := 40539, stop := 40560 ), rec( content := ", then " , count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 40584, root := ~, start := 40568, stop := 40583 ), rec( content := "\n re\ turns the functional digraph with ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 40633, root := ~, start := 40625, stop := 40632 ), rec( content := " vertice\ s defined by\n ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 40670, root := ~, start := 40658, stop := 40669 ), rec( content := ". See ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsFunct\ ionalDigraph" ), content := 0, count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 40709, root := ~, start := 40676, stop := 40708 ), rec( content := ".\n " , count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "P", next := 40719, root := ~, start := 40715, stop := 40718 ), rec( content := "\n\n \ Specifically, the digraph returned by ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 40779, root := ~, start := 40763, stop := 40778 ), rec( content := " has ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "A", next := 40793, root := ~, start := 40785, stop := 40792 ), rec( content := " edges:\\ n for each vertex ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 40829, root := ~, start := 40821, stop := 40828 ), rec( content := " in ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := \ "Text", name := "A", next := 40850, root := ~, start := 40842, stop := 40849 ) , rec( content := "\ ]", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 40855, root := ~, start := 40833, stop := 40854 ), rec( content := ", there \ is a unique edge\n with source ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 40904, root := ~, start := 40896, stop := 40903 ), rec( content := "; this e\ dge has range ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ^", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "trans", count := [ 3, 2, 2, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := \ "Text", name := "A", next := 40943, root := ~, start := 40931, stop := 40942 ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 40947, root := ~, start := 40926, stop := 40946 ), rec( content := ".\n " , count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "P", next := 40957, root := ~, start := 40953, stop := 40956 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 41003, root := ~, start := 40995, stop := 41002 ), rec( content := " is not \ supplied, then the degree of\n the transformation ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 41075, root := ~, start := 41063, stop := 41074 ), rec( content := " is used\ by default. If the restriction of\n ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 41135, root := ~, start := 41123, stop := 41134 ), rec( content := " to ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 41156, root := ~, start := 41148, stop := 41155 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 41161, root := ~, start := 41139, stop := 41160 ), rec( content := " does no\ t define a transformation of\n ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 41219, root := ~, start := 41211, stop := 41218 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 41224, root := ~, start := 41202, stop := 41223 ), rec( content := ", then " , count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph(", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "trans", count := [ 3, 2, 2, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 41256, root := ~, start := 41244, stop := 41255 ) , rec( content := "\ , ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 41266, root := ~, start := 41258, stop := 41265 ) , rec( content := "\ )", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 41271, root := ~, start := 41231, stop := 41270 ), rec( content := "\n re\ turns ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "K", next := 41295, root := ~, start := 41284, stop := 41294 ), rec( content := ".\n\n \ ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTransformation( [\ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )\ngap> AsDigraph(f);\n\ngap> AsDigraph(f, 4);\n\ngap> AsDigraph(f, 5);\nfail\ngap> b := BinaryRelationOnPoi\ nts(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation on 5 po\ ints\ngap> D := AsDigraph(b);\n\n\ ", count := [ 3, 2, 2, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 7 ], mathmode := "Text", name := "Example", next := 41788, root := ~, start := 41302, stop := 41787 ), rec( content := "\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 41805, root := ~, start := 39838, stop := 41804 ) ], count := [ 3, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 41819, root := ~, start := 39723, stop := 41818 ), rec( content := "\n\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph" ), content := 0, count := [ 3, 2, 3, 2 ], mathmode := "Text", name := "Oper", next := 41874, root := ~, start := 41840, stop := 41873 ), rec( attributes := rec( ), content := [ rec( content := "A ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 3, 2, 3, 3 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package\ graph.", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 41920, root := ~, start := 41877, stop := 41919 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "A", next := 41958, root := ~, start := 41944, stop := 41957 ), rec( content := " is a mu\ table or immutable digraph without multiple edges, then this operation\n re\ turns a ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 3, 2, 3, 3 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package\ graph that is isomorphic to ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "A", next := 42109, root := ~, start := 42095, stop := 42108 ), rec( content := ". ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "P", next := 42115, root := ~, start := 42111, stop := 42114 ), rec( content := "\n\n \ If ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "A", next := 42138, root := ~, start := 42124, stop := 42137 ), rec( content := " is a mu\ ltidigraph, then since ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 3, 2, 3, 4 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 4 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " does no\ t support\n multiple edges, the multiple edges will be reduced to a single \ edge in the\n result. In order words, for a multidigraph this operation wil\ l return the\n same as ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raph(DigraphRemoveAllMultipleEdges(", count := [ 3, 2, 3, 4 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "C", next := 42406, root := ~, start := 42363, stop := 42405 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "A", next := 42420, root := ~, start := 42406, stop := 42419 ), rec( attributes := rec( ), content := [ rec( content := ")\ )", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "C", next := 42429, root := ~, start := 42420, stop := 42428 ), rec( content := ".\n\n \ ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y)\ return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nrec(\n adja\ cencies := [ [ 3, 5, 8 ] ],\n group := \n Group( [ ( 1, 2, 3, 5, 7)( 4, 6, \ 8, 9,10), ( 2, 4)( 6, 9)( 7,10) \n ] ),\n isGraph := true,\n names := [ \ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], \n [ 2, 4 ], [ 1, 5 ], \ [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],\n order := 10,\n representatives := [ 1 ],\n\ schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digraph(Petersen\ );\n\ngap> Graph(last) = Peterse\ n;\ntrue", count := [ 3, 2, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 5 ], mathmode := "Text", name := "Example", next := 43082, root := ~, start := 42436, stop := 43081 ), rec( content := "\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 43099, root := ~, start := 41923, stop := 43098 ) ], count := [ 3, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 43113, root := ~, start := 41825, stop := 43112 ), rec( content := "\n\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsGraph" ), content := 0, count := [ 3, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 43170, root := ~, start := 43134, stop := 43169 ), rec( attributes := rec( ), content := [ rec( content := "A ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "https://gap-packages.gi\ thub.io/grape", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 80, root := ~, start := 17, stop := 79 ) ], count := [ 3, 2, 4, 3 ], name := "Alt", next := 86, root := ~, start := 1, stop := 85 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 4, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], name := "Package", next := 130, root := ~, start := 106 , stop := 129 ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 136, root := ~, start := 86, stop := 135 ), rec( content := " package\ graph.", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 43216, root := ~, start := 43173, stop := 43215 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 43254, root := ~, start := 43240, stop := 43253 ), rec( content := " is a di\ graph, then this method returns the same as\n ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ), content := 0, count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 43329, root := ~, start := 43310, stop := 43328 ), rec( content := ", except\ that if ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 43360, root := ~, start := 43346, stop := 43359 ), rec( content := " is immu\ table, then the\n result will be stored as a mutable attribute of ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 43450, root := ~, start := 43436, stop := 43449 ), rec( content := ".\n I\ n this latter case, when ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sGraph(", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "C", next := 43497, root := ~, start := 43482, stop := 43496 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 43511, root := ~, start := 43497, stop := 43510 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "C", next := 43519, root := ~, start := 43511, stop := 43518 ), rec( content := " is call\ ed\n subsequently, the same ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object \ will be returned as before.\n\n ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], []]);\n\ngap> G := AsGraph(D);\nrec( adjacencies := [ [ 1, 2 ], [ 3 ], [ ] ], \ group := Group(()), \n isGraph := true, names := [ 1 .. 3 ], order := 3, \n \ representatives := [ 1, 2, 3 ], schreierVector := [ -1, -2, -3 ] )\n", count := [ 3, 2, 4, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 4, 4 ], mathmode := "Text", name := "Example", next := 43930, root := ~, start := 43603, stop := 43929 ), rec( content := "\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 43947, root := ~, start := 43219, stop := 43946 ) ], count := [ 3, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 43961, root := ~, start := 43119, stop := 43960 ), rec( content := "\n\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsTransformatio\ n" ), content := 0, count := [ 3, 2, 5, 2 ], mathmode := "Text", name := "Attr", next := 44027, root := ~, start := 43982, stop := 44026 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "K", next := 44071, root := ~, start := 44060, stop := 44070 ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 44081, root := ~, start := 44030, stop := 44080 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 44119, root := ~, start := 44105, stop := 44118 ), rec( content := " is a fu\ nctional digraph, then ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 44173, root := ~, start := 44150, stop := 44172 ), rec( content := "\n re\ turns the transformation which is defined by ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 44239, root := ~, start := 44225, stop := 44238 ), rec( content := ". See " , count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsFunct\ ionalDigraph" ), content := 0, count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Ref", next := 44285, root := ~, start := 44246, stop := 44284 ), rec( content := ". Otherw\ ise,\n ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation(", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 44326, root := ~, start := 44302, stop := 44325 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 44340, root := ~, start := 44326, stop := 44339 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 44348, root := ~, start := 44340, stop := 44347 ), rec( content := " returns\ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "K", next := 44368, root := ~, start := 44357, stop := 44367 ), rec( content := ". ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "P", next := 44375, root := ~, start := 44371, stop := 44374 ), rec( content := "\n\n \ If ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 44398, root := ~, start := 44384, stop := 44397 ), rec( content := " is a fu\ nctional digraph with ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 44436, root := ~, start := 44428, stop := 44435 ), rec( content := " vertice\ s, then\n ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation(", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 44480, root := ~, start := 44456, stop := 44479 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 44494, root := ~, start := 44480, stop := 44493 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 44502, root := ~, start := 44494, stop := 44501 ), rec( content := " will re\ turn the\n transformation ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 44546, root := ~, start := 44538, stop := 44545 ), rec( content := " of degr\ ee at most ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 44573, root := ~, start := 44565, stop := 44572 ), rec( content := " where f\ or each ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ \n \\leq i \\leq n", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 44617, root := ~, start := 44589, stop := 44616 ), rec( content := ", ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ^ f", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 44631, root := ~, start := 44619, stop := 44630 ), rec( content := " is equa\ l to the unique out-neighbour of\n vertex ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 44691, root := ~, start := 44683, stop := 44690 ), rec( content := " in ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 44709, root := ~, start := 44695, stop := 44708 ), rec( content := ".\n\n \ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [3], [2]]);\n\ngap> AsTransformation(D);\nTransformation( [ 1, 3, 2 ] )\ngap> D := Cycl\ eDigraph(IsMutableDigraph, 3);\n\nga\ p> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] )\ngap> AsPermutation(las\ t);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> AsTransformation(D);\nfail", count := [ 3, 2, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 5, 5 ], mathmode := "Text", name := "Example", next := 45175, root := ~, start := 44716, stop := 45174 ), rec( content := "\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 45192, root := ~, start := 44084, stop := 45191 ) ], count := [ 3, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 45206, root := ~, start := 43967, stop := 45205 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 45220, root := ~, start := 39022, stop := 45219 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "New digraphs from\ old", count := [ 3, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 45273, root := ~, start := 45233, stop := 45272 ), rec( content := "\n ", count := [ 3, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopy" ), content := 0, count := [ 3, 3, 1, 2 ], mathmode := "Text", name := "Oper", next := 45342, root := ~, start := 45293, stop := 45341 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opy" ), content := 0, count := [ 3, 3, 1, 4 ], mathmode := "Text", name := "Oper", next := 45392, root := ~, start := 45345, stop := 45391 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopySame\ Mutability" ), content := 0, count := [ 3, 3, 1, 6 ], mathmode := "Text", name := "Oper", next := 45449, root := ~, start := 45395, stop := 45448 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopy" ), content := 0, count := [ 3, 3, 1, 8 ], mathmode := "Text", name := "Oper", next := 45492, root := ~, start := 45452, stop := 45491 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Returns", next := 45524, root := ~, start := 45495, stop := 45523 ), rec( attributes := rec( ), content := [ rec( content := "\n Eac\ h of these operations returns a new copy of ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "A", next := 45606, root := ~, start := 45592, stop := 45605 ), rec( content := ",\n o\ f the appropriate mutability, retaining none of the attributes or\n propert\ ies of ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "A", next := 45711, root := ~, start := 45697, stop := 45710 ), rec( content := ". ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "P", next := 45717, root := ~, start := 45713, stop := 45716 ), rec( content := "\n\n \ ", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphCopy", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "C", next := 45741, root := ~, start := 45723, stop := 45740 ), rec( content := " is a sy\ nonym for ", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCopySameMutability", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "C", next := 45791, root := ~, start := 45759, stop := 45790 ), rec( content := ".\n", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(10);\n\ngap>\ DigraphCopy(D) = D;\ntrue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\nga\ p> DigraphMutableCopy(D);\n\n", count := [ 3, 3, 1, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 1, 11 ], mathmode := "Text", name := "Example", next := 46045, root := ~, start := 45793, stop := 46044 ), rec( content := "\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Description", next := 46062, root := ~, start := 45527, stop := 46061 ) ], count := [ 3, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 46076, root := ~, start := 45278, stop := 46075 ), rec( content := "\n\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfImmutable" ), content := 0, count := [ 3, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 46157, root := ~, start := 46097, stop := 46156 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfMutable" ), content := 0, count := [ 3, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 46218, root := ~, start := 46160, stop := 46217 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfMutable" ), content := 0, count := [ 3, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 46277, root := ~, start := 46221, stop := 46276 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfImmutable" ), content := 0, count := [ 3, 3, 2, 8 ], mathmode := "Text", name := "Oper", next := 46338, root := ~, start := 46280, stop := 46337 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Returns", next := 46370, root := ~, start := 46341, stop := 46369 ), rec( attributes := rec( ), content := [ rec( content := "\n Eac\ h of these operations returns either the original argument\n ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 46471, root := ~, start := 46457, stop := 46470 ), rec( content := ", or a n\ ew copy of ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 46504, root := ~, start := 46490, stop := 46503 ), rec( content := " of the \ appropriate\n mutability, according to the mutability of ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 46585, root := ~, start := 46571, stop := 46584 ), rec( content := ".\n", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> C := CycleDigraph(10);\n\ngap>\ D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap> D := D\ igraphImmutableCopyIfMutable(C);\n\n\ gap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D := DigraphMutableC\ opyIfMutable(C);\n\ngap> IsMutableDi\ graph(D);\nfalse\ngap> D := DigraphMutableCopyIfImmutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> C := Cycl\ eDigraph(IsMutableDigraph, 10);\n\ \ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D :\ = DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap> D := Digraph\ MutableCopyIfMutable(C);\n\ngap> I\ sMutableDigraph(D);\ntrue\ngap> D := DigraphMutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap>\ IsMutableDigraph(D);\ntrue", count := [ 3, 3, 2, 10 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 10 ], mathmode := "Text", name := "Example", next := 47847, root := ~, start := 46587, stop := 47846 ), rec( content := "\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Description", next := 47864, root := ~, start := 46373, stop := 47863 ) ], count := [ 3, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 47878, root := ~, start := 46082, stop := 47877 ), rec( content := "\n\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts", Name := "InducedSubdigra\ ph" ), content := 0, count := [ 3, 3, 3, 2 ], mathmode := "Text", name := "Oper", next := 47952, root := ~, start := 47899, stop := 47951 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 47984, root := ~, start := 47955, stop := 47983 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 48022, root := ~, start := 48008, stop := 48021 ), rec( content := " is a di\ graph, and ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 48053, root := ~, start := 48041, stop := 48052 ), rec( content := " is a su\ bset of the\n vertices of ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 48103, root := ~, start := 48089, stop := 48102 ), rec( content := ", then t\ his operation returns a digraph\n constructed from ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 48178, root := ~, start := 48164, stop := 48177 ), rec( content := " by reta\ ining precisely those vertices in\n ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 48236, root := ~, start := 48224, stop := 48235 ), rec( content := ", and th\ ose edges whose source and range vertices are both\n contained in ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 48324, root := ~, start := 48312, stop := 48323 ), rec( content := ". ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "P", next := 48330, root := ~, start := 48326, stop := 48329 ), rec( content := "\n\n \ The vertices of the induced subdigraph are\n ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..Length(verts)]", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "C", next := 48408, root := ~, start := 48383, stop := 48407 ), rec( content := " but the\ original vertex labels can be\n accessed via ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "Ref", next := 48497, root := ~, start := 48464, stop := 48496 ), rec( content := ". ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "P", next := 48503, root := ~, start := 48499, stop := 48502 ), rec( content := "\n\n \ If ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 48526, root := ~, start := 48512, stop := 48525 ), rec( content := " belongs\ to ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 48568, root := ~, start := 48538, stop := 48567 ), rec( content := ", then\n\ ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 48593, root := ~, start := 48579, stop := 48592 ), rec( content := " is modi\ fied in place. If ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 48633, root := ~, start := 48619, stop := 48632 ), rec( content := " belongs\ to ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 48683, root := ~, start := 48645, stop := 48682 ), rec( content := ", a new \ immutable digraph containing the\n appropriate\n vertices and edges is r\ eturned.", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 6 ], mathmode := "Text", name := "P", next := 48779, root := ~, start := 48775, stop := 48778 ), rec( content := "\n\n \ ", count := [ 3, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 1, 2, 3, 4, 4], [1, 3, 4], [3, 1], [1, 1]]);\n\ngap> InducedSubdigraph(D, [1, 3, 4\ ]);\n\ngap> DigraphVertices(l\ ast);\n[ 1 .. 3 ]\ngap> D := DigraphMutableCopy(D);\n\ngap> new := InducedSubdigraph(D, [1, 3, 4]);\n\ngap> D = new;\ntrue\n", count := [ 3, 3, 3, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 3, 7 ], mathmode := "Text", name := "Example", next := 49257, root := ~, start := 48785, stop := 49256 ), rec( content := "\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 49274, root := ~, start := 47987, stop := 49273 ) ], count := [ 3, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 49288, root := ~, start := 47884, stop := 49287 ), rec( content := "\n\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraph"\ ), content := 0, count := [ 3, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 49352, root := ~, start := 49309, stop := 49351 ), rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraphA\ ttr" ), content := 0, count := [ 3, 3, 4, 4 ], mathmode := "Text", name := "Attr", next := 49402, root := ~, start := 49355, stop := 49401 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Returns", next := 49434, root := ~, start := 49405, stop := 49433 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a digraph isomorphic to the subdigraph of\n ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 49537, root := ~, start := 49523, stop := 49536 ), rec( content := " induced\ by the set of non-isolated vertices,\n i.e. the set of those vertices of " , count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 49635, root := ~, start := 49621, stop := 49634 ), rec( content := " which a\ re the source or\n range of some edge in ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 49700, root := ~, start := 49686, stop := 49699 ), rec( content := ".\n S\ ee ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Ref", next := 49741, root := ~, start := 49710, stop := 49740 ), rec( content := ".\n " , count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "P", next := 49751, root := ~, start := 49747, stop := 49750 ), rec( content := "\n\n \ The ordering of the remaining vertices of ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "A", next := 49813, root := ~, start := 49799, stop := 49812 ), rec( content := " is pres\ erved, as are\n the labels of the remaining vertices and edges;\n see ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 49928, root := ~, start := 49895, stop := 49927 ), rec( content := " and ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 49964, root := ~, start := 49933, stop := 49963 ), rec( content := ".\n T\ his can allow one to match a vertex in the reduced digraph to the\n corresp\ onding vertex in ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "A", next := 50079, root := ~, start := 50065, stop := 50078 ), rec( content := ".\n " , count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "P", next := 50089, root := ~, start := 50085, stop := 50088 ), rec( content := "\n\n \ If ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "A", next := 50112, root := ~, start := 50098, stop := 50111 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the isolated \ vertices of the mutable digraph ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "A", next := 50243, root := ~, start := 50229, stop := 50242 ), rec( content := " are\n \ removed in-place.\n ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [], [], [1, 4], []]);\n\ngap> R := ReducedDigraph(D);\n\ngap> OutNeighbours(R);\n[ [ 1, 2 ], [ ], [ 1, 3 ] ]\ngap> Di\ graphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ]\ngap> DigraphEdges(\ R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]\ngap> DigraphVertexLabel(R, 3)\ ;\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap> D := Digraph(IsMutableDigraph, [\ [], [3], [2]]);\n\ngap> ReducedDigra\ ph(D);\n\ngap> D;\n\n", count := [ 3, 3, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 8 ], mathmode := "Text", name := "Example", next := 50936, root := ~, start := 50274, stop := 50935 ), rec( content := "\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Description", next := 50953, root := ~, start := 49437, stop := 50952 ) ], count := [ 3, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 50967, root := ~, start := 49294, stop := 50966 ), rec( content := "\n\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraph" ), content := 0, count := [ 3, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 51043, root := ~, start := 50988, stop := 51042 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphAttr" ), content := 0, count := [ 3, 3, 5, 4 ], mathmode := "Text", name := "Attr", next := 51105, root := ~, start := 51046, stop := 51104 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoops" ), content := 0, count := [ 3, 3, 5, 6 ], mathmode := "Text", name := "Oper", next := 51175, root := ~, start := 51108, stop := 51174 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoopsAttr" ), content := 0, count := [ 3, 3, 5, 8 ], mathmode := "Text", name := "Attr", next := 51249, root := ~, start := 51178, stop := 51248 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Returns", next := 51281, root := ~, start := 51252, stop := 51280 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 51319, root := ~, start := 51305, stop := 51318 ), rec( content := " is a di\ graph, then ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalSymmetricSubdigraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 51372, root := ~, start := 51339, stop := 51371 ), rec( content := "\n re\ turns a symmetric digraph without multiple edges which has the same\n verte\ x set as ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 51479, root := ~, start := 51465, stop := 51478 ), rec( content := ", and wh\ ose edge list is formed from\n ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 51534, root := ~, start := 51520, stop := 51533 ), rec( content := " by igno\ ring the multiplicity of edges, and by ignoring\n edges ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,v]", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 51612, root := ~, start := 51600, stop := 51611 ), rec( content := " for whi\ ch there does not exist an edge ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,u]", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 51664, root := ~, start := 51652, stop := 51663 ), rec( content := ".\n " , count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "P", next := 51674, root := ~, start := 51670, stop := 51673 ), rec( content := "\n\n \ The digraph returned by ", count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalSymmetricSubdigraphWithoutLoops", count := [ 3, 3, 5, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "C", next := 51749, root := ~, start := 51704, stop := 51748 ), rec( content := " is\n \ the same, except that loops are removed.", count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "P", next := 51801, root := ~, start := 51797, stop := 51800 ), rec( content := "\n \n\ If ", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "A", next := 51828, root := ~, start := 51814, stop := 51827 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "A", next := 51934, root := ~, start := 51920, stop := 51933 ), rec( content := " is chan\ ged in-place into such\n a digraph described above.", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "P", next := 51999, root := ~, start := 51995, stop := 51998 ), rec( content := "\n\n \ See ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 52041, root := ~, start := 52009, stop := 52040 ), rec( content := ", ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 52071, root := ~, start := 52043, stop := 52070 ), rec( content := ", and\n \ ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 52110, root := ~, start := 52081, stop := 52109 ), rec( content := " for mor\ e information.\n ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and IsMultiDigraph(D);\ \ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap> \ S := MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigraph(S);\ntrue\nga\ p> OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Maximal\ SymmetricSubdigraph(D);\n\ngap> D;\n\n", count := [ 3, 3, 5, 13 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 5, 13 ], mathmode := "Text", name := "Example", next := 52812, root := ~, start := 52137, stop := 52811 ), rec( content := "\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Description", next := 52829, root := ~, start := 51284, stop := 52828 ) ], count := [ 3, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 52843, root := ~, start := 50973, stop := 52842 ), rec( content := "\n\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraph" ), content := 0, count := [ 3, 3, 6, 2 ], mathmode := "Text", name := "Oper", next := 52923, root := ~, start := 52864, stop := 52922 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraphAttr" ), content := 0, count := [ 3, 3, 6, 4 ], mathmode := "Text", name := "Attr", next := 52989, root := ~, start := 52926, stop := 52988 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Returns", next := 53021, root := ~, start := 52992, stop := 53020 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 53059, root := ~, start := 53045, stop := 53058 ), rec( content := " is a di\ graph, then ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalAntiSymmetricSubdigraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53116, root := ~, start := 53079, stop := 53115 ), rec( content := "\n re\ turns an anti-symmetric subdigraph of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 53175, root := ~, start := 53161, stop := 53174 ), rec( content := " formed \ by\n retaining the vertices of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 53230, root := ~, start := 53216, stop := 53229 ), rec( content := ", discar\ ding any duplicate edges,\n and discarding any edge ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53304, root := ~, start := 53292, stop := 53303 ), rec( content := " of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 53322, root := ~, start := 53308, stop := 53321 ), rec( content := " where " , count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ > j", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53341, root := ~, start := 53329, stop := 53340 ), rec( content := "\n an\ d the reverse edge ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,i]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53379, root := ~, start := 53367, stop := 53378 ), rec( content := " is an e\ dge of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 53408, root := ~, start := 53394, stop := 53407 ), rec( content := ".\n I\ n other words, for every\n symmetric pair of edges ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53480, root := ~, start := 53468, stop := 53479 ), rec( content := " and ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,i]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53497, root := ~, start := 53485, stop := 53496 ), rec( content := " in ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 53515, root := ~, start := 53501, stop := 53514 ), rec( content := ", where \ \n ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53536, root := ~, start := 53528, stop := 53535 ), rec( content := " and ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 53549, root := ~, start := 53541, stop := 53548 ), rec( content := " are dis\ tinct, it discards the edge\n ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ \\max(i,j),\\min(i,j)]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "M", next := 53617, root := ~, start := 53589, stop := 53616 ), rec( content := ".\n " , count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "P", next := 53627, root := ~, start := 53623, stop := 53626 ), rec( content := "\n\n \ If ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "A", next := 53650, root := ~, start := 53636, stop := 53649 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "A", next := 53756, root := ~, start := 53742, stop := 53755 ), rec( content := " is chan\ ged in-place. ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "P", next := 53782, root := ~, start := 53778, stop := 53781 ), rec( content := "\n\n \ See ", count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAntis\ ymmetricDigraph" ), content := 0, count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "Ref", next := 53828, root := ~, start := 53792, stop := 53827 ), rec( content := " for mor\ e information.\n ", count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph\ (D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ng\ ap> D := MaximalAntiSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigra\ ph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D :\ = Digraph(IsMutableDigraph, [[2], [1]]);\n\ngap> MaximalAntiSymmetricSubdigraph(D);\n\ngap> D;\n\n", count := [ 3, 3, 6, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 8 ], mathmode := "Text", name := "Example", next := 54558, root := ~, start := 53855, stop := 54557 ), rec( content := "\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Description", next := 54575, root := ~, start := 53024, stop := 54574 ) ], count := [ 3, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 54589, root := ~, start := 52849, stop := 54588 ), rec( content := "\n\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForest" ), content := 0, count := [ 3, 3, 7, 2 ], mathmode := "Text", name := "Oper", next := 54663, root := ~, start := 54610, stop := 54662 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForestAttr" ), content := 0, count := [ 3, 3, 7, 4 ], mathmode := "Text", name := "Attr", next := 54723, root := ~, start := 54666, stop := 54722 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTree" ), content := 0, count := [ 3, 3, 7, 6 ], mathmode := "Text", name := "Oper", next := 54777, root := ~, start := 54726, stop := 54776 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTreeAttr" ), content := 0, count := [ 3, 3, 7, 8 ], mathmode := "Text", name := "Attr", next := 54835, root := ~, start := 54780, stop := 54834 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "K", next := 54872, root := ~, start := 54861, stop := 54871 ), rec( content := ".", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Returns", next := 54883, root := ~, start := 54838, stop := 54882 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "A", next := 54921, root := ~, start := 54907, stop := 54920 ), rec( content := " is a di\ graph with at least one vertex, then\n ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningForest", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "C", next := 55001, root := ~, start := 54970, stop := 55000 ), rec( content := " returns\ an undirected spanning forest of\n ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "A", next := 55061, root := ~, start := 55047, stop := 55060 ), rec( content := ", otherw\ ise this attribute returns ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "K", next := 55107, root := ~, start := 55096, stop := 55106 ), rec( content := ". See ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningForest" ), content := 0, count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Ref", next := 55160, root := ~, start := 55113, stop := 55159 ), rec( content := " for the\ definition of an undirected\n spanning forest.", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "P", next := 55221, root := ~, start := 55217, stop := 55220 ), rec( content := "\n\n \ If ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "A", next := 55244, root := ~, start := 55230, stop := 55243 ), rec( content := " is a di\ graph with at least one vertex and whose ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 55339, root := ~, start := 55293, stop := 55338 ), rec( content := " is conn\ ected (see ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 55397, root := ~, start := 55358, stop := 55396 ), rec( content := "), then \ ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningTree", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "C", next := 55434, root := ~, start := 55405, stop := 55433 ), rec( content := " returns\ \n an undirected spanning tree of ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "A", next := 55492, root := ~, start := 55478, stop := 55491 ), rec( content := ", otherw\ ise this attribute\n returns ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "K", next := 55542, root := ~, start := 55531, stop := 55541 ), rec( content := ". See " , count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 55588, root := ~, start := 55549, stop := 55587 ), rec( content := " for the\ \n definition of an undirected spanning tree.", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "P", next := 55647, root := ~, start := 55643, stop := 55646 ), rec( content := "\n\n \ If ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 55670, root := ~, start := 55656, stop := 55669 ), rec( content := " is immu\ table, then an immutable digraph is returned.\n Otherwise, the mutable digr\ aph ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 55773, root := ~, start := 55759, stop := 55772 ), rec( content := " is chan\ ged in-place into an\n undirected spanning tree of ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 55848, root := ~, start := 55834, stop := 55847 ), rec( content := ".", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "P", next := 55853, root := ~, start := 55849, stop := 55852 ), rec( content := "\n\n \ Note that for an immutable digraph that has known undirected spanning tree,\n \ the attribute ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningTree", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "C", next := 55982, root := ~, start := 55953, stop := 55981 ), rec( content := " returns\ the same digraph as\n the attribute ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningForest", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "C", next := 56060, root := ~, start := 56029, stop := 56059 ), rec( content := ".\n\n \ ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);\n\ngap> UndirectedSpanningTree(D);\nfail\ngap> for\ est := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\nga\ p> IsUndirectedSpanningForest(D, forest);\ntrue\ngap> DigraphConnectedComponen\ ts(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> DigraphConnectedComponents(Max\ imalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> UndirectedSpa\ nningForest(MaximalSymmetricSubdigraph(D))\n> = forest;\ntrue\ngap> D := Compl\ eteDigraph(4);\n\ngap> tree := Und\ irectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> tree = UndirectedSpanni\ ngForest(D);\ntrue\ngap> UndirectedSpanningForest(EmptyDigraph(0));\nfail\ngap\ > D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n", count := [ 3, 3, 7, 13 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 13 ], mathmode := "Text", name := "Example", next := 57222, root := ~, start := 56067, stop := 57221 ), rec( content := "\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Description", next := 57239, root := ~, start := 54886, stop := 57238 ) ], count := [ 3, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 57253, root := ~, start := 54595, stop := 57252 ), rec( content := "\n\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, p", Name := "QuotientDigraph\ " ), content := 0, count := [ 3, 3, 8, 2 ], mathmode := "Text", name := "Oper", next := 57321, root := ~, start := 57274, stop := 57320 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 57353, root := ~, start := 57324, stop := 57352 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 57391, root := ~, start := 57377, stop := 57390 ), rec( content := " is a di\ graph, and ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 57418, root := ~, start := 57410, stop := 57417 ), rec( content := " is a pa\ rtition of the vertices\n of ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 57471, root := ~, start := 57457, stop := 57470 ), rec( content := ", then t\ his operation returns a digraph constructed by\n amalgamating all vertices \ of ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 57573, root := ~, start := 57559, stop := 57572 ), rec( content := " which l\ ie in the same part of\n ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 57616, root := ~, start := 57608, stop := 57615 ), rec( content := ".\n " , count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "P", next := 57626, root := ~, start := 57622, stop := 57625 ), rec( content := "\n\n \ A partition of the vertices of ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "A", next := 57677, root := ~, start := 57663, stop := 57676 ), rec( content := " is a li\ st of non-empty\n disjoint lists, such that the union of all the sub-lists \ is equal\n to vertex set of ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "A", next := 57806, root := ~, start := 57792, stop := 57805 ), rec( content := ". In par\ ticular, each vertex must appear\n in precisely one sub-list.\n ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "P", next := 57886, root := ~, start := 57882, stop := 57885 ), rec( content := "\n\n \ The vertices of ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 57922, root := ~, start := 57908, stop := 57921 ), rec( content := " in part\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 57939, root := ~, start := 57931, stop := 57938 ), rec( content := " of ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 57951, root := ~, start := 57943, stop := 57950 ), rec( content := " will be\ come\n vertex ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 57983, root := ~, start := 57975, stop := 57982 ), rec( content := " in the \ quotient, and there exists some edge in ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 58045, root := ~, start := 58031, stop := 58044 ), rec( content := "\n wi\ th source in part ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 58078, root := ~, start := 58070, stop := 58077 ), rec( content := " and ran\ ge in part ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 58105, root := ~, start := 58097, stop := 58104 ), rec( content := " if and \ only if there\n is an edge from ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 58155, root := ~, start := 58147, stop := 58154 ), rec( content := " to ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 58167, root := ~, start := 58159, stop := 58166 ), rec( content := " in the \ quotient.\n In particular, this means that the quotient of a digraph has no\ multiple edges.\n which was a change introduced in version 1.0.0 of the ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "P", next := 58355, root := ~, start := 58351, stop := 58354 ), rec( content := "\n\n \ If ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 58378, root := ~, start := 58364, stop := 58377 ), rec( content := " belongs\ to ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 58420, root := ~, start := 58390, stop := 58419 ), rec( content := ", then\n\ ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 58445, root := ~, start := 58431, stop := 58444 ), rec( content := " is modi\ fied in place. If ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 58485, root := ~, start := 58471, stop := 58484 ), rec( content := " belongs\ to ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 58535, root := ~, start := 58497, stop := 58534 ), rec( content := ", a new \ immutable digraph with the above\n properties is returned.", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 7 ], mathmode := "Text", name := "P", next := 58607, root := ~, start := 58603, stop := 58606 ), rec( content := "\n\n \ ", count := [ 3, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 1], [4], [1], [1, 3, 4]]);\n\ngap> DigraphVertices(D);\n[ 1 .. 4 ]\ngap> DigraphEdges(D)\ ;\n[ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], \n [ 4, 4 ] \ ]\ngap> p := [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := Quot\ ientDigraph(D, p);\n\ngap> Digraph\ Vertices(quo);\n[ 1 .. 3 ]\ngap> DigraphEdges(quo);\n[ [ 1, 1 ], [ 1, 2 ], [ 2\ , 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> QuotientDigraph(EmptyDigraph(0), [\ ]);\n\n", count := [ 3, 3, 8, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 8, 8 ], mathmode := "Text", name := "Example", next := 59221, root := ~, start := 58613, stop := 59220 ), rec( content := "\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 59238, root := ~, start := 57356, stop := 59237 ) ], count := [ 3, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 59252, root := ~, start := 57259, stop := 59251 ), rec( content := "\n\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphReverse"\ ), content := 0, count := [ 3, 3, 9, 2 ], mathmode := "Text", name := "Oper", next := 59316, root := ~, start := 59273, stop := 59315 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReverseA\ ttr" ), content := 0, count := [ 3, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 59366, root := ~, start := 59319, stop := 59365 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 59398, root := ~, start := 59369, stop := 59397 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ reverse of a digraph is the digraph formed by reversing the orientation\n \ of each of its edges, i.e. for every edge ", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "C", next := 59554, root := ~, start := 59541, stop := 59553 ), rec( content := " of a di\ graph, the\n reverse contains the corresponding edge ", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, i]", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "C", next := 59630, root := ~, start := 59617, stop := 59629 ), rec( content := ".", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "P", next := 59635, root := ~, start := 59631, stop := 59634 ), rec( content := "\n\n \ ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverse", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "C", next := 59662, root := ~, start := 59641, stop := 59661 ), rec( content := " returns\ the reverse of the digraph ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 59712, root := ~, start := 59698, stop := 59711 ), rec( content := ". If\n \ ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 59735, root := ~, start := 59721, stop := 59734 ), rec( content := " is immu\ table, then a new immutable digraph is returned. Otherwise,\n the mutable d\ igraph ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 59841, root := ~, start := 59827, stop := 59840 ), rec( content := " is chan\ ged in-place into its reverse.\n ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 \ ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ]\ngap> D := Digraph([[2, 4], [1], [4], [3, 4]\ ]);\n\ngap> DigraphEdges(D);\n[ [ \ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> DigraphEdges(\ DigraphReverse(D));\n[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, \ 4 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> Digra\ phReverse(D);\n\ngap> OutNeighbours(\ D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n", count := [ 3, 3, 9, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 7 ], mathmode := "Text", name := "Example", next := 60705, root := ~, start := 59884, stop := 60704 ), rec( content := "\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 60722, root := ~, start := 59401, stop := 60721 ) ], count := [ 3, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 60736, root := ~, start := 59258, stop := 60735 ), rec( content := "\n\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDual" ), content := 0, count := [ 3, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 60797, root := ~, start := 60757, stop := 60796 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphDualAttr\ " ), content := 0, count := [ 3, 3, 10, 4 ], mathmode := "Text", name := "Attr", next := 60844, root := ~, start := 60800, stop := 60843 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Returns", next := 60876, root := ~, start := 60847, stop := 60875 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ual", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 60912, root := ~, start := 60901, stop := 60911 ), rec( content := " of ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 60930, root := ~, start := 60916, stop := 60929 ), rec( content := " has the\ same vertices as ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 60970, root := ~, start := 60956, stop := 60969 ), rec( content := ",\n a\ nd there is an edge in the dual from ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 61022, root := ~, start := 61014, stop := 61021 ), rec( content := " to ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 61034, root := ~, start := 61026, stop := 61033 ), rec( content := "\n wh\ enever there is no edge from ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 61078, root := ~, start := 61070, stop := 61077 ), rec( content := " to ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 61090, root := ~, start := 61082, stop := 61089 ), rec( content := " in ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 61108, root := ~, start := 61094, stop := 61107 ), rec( content := ".\n T\ he ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ual", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 61129, root := ~, start := 61118, stop := 61128 ), rec( content := " is some\ times called the ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplement", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 61171, root := ~, start := 61154, stop := 61170 ), rec( content := ".", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "P", next := 61176, root := ~, start := 61172, stop := 61175 ), rec( content := "\n\n \ ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphDual", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "C", next := 61200, root := ~, start := 61182, stop := 61199 ), rec( content := " returns\ the dual of the digraph ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 61247, root := ~, start := 61233, stop := 61246 ), rec( content := ". If\n \ ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 61270, root := ~, start := 61256, stop := 61269 ), rec( content := " is an i\ mmutable digraph, then a new immutable digraph is\n returned. Otherwise, th\ e mutable digraph ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 61387, root := ~, start := 61373, stop := 61386 ), rec( content := " is chan\ ged in-place\n into its dual.\n ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9], [], [], []]);\n\ \ngap> DigraphDual(D);\n\ngap> D := CycleDigraph(IsMutableDigraph, \ 3);\n\ngap> DigraphDual(D);\n\ngap> D;\n\n", count := [ 3, 3, 10, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 7 ], mathmode := "Text", name := "Example", next := 61853, root := ~, start := 61431, stop := 61852 ), rec( content := "\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Description", next := 61870, root := ~, start := 60879, stop := 61869 ) ], count := [ 3, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 61884, root := ~, start := 60742, stop := 61883 ), rec( content := "\n\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosure" ), content := 0, count := [ 3, 3, 11, 2 ], mathmode := "Text", name := "Oper", next := 61957, root := ~, start := 61905, stop := 61956 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosureAttr" ), content := 0, count := [ 3, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 62016, root := ~, start := 61960, stop := 62015 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 62048, root := ~, start := 62019, stop := 62047 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "A", next := 62086, root := ~, start := 62072, stop := 62085 ), rec( content := " is a di\ graph, then this attribute gives the minimal\n symmetric digraph which has \ the same vertices and contains all the edges of\n ", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "A", next := 62237, root := ~, start := 62223, stop := 62236 ), rec( content := ".", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "P", next := 62242, root := ~, start := 62238, stop := 62241 ), rec( content := "\n \n\ A digraph is ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ymmetric", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "E", next := 62281, root := ~, start := 62265, stop := 62280 ), rec( content := " if its \ adjacency matrix \n ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 62340, root := ~, start := 62311, stop := 62339 ), rec( content := " is symm\ etric. For a digraph with multiple\n edges this means that there are the sa\ me number of edges from a vertex\n ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 62470, root := ~, start := 62462, stop := 62469 ), rec( content := " to a ve\ rtex ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 62491, root := ~, start := 62483, stop := 62490 ), rec( content := " as ther\ e are from ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 62518, root := ~, start := 62510, stop := 62517 ), rec( content := " to ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 62530, root := ~, start := 62522, stop := 62529 ), rec( content := "; \n \ see ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 62573, root := ~, start := 62541, stop := 62572 ), rec( content := ".", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "P", next := 62578, root := ~, start := 62574, stop := 62577 ), rec( content := "\n\n \ If ", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "A", next := 62601, root := ~, start := 62587, stop := 62600 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "A", next := 62707, root := ~, start := 62693, stop := 62706 ), rec( content := " is chan\ ged in-place into its\n symmetric closure.\n", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> IsSymmetricDigraph(D);\ntrue\nga\ p> List(OutNeighbours(D), AsSet);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, \ 3, 4 ] ]\ngap> D := Digraph([[2, 2], [1]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ \ 1, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSymmetricClosure(D);\n\ngap> D;\n\n", count := [ 3, 3, 11, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 11, 8 ], mathmode := "Text", name := "Example", next := 63548, root := ~, start := 62760, stop := 63547 ), rec( content := "\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 63565, root := ~, start := 62051, stop := 63564 ) ], count := [ 3, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 63579, root := ~, start := 61890, stop := 63578 ), rec( content := "\n\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosure" ), content := 0, count := [ 3, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 63653, root := ~, start := 63600, stop := 63652 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosureAttr" ), content := 0, count := [ 3, 3, 12, 4 ], mathmode := "Text", name := "Attr", next := 63713, root := ~, start := 63656, stop := 63712 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosure" ), content := 0, count := [ 3, 3, 12, 6 ], mathmode := "Text", name := "Oper", next := 63778, root := ~, start := 63716, stop := 63777 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosureAttr" ), content := 0, count := [ 3, 3, 12, 8 ], mathmode := "Text", name := "Attr", next := 63847, root := ~, start := 63781, stop := 63846 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Returns", next := 63879, root := ~, start := 63850, stop := 63878 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "A", next := 63917, root := ~, start := 63903, stop := 63916 ), rec( content := " is a di\ graph with no multiple edges, then these\n attributes return the (reflexive\ ) transitive closure of ", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "A", next := 64040, root := ~, start := 64026, stop := 64039 ), rec( content := ".", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "P", next := 64045, root := ~, start := 64041, stop := 64044 ), rec( content := "\n\n \ A digraph is ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eflexive", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 64080, root := ~, start := 64064, stop := 64079 ), rec( content := " if it h\ as a loop at every vertex, and it is\n ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransitive", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 64146, root := ~, start := 64129, stop := 64145 ), rec( content := " if when\ ever ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 64171, root := ~, start := 64159, stop := 64170 ), rec( content := " and ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,k]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 64188, root := ~, start := 64176, stop := 64187 ), rec( content := " are edg\ es of\n ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 64220, root := ~, start := 64206, stop := 64219 ), rec( content := ", ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,k]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 64234, root := ~, start := 64222, stop := 64233 ), rec( content := " is also\ an edge. The ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ reflexive)\n transitive closure", count := [ 3, 3, 12, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 64300, root := ~, start := 64257, stop := 64299 ), rec( content := " of a di\ graph ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 64328, root := ~, start := 64314, stop := 64327 ), rec( content := " is the \ least\n (reflexive and) transitive digraph containing ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 64406, root := ~, start := 64392, stop := 64405 ), rec( content := ".", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "P", next := 64411, root := ~, start := 64407, stop := 64410 ), rec( content := "\n\n \ If ", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "A", next := 64434, root := ~, start := 64420, stop := 64433 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "A", next := 64540, root := ~, start := 64526, stop := 64539 ), rec( content := " is chan\ ged in-place into its\n (reflexive) transitive closure.", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "P", next := 64609, root := ~, start := 64605, stop := 64608 ), rec( content := "\n\n \ Let ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 64627, root := ~, start := 64619, stop := 64626 ), rec( content := " be the \ number of vertices of ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "A", next := 64671, root := ~, start := 64657, stop := 64670 ), rec( content := ", and le\ t\n ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 64693, root := ~, start := 64685, stop := 64692 ), rec( content := " be the \ number of edges. For an arbitrary digraph, these\n attributes will use a v\ ersion of the Floyd-Warshall algorithm, with\n complexity ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 64851, root := ~, start := 64838, stop := 64850 ), rec( content := ".\n\n \ However, for a topologically sortable digraph [see ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "Ref", next := 64951, root := ~, start := 64909, stop := 64950 ), rec( content := "], these\ attributes will use methods\n with complexity ", count := [ 3, 3, 12, 12 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 65035, root := ~, start := 65008, stop := 65034 ), rec( content := " when th\ is is faster.", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 13 ], mathmode := "Text", name := "P", next := 65060, root := ~, start := 65056, stop := 65059 ), rec( content := "\n\n \ ", count := [ 3, 3, 12, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".H`eOWR`Ul^\");\n\ngap> IsReflexiveDigraph(D) or IsTransitiveDigraph(D);\ \nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1, 3 ], [ ], [ 5 ], [ ], [ 7,\ 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := DigraphTransitiveClosure(D);\n\ngap> OutNeighbours(T);\n[\ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [ ], [ 5 ], \n [ ], [ 7,\ 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphReflexiveTransitiveClosure(D);\ \n\ngap> OutNeighbours(R\ T);\n[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7, 8, 9, 2 ], [ 3 ], \n [ 5,\ 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]\ngap> D := CycleDigraph(IsM\ utableDigraph, 3);\n\ngap> DigraphRe\ flexiveTransitiveClosure(D);\n\ngap>\ D;\n\n", count := [ 3, 3, 12, 14 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 12, 14 ], mathmode := "Text", name := "Example", next := 66056, root := ~, start := 65066, stop := 66055 ), rec( content := "\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Description", next := 66073, root := ~, start := 63882, stop := 66072 ) ], count := [ 3, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 66087, root := ~, start := 63585, stop := 66086 ), rec( content := "\n\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReduction" ), content := 0, count := [ 3, 3, 13, 2 ], mathmode := "Text", name := "Oper", next := 66163, root := ~, start := 66108, stop := 66162 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReductionAttr" ), content := 0, count := [ 3, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 66225, root := ~, start := 66166, stop := 66224 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReduction" ), content := 0, count := [ 3, 3, 13, 6 ], mathmode := "Text", name := "Oper", next := 66292, root := ~, start := 66228, stop := 66291 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReductionAttr" ), content := 0, count := [ 3, 3, 13, 8 ], mathmode := "Text", name := "Attr", next := 66363, root := ~, start := 66295, stop := 66362 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Returns", next := 66395, root := ~, start := 66366, stop := 66394 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "A", next := 66433, root := ~, start := 66419, stop := 66432 ), rec( content := " is a to\ pologically sortable digraph\n [see ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Ref", next := 66515, root := ~, start := 66479, stop := 66514 ), rec( content := "]\n w\ ith no multiple edges, then these operations return the\n (reflexive) tran\ sitive reduction of ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "A", next := 66633, root := ~, start := 66619, stop := 66632 ), rec( content := ".", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "P", next := 66638, root := ~, start := 66634, stop := 66637 ), rec( content := "\n\n \ The (reflexive) transitive reduction of such a digraph is the unique\n leas\ t subgraph such that the (reflexive) transitive closure of the\n subgraph i\ s equal to the (reflexive) transitive closure of ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 66861, root := ~, start := 66847, stop := 66860 ), rec( content := "\n [s\ ee ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveClosure" ), content := 0, count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "Ref", next := 66918, root := ~, start := 66871, stop := 66917 ), rec( content := "].\n \ In order words, it is the least subgraph of ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 66983, root := ~, start := 66969, stop := 66982 ), rec( content := " which\n\ retains the same reachability as ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 67041, root := ~, start := 67027, stop := 67040 ), rec( content := ".", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "P", next := 67046, root := ~, start := 67042, stop := 67045 ), rec( content := "\n\n \ If ", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "A", next := 67069, root := ~, start := 67055, stop := 67068 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "A", next := 67175, root := ~, start := 67161, stop := 67174 ), rec( content := " is chan\ ged in-place into its\n (reflexive) transitive reduction.", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "P", next := 67246, root := ~, start := 67242, stop := 67245 ), rec( content := "\n\n \ Let ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 67264, root := ~, start := 67256, stop := 67263 ), rec( content := " be the \ number of vertices of an arbitrary digraph, and let\n ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 67340, root := ~, start := 67332, stop := 67339 ), rec( content := " be the \ number of edges.\n Then these operations\n use methods with complexity " , count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 67450, root := ~, start := 67423, stop := 67449 ), rec( content := ".\n " , count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 13 ], mathmode := "Text", name := "P", next := 67460, root := ~, start := 67456, stop := 67459 ), rec( content := "\n ", count := [ 3, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(D);\ntrue\ng\ ap> D1 := DigraphReflexiveTransitiveReduction(D);\n\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeighbours(D1);\n\ [ [ 2 ], [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\n\ngap> DigraphHasLoops(D2);\ntrue\ngap> OutNe\ ighbours(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveTransitiveClos\ ure(D)\n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> DigraphTransit\ iveClosure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D := Digraph(IsM\ utableDigraph, [[1], [1], [1, 2, 3]]);\n\ngap> DigraphReflexiveTransitiveReduction(D);\n\ngap> D;\n\n", count := [ 3, 3, 13, 14 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 13, 14 ], mathmode := "Text", name := "Example", next := 68320, root := ~, start := 67465, stop := 68319 ), rec( content := "\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Description", next := 68337, root := ~, start := 66398, stop := 68336 ) ], count := [ 3, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 68351, root := ~, start := 66093, stop := 68350 ), rec( content := "\n\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, label \ ]", Name := "DigraphAddVertex" ), content := 0, count := [ 3, 3, 14, 2 ], mathmode := "Text", name := "Oper", next := 68427, root := ~, start := 68372, stop := 68426 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 68459, root := ~, start := 68430, stop := 68458 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ operation returns a digraph constructed from ", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "A", next := 68543, root := ~, start := 68529, stop := 68542 ), rec( content := " by addi\ ng a\n single new vertex, and no new edges. ", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "P", next := 68601, root := ~, start := 68597, stop := 68600 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abel", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "A", next := 68651, root := ~, start := 68639, stop := 68650 ), rec( content := " is a ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object,\ \n then the new vertex will be labelled ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abel", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "A", next := 68724, root := ~, start := 68712, stop := 68723 ), rec( content := ".\n " , count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "P", next := 68734, root := ~, start := 68730, stop := 68733 ), rec( content := "\n\n \ If ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 68757, root := ~, start := 68743, stop := 68756 ), rec( content := " belongs\ to ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 68799, root := ~, start := 68769, stop := 68798 ), rec( content := ", then t\ he\n vertex is added directly to ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 68856, root := ~, start := 68842, stop := 68855 ), rec( content := ". If ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 68875, root := ~, start := 68861, stop := 68874 ), rec( content := " belongs\ to\n ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 68923, root := ~, start := 68891, stop := 68922 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 68960, root := ~, start := 68946, stop := 68959 ), rec( content := " with\n \ the additional vertex is returned.", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 6 ], mathmode := "Text", name := "P", next := 69008, root := ~, start := 69004, stop := 69007 ), rec( content := "\n\n \ ", count := [ 3, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> new := DigraphAddVertex(D);\n\ \ngap> D = new;\nfalse\ngap> DigraphVertices(new);\n[ 1 .. 4 ]\ngap> new := Di\ graphAddVertex(D, Group([(1, 2)]));\n\ngap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := \ CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> new := DigraphAddVertex(D);\n\ngap> D = new;\ntrue\n", count := [ 3, 3, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 69613, root := ~, start := 69014, stop := 69612 ), rec( content := "\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 69630, root := ~, start := 68462, stop := 69629 ) ], count := [ 3, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 69644, root := ~, start := 68357, stop := 69643 ), rec( content := "\n\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, m", Label := "for a digraph \ and an integer", Name := "DigraphAddVertices" ), content := 0, count := [ 3, 3, 15, 2 ], mathmode := "Text", name := "Oper", next := 69756, root := ~, start := 69665, stop := 69755 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of labels", Name := "DigraphAddVertices" ), content := 0, count := [ 3, 3, 15, 4 ], mathmode := "Text", name := "Oper", next := 69861, root := ~, start := 69759, stop := 69860 ), rec( content := " The 3-argument \ version is deliberately undocumented. It is redundant,\n but it is inclu\ ded for backwards compatibility. ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Returns", next := 70029, root := ~, start := 70000, stop := 70028 ), rec( attributes := rec( ), content := [ rec( content := "\n For\ a non-negative integer ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 70085, root := ~, start := 70077, stop := 70084 ), rec( content := ", this o\ peration returns a digraph\n constructed from ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 70155, root := ~, start := 70141, stop := 70154 ), rec( content := " by addi\ ng ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 70174, root := ~, start := 70166, stop := 70173 ), rec( content := " new ver\ tices.\n ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "P", next := 70197, root := ~, start := 70193, stop := 70196 ), rec( content := "\n\n \ Otherwise, if ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "A", next := 70230, root := ~, start := 70217, stop := 70229 ), rec( content := " is a li\ st\n consisting of ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "C", next := 70267, root := ~, start := 70259, stop := 70266 ), rec( content := " ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ , then this operation returns a\n digraph constructed from ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "A", next := 70356, root := ~, start := 70342, stop := 70355 ), rec( content := " by addi\ ng ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "C", next := 70375, root := ~, start := 70367, stop := 70374 ), rec( content := " new ver\ tices,\n which are labelled according to this list.\n ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "P", next := 70445, root := ~, start := 70441, stop := 70444 ), rec( content := "\n\n \ If ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 70468, root := ~, start := 70454, stop := 70467 ), rec( content := " belongs\ to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 70510, root := ~, start := 70480, stop := 70509 ), rec( content := ", then t\ he\n vertices are added directly to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 70570, root := ~, start := 70556, stop := 70569 ), rec( content := ", which \ is changed in-place.\n If ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 70620, root := ~, start := 70606, stop := 70619 ), rec( content := " belongs\ to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 70664, root := ~, start := 70632, stop := 70663 ), rec( content := ", then\n\ ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 70689, root := ~, start := 70675, stop := 70688 ), rec( content := " itself \ is returned if no vertices are added\n (i.e. ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "m", count := [ 3, 3, 15, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := \ "Text", name := "A", next := 70755, root := ~, start := 70747, stop := 70754 ) , rec( content := "\ =0", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "C", next := 70761, root := ~, start := 70744, stop := 70760 ), rec( content := " or ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 70778, root := ~, start := 70765, stop := 70777 ), rec( content := " is empt\ y), otherwise\n the result is a new immutable digraph.\n ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 8 ], mathmode := "Text", name := "P", next := 70851, root := ~, start := 70847, stop := 70850 ), rec( content := "\n\n \ ", count := [ 3, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> new := DigraphAddVertices(D, 3);\n\ngap> DigraphVertices(new);\n[ 1 .. 6 ]\ngap> new := DigraphAddVertices(\ D, [Group([(1, 2)]), \"d\"]);\n\ng\ ap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]), \"d\" ]\ngap> Digr\ aphAddVertices(D, 0) = D;\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableD\ igraph, 2, 3);\n\ngap> new := Digra\ phAddVertices(D, 4);\n\ngap> D = ne\ w;\ntrue\n", count := [ 3, 3, 15, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 9 ], mathmode := "Text", name := "Example", next := 71500, root := ~, start := 70857, stop := 71499 ), rec( content := "\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Description", next := 71517, root := ~, start := 70032, stop := 71516 ) ], count := [ 3, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 71531, root := ~, start := 69650, stop := 71530 ), rec( content := "\n\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphAddEdge" ), content := 0, count := [ 3, 3, 16, 2 ], mathmode := "Text", name := "Oper", next := 71639, root := ~, start := 71552, stop := 71638 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphAddEdge" ), content := 0, count := [ 3, 3, 16, 4 ], mathmode := "Text", name := "Oper", next := 71740, root := ~, start := 71642, stop := 71739 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 71772, root := ~, start := 71743, stop := 71771 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 71807, root := ~, start := 71796, stop := 71806 ), rec( content := " is a pa\ ir of vertices of ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 71847, root := ~, start := 71833, stop := 71846 ), rec( content := ", or ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 71862, root := ~, start := 71852, stop := 71861 ), rec( content := " and\n \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 71881, root := ~, start := 71871, stop := 71880 ), rec( content := " are ver\ tices of ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 71912, root := ~, start := 71898, stop := 71911 ), rec( content := ",\n t\ hen this operation returns a digraph constructed from ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 71987, root := ~, start := 71973, stop := 71986 ), rec( content := "\n by\ adding a new edge with source ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 72036, root := ~, start := 72025, stop := 72035 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "C", next := 72046, root := ~, start := 72036, stop := 72045 ), rec( content := " [", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 72058, root := ~, start := 72048, stop := 72057 ), rec( content := "] and ra\ nge\n ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 72085, root := ~, start := 72074, stop := 72084 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "C", next := 72095, root := ~, start := 72085, stop := 72094 ), rec( content := " [", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 72107, root := ~, start := 72097, stop := 72106 ), rec( content := "].\n \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "P", next := 72118, root := ~, start := 72114, stop := 72117 ), rec( content := "\n \n\ If ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 72145, root := ~, start := 72131, stop := 72144 ), rec( content := " belongs\ to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "Ref", next := 72187, root := ~, start := 72157, stop := 72186 ), rec( content := ", then t\ he edge\n is added directly to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 72242, root := ~, start := 72228, stop := 72241 ), rec( content := ". If ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 72261, root := ~, start := 72247, stop := 72260 ), rec( content := " belongs\ to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "Ref", next := 72311, root := ~, start := 72273, stop := 72310 ), rec( content := ", then a\ n immutable copy of ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 72353, root := ~, start := 72339, stop := 72352 ), rec( content := " with th\ e\n additional edge is returned. ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 16, 7 ], mathmode := "Text", name := "P", next := 72401, root := ~, start := 72397, stop := 72400 ), rec( content := "\n\n \ ", count := [ 3, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([[2], [3], []]);\n\ngap> DigraphEdges(D1);\n[ [ 1, 2 ], [ 2, 3 ] ]\ngap> D2 := DigraphAddEdg\ e(D1, [3, 1]);\n\ngap> DigraphEdge\ s(D2);\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D3 := DigraphAddEdge(D2, [2, 3]\ );\n\ngap> DigraphEdges(D3);\ \n[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D := CycleDigraph(IsMutable\ Digraph, 4);\n\ngap> new := DigraphA\ ddEdge(D, [1, 3]);\n\ngap> DigraphEd\ ges(new);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ]\ngap> D = new;\ \ntrue\n", count := [ 3, 3, 16, 8 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 8 ], mathmode := "Text", name := "Example", next := 73127, root := ~, start := 72407, stop := 73126 ), rec( content := "\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 73144, root := ~, start := 71775, stop := 73143 ) ], count := [ 3, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 73158, root := ~, start := 71537, stop := 73157 ), rec( content := "\n\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Name := "DigraphAddEdgeO\ rbit" ), content := 0, count := [ 3, 3, 17, 2 ], mathmode := "Text", name := "Oper", next := 73233, root := ~, start := 73179, stop := 73232 ), rec( attributes := rec( ), content := [ rec( content := "\n A n\ ew digraph.\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 73277, root := ~, start := 73236, stop := 73276 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a new digraph with the same vertices and edges as\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73389, root := ~, start := 73375, stop := 73388 ), rec( content := " and wit\ h additional edges consisting of the orbit of the\n edge ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73467, root := ~, start := 73456, stop := 73466 ), rec( content := " under t\ he action of the ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 73518, root := ~, start := 73492, stop := 73517 ), rec( content := " of\n \ ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73540, root := ~, start := 73526, stop := 73539 ), rec( content := ". If ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73557, root := ~, start := 73546, stop := 73556 ), rec( content := " is alre\ ady an edge in ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73594, root := ~, start := 73580, stop := 73593 ), rec( content := ", then\n\ ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73619, root := ~, start := 73605, stop := 73618 ), rec( content := " is retu\ rned unchanged. The argument ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 73670, root := ~, start := 73656, stop := 73669 ), rec( content := " must be\ \n an immutable digraph.\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "P", next := 73713, root := ~, start := 73709, stop := 73712 ), rec( content := "\n\n \ An edge is simply a pair of vertices of ", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "A", next := 73773, root := ~, start := 73759, stop := 73772 ), rec( content := ".\n\n \ ", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ],\ [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 \ ], [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := Digraph\ RemoveEdgeOrbit(gr2, [1, 8]);\n\n\ gap> gr3 = gr1;\ntrue\n", count := [ 3, 3, 17, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 74749, root := ~, start := 73780, stop := 74748 ), rec( content := "\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 74766, root := ~, start := 73280, stop := 74765 ) ], count := [ 3, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 74780, root := ~, start := 73164, stop := 74779 ), rec( content := "\n\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Name := "DigraphAddEdges\ " ), content := 0, count := [ 3, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 74852, root := ~, start := 74801, stop := 74851 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 74884, root := ~, start := 74855, stop := 74883 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 74920, root := ~, start := 74908, stop := 74919 ), rec( content := " is a (p\ ossibly empty) list of pairs of vertices of\n ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 74990, root := ~, start := 74976, stop := 74989 ), rec( content := ", then t\ his operation returns a digraph constructed from\n ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 75065, root := ~, start := 75051, stop := 75064 ), rec( content := " by addi\ ng the edges specified by ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 75111, root := ~, start := 75099, stop := 75110 ), rec( content := ". More\\ n precisely, for every ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 75155, root := ~, start := 75144, stop := 75154 ), rec( content := " in ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 75171, root := ~, start := 75159, stop := 75170 ), rec( content := ", a new \ edge\n will be added with source ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge[1]", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 75228, root := ~, start := 75214, stop := 75227 ), rec( content := " and ran\ ge ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges[2]", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 75254, root := ~, start := 75239, stop := 75253 ), rec( content := ".\n " , count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "P", next := 75264, root := ~, start := 75260, stop := 75263 ), rec( content := "\n\n \ If an edge is included in ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 75308, root := ~, start := 75296, stop := 75307 ), rec( content := " with mu\ ltiplicity ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "C", next := 75335, root := ~, start := 75327, stop := 75334 ), rec( content := ",\n t\ hen it will be added ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "C", next := 75371, root := ~, start := 75363, stop := 75370 ), rec( content := " times.\\ n\n If ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 75401, root := ~, start := 75387, stop := 75400 ), rec( content := " belongs\ to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "Ref", next := 75443, root := ~, start := 75413, stop := 75442 ), rec( content := ", then t\ he edges\n are added directly to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 75500, root := ~, start := 75486, stop := 75499 ), rec( content := ". If ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 75519, root := ~, start := 75505, stop := 75518 ), rec( content := " belongs\ to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "Ref", next := 75569, root := ~, start := 75531, stop := 75568 ), rec( content := ", then t\ he result is returned as an immutable\n digraph.\n ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 18, 5 ], mathmode := "Text", name := "P", next := 75636, root := ~, start := 75632, stop := 75635 ), rec( content := "\n\n \ ", count := [ 3, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> func := function(n)\n> local source, range, i;\n> source := [];\n> ra\ nge := [];\n> for i in [1 .. n - 2] do\n> Add(source, i);\n> Add(range\ , i + 1);\n> od;\n> return Digraph(n, source, range);\n> end;;\ngap> D := fu\ nc(1024);\n\ngap> new := Dig\ raphAddEdges(D,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n\ngap> D = new;\nfalse\ngap>\ D2 := DigraphMutableCopy(func(1024));\n\ngap> new := DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1024], [1023\ , 1024], [1024, 1]]);\n\n\ gap> D2 = new;\ntrue\n", count := [ 3, 3, 18, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 6 ], mathmode := "Text", name := "Example", next := 76373, root := ~, start := 75642, stop := 76372 ), rec( content := "\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 76390, root := ~, start := 74887, stop := 76389 ) ], count := [ 3, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 76404, root := ~, start := 74786, stop := 76403 ), rec( content := "\n\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphRemoveVe\ rtex" ), content := 0, count := [ 3, 3, 19, 2 ], mathmode := "Text", name := "Oper", next := 76476, root := ~, start := 76425, stop := 76475 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 76508, root := ~, start := 76479, stop := 76507 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 76540, root := ~, start := 76532, stop := 76539 ), rec( content := " is a ve\ rtex of ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 76570, root := ~, start := 76556, stop := 76569 ), rec( content := ", then t\ his operation returns a\n digraph constructed from ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 76645, root := ~, start := 76631, stop := 76644 ), rec( content := " by remo\ ving vertex ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 76673, root := ~, start := 76665, stop := 76672 ), rec( content := ",\n a\ long with any edge whose source or range vertex is ", count := [ 3, 3, 19, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 76739, root := ~, start := 76731, stop := 76738 ), rec( content := ".", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "P", next := 76744, root := ~, start := 76740, stop := 76743 ), rec( content := "\n\n \ If ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "A", next := 76767, root := ~, start := 76753, stop := 76766 ), rec( content := " has ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "C", next := 76780, root := ~, start := 76772, stop := 76779 ), rec( content := " vertice\ s, then the vertices of the returned\n digraph are ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..n-1]", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "C", next := 76856, root := ~, start := 76841, stop := 76855 ), rec( content := ", but th\ e original labels can be\n accessed via ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "Ref", next := 76939, root := ~, start := 76906, stop := 76938 ), rec( content := ". ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "P", next := 76945, root := ~, start := 76941, stop := 76944 ), rec( content := "\n\n \ If ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 76968, root := ~, start := 76954, stop := 76967 ), rec( content := " belongs\ to ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "Ref", next := 77010, root := ~, start := 76980, stop := 77009 ), rec( content := ", then t\ he\n vertex is removed directly from ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 77071, root := ~, start := 77057, stop := 77070 ), rec( content := ". If ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 77090, root := ~, start := 77076, stop := 77089 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "Ref", next := 77138, root := ~, start := 77106, stop := 77137 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 77175, root := ~, start := 77161, stop := 77174 ), rec( content := "\n wi\ thout the vertex is returned.", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 6 ], mathmode := "Text", name := "P", next := 77215, root := ~, start := 77211, stop := 77214 ), rec( content := "\n\n \ ", count := [ 3, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"a\", \ \"b\", \"c\", \"c\"],\n> [\"b\", \"c\", \"a\", \"a\", \"c\"])\ ;\n\ngap> DigraphVertexLabels(D);\ \n[ \"a\", \"b\", \"c\" ]\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1\ ], [ 3, 1 ], [ 3, 3 ] ]\ngap> new := DigraphRemoveVertex(D, 2);\n\ngap> DigraphVertexLabels(new);\n[ \"a\", \"c\ \" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertex(D, 1);\n\ngap> DigraphVertexLabels(D);\n[ 2, 3, 4, 5 ]\ngap> D =\ new;\ntrue\n", count := [ 3, 3, 19, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 7 ], mathmode := "Text", name := "Example", next := 77909, root := ~, start := 77221, stop := 77908 ), rec( content := "\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 77926, root := ~, start := 76511, stop := 77925 ) ], count := [ 3, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 77940, root := ~, start := 76410, stop := 77939 ), rec( content := "\n\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts", Name := "DigraphRemoveVe\ rtices" ), content := 0, count := [ 3, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 78018, root := ~, start := 77961, stop := 78017 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 78050, root := ~, start := 78021, stop := 78049 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 78086, root := ~, start := 78074, stop := 78085 ), rec( content := " is a (p\ ossibly empty) duplicate-free list of vertices of\n ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 78162, root := ~, start := 78148, stop := 78161 ), rec( content := ", then t\ his operation returns a digraph constructed from\n ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 78237, root := ~, start := 78223, stop := 78236 ), rec( content := " by remo\ ving every vertex in ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 78278, root := ~, start := 78266, stop := 78277 ), rec( content := ", along \ with any edge\n whose source or range vertex is in ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 78351, root := ~, start := 78339, stop := 78350 ), rec( content := ".", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "P", next := 78356, root := ~, start := 78352, stop := 78355 ), rec( content := "\n\n \ If ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "A", next := 78379, root := ~, start := 78365, stop := 78378 ), rec( content := " has ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 78392, root := ~, start := 78384, stop := 78391 ), rec( content := " vertice\ s, then the vertices of the new\n digraph are ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. n-Length(", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "vert\ s", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := \ "Text", name := "A", next := 78478, root := ~, start := 78466, stop := 78477 ) , rec( content := "\ )]", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 78484, root := ~, start := 78448, stop := 78483 ), rec( content := ", but th\ e\n original labels can be accessed via ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "Ref", next := 78567, root := ~, start := 78534, stop := 78566 ), rec( content := ". ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "P", next := 78573, root := ~, start := 78569, stop := 78572 ), rec( content := "\n\n \ If ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 78596, root := ~, start := 78582, stop := 78595 ), rec( content := " belongs\ to ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "Ref", next := 78638, root := ~, start := 78608, stop := 78637 ), rec( content := ", then t\ he\n vertices are removed directly from ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 78702, root := ~, start := 78688, stop := 78701 ), rec( content := ". If ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 78721, root := ~, start := 78707, stop := 78720 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "Ref", next := 78769, root := ~, start := 78737, stop := 78768 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 78806, root := ~, start := 78792, stop := 78805 ), rec( content := "\n wi\ thout the vertices is returned.", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 6 ], mathmode := "Text", name := "P", next := 78848, root := ~, start := 78844, stop := 78847 ), rec( content := "\n\n \ ", count := [ 3, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> SetDigraphVertexLabels(D, [\"a\", \"b\ \", \"c\", \"d\", \"e\"]);\ngap> new := DigraphRemoveVertices(D, [2, 4]);\n\ngap> DigraphVertexLabels(new);\n[ \ \"a\", \"c\", \"e\" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertices(D, [1, 3]\ );\n\ngap> DigraphVertexLabels(D);\n[\ 2, 4, 5 ]\ngap> D = new;\ntrue\n", count := [ 3, 3, 20, 7 ], name := "PCDATA" , root := ~ ) ], count := [ 3, 3, 20, 7 ], mathmode := "Text", name := "Example", next := 79431, root := ~, start := 78854, stop := 79430 ), rec( content := "\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 79448, root := ~, start := 78053, stop := 79447 ) ], count := [ 3, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 79462, root := ~, start := 77946, stop := 79461 ), rec( content := "\n\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphRemoveEdge" ), content := 0, count := [ 3, 3, 21, 2 ], mathmode := "Text", name := "Oper", next := 79573, root := ~, start := 79483, stop := 79572 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphRemoveEdge" ) , content := 0, count := [ 3, 3, 21, 4 ], mathmode := "Text", name := "Oper", next := 79677, root := ~, start := 79576, stop := 79676 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Returns", next := 79709, root := ~, start := 79680, stop := 79708 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79747, root := ~, start := 79733, stop := 79746 ), rec( content := " is a di\ graph with no multiple edges and ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79799, root := ~, start := 79788, stop := 79798 ), rec( content := " is a\n \ pair of vertices of ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79843, root := ~, start := 79829, stop := 79842 ), rec( content := ", or ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79858, root := ~, start := 79848, stop := 79857 ), rec( content := " and ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79873, root := ~, start := 79863, stop := 79872 ), rec( content := " are\n \ vertices of ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79908, root := ~, start := 79894, stop := 79907 ), rec( content := ", then t\ his operation returns a digraph\n constructed from ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 79983, root := ~, start := 79969, stop := 79982 ), rec( content := " by remo\ ving the edge specified by\n ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 80033, root := ~, start := 80022, stop := 80032 ), rec( content := " or ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ src, ran]", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 80054, root := ~, start := 80037, stop := 80053 ), rec( content := ". ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "P", next := 80060, root := ~, start := 80056, stop := 80059 ), rec( content := " \n\n \ If ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 80085, root := ~, start := 80071, stop := 80084 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "Ref", next := 80133, root := ~, start := 80097, stop := 80132 ), rec( content := ", then t\ he edge is removed directly from\n ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 80192, root := ~, start := 80178, stop := 80191 ), rec( content := ". If ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 80211, root := ~, start := 80197, stop := 80210 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "Ref", next := 80261, root := ~, start := 80223, stop := 80260 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 80298, root := ~, start := 80284, stop := 80297 ), rec( content := " without\ \n the edge is returned.\n ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "P", next := 80341, root := ~, start := 80337, stop := 80340 ), rec( content := "\n\n \ Note that if ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 80374, root := ~, start := 80360, stop := 80373 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "Ref", next := 80418, root := ~, start := 80386, stop := 80417 ), rec( content := ",\n t\ hen a new copy of ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 80457, root := ~, start := 80443, stop := 80456 ), rec( content := " will be\ returned even if ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 80494, root := ~, start := 80483, stop := 80493 ), rec( content := " or \n \ ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ src, ran]", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 80520, root := ~, start := 80503, stop := 80519 ), rec( content := " does no\ t define an edge of ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 80562, root := ~, start := 80548, stop := 80561 ), rec( content := ".", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 8 ], mathmode := "Text", name := "P", next := 80567, root := ~, start := 80563, stop := 80566 ), rec( content := "\n\n \ ", count := [ 3, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdge(D, [250000, 1]);\n\ngap> new := DigraphRemoveEdge(D, [25000, 2]);;\nga\ p> new = D;\ntrue\ngap> IsIdenticalObj(new, D);\nfalse\ngap> D := DigraphMutab\ leCopy(D);;\ngap> new := DigraphRemoveEdge(D, 2500, 2);;\ngap> IsIdenticalObj(\ new, D);\ntrue\n", count := [ 3, 3, 21, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 9 ], mathmode := "Text", name := "Example", next := 80997, root := ~, start := 80573, stop := 80996 ), rec( content := "\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Description", next := 81014, root := ~, start := 79712, stop := 81013 ) ], count := [ 3, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 81028, root := ~, start := 79468, stop := 81027 ), rec( content := "\n\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Name := "DigraphRemoveEd\ geOrbit" ), content := 0, count := [ 3, 3, 22, 2 ], mathmode := "Text", name := "Oper", next := 81106, root := ~, start := 81049, stop := 81105 ), rec( attributes := rec( ), content := [ rec( content := "\n A n\ ew digraph.\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 81150, root := ~, start := 81109, stop := 81149 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a new digraph with the same vertices as\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81252, root := ~, start := 81238, stop := 81251 ), rec( content := " and wit\ h the orbit of the edge ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81295, root := ~, start := 81284, stop := 81294 ), rec( content := " (under \ the action\n of the ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 81351, root := ~, start := 81325, stop := 81350 ), rec( content := " of ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81369, root := ~, start := 81355, stop := 81368 ), rec( content := ") remove\ d.\n If ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81398, root := ~, start := 81387, stop := 81397 ), rec( content := " is not \ an edge in ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81431, root := ~, start := 81417, stop := 81430 ), rec( content := ", then " , count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81452, root := ~, start := 81438, stop := 81451 ), rec( content := " is\n \ returned unchanged. The argument ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 81507, root := ~, start := 81493, stop := 81506 ), rec( content := " must be\ an immutable\n digraph.\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "P", next := 81550, root := ~, start := 81546, stop := 81549 ), rec( content := "\n\n \ An edge is simply a pair of vertices of ", count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "A", next := 81610, root := ~, start := 81596, stop := 81609 ), rec( content := ".\n " , count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ],\ [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 \ ], [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := Digraph\ RemoveEdgeOrbit(gr2, [1, 8]);\n\n\ gap> gr3 = gr1;\ntrue\n", count := [ 3, 3, 22, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 5 ], mathmode := "Text", name := "Example", next := 82585, root := ~, start := 81616, stop := 82584 ), rec( content := "\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 82602, root := ~, start := 81153, stop := 82601 ) ], count := [ 3, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 82616, root := ~, start := 81034, stop := 82615 ), rec( content := "\n\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Name := "DigraphRemoveEd\ ges" ), content := 0, count := [ 3, 3, 23, 2 ], mathmode := "Text", name := "Oper", next := 82691, root := ~, start := 82637, stop := 82690 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 82723, root := ~, start := 82694, stop := 82722 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ one of the following holds:\n ", count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 3, 3, 23, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph", count := [ 3, 3, 23, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], mathmode := "Te\ xt", name := "A", next := 82821, root := ~, start := 82807, stop := 82820 ), rec( cont\ ent := " is a digraph with no multiple edges, and\n ", count := [ 3, 3,\ 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "edges", count := [ 3, 3, 23, 5 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], math\ mode := "Text", name := "A", next := 82883, root := ~, start := 82871, stop :=\ 82882 ), rec( content := " is a list of pairs of vertices of ", count := [ 3,\ 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "digraph", count := [ 3, 3, 23, 5 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ] , math\ mode := "Text", name := "A", next := 82932, root := ~, start := 82918, stop :=\ 82931 ), rec( content := ", or\n ", count := [ 3, 3, 23, 5 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], name := "Ite\ m", next := 82950, root := ~, start := 82792, stop := 82949 ), rec( attributes :\ = rec( ), content := [ rec( content := " ", count := [ 3, 3, 23, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "digraph", count := [ 3, 3, 23, 7 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 7 ], mathmode := "Text\ ", name := "A", next := 82978, root := ~, start := 82964, stop := 82977 ), rec( cont\ ent := " is a digraph and ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "edges", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "A", next\ := 83008, root := ~, start := 82996, stop := 83007 ), rec( content := " is an\ empty list ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 3, 23, 7 ], name := "Item", next := 83033, root := ~, start := 829\ 57, stop := 83032 ) ], count := [ 3, 3, 23, 4 ], mathmode := "Text", name := "List", next := 83045, root := ~, start := 82779, stop := 83044 ), rec( content := "\n\n \ then this operation returns a digraph constructed from ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "A", next := 83120, root := ~, start := 83106, stop := 83119 ), rec( content := "\n by\ removing all of the edges specified by ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "A", next := 83179, root := ~, start := 83167, stop := 83178 ), rec( content := "\n (s\ ee ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and an edge", Oper := "DigraphRemoveEdge" ), content := 0, count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "Ref", next := 83254, root := ~, start := 83189, stop := 83253 ), rec( content := ").\n \ ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "P", next := 83265, root := ~, start := 83261, stop := 83264 ), rec( content := "\n\n \ If ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 83288, root := ~, start := 83274, stop := 83287 ), rec( content := " belongs\ to ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "Ref", next := 83330, root := ~, start := 83300, stop := 83329 ), rec( content := ", then t\ he edge\n is removed directly from ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 83389, root := ~, start := 83375, stop := 83388 ), rec( content := ". If ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 83408, root := ~, start := 83394, stop := 83407 ), rec( content := " belongs\ to ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "Ref", next := 83458, root := ~, start := 83420, stop := 83457 ), rec( content := ", the ed\ ge is removed from an immutable copy of\n ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 83524, root := ~, start := 83510, stop := 83523 ), rec( content := " and thi\ s new digraph is returned.", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "P", next := 83562, root := ~, start := 83558, stop := 83561 ), rec( content := "\n\n \ Note that if ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 83593, root := ~, start := 83581, stop := 83592 ), rec( content := " is empt\ y, then this operation\n will always return ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 83661, root := ~, start := 83647, stop := 83660 ), rec( content := " rather \ than a copy. Also, if any element\n of ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 83722, root := ~, start := 83710, stop := 83721 ), rec( content := " is inva\ lid (i.e. does not define an edge of ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 83781, root := ~, start := 83767, stop := 83780 ), rec( content := ")\n t\ hen that element will simply be ignored.\n\n ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> D := DigraphMutableCopy(D);\n\ngap> new := DigraphRemoveEdges(D, [[1\ , 2], [2, 3], [3, 100]]);\n\ngap> new = D;\ntrue\n", count := [ 3, 3, 23, 12 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 12 ], mathmode := "Text", name := "Example", next := 84269, root := ~, start := 83834, stop := 84268 ), rec( content := "\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 84286, root := ~, start := 82726, stop := 84285 ) ], count := [ 3, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 84300, root := ~, start := 82622, stop := 84299 ), rec( content := "\n\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ ops" ), content := 0, count := [ 3, 3, 24, 2 ], mathmode := "Text", name := "Oper", next := 84368, root := ~, start := 84321, stop := 84367 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ opsAttr" ), content := 0, count := [ 3, 3, 24, 4 ], mathmode := "Text", name := "Attr", next := 84422, root := ~, start := 84371, stop := 84421 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Returns", next := 84454, root := ~, start := 84425, stop := 84453 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "A", next := 84492, root := ~, start := 84478, stop := 84491 ), rec( content := " is a di\ graph, then this operation returns a\n digraph constructed from ", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "A", next := 84581, root := ~, start := 84567, stop := 84580 ), rec( content := " by remo\ ving every loop. A loop\n is an edge with equal source and range.", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "P", next := 84660, root := ~, start := 84656, stop := 84659 ), rec( content := "\n\n \ If ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "A", next := 84683, root := ~, start := 84669, stop := 84682 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ removed from the mutable digraph ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "A", next := 84816, root := ~, start := 84802, stop := 84815 ), rec( content := "\n in\ -place.\n ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);\n\ngap> DigraphRemoveLoops(D);\n\ngap> D := Digraph(IsMutableDigraph, [[1, 2]\ , [1]]);\n\ngap> DigraphRemoveLoops(\ D);\n\ngap> D;\n\n", count := [ 3, 3, 24, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 7 ], mathmode := "Text", name := "Example", next := 85270, root := ~, start := 84835, stop := 85269 ), rec( content := "\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Description", next := 85287, root := ~, start := 84457, stop := 85286 ) ], count := [ 3, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 85301, root := ~, start := 84306, stop := 85300 ), rec( content := "\n\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdges" ), content := 0, count := [ 3, 3, 25, 2 ], mathmode := "Text", name := "Oper", next := 85380, root := ~, start := 85322, stop := 85379 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdgesAttr" ), content := 0, count := [ 3, 3, 25, 4 ], mathmode := "Text", name := "Attr", next := 85445, root := ~, start := 85383, stop := 85444 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Returns", next := 85477, root := ~, start := 85448, stop := 85476 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 85515, root := ~, start := 85501, stop := 85514 ), rec( content := " is a di\ graph, then this operation returns a\n digraph constructed from ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 85604, root := ~, start := 85590, stop := 85603 ), rec( content := " by remo\ ving all multiple edges.\n The result is the largest subdigraph of ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 85695, root := ~, start := 85681, stop := 85694 ), rec( content := " which d\ oes not\n contain multiple edges.", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "P", next := 85742, root := ~, start := 85738, stop := 85741 ), rec( content := "\n\n \ If ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "A", next := 85765, root := ~, start := 85751, stop := 85764 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the multiple \ edges of the mutable digraph ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "A", next := 85893, root := ~, start := 85879, stop := 85892 ), rec( content := " are\n \ removed in-place.\n ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);\n\ngap> D2 := DigraphRemoveAllMultipleEdges(D1)\ ;\n\ngap> OutNeighbours(D2);\n[ [ \ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 2], [1]\ ]);\n\ngap> DigraphRemoveAllMul\ tipleEdges(D);\n\ngap> D;\n\n", count := [ 3, 3, 25, 7 ], name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 25, 7 ], mathmode := "Text", name := "Example", next := 86446, root := ~, start := 85924, stop := 86445 ), rec( content := "\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Description", next := 86463, root := ~, start := 85480, stop := 86462 ) ], count := [ 3, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 86477, root := ~, start := 85307, stop := 86476 ), rec( content := "\n\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Label := "for a digraph \ and a list of edges", Name := "DigraphReverseEdges" ), content := 0, count := [ 3, 3, 26, 2 ], mathmode := "Text", name := "Oper", next := 86599, root := ~, start := 86498, stop := 86598 ), rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphReverseEdge" ), content := 0, count := [ 3, 3, 26, 4 ], mathmode := "Text", name := "Oper", next := 86693, root := ~, start := 86602, stop := 86692 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphReverseEdge" ), content := 0, count := [ 3, 3, 26, 6 ], mathmode := "Text", name := "Oper", next := 86798, root := ~, start := 86696, stop := 86797 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Returns", next := 86830, root := ~, start := 86801, stop := 86829 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 86868, root := ~, start := 86854, stop := 86867 ), rec( content := " is a di\ graph without multiple edges, and ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 86922, root := ~, start := 86910, stop := 86921 ), rec( content := " is a\n \ list of pairs of vertices of ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 86975, root := ~, start := 86961, stop := 86974 ), rec( content := " (the en\ tries of each pair\n corresponding to the source and the range of an edge, \ respectively),\n then ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverseEdges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "C", next := 87110, root := ~, start := 87084, stop := 87109 ), rec( content := " returns\ a digraph constructed from\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 87164, root := ~, start := 87150, stop := 87163 ), rec( content := " by reve\ rsing the orientation of every edge specified by\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 87237, root := ~, start := 87225, stop := 87236 ), rec( content := ". If onl\ y one edge is to be reversed, then\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverseEdge", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "C", next := 87309, root := ~, start := 87284, stop := 87308 ), rec( content := " can be \ used instead. In this case, the second\n argument should just be a single v\ ertex-pair, or the second and third\n arguments should be the source and ra\ nge of an edge respectively. ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "P", next := 87504, root := ~, start := 87500, stop := 87503 ), rec( content := "\n\n \ Note that even though ", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "A", next := 87546, root := ~, start := 87532, stop := 87545 ), rec( content := " cannot \ have multiple edges, the\n output may have multiple edges. ", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "P", next := 87619, root := ~, start := 87615, stop := 87618 ), rec( content := "\n\n \ If ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 87642, root := ~, start := 87628, stop := 87641 ), rec( content := " belongs\ to ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "Ref", next := 87684, root := ~, start := 87654, stop := 87683 ), rec( content := ", then t\ he edges\n are reversed in ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 87735, root := ~, start := 87721, stop := 87734 ), rec( content := ". If ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 87754, root := ~, start := 87740, stop := 87753 ), rec( content := " belongs\ to ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "Ref", next := 87804, root := ~, start := 87766, stop := 87803 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 87841, root := ~, start := 87827, stop := 87840 ), rec( content := " with th\ e\n specified edges reversed is returned.", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 10 ], mathmode := "Text", name := "P", next := 87896, root := ~, start := 87892, stop := 87895 ), rec( content := "\n ", count := [ 3, 3, 26, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".Tg?i@s?t_e?_qEsC\");\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 7 ], [\ 1, 8 ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], \n [ 11, 2 ], [ 21, 1 ] ]\ngap> new \ := DigraphReverseEdge(D, [7, 19]);\n\ngap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, \ 1 ], [ 11, 2 ], \n [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := DigraphMutableCopy(new)\ ;;\ngap> new := DigraphReverseEdges(D2, [[19, 7]]);;\ngap> D2 = new;\ntrue\nga\ p> D = new;\ntrue\n", count := [ 3, 3, 26, 11 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 11 ], mathmode := "Text", name := "Example", next := 88479, root := ~, start := 87901, stop := 88478 ), rec( content := "\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Description", next := 88496, root := ~, start := 86833, stop := 88495 ) ], count := [ 3, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 88510, root := ~, start := 86483, stop := 88509 ), rec( content := "\n\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for an\n ar\ bitrary number of digraphs", Name := "DigraphDisjointUnion" ), content := 0, count := [ 3, 3, 27, 2 ], mathmode := "Text", name := "Func", next := 88632, root := ~, start := 88531, stop := 88631 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphDisjointUnion" ), content := 0, count := [ 3, 3, 27, 4 ], mathmode := "Text", name := "Func", next := 88712, root := ~, start := 88635, stop := 88711 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Returns", next := 88744, root := ~, start := 88715, stop := 88743 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 88796, root := ~, start := 88787, stop := 88795 ), rec( content := ", ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 88807, root := ~, start := 88798, stop := 88806 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDisjointUnion", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "C", next := 88864, root := ~, start := 88837, stop := 88863 ), rec( content := " returns\ their disjoint union.\n\n In the second form, if ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 88934, root := ~, start := 88923, stop := 88933 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDisjointUnion", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "C", next := 89004, root := ~, start := 88977, stop := 89003 ), rec( content := " returns\ the disjoint union of the digraphs\n contained in the list. ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 6 ], mathmode := "Text", name := "P", next := 89080, root := ~, start := 89076, stop := 89079 ), rec( content := "\n\n \ For a disjoint union of digraphs, the vertex set is the disjoint union of\n \ the vertex sets, and the edge list is the disjoint union of the edge lists.\n\ ", count := [ 3, 3, 27, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "P", next := 89248, root := ~, start := 89244, stop := 89247 ), rec( content := "\n\n \ More specifically, for a collection of digraphs ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89311, root := ~, start := 89302, stop := 89310 ), rec( content := ", ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89322, root := ~, start := 89313, stop := 89321 ), rec( content := ",\n " , count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89338, root := ~, start := 89328, stop := 89337 ), rec( content := ", the di\ sjoint union with have\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89398, root := ~, start := 89373, stop := 89397 ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89407, root := ~, start := 89398, stop := 89406 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89415, root := ~, start := 89407, stop := 89414 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89424, root := ~, start := 89416, stop := 89423 ), rec( content := "\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89454, root := ~, start := 89429, stop := 89453 ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89463, root := ~, start := 89454, stop := 89462 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89471, root := ~, start := 89463, stop := 89470 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89480, root := ~, start := 89472, stop := 89479 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89491, root := ~, start := 89481, stop := 89490 ), rec( content := " vertice\ s.\n The edges of ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89528, root := ~, start := 89519, stop := 89527 ), rec( content := " will re\ main unchanged, whilst the edges of the\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89588, root := ~, start := 89580, stop := 89587 ), rec( content := "th digra\ ph, ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89608, root := ~, start := 89600, stop := 89607 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89618, root := ~, start := 89608, stop := 89617 ), rec( content := ", will b\ e changed so that they\n belong to the vertices of the disjoint union corre\ sponding to\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89727, root := ~, start := 89719, stop := 89726 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89737, root := ~, start := 89727, stop := 89736 ), rec( content := ". In pa\ rticular, the edges of ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89776, root := ~, start := 89768, stop := 89775 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89786, root := ~, start := 89776, stop := 89785 ), rec( content := " will\n \ have their source and range increased by\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89866, root := ~, start := 89841, stop := 89865 ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89875, root := ~, start := 89866, stop := 89874 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89883, root := ~, start := 89875, stop := 89882 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89892, root := ~, start := 89884, stop := 89891 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89903, root := ~, start := 89893, stop := 89902 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89912, root := ~, start := 89904, stop := 89911 ), rec( content := "\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89942, root := ~, start := 89917, stop := 89941 ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 89950, root := ~, start := 89942, stop := 89949 ), rec( attributes := rec( ), content := [ rec( content := "[\ i-1])", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 89963, root := ~, start := 89950, stop := 89962 ), rec( content := ".\n\n \ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "P", next := 89974, root := ~, start := 89970, stop := 89973 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "Ref", next := 90038, root := ~, start := 90005, stop := 90037 ), rec( content := " will be\ lost.", count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "P", next := 90056, root := ~, start := 90052, stop := 90055 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 90092, root := ~, start := 90083, stop := 90091 ), rec( content := " [", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 90108, root := ~, start := 90094, stop := 90107 ), rec( content := "] belong\ s to ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "Ref", next := 90155, root := ~, start := 90121, stop := 90154 ), rec( content := ", then " , count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 90171, root := ~, start := 90162, stop := 90170 ), rec( content := " [", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 90187, root := ~, start := 90173, stop := 90186 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 90278, root := ~, start := 90264, stop := 90277 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "Ref", next := 90326, root := ~, start := 90294, stop := 90325 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := CycleDigraph(3);\n\ngap> \ OutNeighbours(D1);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> D2 := CompleteDigraph(3);\n<\ immutable complete digraph with 3 vertices>\ngap> OutNeighbours(D2);\n[ [ 2, 3\ ], [ 1, 3 ], [ 1, 2 ] ]\ngap> union := DigraphDisjointUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2 ], [ \ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ]\n", count := [ 3, 3, 27, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 10 ], mathmode := "Text", name := "Example", next := 90868, root := ~, start := 90415, stop := 90867 ), rec( content := "\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Description", next := 90885, root := ~, start := 88747, stop := 90884 ) ], count := [ 3, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 90899, root := ~, start := 88516, stop := 90898 ), rec( content := "\n\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphEdgeUnion" ), content := 0, count := [ 3, 3, 28, 2 ], mathmode := "Text", name := "Func", next := 91015, root := ~, start := 90920, stop := 91014 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphEdgeUnion" ), content := 0, count := [ 3, 3, 28, 4 ], mathmode := "Text", name := "Func", next := 91091, root := ~, start := 91018, stop := 91090 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Returns", next := 91123, root := ~, start := 91094, stop := 91122 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 91175, root := ~, start := 91166, stop := 91174 ), rec( content := ", ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 91186, root := ~, start := 91177, stop := 91185 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeUnion", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "C", next := 91239, root := ~, start := 91216, stop := 91238 ), rec( content := " returns\ their edge union.\n\n In the second form, if ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 91305, root := ~, start := 91294, stop := 91304 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeUnion", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "C", next := 91371, root := ~, start := 91348, stop := 91370 ), rec( content := " returns\ the edge union of the digraphs contained\n in the list. ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "P", next := 91443, root := ~, start := 91439, stop := 91442 ), rec( content := "\n\n \ The vertex set of the edge union of a collection of digraphs is the\n ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ nion", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 91533, root := ~, start := 91521, stop := 91532 ), rec( content := " of the \ vertex sets, whilst the edge list of the edge union is\n the ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oncatenation", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 91624, root := ~, start := 91604, stop := 91623 ), rec( content := " of the \ edge lists. The number of vertices of the\n edge union is equal to the ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximum", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 91720, root := ~, start := 91706, stop := 91719 ), rec( content := " number \ of vertices of one of the\n digraphs, whilst the number of edges of the edg\ e union will equal the\n ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ um", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 91842, root := ~, start := 91832, stop := 91841 ), rec( content := " of the \ number of edges of each digraph. ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "P", next := 91887, root := ~, start := 91883, stop := 91886 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "Ref", next := 91951, root := ~, start := 91918, stop := 91950 ), rec( content := " will be\ lost.\n ", count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "P", next := 91974, root := ~, start := 91970, stop := 91973 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 92010, root := ~, start := 92001, stop := 92009 ), rec( content := " [", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 92026, root := ~, start := 92012, stop := 92025 ), rec( content := "] belong\ s to ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "Ref", next := 92073, root := ~, start := 92039, stop := 92072 ), rec( content := ", then " , count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 92089, root := ~, start := 92080, stop := 92088 ), rec( content := " [", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 92105, root := ~, start := 92091, stop := 92104 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 92196, root := ~, start := 92182, stop := 92195 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "Ref", next := 92244, root := ~, start := 92212, stop := 92243 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(10);\n\ngap>\ DigraphEdgeUnion(D, D);\n\ \ngap> D1 := Digraph([[2], [1]]);\n\ngap> D2 := Digraph([[2, 3], [2], [1]]);\n\ngap> union := DigraphEdgeUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2, 2, 3 ], [ 1, 2 ],\ [ 1 ] ]\ngap> union = DigraphByEdges(\n> Concatenation(DigraphEdges(D1), Digr\ aphEdges(D2)));\ntrue\n", count := [ 3, 3, 28, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 9 ], mathmode := "Text", name := "Example", next := 92917, root := ~, start := 92333, stop := 92916 ), rec( content := "\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Description", next := 92934, root := ~, start := 91126, stop := 92933 ) ], count := [ 3, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 92948, root := ~, start := 90905, stop := 92947 ), rec( content := "\n\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphJoin" ), content := 0, count := [ 3, 3, 29, 2 ], mathmode := "Text", name := "Func", next := 93059, root := ~, start := 92969, stop := 93058 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphJoin" ), content := 0, count := [ 3, 3, 29, 4 ], mathmode := "Text", name := "Func", next := 93130, root := ~, start := 93062, stop := 93129 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Returns", next := 93162, root := ~, start := 93133, stop := 93161 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 93214, root := ~, start := 93205, stop := 93213 ), rec( content := ", ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 93225, root := ~, start := 93216, stop := 93224 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphJoin", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "C", next := 93273, root := ~, start := 93255, stop := 93272 ), rec( content := " returns\ their join.\n\n In the second form, if ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 93333, root := ~, start := 93322, stop := 93332 ), rec( content := " is a no\ n-empty list of digraphs,\n then ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphJoin", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "C", next := 93394, root := ~, start := 93376, stop := 93393 ), rec( content := " returns\ the join of the digraphs contained in the\n list. ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "P", next := 93459, root := ~, start := 93455, stop := 93458 ), rec( content := "\n\n \ The join of a collection of digraphs ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 93511, root := ~, start := 93502, stop := 93510 ), rec( content := ", ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 93522, root := ~, start := 93513, stop := 93521 ), rec( content := ", ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 93534, root := ~, start := 93524, stop := 93533 ), rec( content := " is\n \ formed by first taking the ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "Ref", next := 93640, root := ~, start := 93569, stop := 93639 ), rec( content := " of the \ collection.\n\n In the disjoint union, if ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ \\neq j", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "M", next := 93706, root := ~, start := 93691, stop := 93705 ), rec( content := " then th\ ere are no edges between\n vertices corresponding to digraphs ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 93786, root := ~, start := 93778, stop := 93785 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 93796, root := ~, start := 93786, stop := 93795 ), rec( content := " and\n \ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 93813, root := ~, start := 93805, stop := 93812 ), rec( attributes := rec( ), content := [ rec( content := "[\ j]", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 93823, root := ~, start := 93813, stop := 93822 ), rec( content := " in the \ collection; the join is created by including\n all such edges.", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 7 ], mathmode := "Text", name := "P", next := 93899, root := ~, start := 93895, stop := 93898 ), rec( content := "\n\n \ For example, the join of two empty digraphs is a complete bipartite\n digra\ ph.", count := [ 3, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "P", next := 93989, root := ~, start := 93985, stop := 93988 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "Ref", next := 94053, root := ~, start := 94020, stop := 94052 ), rec( content := " will be\ lost.\n ", count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "P", next := 94076, root := ~, start := 94072, stop := 94075 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 94112, root := ~, start := 94103, stop := 94111 ), rec( content := " [", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 94128, root := ~, start := 94114, stop := 94127 ), rec( content := "] belong\ s to ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "Ref", next := 94175, root := ~, start := 94141, stop := 94174 ), rec( content := ", then " , count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 94191, root := ~, start := 94182, stop := 94190 ), rec( content := " [", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 94207, root := ~, start := 94193, stop := 94206 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 94298, root := ~, start := 94284, stop := 94297 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "Ref", next := 94346, root := ~, start := 94314, stop := 94345 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> IsCompleteDigraph(DigraphJoin(D, D));\ntrue\ngap> D2 := CycleDigraph(3);\ \n\ngap> DigraphJoin(D, D2);\n\n", count := [ 3, 3, 29, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 29, 10 ], mathmode := "Text", name := "Example", next := 94731, root := ~, start := 94435, stop := 94730 ), rec( content := "\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Description", next := 94748, root := ~, start := 93165, stop := 94747 ) ], count := [ 3, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 94762, root := ~, start := 92954, stop := 94761 ), rec( content := "\n\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "gr1, gr2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphCartesianProduct" ), content := 0, count := [ 3, 3, 30, 2 ], mathmode := "Text", name := "Func", next := 94887, root := ~, start := 94783, stop := 94886 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphCartesianProduct" ), content := 0, count := [ 3, 3, 30, 4 ], mathmode := "Text", name := "Func", next := 94974, root := ~, start := 94890, stop := 94973 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Returns", next := 95006, root := ~, start := 94977, stop := 95005 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "A", next := 95059, root := ~, start := 95049, stop := 95058 ), rec( content := ", ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "A", next := 95071, root := ~, start := 95061, stop := 95070 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "C", next := 95131, root := ~, start := 95101, stop := 95130 ), rec( content := " returns\ a digraph isomorphic to their\n Cartesian product.\n ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "P", next := 95201, root := ~, start := 95197, stop := 95200 ), rec( content := "\n\n \ In the second form, if ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "A", next := 95241, root := ~, start := 95230, stop := 95240 ), rec( content := " is a no\ n-empty list of digraphs,\n then ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "C", next := 95314, root := ~, start := 95284, stop := 95313 ), rec( content := " returns\ a digraph isomorphic to the \n Cartesian product of the digraphs contained\ in the list.\n ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "P", next := 95421, root := ~, start := 95417, stop := 95420 ), rec( content := "\n \n \ Mathematically, the Cartesian product of two digraphs ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95490, root := ~, start := 95482, stop := 95489 ), rec( content := ", ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95500, root := ~, start := 95492, stop := 95499 ), rec( content := "\n is\ a digraph with vertex set ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ artesian(DigraphVertices(G),\n DigraphVertices(H))", count := [ 3, 3, 30, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95594, root := ~, start := 95534, stop := 95593 ), rec( content := " such th\ at there is an edge from ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, u']", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95641, root := ~, start := 95627, stop := 95640 ), rec( content := " to\n \ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, v']", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95663, root := ~, start := 95649, stop := 95662 ), rec( content := " iff ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ u = v ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95682, root := ~, start := 95668, stop := 95681 ), rec( content := " and the\ re is an edge from ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95718, root := ~, start := 95709, stop := 95717 ), rec( content := " to\n \ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ '", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95735, root := ~, start := 95726, stop := 95734 ), rec( content := " in ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95747, root := ~, start := 95739, stop := 95746 ), rec( content := " or ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ u' = v'", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95766, root := ~, start := 95751, stop := 95765 ), rec( content := " and the\ re is an edge from\n ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95805, root := ~, start := 95797, stop := 95804 ), rec( content := " to ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95817, root := ~, start := 95809, stop := 95816 ), rec( content := " in ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 95829, root := ~, start := 95821, stop := 95828 ), rec( content := ".\n " , count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "P", next := 95839, root := ~, start := 95835, stop := 95838 ), rec( content := "\n\n \ Due to technical reasons, the digraph ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 95891, root := ~, start := 95883, stop := 95890 ), rec( content := " returne\ d by\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 95938, root := ~, start := 95908, stop := 95937 ), rec( content := " has ver\ tex set\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]", count := [ 3, 3, 30, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96014, root := ~, start := 95958, stop := 96013 ), rec( content := " instead\ , and the\n exact\n method of encoding pairs of vertices into integers \ is implementation\n specific. The original vertex pair can be somewhat reg\ ained by using\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ CartesianProductProjections" ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "Ref", next := 96242, root := ~, start := 96194, stop := 96241 ), rec( content := ". In add\ ition,\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "Ref", next := 96294, root := ~, start := 96261, stop := 96293 ), rec( content := " are pre\ served: if vertex pair\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,u']", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96342, root := ~, start := 96329, stop := 96341 ), rec( content := " was enc\ oded as ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96366, root := ~, start := 96358, stop := 96365 ), rec( content := " then th\ e vertex label of ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96400, root := ~, start := 96392, stop := 96399 ), rec( content := "\n wi\ ll be the pair of vertex labels of ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96450, root := ~, start := 96442, stop := 96449 ), rec( content := " and ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96464, root := ~, start := 96455, stop := 96463 ), rec( content := " i.e.\n \ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H\ ,u')]", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 96564, root := ~, start := 96474, stop := 96563 ), rec( content := ".\n " , count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "P", next := 96574, root := ~, start := 96570, stop := 96573 ), rec( content := "\n\n \ As the Cartesian product is associative, the Cartesian product of a collection\ \n of digraphs ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "A", next := 96686, root := ~, start := 96676, stop := 96685 ), rec( content := ", ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "A", next := 96698, root := ~, start := 96688, stop := 96697 ), rec( content := ", ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "C", next := 96710, root := ~, start := 96700, stop := 96709 ), rec( content := " is comp\ uted in the obvious \n fashion. \n ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 10 ], mathmode := "Text", name := "P", next := 96761, root := ~, start := 96757, stop := 96760 ), rec( content := "\n\n", count := [ 3, 3, 30, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(4);\n\ngap> \ gr2 := CycleDigraph(3);\n\ngap> gr3 :\ = DigraphCartesianProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[2, 5], [3, 6], [4, 7], [8],\ \n> [6, 9], [7, 10], [8, 11], [12],\n> [10, 1], [11, 2], [1\ 2, 3], [4]]));\ntrue\n", count := [ 3, 3, 30, 11 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 11 ], mathmode := "Text", name := "Example", next := 97195, root := ~, start := 96763, stop := 97194 ), rec( content := "\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 97212, root := ~, start := 95009, stop := 97211 ) ], count := [ 3, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 97226, root := ~, start := 94768, stop := 97225 ), rec( content := "\n\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "gr1, gr2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphDirectProduct" ), content := 0, count := [ 3, 3, 31, 2 ], mathmode := "Text", name := "Func", next := 97348, root := ~, start := 97247, stop := 97347 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphDirectProduct" ), content := 0, count := [ 3, 3, 31, 4 ], mathmode := "Text", name := "Func", next := 97432, root := ~, start := 97351, stop := 97431 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Returns", next := 97464, root := ~, start := 97435, stop := 97463 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "A", next := 97517, root := ~, start := 97507, stop := 97516 ), rec( content := ", ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "A", next := 97529, root := ~, start := 97519, stop := 97528 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "C", next := 97586, root := ~, start := 97559, stop := 97585 ), rec( content := " returns\ a digraph isomorphic to their direct\n product.\n ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "P", next := 97653, root := ~, start := 97649, stop := 97652 ), rec( content := "\n\n \ In the second form, if ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "A", next := 97693, root := ~, start := 97682, stop := 97692 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "C", next := 97763, root := ~, start := 97736, stop := 97762 ), rec( content := " returns\ a digraph isomorphic to the direct\n product of the digraphs contained in \ the list.\n ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "P", next := 97866, root := ~, start := 97862, stop := 97865 ), rec( content := "\n\n \ Mathematically, the direct product of two digraphs ", count := [ 3, 3, 31, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 97931, root := ~, start := 97923, stop := 97930 ), rec( content := ", ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 97941, root := ~, start := 97933, stop := 97940 ), rec( content := " is a\n \ digraph with vertex set ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ artesian(DigraphVertices(G),\n DigraphVertices(H))", count := [ 3, 3, 31, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98035, root := ~, start := 97975, stop := 98034 ), rec( content := " such th\ at there is an edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, u']", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98082, root := ~, start := 98068, stop := 98081 ), rec( content := "\n to\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, v']", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98104, root := ~, start := 98090, stop := 98103 ), rec( content := " iff the\ re is an edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98139, root := ~, start := 98131, stop := 98138 ), rec( content := " to ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98151, root := ~, start := 98143, stop := 98150 ), rec( content := " in\n \ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98167, root := ~, start := 98159, stop := 98166 ), rec( content := " and an \ edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98194, root := ~, start := 98185, stop := 98193 ), rec( content := " to ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ '", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98207, root := ~, start := 98198, stop := 98206 ), rec( content := " in ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 98219, root := ~, start := 98211, stop := 98218 ), rec( content := ". \n \ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "P", next := 98230, root := ~, start := 98226, stop := 98229 ), rec( content := " \n\n \ Due to technical reasons, the digraph ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98283, root := ~, start := 98275, stop := 98282 ), rec( content := " returne\ d by\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98327, root := ~, start := 98300, stop := 98326 ), rec( content := " has ver\ tex set\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]", count := [ 3, 3, 31, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98403, root := ~, start := 98347, stop := 98402 ), rec( content := " instead\ , and the\n exact method of encoding pairs of vertices into integers is imp\ lementation\n specific. The original vertex pair can be somewhat regained \ by using\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ DirectProductProjections" ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "Ref", next := 98623, root := ~, start := 98578, stop := 98622 ), rec( content := ". In add\ ition\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "Ref", next := 98674, root := ~, start := 98641, stop := 98673 ), rec( content := " are pre\ served: if vertex pair\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,u']", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98722, root := ~, start := 98709, stop := 98721 ), rec( content := " was enc\ oded as ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98746, root := ~, start := 98738, stop := 98745 ), rec( content := " then th\ e vertex label of ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98780, root := ~, start := 98772, stop := 98779 ), rec( content := "\n wi\ ll be the pair of vertex labels of ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98830, root := ~, start := 98822, stop := 98829 ), rec( content := " and ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98844, root := ~, start := 98835, stop := 98843 ), rec( content := " i.e.\n \ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u\ ')]", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 98942, root := ~, start := 98854, stop := 98941 ), rec( content := ".\n " , count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "P", next := 98952, root := ~, start := 98948, stop := 98951 ), rec( content := "\n\n \ As the direct product is associative, the direct product of a collection \n \ of digraphs ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "A", next := 99058, root := ~, start := 99048, stop := 99057 ), rec( content := ", ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "A", next := 99070, root := ~, start := 99060, stop := 99069 ), rec( content := ", ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "C", next := 99082, root := ~, start := 99072, stop := 99081 ), rec( content := " is comp\ uted in the obvious \n fashion. \n ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 10 ], mathmode := "Text", name := "P", next := 99133, root := ~, start := 99129, stop := 99132 ), rec( content := "\n\n", count := [ 3, 3, 31, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(4);\n\ngap> \ gr2 := CycleDigraph(3);\n\ngap> gr3 :\ = DigraphDirectProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[6], [7], [8], [], \n> \ [10], [11], [12], [],\n> [2], [3], [4], []]));\ntrue\n", count := [ 3, 3, 31, 11 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 11 ], mathmode := "Text", name := "Example", next := 99530, root := ~, start := 99135, stop := 99529 ), rec( content := "\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 99547, root := ~, start := 97467, stop := 99546 ) ], count := [ 3, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 99561, root := ~, start := 97232, stop := 99560 ), rec( content := "\n\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphCartesia\ nProductProjections" ), content := 0, count := [ 3, 3, 32, 2 ], mathmode := "Text", name := "Attr", next := 99645, root := ~, start := 99582, stop := 99644 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Returns", next := 99693, root := ~, start := 99648, stop := 99692 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "A", next := 99731, root := ~, start := 99717, stop := 99730 ), rec( content := " is a Ca\ rtesian product digraph, \n ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph = DigraphCartesianProduct(gr_1, gr_2, ... )", count := [ 3, 3, 32, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 99827, root := ~, start := 99769, stop := 99826 ), rec( content := ", then\n\ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProductProjections", count := [ 3, 3, 32, 3 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 99879, root := ~, start := 99838, stop := 99878 ), rec( content := " returns\ a list ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 99906, root := ~, start := 99895, stop := 99905 ), rec( content := "\n su\ ch that ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 99935, root := ~, start := 99921, stop := 99934 ), rec( content := " is the \ projection onto the ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 99971, root := ~, start := 99963, stop := 99970 ), rec( content := "-th\n \ coordinate of the product.\n ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "P", next := 100014, root := ~, start := 100010, stop := 100013 ), rec( content := "\n\n \ A projection is an idempotent endomorphism of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "A", next := 100080, root := ~, start := 100066, stop := 100079 ), rec( content := ". If\n \ ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1, gr2, ...", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 100109, root := ~, start := 100089, stop := 100108 ), rec( content := " are all\ loopless digraphs, then the induced \n subdigraph of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "A", next := 100187, root := ~, start := 100173, stop := 100186 ), rec( content := " on the \ image of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 100218, root := ~, start := 100204, stop := 100217 ), rec( content := " is \n \ isomorphic to ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r_i", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 100252, root := ~, start := 100241, stop := 100251 ), rec( content := ".\n " , count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "P", next := 100262, root := ~, start := 100258, stop := 100261 ), rec( content := "\n\n \ Currently this attribute is only set upon creating an immutable \n digraph \ via ", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "C", next := 100379, root := ~, start := 100349, stop := 100378 ), rec( content := " and the\ re is no way \n of calculating it for other digraphs.\n ", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "P", next := 100451, root := ~, start := 100447, stop := 100450 ), rec( content := "\n\n \ For more information see\n ", count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ CartesianProduct", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "Ref", next := 100554, root := ~, start := 100486, stop := 100553 ), rec( content := " \n\n \ ", count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphCartesianProduct(ChainDigraph(3), CycleDigraph(4),\n> Digrap\ h([[2], [2]]));;\ngap> HasDigraphCartesianProductProjections(D);\ntrue\ngap> p\ roj := DigraphCartesianProductProjections(D);; Length(proj);\n3\ngap> IsIdempo\ tent(proj[2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> S := ImageS\ etOfTransformation(proj[2]);;\ngap> IsIsomorphicDigraph(CycleDigraph(4), Induc\ edSubdigraph(D, S));\ntrue\n", count := [ 3, 3, 32, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 7 ], mathmode := "Text", name := "Example", next := 100997, root := ~, start := 100561, stop := 100996 ), rec( content := "\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Description", next := 101014, root := ~, start := 99696, stop := 101013 ) ], count := [ 3, 3, 32, 1 ], mathmode := "Text", name := "ManSection", next := 101028, root := ~, start := 99567, stop := 101027 ), rec( content := "\n\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDirectPr\ oductProjections" ), content := 0, count := [ 3, 3, 33, 2 ], mathmode := "Text", name := "Attr", next := 101109, root := ~, start := 101049, stop := 101108 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Returns", next := 101157, root := ~, start := 101112, stop := 101156 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "A", next := 101195, root := ~, start := 101181, stop := 101194 ), rec( content := " is a di\ rect product digraph, \n ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph = DigraphDirectProduct(gr_1, gr_2, ... )", count := [ 3, 3, 33, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 101285, root := ~, start := 101230, stop := 101284 ), rec( content := ", then\n\ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProductProjections", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 101334, root := ~, start := 101296, stop := 101333 ), rec( content := " returns\ a list ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 101361, root := ~, start := 101350, stop := 101360 ), rec( content := "\n su\ ch that ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 101390, root := ~, start := 101376, stop := 101389 ), rec( content := " is the \ projection onto the ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 101426, root := ~, start := 101418, stop := 101425 ), rec( content := "-th\n \ coordinate of the product.\n ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "P", next := 101469, root := ~, start := 101465, stop := 101468 ), rec( content := "\n\n \ A projection is an idempotent endomorphism of ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "A", next := 101535, root := ~, start := 101521, stop := 101534 ), rec( content := ". If\n \ ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1, gr2, ...", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 101564, root := ~, start := 101544, stop := 101563 ), rec( content := " are all\ loopless digraphs, then the image \n of ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "A", next := 101629, root := ~, start := 101615, stop := 101628 ), rec( content := " under " , count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 101650, root := ~, start := 101636, stop := 101649 ), rec( content := " is isom\ orphic to ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r_i", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 101679, root := ~, start := 101668, stop := 101678 ), rec( content := ".\n " , count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "P", next := 101689, root := ~, start := 101685, stop := 101688 ), rec( content := "\n\n \ Currently this attribute is only set upon creating an immutable\n digraph v\ ia ", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "C", next := 101802, root := ~, start := 101775, stop := 101801 ), rec( content := " and the\ re is no way of \n calculating it for other digraphs.\n ", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "P", next := 101874, root := ~, start := 101870, stop := 101873 ), rec( content := "\n\n \ For more information, see\n ", count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DirectProduct", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "Ref", next := 101975, root := ~, start := 101910, stop := 101974 ), rec( content := " \n\n \ ", count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphDirectProduct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([\ [2], [2]]));;\ngap> HasDigraphDirectProductProjections(D);\ntrue\ngap> proj :=\ DigraphDirectProductProjections(D);; Length(proj);\n3\ngap> IsIdempotent(proj\ [2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> P := DigraphRemoveAl\ lMultipleEdges(\n> ReducedDigraph(OnDigraphs(D, proj[2])));; \ngap> IsIsomorph\ icDigraph(CycleDigraph(4), P);\ntrue\n", count := [ 3, 3, 33, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 7 ], mathmode := "Text", name := "Example", next := 102427, root := ~, start := 101982, stop := 102426 ), rec( content := "\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Description", next := 102444, root := ~, start := 101160, stop := 102443 ) ], count := [ 3, 3, 33, 1 ], mathmode := "Text", name := "ManSection", next := 102458, root := ~, start := 101034, stop := 102457 ), rec( content := "\n\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LineDigraph" ), content := 0, count := [ 3, 3, 34, 2 ], mathmode := "Text", name := "Oper", next := 102519, root := ~, start := 102479, stop := 102518 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeDigraph" ), content := 0, count := [ 3, 3, 34, 4 ], mathmode := "Text", name := "Oper", next := 102562, root := ~, start := 102522, stop := 102561 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Returns", next := 102594, root := ~, start := 102565, stop := 102593 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a digraph ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "A", next := 102645, root := ~, start := 102631, stop := 102644 ), rec( content := ", the op\ eration returns the digraph\n obtained by associating a vertex with each ed\ ge of ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "A", next := 102750, root := ~, start := 102736, stop := 102749 ), rec( content := ", and\n \ creating an edge from a vertex ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 102799, root := ~, start := 102791, stop := 102798 ), rec( content := " to a ve\ rtex ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 102820, root := ~, start := 102812, stop := 102819 ), rec( content := " if and \ only\n if the terminal vertex of the edge associated with ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 102896, root := ~, start := 102888, stop := 102895 ), rec( content := " is the \ start\n vertex of the edge associated with ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 102957, root := ~, start := 102949, stop := 102956 ), rec( content := ".", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "P", next := 102962, root := ~, start := 102958, stop := 102961 ), rec( content := "\n\n \ Note that the returned digraph is always a new immutable digraph, and the\n \ argument ", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "A", next := 103069, root := ~, start := 103055, stop := 103068 ), rec( content := " is neve\ r modified.\n ", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> LineDigraph(CompleteDigraph(3));\n\ngap> LineDigraph(ChainDigraph(3));\n\n", count := [ 3, 3, 34, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 7 ], mathmode := "Text", name := "Example", next := 103288, root := ~, start := 103093, stop := 103287 ), rec( content := "\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Description", next := 103305, root := ~, start := 102597, stop := 103304 ) ], count := [ 3, 3, 34, 1 ], mathmode := "Text", name := "ManSection", next := 103319, root := ~, start := 102464, stop := 103318 ), rec( content := "\n\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LineUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 2 ], mathmode := "Text", name := "Oper", next := 103390, root := ~, start := 103340, stop := 103389 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 4 ], mathmode := "Text", name := "Oper", next := 103443, root := ~, start := 103393, stop := 103442 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Returns", next := 103475, root := ~, start := 103446, stop := 103474 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a symmetric digraph ", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "A", next := 103536, root := ~, start := 103522, stop := 103535 ), rec( content := ", the op\ eration returns the\n symmetric digraph obtained by associating a vertex wi\ th each edge of\n ", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "A", next := 103655, root := ~, start := 103641, stop := 103654 ), rec( content := ", ignori\ ng directions and multiplicites, and adding an edge\n between two vertices \ if and only if the corresponding edges have a vertex in\n common.", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "P", next := 103811, root := ~, start := 103807, stop := 103810 ), rec( content := "\n\n \ Note that the returned digraph is always a new immutable digraph, and the\n \ argument ", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "A", next := 103918, root := ~, start := 103904, stop := 103917 ), rec( content := " is neve\ r modified.\n ", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> LineUndirectedDigraph(CompleteDigraph(3));\n\ngap> LineUndirectedDigraph(DigraphSymmetricClosure(ChainDigr\ aph(3)));\n\n", count := [ 3, 3, 35, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 35, 7 ], mathmode := "Text", name := "Example", next := 104182, root := ~, start := 103942, stop := 104181 ), rec( content := "\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Description", next := 104199, root := ~, start := 103478, stop := 104198 ) ], count := [ 3, 3, 35, 1 ], mathmode := "Text", name := "ManSection", next := 104213, root := ~, start := 103325, stop := 104212 ), rec( content := "\n\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DoubleDigraph" ), content := 0, count := [ 3, 3, 36, 2 ], mathmode := "Text", name := "Oper", next := 104276, root := ~, start := 104234, stop := 104275 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Returns", next := 104308, root := ~, start := 104279, stop := 104307 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 104347, root := ~, start := 104333, stop := 104346 ), rec( content := " be a di\ graph with vertex set ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 104385, root := ~, start := 104377, stop := 104384 ), rec( content := ". This f\ unction\n returns the double digraph of ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 104449, root := ~, start := 104435, stop := 104448 ), rec( content := ". The ve\ rtex set of the double\n digraph is the orginal vertex set together with a \ duplicate. The edges are\n ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_1, v_2]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 104580, root := ~, start := 104563, stop := 104579 ), rec( content := " and ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_2, v_1]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 104602, root := ~, start := 104585, stop := 104601 ), rec( content := " if and \ only if ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 104631, root := ~, start := 104618, stop := 104630 ), rec( content := " is an\n\ edge in ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 104664, root := ~, start := 104650, stop := 104663 ), rec( content := ", togeth\ er with the original edges and their\n duplicates.", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "P", next := 104728, root := ~, start := 104724, stop := 104727 ), rec( content := "\n\n \ If ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 104751, root := ~, start := 104737, stop := 104750 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 104783, root := ~, start := 104769, stop := 104782 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 104827, root := ~, start := 104813, stop := 104826 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gamma := Digraph([[2], [3], [1]]);\n\ngap> DoubleDigraph(gamma);\n\n", count := [ 3, 3, 36, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 5 ], mathmode := "Text", name := "Example", next := 105113, root := ~, start := 104923, stop := 105112 ), rec( content := "\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Description", next := 105130, root := ~, start := 104311, stop := 105129 ) ], count := [ 3, 3, 36, 1 ], mathmode := "Text", name := "ManSection", next := 105144, root := ~, start := 104219, stop := 105143 ), rec( content := "\n\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BipartiteDouble\ Digraph" ), content := 0, count := [ 3, 3, 37, 2 ], mathmode := "Text", name := "Oper", next := 105216, root := ~, start := 105165, stop := 105215 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Returns", next := 105248, root := ~, start := 105219, stop := 105247 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 105287, root := ~, start := 105273, stop := 105286 ), rec( content := " be a di\ graph with vertex set ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 105325, root := ~, start := 105317, stop := 105324 ), rec( content := ". This f\ unction\n returns the bipartite double digraph of ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 105399, root := ~, start := 105385, stop := 105398 ), rec( content := ". The ve\ rtex set of\n the double digraph is the original vertex set together with a\ duplicate. The\n edges are ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_1, v_2]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 105531, root := ~, start := 105514, stop := 105530 ), rec( content := " and ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_2, v_1]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 105553, root := ~, start := 105536, stop := 105552 ), rec( content := " if and \ only if ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,\n v]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 105588, root := ~, start := 105569, stop := 105587 ), rec( content := " is an e\ dge in ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 105617, root := ~, start := 105603, stop := 105616 ), rec( content := ". The re\ sulting graph is bipartite,\n since the orignal edges are not included in t\ he resulting digraph.", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "P", next := 105727, root := ~, start := 105723, stop := 105726 ), rec( content := "\n\n \ If ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 105750, root := ~, start := 105736, stop := 105749 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 105782, root := ~, start := 105768, stop := 105781 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 105826, root := ~, start := 105812, stop := 105825 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gamma := Digraph([[2], [3], [1]]);\n\ngap> BipartiteDoubleDigraph(gamma);\n\n", count := [ 3, 3, 37, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 5 ], mathmode := "Text", name := "Example", next := 106120, root := ~, start := 105922, stop := 106119 ), rec( content := "\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Description", next := 106137, root := ~, start := 105251, stop := 106136 ) ], count := [ 3, 3, 37, 1 ], mathmode := "Text", name := "ManSection", next := 106151, root := ~, start := 105150, stop := 106150 ), rec( content := "\n\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ ops" ), content := 0, count := [ 3, 3, 38, 2 ], mathmode := "Text", name := "Oper", next := 106219, root := ~, start := 106172, stop := 106218 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ opsAttr" ), content := 0, count := [ 3, 3, 38, 4 ], mathmode := "Text", name := "Attr", next := 106273, root := ~, start := 106222, stop := 106272 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Returns", next := 106305, root := ~, start := 106276, stop := 106304 ), rec( attributes := rec( ), content := [ rec( content := "\n For\ a digraph ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 106354, root := ~, start := 106340, stop := 106353 ), rec( content := " this op\ eration returns a new digraph constructed\n from ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 106427, root := ~, start := 106413, stop := 106426 ), rec( content := ", such t\ hat a loop is added for every vertex which did not\n have a loop in ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 106519, root := ~, start := 106505, stop := 106518 ), rec( content := ".", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "P", next := 106524, root := ~, start := 106520, stop := 106523 ), rec( content := "\n\n \ If ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "A", next := 106547, root := ~, start := 106533, stop := 106546 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ added to the loopless vertices of the mutable digraph\n ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "A", next := 106705, root := ~, start := 106691, stop := 106704 ), rec( content := " in-plac\ e.\n ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := EmptyDigraph(13);\n\ngap>\ D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], \ [ 7 ], [ 8 ], [ 9 ], \n [ 10 ], [ 11 ], [ 12 ], [ 13 ] ]\ngap> D := Digraph([\ [1, 2, 3], [1, 3], [1]]);\n\ngap> \ D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ]\ngap> D \ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAddAllLoops(D);\n\n\ gap> D;\n\n", count := [ 3, 3, 38, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 38, 7 ], mathmode := "Text", name := "Example", next := 107487, root := ~, start := 106720, stop := 107486 ), rec( content := "\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Description", next := 107504, root := ~, start := 106308, stop := 107503 ) ], count := [ 3, 3, 38, 1 ], mathmode := "Text", name := "ManSection", next := 107518, root := ~, start := 106157, stop := 107517 ), rec( content := "\n\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Label := "for digraph an\ d int", Name := "DistanceDigraph" ), content := 0, count := [ 3, 3, 39, 2 ], mathmode := "Text", name := "Oper", next := 107622, root := ~, start := 107539, stop := 107621 ), rec( attributes := rec( Arg := "digraph, list", Label := "for digraph an\ d list", Name := "DistanceDigraph" ), content := 0, count := [ 3, 3, 39, 4 ], mathmode := "Text", name := "Oper", next := 107712, root := ~, start := 107625, stop := 107711 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Returns", next := 107744, root := ~, start := 107715, stop := 107743 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ first argument is a digraph, the second argument is a non-negative\n integ\ er or a list of positive integers. This operation returns a digraph on\n th\ e same set of vertices as ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 107962, root := ~, start := 107948, stop := 107961 ), rec( content := ", with t\ wo vertices being adjacent\n if and only if the distance between them in ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 108059, root := ~, start := 108045, stop := 108058 ), rec( content := " equals \ ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 108075, root := ~, start := 108067, stop := 108074 ), rec( content := "\n or\ is a number in ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 108109, root := ~, start := 108098, stop := 108108 ), rec( content := ". See " , count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and two vertices", Oper := "DigraphShortestDistance" ), content := 0, count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Ref", next := 108196, root := ~, start := 108116, stop := 108195 ), rec( content := ". ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "P", next := 108202, root := ~, start := 108198, stop := 108201 ), rec( content := "\n\n \ If ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 108225, root := ~, start := 108211, stop := 108224 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 108257, root := ~, start := 108243, stop := 108256 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 108301, root := ~, start := 108287, stop := 108300 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := DigraphFromSparse6String(\n> \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL\ _@McDHfILaBJfHMjKM\");\n\ngap> DistanceDigraph(digraph, 1);\n\ngap> DistanceDigraph(digraph, [1, 2]);\n\n", count := [ 3, 3, 39, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 7 ], mathmode := "Text", name := "Example", next := 108751, root := ~, start := 108397, stop := 108750 ), rec( content := "\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Description", next := 108768, root := ~, start := 107747, stop := 108767 ) ], count := [ 3, 3, 39, 1 ], mathmode := "Text", name := "ManSection", next := 108782, root := ~, start := 107524, stop := 108781 ), rec( content := "\n\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, k", Name := "DigraphClosure"\ ), content := 0, count := [ 3, 3, 40, 2 ], mathmode := "Text", name := "Oper", next := 108849, root := ~, start := 108803, stop := 108848 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Returns", next := 108881, root := ~, start := 108852, stop := 108880 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a symmetric loopless digraph with no multiple edges ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 108974, root := ~, start := 108960, stop := 108973 ), rec( content := ",\n t\ he ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 3, 40, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := \ "Text", name := "A", next := 108995, root := ~, start := 108987, stop := 10899\ 4 ), rec( content := "-closure of ", count := [ 3, 3, 40, 3 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "digraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 1090\ 21, root := ~, start := 109007, stop := 109020 ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "E", next := 109025, root := ~, start := 108984, stop := 109024 ), rec( content := " is defi\ ned to be the unique\n smallest symmetric loopless digraph ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "C", next := 109102, root := ~, start := 109094, stop := 109101 ), rec( content := " with no\ multiple edges on the\n vertices of ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 109163, root := ~, start := 109149, stop := 109162 ), rec( content := " that co\ ntains all the edges of ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 109209, root := ~, start := 109195, stop := 109208 ), rec( content := "\n an\ d satsifies the property that the sum of the degrees of every two\n non-adj\ acenct vertices in ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "C", next := 109320, root := ~, start := 109312, stop := 109319 ), rec( content := " is less\ than ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 109342, root := ~, start := 109334, stop := 109341 ), rec( content := ". See ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 109384, root := ~, start := 109348, stop := 109383 ), rec( content := ", ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 109415, root := ~, start := 109386, stop := 109414 ), rec( content := ", ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 109449, root := ~, start := 109417, stop := 109448 ), rec( content := ", and ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 109486, root := ~, start := 109455, stop := 109485 ), rec( content := ". ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "P", next := 109492, root := ~, start := 109488, stop := 109491 ), rec( content := "\n\n \ The operation ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphClosure", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "C", next := 109533, root := ~, start := 109512, stop := 109532 ), rec( content := " returns\ the ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "A", next := 109554, root := ~, start := 109546, stop := 109553 ), rec( content := "-closure\ of\n ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "A", next := 109584, root := ~, start := 109570, stop := 109583 ), rec( content := ".\n " , count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(6);\n\n\ gap> D := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);\n\ngap> closure := DigraphClosure(D, 6);\n\ngap> IsCompleteDigraph(closure);\ntrue\n", count := [ 3, 3, 40, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 40, 5 ], mathmode := "Text", name := "Example", next := 109916, root := ~, start := 109590, stop := 109915 ), rec( content := "\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Description", next := 109933, root := ~, start := 108884, stop := 109932 ) ], count := [ 3, 3, 40, 1 ], mathmode := "Text", name := "ManSection", next := 109947, root := ~, start := 108788, stop := 109946 ), rec( content := "\n\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ian" ), content := 0, count := [ 3, 3, 41, 2 ], mathmode := "Text", name := "Oper", next := 110015, root := ~, start := 109968, stop := 110014 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ianAttr" ), content := 0, count := [ 3, 3, 41, 4 ], mathmode := "Text", name := "Attr", next := 110069, root := ~, start := 110018, stop := 110068 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Returns", next := 110101, root := ~, start := 110072, stop := 110100 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "A", next := 110139, root := ~, start := 110125, stop := 110138 ), rec( content := " is a sy\ mmetric digraph, then\n ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphMycielskian", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "C", next := 110198, root := ~, start := 110173, stop := 110197 ), rec( content := " returns\ the Mycielskian of ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "A", next := 110240, root := ~, start := 110226, stop := 110239 ), rec( content := ".", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "P", next := 110245, root := ~, start := 110241, stop := 110244 ), rec( content := "\n \n\ The Mycielskian of a symmetric digraph is a larger symmetric digraph\n \ constructed from it, which has a larger chromatic number. For further\n inf\ ormation, see ", count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Mycielskian", count := [ 3, 3, 41, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "URL", next := 110471, root := ~, start := 110419, stop := 110470 ), rec( content := ".", count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "P", next := 110476, root := ~, start := 110472, stop := 110475 ), rec( content := "\n\n \ If ", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "A", next := 110499, root := ~, start := 110485, stop := 110498 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "A", next := 110605, root := ~, start := 110591, stop := 110604 ), rec( content := " is chan\ ged in-place into\n its Mycielskian.", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 8 ], mathmode := "Text", name := "P", next := 110655, root := ~, start := 110651, stop := 110654 ), rec( content := "\n\n \ ", count := [ 3, 3, 41, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(2);\n\ngap> C\ hromaticNumber(D);\n2\ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n3\ngap> D := DigraphMyciel\ skian(D);\n\ngap> ChromaticNumbe\ r(D);\n4\ngap> D := CompleteBipartiteDigraph(IsMutable, 2, 3);\n\ngap> DigraphMycielskian(D);\n\ngap> D;\n\n", count := [ 3, 3, 41, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 9 ], mathmode := "Text", name := "Example", next := 111225, root := ~, start := 110661, stop := 111224 ), rec( content := "\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Description", next := 111242, root := ~, start := 110104, stop := 111241 ) ], count := [ 3, 3, 41, 1 ], mathmode := "Text", name := "ManSection", next := 111256, root := ~, start := 109953, stop := 111255 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 111270, root := ~, start := 45224, stop := 111269 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Random digraphs", count := [ 3, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 111317, root := ~, start := 111283, stop := 111316 ), rec( content := "\n ", count := [ 3, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n[, p]", Name := "RandomDigraph" ), content := 0, count := [ 3, 4, 1, 2 ], mathmode := "Text", name := "Oper", next := 111386, root := ~, start := 111337, stop := 111385 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 111418, root := ~, start := 111389, stop := 111417 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 111481, root := ~, start := 111470, stop := 111480 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 111627, root := ~, start := 111616, stop := 111626 ), rec( content := " is ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 111661, root := ~, start := 111631, stop := 111660 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 111729, root := ~, start := 111718, stop := 111728 ), rec( content := " is ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 111769, root := ~, start := 111733, stop := 111768 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 111854, root := ~, start := 111843, stop := 111853 ), rec( content := " is not \ present, then ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 111914, root := ~, start := 111876, stop := 111913 ), rec( content := " is used\ by default.", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "P", next := 111938, root := ~, start := 111934, stop := 111937 ), rec( content := "\n\n \ If ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "A", next := 111955, root := ~, start := 111947, stop := 111954 ), rec( content := " is a po\ sitive integer, then this function returns a random\n digraph with ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "A", next := 112040, root := ~, start := 112032, stop := 112039 ), rec( content := " vertice\ s and without multiple edges. The result\n may or may not have loops.", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "P", next := 112123, root := ~, start := 112119, stop := 112122 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 112169, root := ~, start := 112161, stop := 112168 ), rec( content := " is a fl\ oat with value\n ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ \\leq ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "M", next := 112210, root := ~, start := 112196, stop := 112209 ), rec( content := " ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ p ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 112221, root := ~, start := 112211, stop := 112220 ), rec( content := " ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ \\leq 1", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "M", next := 112236, root := ~, start := 112222, stop := 112235 ), rec( content := ", then a\ n edge will exist between each\n pair of vertices with probability approxim\ ately ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 112335, root := ~, start := 112327, stop := 112334 ), rec( content := ".\n I\ f ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 112352, root := ~, start := 112344, stop := 112351 ), rec( content := " is not \ specified, then a random probability will be assumed\n (chosen with uniform\ probability).\n ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2);\n\n", count := [ 3, 4, 1, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 4, 1, 6 ], mathmode := "Text", name := "Log", next := 112750, root := ~, start := 112456, stop := 112749 ), rec( content := "\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 112767, root := ~, start := 111421, stop := 112766 ) ], count := [ 3, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 112781, root := ~, start := 111322, stop := 112780 ), rec( content := "\n\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "n[, m]", Name := "RandomMultiDigr\ aph" ), content := 0, count := [ 3, 4, 2, 2 ], mathmode := "Text", name := "Oper", next := 112848, root := ~, start := 112802, stop := 112847 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 112880, root := ~, start := 112851, stop := 112879 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 112912, root := ~, start := 112904, stop := 112911 ), rec( content := " is a po\ sitive integer, then this function returns a random\n digraph with ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 112997, root := ~, start := 112989, stop := 112996 ), rec( content := " vertice\ s. If the optional second argument\n ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 113052, root := ~, start := 113044, stop := 113051 ), rec( content := " is a po\ sitive integer, then the digraph will have ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 113111, root := ~, start := 113103, stop := 113110 ), rec( content := "\n ed\ ges. If ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 113134, root := ~, start := 113126, stop := 113133 ), rec( content := " is not \ specified, then the number of edges will be\n chosen randomly (with uniform\ probability) from the range\n ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "C", next := 113265, root := ~, start := 113252, stop := 113264 ), rec( attributes := rec( ), content := [ rec( content := "{\ n \\choose 2}", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "M", next := 113285, root := ~, start := 113265, stop := 113284 ), rec( attributes := rec( ), content := [ rec( content := "]\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "C", next := 113293, root := ~, start := 113285, stop := 113292 ), rec( content := ".\n " , count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 2, 4 ], mathmode := "Text", name := "P", next := 113303, root := ~, start := 113299, stop := 113302 ), rec( content := "\n\n \ The method used by this function chooses each edge from the set of\n all po\ ssible edges with uniform probability. No effort is made to avoid\n creatin\ g multiple edges, so it is possible (but not guaranteed) that the\n result \ will have multiple edges. The result may or may not have loops.", count := [ 3, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 2, 5 ], mathmode := "Text", name := "P", next := 113607, root := ~, start := 113603, stop := 113606 ), rec( content := "\n ", count := [ 3, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomMultiDigraph(1000);\n\ngap> RandomMultiDigraph(1000, 950);\n\n", count := [ 3, 4, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 6 ], mathmode := "Text", name := "Log", next := 113816, root := ~, start := 113612, stop := 113815 ), rec( content := "\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 113833, root := ~, start := 112883, stop := 113832 ) ], count := [ 3, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 113847, root := ~, start := 112787, stop := 113846 ), rec( content := "\n\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "RandomTournamen\ t" ), content := 0, count := [ 3, 4, 3, 2 ], mathmode := "Text", name := "Oper", next := 113915, root := ~, start := 113868, stop := 113914 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Returns", next := 113947, root := ~, start := 113918, stop := 113946 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 114010, root := ~, start := 113999, stop := 114009 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 114156, root := ~, start := 114145, stop := 114155 ), rec( content := " is ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 114190, root := ~, start := 114160, stop := 114189 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 114258, root := ~, start := 114247, stop := 114257 ), rec( content := " is ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 114298, root := ~, start := 114262, stop := 114297 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 114383, root := ~, start := 114372, stop := 114382 ), rec( content := " is not \ present, then ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 114443, root := ~, start := 114405, stop := 114442 ), rec( content := " is used\ by default.", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "P", next := 114467, root := ~, start := 114463, stop := 114466 ), rec( content := "\n\n \ If ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "A", next := 114484, root := ~, start := 114476, stop := 114483 ), rec( content := " is a no\ n-negative integer, this function returns a random\n tournament with ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "A", next := 114571, root := ~, start := 114563, stop := 114570 ), rec( content := " vertice\ s. See ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "Ref", next := 114612, root := ~, start := 114586, stop := 114611 ), rec( content := ". ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 3, 5 ], mathmode := "Text", name := "P", next := 114618, root := ~, start := 114614, stop := 114617 ), rec( content := "\n ", count := [ 3, 4, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomTournament(10);\n\ngap> Ran\ domTournament(IsMutableDigraph, 10);\n\n", count := [ 3, 4, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 6 ], mathmode := "Text", name := "Log", next := 114810, root := ~, start := 114623, stop := 114809 ), rec( content := "\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Description", next := 114827, root := ~, start := 113950, stop := 114826 ) ], count := [ 3, 4, 3, 1 ], mathmode := "Text", name := "ManSection", next := 114841, root := ~, start := 113853, stop := 114840 ), rec( content := "\n\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "n", Name := "RandomLattice" ), content := 0, count := [ 3, 4, 4, 2 ], mathmode := "Text", name := "Oper", next := 114898, root := ~, start := 114862, stop := 114897 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Returns", next := 114930, root := ~, start := 114901, stop := 114929 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 114993, root := ~, start := 114982, stop := 114992 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 115139, root := ~, start := 115128, stop := 115138 ), rec( content := " is ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 115173, root := ~, start := 115143, stop := 115172 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 115241, root := ~, start := 115230, stop := 115240 ), rec( content := " is ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 115281, root := ~, start := 115245, stop := 115280 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 115366, root := ~, start := 115355, stop := 115365 ), rec( content := " is not \ present, then ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 115426, root := ~, start := 115388, stop := 115425 ), rec( content := " is used\ by default.", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "P", next := 115450, root := ~, start := 115446, stop := 115449 ), rec( content := "\n\n \ If ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "A", next := 115467, root := ~, start := 115459, stop := 115466 ), rec( content := " is a po\ sitive integer, this function return a random lattice\n with ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 115546, root := ~, start := 115538, stop := 115545 ), rec( content := " vertice\ s, where it is guaranteed that ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 115593, root := ~, start := 115585, stop := 115592 ), rec( content := " is betw\ een\n ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "A", next := 115617, root := ~, start := 115609, stop := 115616 ), rec( content := " and ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ * ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 4, 4, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := \ "Text", name := "A", next := 115637, root := ~, start := 115629, stop := 11563\ 6 ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 115641, root := ~, start := 115622, stop := 115640 ), rec( content := ". See ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "Ref", next := 115677, root := ~, start := 115647, stop := 115676 ), rec( content := ". ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 4, 5 ], mathmode := "Text", name := "P", next := 115683, root := ~, start := 115679, stop := 115682 ), rec( content := "\n ", count := [ 3, 4, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomLattice(10);\n\ngap> RandomLattice(IsMutableDigraph, 10);\n\n", count := [ 3, 4, 4, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 6 ], mathmode := "Text", name := "Log", next := 115878, root := ~, start := 115688, stop := 115877 ), rec( content := "\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Description", next := 115895, root := ~, start := 114933, stop := 115894 ) ], count := [ 3, 4, 4, 1 ], mathmode := "Text", name := "ManSection", next := 115909, root := ~, start := 114847, stop := 115908 ), rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 115923, root := ~, start := 111274, stop := 115922 ) , rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Standard examples\ ", count := [ 3, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 115972, root := ~, start := 115936, stop := 115971 ), rec( content := "\n ", count := [ 3, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "ChainDigraph" ) , content := 0, count := [ 3, 5, 1, 2 ], mathmode := "Text", name := "Oper", next := 116035, root := ~, start := 115992, stop := 116034 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 116067, root := ~, start := 116038, stop := 116066 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "A", next := 116099, root := ~, start := 116091, stop := 116098 ), rec( content := " is a po\ sitive integer, this function returns a chain\n with ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "A", next := 116170, root := ~, start := 116162, stop := 116169 ), rec( content := " vertice\ s and ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 5, 1, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := \ "Text", name := "A", next := 116195, root := ~, start := 116187, stop := 11619\ 4 ), rec( content := " - 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 116203, root := ~, start := 116184, stop := 116202 ), rec( content := " edges.\\ n Specifically,\n for each vertex ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 116257, root := ~, start := 116249, stop := 116256 ), rec( content := " (with " , count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 116272, root := ~, start := 116264, stop := 116271 ), rec( content := " < ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 116286, root := ~, start := 116278, stop := 116285 ), rec( content := "), there\ is a directed\n edge with source ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 116338, root := ~, start := 116330, stop := 116337 ), rec( content := " and ran\ ge ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 116361, root := ~, start := 116349, stop := 116360 ), rec( content := ". ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "P", next := 116367, root := ~, start := 116363, stop := 116366 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 116415, root := ~, start := 116404, stop := 116414 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 116561, root := ~, start := 116550, stop := 116560 ), rec( content := " is ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 116595, root := ~, start := 116565, stop := 116594 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 116663, root := ~, start := 116652, stop := 116662 ), rec( content := " is ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 116703, root := ~, start := 116667, stop := 116702 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 116788, root := ~, start := 116777, stop := 116787 ), rec( content := " is not \ present, then ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 116848, root := ~, start := 116810, stop := 116847 ), rec( content := " is used\ by default.", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "P", next := 116872, root := ~, start := 116868, stop := 116871 ), rec( content := "\n\n \ The ", count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveClosure" ), content := 0, count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 116929, root := ~, start := 116882, stop := 116928 ), rec( content := " of a ch\ ain represents a\n total order.\n ", count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChainDigraph(42);\n\ngap> Chai\ nDigraph(IsMutableDigraph, 10);\n\n\ ", count := [ 3, 5, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 6 ], mathmode := "Text", name := "Example", next := 117158, root := ~, start := 116975, stop := 117157 ), rec( content := "\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 117175, root := ~, start := 116070, stop := 117174 ) ], count := [ 3, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 117189, root := ~, start := 115977, stop := 117188 ), rec( content := "\n\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "CompleteDigraph\ " ), content := 0, count := [ 3, 5, 2, 2 ], mathmode := "Text", name := "Oper", next := 117256, root := ~, start := 117210, stop := 117255 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 117288, root := ~, start := 117259, stop := 117287 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "A", next := 117320, root := ~, start := 117312, stop := 117319 ), rec( content := " is a no\ n-negative integer, this function returns the complete\n digraph with ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "A", next := 117408, root := ~, start := 117400, stop := 117407 ), rec( content := " vertice\ s. See ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsCompl\ eteDigraph" ), content := 0, count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 117454, root := ~, start := 117423, stop := 117453 ), rec( content := ". ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "P", next := 117460, root := ~, start := 117456, stop := 117459 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 117508, root := ~, start := 117497, stop := 117507 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 117654, root := ~, start := 117643, stop := 117653 ), rec( content := " is ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 117688, root := ~, start := 117658, stop := 117687 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 117756, root := ~, start := 117745, stop := 117755 ), rec( content := " is ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 117796, root := ~, start := 117760, stop := 117795 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 117881, root := ~, start := 117870, stop := 117880 ), rec( content := " is not \ present, then ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 117941, root := ~, start := 117903, stop := 117940 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteDigraph(20);\n\ngap\ > CompleteDigraph(IsMutableDigraph, 10);\n\n", count := [ 3, 5, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 5 ], mathmode := "Text", name := "Example", next := 118159, root := ~, start := 117966, stop := 118158 ), rec( content := "\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 118176, root := ~, start := 117291, stop := 118175 ) ], count := [ 3, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 118190, root := ~, start := 117195, stop := 118189 ), rec( content := "\n\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]m, n", Name := "CompleteBiparti\ teDigraph" ), content := 0, count := [ 3, 5, 3, 2 ], mathmode := "Text", name := "Oper", next := 118269, root := ~, start := 118211, stop := 118268 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 118301, root := ~, start := 118272, stop := 118300 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ omplete bipartite digraph is a digraph whose vertices\n can be partitioned \ into two non-empty vertex sets, such there exists\n a unique edge with sour\ ce ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 118490, root := ~, start := 118482, stop := 118489 ), rec( content := " and ran\ ge ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 118509, root := ~, start := 118501, stop := 118508 ), rec( content := " if and \ only if\n ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 118537, root := ~, start := 118529, stop := 118536 ), rec( content := " and ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 118550, root := ~, start := 118542, stop := 118549 ), rec( content := " lie in \ different vertex sets. ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "P", next := 118585, root := ~, start := 118581, stop := 118584 ), rec( content := "\n\n \ If ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 118602, root := ~, start := 118594, stop := 118601 ), rec( content := " and ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 118615, root := ~, start := 118607, stop := 118614 ), rec( content := " are pos\ itive integers, this function\n returns the complete bipartite digraph with\ vertex sets of sizes\n ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 118734, root := ~, start := 118726, stop := 118733 ), rec( content := " (contai\ ning the vertices ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. m]", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "C", next := 118775, root := ~, start := 118760, stop := 118774 ), rec( content := ") and\n \ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 118793, root := ~, start := 118785, stop := 118792 ), rec( content := " (contai\ ning the vertices ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ m + 1 .. m + n]", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "C", next := 118842, root := ~, start := 118819, stop := 118841 ), rec( content := ").\n \ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "P", next := 118853, root := ~, start := 118849, stop := 118852 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 118901, root := ~, start := 118890, stop := 118900 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 119047, root := ~, start := 119036, stop := 119046 ), rec( content := " is ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 119081, root := ~, start := 119051, stop := 119080 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 119149, root := ~, start := 119138, stop := 119148 ), rec( content := " is ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 119189, root := ~, start := 119153, stop := 119188 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 119274, root := ~, start := 119263, stop := 119273 ), rec( content := " is not \ present, then ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 119334, root := ~, start := 119296, stop := 119333 ), rec( content := " is used\ by default.\n\n ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteBipartiteDigraph(2, 3);\n\ngap> CompleteBipartiteDigraph(IsMutableDigraph\ , 3, 2);\n\n", count := [ 3, 5, 3, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 3, 6 ], mathmode := "Text", name := "Example", next := 119598, root := ~, start := 119360, stop := 119597 ), rec( content := "\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 119615, root := ~, start := 118304, stop := 119614 ) ], count := [ 3, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 119629, root := ~, start := 118196, stop := 119628 ), rec( content := "\n\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]orders", Name := "CompleteMultipa\ rtiteDigraph" ), content := 0, count := [ 3, 5, 4, 2 ], mathmode := "Text", name := "Oper", next := 119719, root := ~, start := 119654, stop := 119718 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Returns", next := 119753, root := ~, start := 119724, stop := 119752 ), rec( attributes := rec( ), content := [ rec( content := "\n F\ or a list ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rders", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "A", next := 119802, root := ~, start := 119789, stop := 119801 ), rec( content := " of ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 119814, root := ~, start := 119806, stop := 119813 ), rec( content := " positiv\ e integers,\n this function returns the digraph containing ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 119893, root := ~, start := 119885, stop := 119892 ), rec( content := " indepen\ dent sets\n of vertices of orders ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "l", count := [ 3, 5, 4, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := \ "Text", name := "A", next := 119951, root := ~, start := 119943, stop := 11995\ 0 ), rec( content := "[1] .. ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := \ "Text", name := "A", next := 119966, root := ~, start := 119958, stop := 11996\ 5 ), rec( content := "[n]]", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 119974, root := ~, start := 119939, stop := 119973 ), rec( content := ". Moreov\ er,\n each vertex is adjacent to every other not contained in the\n \ same independent set.\n ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "P", next := 120090, root := ~, start := 120086, stop := 120089 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 120140, root := ~, start := 120129, stop := 120139 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digrap\ h being created will\n belong to. For example, if ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 120290, root := ~, start := 120279, stop := 120289 ), rec( content := " is ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 120324, root := ~, start := 120294, stop := 120323 ), rec( content := ",\n \ then the digraph being created will be mutable, if ", count := [ 3, 5, 4, 4 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 120394, root := ~, start := 120383, stop := 120393 ), rec( content := " is ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 120436, root := ~, start := 120398, stop := 120435 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 120523, root := ~, start := 120512, stop := 120522 ), rec( content := " is not \ present, then ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 120585, root := ~, start := 120545, stop := 120584 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteMultipartiteDigraph([5, 4, 2]);\n\ngap> CompleteMultipartiteDigraph(IsMuta\ bleDigraph, [5, 4, 2]);\n\n", count := [ 3, 5, 4, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 5 ], mathmode := "Text", name := "Example", next := 120866, root := ~, start := 120612, stop := 120865 ), rec( content := "\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Description", next := 120883, root := ~, start := 119758, stop := 120882 ) ], count := [ 3, 5, 4, 1 ], mathmode := "Text", name := "ManSection", next := 120897, root := ~, start := 119637, stop := 120896 ), rec( content := "\n\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "CycleDigraph" ) , content := 0, count := [ 3, 5, 5, 2 ], mathmode := "Text", name := "Oper", next := 120961, root := ~, start := 120918, stop := 120960 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Returns", next := 120993, root := ~, start := 120964, stop := 120992 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 121025, root := ~, start := 121017, stop := 121024 ), rec( content := " is a po\ sitive integer, this function returns a ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "E", next := 121085, root := ~, start := 121073, stop := 121084 ), rec( content := "\n di\ graph with ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 121111, root := ~, start := 121103, stop := 121110 ), rec( content := " vertice\ s and ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 121133, root := ~, start := 121125, stop := 121132 ), rec( content := " edges. \ Specifically,\n for each vertex ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121183, root := ~, start := 121175, stop := 121182 ), rec( content := " (with " , count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121198, root := ~, start := 121190, stop := 121197 ), rec( content := " < ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121212, root := ~, start := 121204, stop := 121211 ), rec( content := "), there\ is a directed\n edge with source ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121264, root := ~, start := 121256, stop := 121263 ), rec( content := " and ran\ ge ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121287, root := ~, start := 121275, stop := 121286 ), rec( content := ". In add\ ition, there is\n an edge with source ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121343, root := ~, start := 121335, stop := 121342 ), rec( content := " and ran\ ge ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 121362, root := ~, start := 121354, stop := 121361 ), rec( content := ". ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "P", next := 121368, root := ~, start := 121364, stop := 121367 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 121416, root := ~, start := 121405, stop := 121415 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 121562, root := ~, start := 121551, stop := 121561 ), rec( content := " is ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 121596, root := ~, start := 121566, stop := 121595 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 121664, root := ~, start := 121653, stop := 121663 ), rec( content := " is ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 121704, root := ~, start := 121668, stop := 121703 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 121789, root := ~, start := 121778, stop := 121788 ), rec( content := " is not \ present, then ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 121847, root := ~, start := 121811, stop := 121846 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CycleDigraph(1);\n\ngap> CycleD\ igraph(123);\n\ngap> CycleDigraph(I\ sMutableDigraph, 10);\n\n", count := [ 3, 5, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 5, 5 ], mathmode := "Text", name := "Example", next := 122120, root := ~, start := 121870, stop := 122119 ), rec( content := "\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Description", next := 122137, root := ~, start := 120996, stop := 122136 ) ], count := [ 3, 5, 5, 1 ], mathmode := "Text", name := "ManSection", next := 122151, root := ~, start := 120903, stop := 122150 ), rec( content := "\n\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "EmptyDigraph" ) , content := 0, count := [ 3, 5, 6, 2 ], mathmode := "Text", name := "Oper", next := 122215, root := ~, start := 122172, stop := 122214 ), rec( attributes := rec( Arg := "[filt, ]n", Name := "NullDigraph" ), content := 0, count := [ 3, 5, 6, 4 ], mathmode := "Text", name := "Oper", next := 122260, root := ~, start := 122218, stop := 122259 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Returns", next := 122292, root := ~, start := 122263, stop := 122291 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "A", next := 122324, root := ~, start := 122316, stop := 122323 ), rec( content := " is a no\ n-negative integer, this function returns the\n ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mpty", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "E", next := 122394, root := ~, start := 122382, stop := 122393 ), rec( content := " or ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ull", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "E", next := 122409, root := ~, start := 122398, stop := 122408 ), rec( content := " digraph\ with ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "A", next := 122431, root := ~, start := 122423, stop := 122430 ), rec( content := " vertice\ s. An empty\n digraph is one with no edges.\n ", count := [ 3, 5, 6, 5 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "P", next := 122493, root := ~, start := 122489, stop := 122492 ), rec( content := "\n \ \n If the optional first argument ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 122547, root := ~, start := 122536, stop := 122546 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 122693, root := ~, start := 122682, stop := 122692 ), rec( content := " is ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 122727, root := ~, start := 122697, stop := 122726 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 122795, root := ~, start := 122784, stop := 122794 ), rec( content := " is ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 122835, root := ~, start := 122799, stop := 122834 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 122920, root := ~, start := 122909, stop := 122919 ), rec( content := " is not \ present, then ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 122980, root := ~, start := 122942, stop := 122979 ), rec( content := " is used\ by default.", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "P", next := 123004, root := ~, start := 123000, stop := 123003 ), rec( content := "\n\n \ ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ullDigraph", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "C", next := 123028, root := ~, start := 123010, stop := 123027 ), rec( content := " is a sy\ nonym for ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ mptyDigraph", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "C", next := 123065, root := ~, start := 123046, stop := 123064 ), rec( content := ".\n\n \ ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> EmptyDigraph(20);\n\ngap> Null\ Digraph(10);\n\ngap> EmptyDigraph(Is\ MutableDigraph, 10);\n", count := [ 3, 5, 6, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 6, 8 ], mathmode := "Text", name := "Example", next := 123316, root := ~, start := 123072, stop := 123315 ), rec( content := "\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Description", next := 123333, root := ~, start := 122295, stop := 123332 ) ], count := [ 3, 5, 6, 1 ], mathmode := "Text", name := "ManSection", next := 123347, root := ~, start := 122157, stop := 123346 ), rec( content := "\n\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n, k", Name := "JohnsonDigraph"\ ), content := 0, count := [ 3, 5, 7, 2 ], mathmode := "Text", name := "Oper", next := 123416, root := ~, start := 123368, stop := 123415 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Returns", next := 123448, root := ~, start := 123419, stop := 123447 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "A", next := 123480, root := ~, start := 123472, stop := 123479 ), rec( content := " and ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "A", next := 123493, root := ~, start := 123485, stop := 123492 ), rec( content := " are non\ -negative integers, then this operation\n returns a symmetric digraph which\ corresponds to the undirected ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnson\n graph", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "E", next := 123633, root := ~, start := 123609, stop := 123632 ), rec( content := " ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ (n, k)", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "M", next := 123648, root := ~, start := 123634, stop := 123647 ), rec( content := ". ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "P", next := 123654, root := ~, start := 123650, stop := 123653 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 123702, root := ~, start := 123691, stop := 123701 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 123848, root := ~, start := 123837, stop := 123847 ), rec( content := " is ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 123882, root := ~, start := 123852, stop := 123881 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 123950, root := ~, start := 123939, stop := 123949 ), rec( content := " is ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 123990, root := ~, start := 123954, stop := 123989 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 124075, root := ~, start := 124064, stop := 124074 ), rec( content := " is not \ present, then ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 124135, root := ~, start := 124097, stop := 124134 ), rec( content := " is used\ by default.", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "P", next := 124159, root := ~, start := 124155, stop := 124158 ), rec( content := "\n\n \ The ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnson graph", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "E", next := 124189, root := ~, start := 124169, stop := 124188 ), rec( content := " ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ (n, k)", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 124204, root := ~, start := 124190, stop := 124203 ), rec( content := " has ver\ tices given by all the\n ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "A", next := 124247, root := ~, start := 124239, stop := 124246 ), rec( content := "-subsets\ of the range ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 5, 7, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := \ "Text", name := "A", next := 124286, root := ~, start := 124278, stop := 12428\ 5 ), rec( content := "]", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "C", next := 124291, root := ~, start := 124269, stop := 124290 ), rec( content := ", and tw\ o vertices are\n connected by an edge iff their intersection has size ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 5, 7, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := \ "Text", name := "A", next := 124384, root := ~, start := 124376, stop := 12438\ 3 ), rec( content := " - 1", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 124392, root := ~, start := 124373, stop := 124391 ), rec( content := ".\n\n \ ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := JohnsonDigraph(3, 1);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> g\ r := JohnsonDigraph(4, 2);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [ 1, 2, 5, 6\ ], [ 1, 2, 5, 6 ], \n [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ]\ngap> JohnsonDigraph(\ 1, 0);\n\ngap> JohnsonDigraph(IsMutable\ Digraph, 1, 0);\n\n", count := [ 3, 5, 7, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 6 ], mathmode := "Text", name := "Example", next := 124941, root := ~, start := 124399, stop := 124940 ), rec( content := "\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Description", next := 124958, root := ~, start := 123451, stop := 124957 ) ], count := [ 3, 5, 7, 1 ], mathmode := "Text", name := "ManSection", next := 124972, root := ~, start := 123353, stop := 124971 ), rec( content := "\n\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt]", Name := "PetersenGraph" ), content := 0, count := [ 3, 5, 8, 2 ], mathmode := "Text", name := "Oper", next := 125034, root := ~, start := 124993, stop := 125033 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Returns", next := 125066, root := ~, start := 125037, stop := 125065 ), rec( attributes := rec( ), content := [ rec( content := "\n\n F\ rom ", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Petersen_graph", count := [ 3, 5, 8, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "URL", next := 125148, root := ~, start := 125093, stop := 125147 ), rec( content := ":", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "P", next := 125153, root := ~, start := 125149, stop := 125152 ), rec( content := "\n\n \ ", count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ he Petersen graph is an undirected graph with 10 vertices and 15 edges.\n I\ t is a small graph that serves as a useful example and counterexample for\n \ many problems in graph theory. The Petersen graph is named after Julius\n \ Petersen, who in 1898 constructed it to be the smallest bridgeless cubic\n \ graph with no three-edge-coloring.", count := [ 3, 5, 8, 4 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "Q", next := 125509, root := ~, start := 125159, stop := 125508 ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "P", next := 125513, root := ~, start := 125509, stop := 125512 ), rec( content := "\n \n\ If the optional first argument ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 125565, root := ~, start := 125554, stop := 125564 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 125711, root := ~, start := 125700, stop := 125710 ), rec( content := " is ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 125745, root := ~, start := 125715, stop := 125744 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 125813, root := ~, start := 125802, stop := 125812 ), rec( content := " is ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 125853, root := ~, start := 125817, stop := 125852 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 125938, root := ~, start := 125927, stop := 125937 ), rec( content := " is not \ present, then ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 125998, root := ~, start := 125960, stop := 125997 ), rec( content := " is used\ by default.", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "P", next := 126022, root := ~, start := 126018, stop := 126021 ), rec( content := "\n\n \ See also ", count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "General\ isedPetersenGraph" ), content := 0, count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "Ref", next := 126075, root := ~, start := 126037, stop := 126074 ), rec( content := ".\n\n", count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChromaticNumber(PetersenGraph());\n3\ngap> PetersenGraph(IsMutableDigrap\ h);\n\n", count := [ 3, 5, 8, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 8, 7 ], mathmode := "Text", name := "Example", next := 126234, root := ~, start := 126078, stop := 126233 ), rec( content := "\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Description", next := 126251, root := ~, start := 125069, stop := 126250 ) ], count := [ 3, 5, 8, 1 ], mathmode := "Text", name := "ManSection", next := 126265, root := ~, start := 124978, stop := 126264 ), rec( content := "\n\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n, k", Name := "GeneralisedPete\ rsenGraph" ), content := 0, count := [ 3, 5, 9, 2 ], mathmode := "Text", name := "Oper", next := 126344, root := ~, start := 126286, stop := 126343 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Returns", next := 126376, root := ~, start := 126347, stop := 126375 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "A", next := 126408, root := ~, start := 126400, stop := 126407 ), rec( content := " is a po\ sitive integer and ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "A", next := 126443, root := ~, start := 126435, stop := 126442 ), rec( content := " is a no\ n-negative\n integer less than ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 5, 9, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := \ "Text", name := "A", next := 126495, root := ~, start := 126487, stop := 12649\ 4 ), rec( content := " / 2", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "C", next := 126503, root := ~, start := 126484, stop := 126502 ), rec( content := ", then t\ his operation returns\n the ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ eneralised Petersen graph", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "E", next := 126574, root := ~, start := 126541, stop := 126573 ), rec( content := " ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ PG(", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 5, 9, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := \ "Text", name := "A", next := 126590, root := ~, start := 126582, stop := 12658\ 9 ), rec( content := ", k)", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "M", next := 126598, root := ~, start := 126575, stop := 126597 ), rec( content := ". ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "P", next := 126604, root := ~, start := 126600, stop := 126603 ), rec( content := "\n\n \ From\n ", count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Generalized_Petersen_graph", count := [ 3, 5, 9, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "URL", next := 126686, root := ~, start := 126619, stop := 126685 ), rec( content := ":\n " , count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "P", next := 126696, root := ~, start := 126692, stop := 126695 ), rec( content := "\n\n \ ", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ he generalized Petersen graphs are a family of cubic\n graphs formed by con\ necting the vertices of a regular polygon to the\n corresponding vertices o\ f a star polygon. They include the Petersen graph\n and generalize one of t\ he ways of constructing the Petersen graph. The\n generalized Petersen grap\ h family was introduced in 1950 by H. S. M.\n Coxeter and was given its nam\ e in 1969 by Mark Watkins.", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "Q", next := 127121, root := ~, start := 126702, stop := 127120 ), rec( content := "\n ", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "P", next := 127130, root := ~, start := 127126, stop := 127129 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 127178, root := ~, start := 127167, stop := 127177 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 127324, root := ~, start := 127313, stop := 127323 ), rec( content := " is ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 127358, root := ~, start := 127328, stop := 127357 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 127426, root := ~, start := 127415, stop := 127425 ), rec( content := " is ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 127466, root := ~, start := 127430, stop := 127465 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 127551, root := ~, start := 127540, stop := 127550 ), rec( content := " is not \ present, then ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 127611, root := ~, start := 127573, stop := 127610 ), rec( content := " is used\ by default.", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "P", next := 127635, root := ~, start := 127631, stop := 127634 ), rec( content := "\n\n \ See also ", count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Peterse\ nGraph" ), content := 0, count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "Ref", next := 127677, root := ~, start := 127650, stop := 127676 ), rec( content := ".\n\n", count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> GeneralisedPetersenGraph(7, 2);\n\ngap> GeneralisedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, 2\ );\n\ngap> IsIsomorphi\ cDigraph(D, PetersenGraph());\ntrue\ngap> GeneralisedPetersenGraph(IsMutableDi\ graph, 9, 4);\n\n", count := [ 3, 5, 9, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 8 ], mathmode := "Text", name := "Example", next := 128152, root := ~, start := 127680, stop := 128151 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Description", next := 128169, root := ~, start := 126379, stop := 128168 ) ], count := [ 3, 5, 9, 1 ], mathmode := "Text", name := "ManSection", next := 128183, root := ~, start := 126271, stop := 128182 ), rec( content := "\n\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 128197, root := ~, start := 115927, stop := 128196 ) , rec( content := "\n\n", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 128209, root := ~, start := 20825, stop := 128208 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " creating ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Operators" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Operators", count := [ 4, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 128286, root := ~, start := 128258, stop := 128285 ) , rec( content := "\n\n ", count := [ 4, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Operators for dig\ raphs", count := [ 4, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 128340, root := ~, start := 128299, stop := 128339 ), rec( content := "\n ", count := [ 4, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 4 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 4 ], mathmode := \ "Text", name := "A", next := 128381, root := ~, start := 128366, stop := 12838\ 0 ), rec( content := " = ", count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "A", next := 1283\ 99, root := ~, start := 128384, stop := 128398 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "C", next := 128403, root := ~, start := 128363, stop := 128402 ) ], count := [ 4, 1, 0, 4 ], name := "Mark", next := 128410, root := ~, start := 128357, stop := 128409 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "=" ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "=", count := [ 4, 1, 0, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "C", next := 128453, root := ~, start := 128445, stop := 12845\ 2 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 5 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "Index", next := 128476, root := ~, start := 128430, stop := 128475 ), rec( content := "\n \ returns ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "K", next := 128503, root := ~, start := 128492, stop := 128502 ), rec( content := " if ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 128522, root := ~, start := 128507, stop := 128521 ), rec( content := " and ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 128542, root := ~, start := 128527, stop := 128541 ), rec( content := " have th\ e same\n vertices, and ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "A", next := 128609, root := ~, start := 128594, stop := 12860\ 8 ), rec( content := ") =\n DigraphEdges(", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph2", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 1286\ 50, root := ~, start := 128635, stop := 128649 ), rec( content := ")", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "C", next := 128655, root := ~, start := 128578, stop := 128654 ), rec( content := ", up to \ some re-ordering of the edge\n lists. ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "P", next := 128710, root := ~, start := 128706, stop := 128709 ), rec( content := "\n\n \ Note that this operator does not compare the vertex labels\n of ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "A", next := 128803, root := ~, start := 128788, stop := 128802 ), rec( content := " and ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "A", next := 128823, root := ~, start := 128808, stop := 128822 ), rec( content := ".\n \ ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], name := "Item", next := 128837, root := ~, start := 128416, stop := 128836 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 7 ], mathmode := \ "Text", name := "A", next := 128868, root := ~, start := 128853, stop := 12886\ 7 ), rec( content := " < ", count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "A", next := 1288\ 89, root := ~, start := 128874, stop := 128888 ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "C", next := 128893, root := ~, start := 128850, stop := 128892 ) ], count := [ 4, 1, 0, 7 ], name := "Mark", next := 128900, root := ~, start := 128844, stop := 128899 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "<" ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "<", count := [ 4, 1, 0, 8 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 4, 1, 0, 8 ] , mathmode := \ "Text", name := "C", next := 128949, root := ~, start := 128938, stop := 12894\ 8 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "Index", next := 128972, root := ~, start := 128920, stop := 128971 ), rec( content := "\n\n \ This operator returns ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "K", next := 129014, root := ~, start := 129003, stop := 129013 ), rec( content := " if one \ of the following holds:\n\n ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n The number ", count := [ \ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "n_1", count := [ 4, 1, 0, 10 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "M", next := 129109, root := ~, start := 129099, stop \ := 129108 ), rec( content := " of vertices in ", count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph1", count := [ 4, 1, 0, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "T\ ext", name := "A", next := 129140, root := ~, start := 129125, stop := 129139 ), rec( cont\ ent := " is less than\n the number ", count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "n_2", count := [ 4, 1, 0, 10 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "Text", name\ := "M", next := 129186, root := ~, start := 129176, stop := 129185 ), rec( cont\ ent := " of vertices in ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "digraph2", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "A", next\ := 129217, root := ~, start := 129202, stop := 129216 ), rec( content := ";\n\ ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 10 ], name := "Item", next := 129235, root := ~, start := 129\ 070, stop := 129234 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n_\ 1 = n_2", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := \ 129279, root := ~, start := 129263, stop := 129278 ), rec( content := ", and t\ he number ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "m_1", count\ := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := 129306, root :=\ ~, start := 129296, stop := 129305 ), rec( content := " of edges in\n \ ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph1", coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "A", next := 129345, root :=\ ~, start := 129330, stop := 129344 ), rec( content := " is less than the numb\ er ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "m_2", count := \ [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := 129380, root := ~, star\ t := 129370, stop := 129379 ), rec( content := " of edges in\n ", coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph2", count := [ 4, 1, 0\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, \ 0, 12 ], mathmode := "Text", name := "A", next := 129419, root := ~, start := \ 129404, stop := 129418 ), rec( content := ";\n ", count := [ 4, 1, 0, \ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 12 ], name := "Ite\ m", next := 129437, root := ~, start := 129245, stop := 129436 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "n_1 = n_2", count := [ 4, 1, 0, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], math\ mode := "Text", name := "M", next := 129481, root := ~, start := 129465, stop \ := 129480 ), rec( content := ", ", count := [ 4, 1, 0, 14 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "m_1 = m_2", count := [ 4, 1, 0, 14 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], mathmode := "Text", name \ := "M", next := 129499, root := ~, start := 129483, stop := 129498 ), rec( cont\ ent := ", and\n ", count := [ 4, 1, 0, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphEdges(", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "digraph1", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "A", next \ := 129547, root := ~, start := 129532, stop := 129546 ), rec( content := ")", \ count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "C", next := 129552, root :=\ ~, start := 129516, stop := 129551 ), rec( content := " is less than\n \ ", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphEdges("\ , count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ),\ rec( attributes := rec( ), content := [ rec( content := "digraph2", count :=\ [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 4, 1, 0, 14 ], mathmode := "Text", name := "A", next := 129608, root := ~,\ start := 129593, stop := 129607 ), rec( content := ")", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, \ 14 ], mathmode := "Text", name := "C", next := 129613, root := ~, start := 129\ 577, stop := 129612 ), rec( content := " after having both of these sets\n \ have been sorted with respect to the lexicographical order.\n " , coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], name := "Ite\ m", next := 129733, root := ~, start := 129447, stop := 129732 ) ], count := [ 4, 1, 0, 9 ], mathmode := "Text", name := "List", next := 129748, root := ~, start := 129054, stop := 129747 ), rec( content := "\n " , count := [ 4, 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], name := "Item", next := 129761, root := ~, start := 128906, stop := 129760 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "List", next := 129772, root := ~, start := 128345, stop := 129771 ), rec( content := "\n\n ", count := [ 4, 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "super, sub", Name := "IsSubdigraph" ) , content := 0, count := [ 4, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 129836, root := ~, start := 129792, stop := 129835 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 129859, root := ~, start := 129848, stop := 129858 ), rec( content := " or ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 129875, root := ~, start := 129863, stop := 129874 ), rec( content := ".", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 129886, root := ~, start := 129839, stop := 129885 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 129922, root := ~, start := 129910, stop := 129921 ), rec( content := " and ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 129937, root := ~, start := 129927, stop := 129936 ), rec( content := " are dig\ raphs, then this operation returns\n ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 129995, root := ~, start := 129984, stop := 129994 ), rec( content := " if ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 130009, root := ~, start := 129999, stop := 130008 ), rec( content := " is a su\ bdigraph of ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 130041, root := ~, start := 130029, stop := 130040 ), rec( content := ", and ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 130059, root := ~, start := 130047, stop := 130058 ), rec( content := "\n if\ it is not. ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "P", next := 130082, root := ~, start := 130078, stop := 130081 ), rec( content := "\n\n \ A digraph ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 130108, root := ~, start := 130098, stop := 130107 ), rec( content := " is a ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ubdigraph", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "E", next := 130131, root := ~, start := 130114, stop := 130130 ), rec( content := " of a di\ graph ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 130157, root := ~, start := 130145, stop := 130156 ), rec( content := " if\n \ ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 130175, root := ~, start := 130165, stop := 130174 ), rec( content := " and ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 130192, root := ~, start := 130180, stop := 130191 ), rec( content := " share t\ he same number of vertices, and the\n collection of edges of ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 130275, root := ~, start := 130263, stop := 130274 ), rec( content := " (includ\ ing repeats) contains the\n collection of edges of ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 130346, root := ~, start := 130336, stop := 130345 ), rec( content := " (includ\ ing repeats). ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "P", next := 130373, root := ~, start := 130369, stop := 130372 ), rec( content := "\n\n \ In other words, ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 130405, root := ~, start := 130395, stop := 130404 ), rec( content := " is a su\ bdigraph of ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 130437, root := ~, start := 130425, stop := 130436 ), rec( content := " if and \ only if\n ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "su\ b", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "A", next := 130488, root := ~, start := 130478, stop := 13048\ 7 ), rec( content := ") = DigraphNrVertices(", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "super", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 1305\ 22, root := ~, start := 130510, stop := 130521 ), rec( content := ")", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 130527, root := ~, start := 130457, stop := 130526 ), rec( content := ", and\n \ for each pair of vertices ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 130571, root := ~, start := 130563, stop := 130570 ), rec( content := " and ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 130584, root := ~, start := 130576, stop := 130583 ), rec( content := ", there \ are at least as many\n edges of the form ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 130648, root := ~, start := 130635, stop := 130647 ), rec( content := " in ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 130664, root := ~, start := 130652, stop := 130663 ), rec( content := " as ther\ e are in ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 130691, root := ~, start := 130681, stop := 130690 ), rec( content := ".\n " , count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 6 ], mathmode := "Text", name := "P", next := 130701, root := ~, start := 130697, stop := 130700 ), rec( content := "\n\n \ ", count := [ 4, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> g := Digraph([[2, 3], [1], [2, 3]]);\n\ngap> h := Digraph([[2, 3], [], [2]]);\n\ngap> IsSubdigraph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nf\ alse\ngap> IsSubdigraph(CompleteDigraph(4), CycleDigraph(4));\ntrue\ngap> IsSu\ bdigraph(CycleDigraph(4), ChainDigraph(4));\ntrue\ngap> g := Digraph([[2, 2], \ [1]]);\n\ngap> h := Digraph([\ [2], [1]]);\n\ngap> IsSubdigraph(g\ , h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse", count := [ 4, 1, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 1, 7 ], mathmode := "Text", name := "Example", next := 131309, root := ~, start := 130707, stop := 131308 ), rec( content := "\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 131326, root := ~, start := 129889, stop := 131325 ) ], count := [ 4, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 131340, root := ~, start := 129777, stop := 131339 ), rec( content := "\n\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningTree" ), content := 0, count := [ 4, 1, 2, 2 ], mathmode := "Text", name := "Oper", next := 131416, root := ~, start := 131360, stop := 131415 ), rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningForest" ), content := 0, count := [ 4, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 131477, root := ~, start := 131419, stop := 131476 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 131500, root := ~, start := 131489, stop := 131499 ), rec( content := " or ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 131516, root := ~, start := 131504, stop := 131515 ), rec( content := ".", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 131527, root := ~, start := 131480, stop := 131526 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ operation ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedSpanningTree", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "C", next := 131593, root := ~, start := 131562, stop := 131592 ), rec( content := " returns\ ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 131613, root := ~, start := 131602, stop := 131612 ), rec( content := " if the\\ n digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 131643, root := ~, start := 131633, stop := 131642 ), rec( content := " is an u\ ndirected spanning tree of the digraph\n ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 131706, root := ~, start := 131694, stop := 131705 ), rec( content := ", and th\ e operation ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedSpanningForest", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "C", next := 131759, root := ~, start := 131726, stop := 131758 ), rec( content := " returns\ \n ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 131783, root := ~, start := 131772, stop := 131782 ), rec( content := " if the \ digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 131809, root := ~, start := 131799, stop := 131808 ), rec( content := " is an u\ ndirected spanning forest of\n the digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 131874, root := ~, start := 131862, stop := 131873 ), rec( content := ". ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "P", next := 131880, root := ~, start := 131876, stop := 131879 ), rec( content := "\n\n \ An ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected spanning tree", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "E", next := 131920, root := ~, start := 131889, stop := 131919 ), rec( content := " of a di\ graph ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "A", next := 131946, root := ~, start := 131934, stop := 131945 ), rec( content := " is a su\ bdigraph\n of ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "A", next := 131982, root := ~, start := 131970, stop := 131981 ), rec( content := " that is\ an undirected tree (see ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsSubdi\ graph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 132042, root := ~, start := 132015, stop := 132041 ), rec( content := "\n an\ d ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsUndir\ ectedTree" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 132082, root := ~, start := 132051, stop := 132081 ), rec( content := "). Note \ that a digraph whose ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 132157, root := ~, start := 132111, stop := 132156 ), rec( content := " is not \ connected has no undirected\n spanning trees (see ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 132249, root := ~, start := 132217, stop := 132248 ), rec( content := "). ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "P", next := 132256, root := ~, start := 132252, stop := 132255 ), rec( content := "\n\n \ An ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected spanning forest", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "E", next := 132298, root := ~, start := 132265, stop := 132297 ), rec( content := " of a di\ graph ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 132324, root := ~, start := 132312, stop := 132323 ), rec( content := " is a\n \ subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 132360, root := ~, start := 132348, stop := 132359 ), rec( content := " that is\ an undirected forest (see ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsSubdi\ graph" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 132427, root := ~, start := 132395, stop := 132426 ), rec( content := " and ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsUndir\ ectedForest" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 132464, root := ~, start := 132432, stop := 132463 ), rec( content := "), and i\ s not\n contained in any larger such subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 132537, root := ~, start := 132525, stop := 132536 ), rec( content := ". Equiv\ alently, an\n undirected spanning forest is a subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 132619, root := ~, start := 132607, stop := 132618 ), rec( content := " whose c\ onnected\n components coincide with those of the ", count := [ 4, 1, 2, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 132724, root := ~, start := 132678, stop := 132723 ), rec( content := " of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 132740, root := ~, start := 132728, stop := 132739 ), rec( content := " (see ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 132792, root := ~, start := 132746, stop := 132791 ), rec( content := "). ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 8 ], mathmode := "Text", name := "P", next := 132799, root := ~, start := 132795, stop := 132798 ), rec( content := "\n\n \ Note that an undirected spanning tree is an undirected spanning forest that\n \ is connected.\n\n ", count := [ 4, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);\n\n\ gap> tree := Digraph([[3], [4], [1, 4], [2, 3]]);\n\ngap> IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntru\ e\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(\ 4);\n\ngap> IsSubdigraph(D, forest) a\ nd IsUndirectedForest(forest);\ntrue\ngap> IsUndirectedSpanningForest(D, fores\ t);\nfalse\ngap> IsSubdigraph(tree, forest);\ntrue\ngap> D := DigraphDisjointU\ nion(CycleDigraph(2), CycleDigraph(2));\n\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUn\ directedSpanningForest(D, D);\ntrue", count := [ 4, 1, 2, 9 ], name := "PCDATA\ ", root := ~ ) ], count := [ 4, 1, 2, 9 ], mathmode := "Text", name := "Example", next := 133651, root := ~, start := 132904, stop := 133650 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 133668, root := ~, start := 131530, stop := 133667 ) ], count := [ 4, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 133682, root := ~, start := 131345, stop := 133681 ), rec( content := "\n\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 133696, root := ~, start := 128290, stop := 133695 ) , rec( content := "\n", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 133707, root := ~, start := 128231, stop := 133706 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " operators/hashs ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Attributes and operations" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Attributes and operations\ ", count := [ 5, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 133823, root := ~, start := 133779, stop := 133822 ), rec( content := "\n ", count := [ 5, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Vertices and edge\ s", count := [ 5, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 133872, root := ~, start := 133835, stop := 133871 ), rec( content := "\n ", count := [ 5, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertices\ " ), content := 0, count := [ 5, 1, 1, 2 ], mathmode := "Text", name := "Attr", next := 133936, root := ~, start := 133892, stop := 133935 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers.", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 133977, root := ~, start := 133939, stop := 133976 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the vertices of the digraph ", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "A", next := 134048, root := ~, start := 134034, stop := 134047 ), rec( content := ". ", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "P", next := 134054, root := ~, start := 134050, stop := 134053 ), rec( content := "\n\n \ Note that the vertices of a digraph are always a range of\n positive intege\ rs from ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "C", next := 134153, root := ~, start := 134145, stop := 134152 ), rec( content := " to the \ number of vertices of the graph.\n ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphVertices(gr);\n[ 1 .. 3 ]\ngap> gr := Digraph\ ([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> \ [2, 7, 5, 3, 7]);\n\ngap> DigraphV\ ertices(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 10\ 0 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\n", count := [ 5, 1, 1, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 1, 5 ], mathmode := "Text", name := "Example", next := 134794, root := ~, start := 134198, stop := 134793 ), rec( content := "\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 134811, root := ~, start := 133980, stop := 134810 ) ], count := [ 5, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 134825, root := ~, start := 133877, stop := 134824 ), rec( content := "\n\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrVertic\ es" ), content := 0, count := [ 5, 1, 2, 2 ], mathmode := "Text", name := "Attr", next := 134892, root := ~, start := 134846, stop := 134891 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 134925, root := ~, start := 134895, stop := 134924 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of vertices of the digraph ", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "A", next := 135006, root := ~, start := 134992, stop := 135005 ), rec( content := ". ", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 2, 4 ], mathmode := "Text", name := "P", next := 135012, root := ~, start := 135008, stop := 135011 ), rec( content := "\n\n \ ", count := [ 5, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Digraph([1, 2,\ 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> [2, 7, \ 5, 3, 7]);\n\ngap> DigraphNrVertic\ es(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(100));\n100\ngap> D := CycleD\ igraph(IsMutableDigraph, 3);\n\ngap>\ DigraphNrVertices(D);\n3\n", count := [ 5, 1, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 5 ], mathmode := "Text", name := "Example", next := 135586, root := ~, start := 135018, stop := 135585 ), rec( content := "\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 135603, root := ~, start := 134928, stop := 135602 ) ], count := [ 5, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 135617, root := ~, start := 134831, stop := 135616 ), rec( content := "\n\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdges" ) , content := 0, count := [ 5, 1, 3, 2 ], mathmode := "Text", name := "Attr", next := 135679, root := ~, start := 135638, stop := 135678 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists.", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 135717, root := ~, start := 135682, stop := 135716 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "C", next := 135757, root := ~, start := 135738, stop := 135756 ), rec( content := " returns\ a list of edges of the digraph ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "A", next := 135811, root := ~, start := 135797, stop := 135810 ), rec( content := ",\n w\ here each edge is a pair of elements of ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 135887, root := ~, start := 135858, stop := 135886 ), rec( content := " of\n \ the form ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ source,range]", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "C", next := 135925, root := ~, start := 135904, stop := 135924 ), rec( content := ".\n " , count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "P", next := 135935, root := ~, start := 135931, stop := 135934 ), rec( content := "\n\n \ The entries of ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 135976, root := ~, start := 135956, stop := 135975 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 135990, root := ~, start := 135976, stop := 135989 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 135998, root := ~, start := 135990, stop := 135997 ), rec( content := " are in \ one-to-one\n correspondence with the edges of ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 136068, root := ~, start := 136054, stop := 136067 ), rec( content := ". Hence\ \n ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 136101, root := ~, start := 136081, stop := 136100 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 136115, root := ~, start := 136101, stop := 136114 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 136123, root := ~, start := 136115, stop := 136122 ), rec( content := " is dupl\ icate-free if and only if\n ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 136175, root := ~, start := 136161, stop := 136174 ), rec( content := " contain\ s no multiple edges. ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "P", next := 136208, root := ~, start := 136204, stop := 136207 ), rec( content := "\n\n \ The entries of ", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "C", next := 136248, root := ~, start := 136229, stop := 136247 ), rec( content := " are gua\ ranteed to be sorted by their\n first component (i.e. by the source of each\ edge), but they are not\n necessarily then sorted by the second component.\ \n ", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := DigraphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n\ngap> edges := ShallowCopy(DigraphEdges(gr))\ ;; Sort(edges);\ngap> edges;\n[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5\ ], [ 2, 1 ], \n [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3, 4 ], [ 3, 5 ], \ \n [ 4, 2 ], [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]\ngap> D := CycleDigraph(IsMutable\ Digraph, 3);\n\ngap> DigraphEdges(D)\ ;\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\n", count := [ 5, 1, 3, 6 ], name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 3, 6 ], mathmode := "Text", name := "Example", next := 136938, root := ~, start := 136415, stop := 136937 ), rec( content := "\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 136955, root := ~, start := 135720, stop := 136954 ) ], count := [ 5, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 136969, root := ~, start := 135623, stop := 136968 ), rec( content := "\n\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrEdges"\ ), content := 0, count := [ 5, 1, 4, 2 ], mathmode := "Text", name := "Attr", next := 137033, root := ~, start := 136990, stop := 137032 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 137066, root := ~, start := 137036, stop := 137065 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the number of edges of the digraph ", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "A", next := 137158, root := ~, start := 137144, stop := 137157 ), rec( content := ".\n " , count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]\ ]);;\ngap> DigraphNrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\ \n> [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\",\ \"a\"]);\n\ngap> DigraphNrEd\ ges(gr);\n3\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrEdges(D);\n3\n", count := [ 5, 1, 4, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 4, 4 ], mathmode := "Text", name := "Example", next := 137605, root := ~, start := 137164, stop := 137604 ), rec( content := "\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 137622, root := ~, start := 137069, stop := 137621 ) ], count := [ 5, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 137636, root := ~, start := 136975, stop := 137635 ), rec( content := "\n\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrLoops"\ ), content := 0, count := [ 5, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 137700, root := ~, start := 137657, stop := 137699 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 137733, root := ~, start := 137703, stop := 137732 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the number of loops of the digraph ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "A", next := 137825, root := ~, start := 137811, stop := 137824 ), rec( content := ". See ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Ref", next := 137860, root := ~, start := 137831, stop := 137859 ), rec( content := ". ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 5, 4 ], mathmode := "Text", name := "P", next := 137866, root := ~, start := 137862, stop := 137865 ), rec( content := "\n ", count := [ 5, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [1, 4], [3, 3, 5], [], [2, 5]]);\n\ngap> DigraphNrLoops(D);\n3\ngap> D := Empty\ Digraph(5);\n\ngap> DigraphNrLoops(D)\ ;\n0\ngap> D := CompleteDigraph(5);\n\ngap> DigraphNrLoops(D);\n0\n", count := [ 5, 1, 5, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "Example", next := 138235, root := ~, start := 137871, stop := 138234 ), rec( content := "\n ", count := [ 5, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 138252, root := ~, start := 137736, stop := 138251 ) ], count := [ 5, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 138266, root := ~, start := 137642, stop := 138265 ), rec( content := "\n\n ", count := [ 5, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSinks" ) , content := 0, count := [ 5, 1, 6, 2 ], mathmode := "Text", name := "Attr", next := 138328, root := ~, start := 138287, stop := 138327 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 138369, root := ~, start := 138331, stop := 138368 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a list of the sinks of the digraph\n ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "A", next := 138465, root := ~, start := 138451, stop := 138464 ), rec( content := ".\n A\ sink of a digraph is a vertex with out-degree zero.\n See ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Ref", next := 138564, root := ~, start := 138533, stop := 138563 ), rec( content := ".\n " , count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSinks(gr);\n[ 3, 5 ]\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSinks(D);\n[ ]\n", count := [ 5, 1, 6, 4 ], name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 6, 4 ], mathmode := "Text", name := "Example", next := 138862, root := ~, start := 138570, stop := 138861 ), rec( content := "\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 138879, root := ~, start := 138372, stop := 138878 ) ], count := [ 5, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 138893, root := ~, start := 138272, stop := 138892 ), rec( content := "\n\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSources"\ ), content := 0, count := [ 5, 1, 7, 2 ], mathmode := "Text", name := "Attr", next := 138957, root := ~, start := 138914, stop := 138956 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Returns", next := 138998, root := ~, start := 138960, stop := 138997 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns an immutable list of the sources of the digraph\n ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "A", next := 139107, root := ~, start := 139093, stop := 139106 ), rec( content := ".\n A\ source of a digraph is a vertex with in-degree zero.\n See ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Ref", next := 139206, root := ~, start := 139176, stop := 139205 ), rec( content := ".\n " , count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSources(gr);\n[ 1, 4 ]\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSources(D);\n[ ]\n", count := [ 5, 1, 7, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "Example", next := 139508, root := ~, start := 139212, stop := 139507 ), rec( content := "\n ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Description", next := 139525, root := ~, start := 139001, stop := 139524 ) ], count := [ 5, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 139539, root := ~, start := 138899, stop := 139538 ), rec( content := "\n\n ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTopologi\ calSort" ), content := 0, count := [ 5, 1, 8, 2 ], mathmode := "Text", name := "Attr", next := 139611, root := ~, start := 139560, stop := 139610 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers, or ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "K", next := 139666, root := ~, start := 139655, stop := 139665 ), rec( content := ".", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 139677, root := ~, start := 139614, stop := 139676 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "A", next := 139715, root := ~, start := 139701, stop := 139714 ), rec( content := " is a di\ graph whose only directed cycles are loops, then\n ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphTopologicalSort", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "C", next := 139805, root := ~, start := 139776, stop := 139804 ), rec( content := " returns\ the vertices of ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "A", next := 139844, root := ~, start := 139830, stop := 139843 ), rec( content := " ordered\ \n so that every edge's source appears no earlier in the list than its rang\ e.\n If the digraph ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "A", next := 139965, root := ~, start := 139951, stop := 139964 ), rec( content := " contain\ s directed cycles of length greater\n than ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "M", next := 140026, root := ~, start := 140018, stop := 140025 ), rec( content := ", then t\ his operation returns ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "K", next := 140067, root := ~, start := 140056, stop := 140066 ), rec( content := ".\n " , count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "P", next := 140077, root := ~, start := 140073, stop := 140076 ), rec( content := "\n\n \ See Section ", count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "Ref", next := 140139, root := ~, start := 140095, stop := 140138 ), rec( content := " for the\ definition\n of a directed cycle, and the definition of a loop.\n\n ", count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "P", next := 140223, root := ~, start := 140219, stop := 140222 ), rec( content := "\n\n \ The method used for this attribute has complexity ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "M", next := 140292, root := ~, start := 140279, stop := 140291 ), rec( content := " where\n\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "M", next := 140311, root := ~, start := 140303, stop := 140310 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "M", next := 140384, root := ~, start := 140376, stop := 140383 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "P", next := 140431, root := ~, start := 140427, stop := 140430 ), rec( content := "\n ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 3], [], [4, 6], [5], [], [7, 8, 9], [], [], []]);\ \n\ngap> DigraphTopologicalSort(D)\ ;\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ngap> D := Digraph(IsMutableDigraph, [[2, 3],\ [3], [4], []]);\n\ngap> DigraphTopo\ logicalSort(D);\n[ 4, 3, 2, 1 ]\n", count := [ 5, 1, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "Example", next := 140802, root := ~, start := 140436, stop := 140801 ), rec( content := "\n ", count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 140819, root := ~, start := 139680, stop := 140818 ) ], count := [ 5, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 140833, root := ~, start := 139545, stop := 140832 ), rec( content := "\n\n ", count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Name := "DigraphVertexLa\ bel" ), content := 0, count := [ 5, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 140906, root := ~, start := 140854, stop := 140905 ), rec( attributes := rec( Arg := "digraph, i, obj" , Name := "SetDigraphVerte\ xLabel" ), content := 0, count := [ 5, 1, 9, 4 ], mathmode := "Text", name := "Oper", next := 140968, root := ~, start := 140909, stop := 140967 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 141006, root := ~, start := 140992, stop := 141005 ), rec( content := " is a di\ graph, then the first operation returns the label\n of the vertex ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 141090, root := ~, start := 141082, stop := 141089 ), rec( content := ". The se\ cond operation can be used to set the label\n of the vertex ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 141168, root := ~, start := 141160, stop := 141167 ), rec( content := " in ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 141186, root := ~, start := 141172, stop := 141185 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 141231, root := ~, start := 141221, stop := 141230 ), rec( content := ". ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "P", next := 141238, root := ~, start := 141234, stop := 141237 ), rec( content := "\n\n \ The label of a vertex can be changed an arbitrary number of times. If no\n \ label has been set for the vertex ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "A", next := 141363, root := ~, start := 141355, stop := 141362 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "A", next := 141403, root := ~, start := 141395, stop := 141402 ), rec( content := ". ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "P", next := 141409, root := ~, start := 141405, stop := 141408 ), rec( content := "\n\n \ If ", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "A", next := 141432, root := ~, start := 141418, stop := 141431 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertices", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "C", next := 141511, root := ~, start := 141489, stop := 141510 ), rec( content := ", then t\ he labels of the vertices are set to\n the value of this component.", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 8 ], mathmode := "Text", name := "P", next := 141592, root := ~, start := 141588, stop := 141591 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs from\ \n old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabel(D, 3);\n3\ngap> D := Digraph([\"a\ \", \"b\", \"c\"], [], []);\n\ngap> D\ igraphVertexLabel(D, 2);\n\"b\"\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap\ > DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := InducedSubdigraph(D, [1, 2]);\n<\ immutable empty digraph with 2 vertices>\ngap> DigraphVertexLabel(D, 2);\n\"d\ \"\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> DigraphVertexLabel(D, 1);\n\"e\"\ngap\ > SetDigraphVertexLabel(D, 1, \"h\");\ngap> DigraphVertexLabel(D, 1);\n\"h\"\n\ gap> InducedSubdigraph(D, [1, 2]);\n\ng\ ap> DigraphVertexLabel(D, 1);\n\"h\"\n", count := [ 5, 1, 9, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 9, 9 ], mathmode := "Text", name := "Example", next := 142528, root := ~, start := 141736, stop := 142527 ), rec( content := "\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Description", next := 142545, root := ~, start := 140971, stop := 142544 ) ], count := [ 5, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 142559, root := ~, start := 140839, stop := 142558 ), rec( content := "\n\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertexLa\ bels" ), content := 0, count := [ 5, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 142628, root := ~, start := 142580, stop := 142627 ), rec( attributes := rec( Arg := "digraph, list", Name := "SetDigraphVerte\ xLabels" ), content := 0, count := [ 5, 1, 10, 4 ], mathmode := "Text", name := "Oper", next := 142688, root := ~, start := 142631, stop := 142687 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142726, root := ~, start := 142712, stop := 142725 ), rec( content := " is a di\ graph, then ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabels", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "C", next := 142772, root := ~, start := 142746, stop := 142771 ), rec( content := " returns\ a\n copy of the labels of the vertices in ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142839, root := ~, start := 142825, stop := 142838 ), rec( content := ".\n " , count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphVertexLabels", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "C", next := 142874, root := ~, start := 142845, stop := 142873 ), rec( content := " can be \ used to set the labels of the vertices\n in ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142942, root := ~, start := 142928, stop := 142941 ), rec( content := " to the \ list of\n arbitrary ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142997, root := ~, start := 142986, stop := 142996 ), rec( content := ", which \ must be of the same length\n as the number of vertices of ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 143079, root := ~, start := 143065, stop := 143078 ), rec( content := ". ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "P", next := 143085, root := ~, start := 143081, stop := 143084 ), rec( content := "\n\n \ If the list ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 143114, root := ~, start := 143103, stop := 143113 ), rec( content := " is immu\ table, then the vertex labels are set to a\n mutable copy of ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 143196, root := ~, start := 143185, stop := 143195 ), rec( content := ". Otherw\ ise, the labels are set to exactly\n ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 143254, root := ~, start := 143243, stop := 143253 ), rec( content := ". ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "P", next := 143260, root := ~, start := 143256, stop := 143259 ), rec( content := "\n\n \ The label of a vertex can be changed an arbitrary number of times. If no\n \ label has been set for the vertex ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "A", next := 143385, root := ~, start := 143377, stop := 143384 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "A", next := 143425, root := ~, start := 143417, stop := 143424 ), rec( content := ". ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "P", next := 143431, root := ~, start := 143427, stop := 143430 ), rec( content := "\n\n \ If ", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "A", next := 143454, root := ~, start := 143440, stop := 143453 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertices", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "C", next := 143533, root := ~, start := 143511, stop := 143532 ), rec( content := ", then t\ he labels of the vertices are set to the\n value of this component. As in t\ he above, if the component is immutable\n then the digraph's vertex labels \ are set to a mutable copy of\n ", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertices", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "C", next := 143750, root := ~, start := 143728, stop := 143749 ), rec( content := ". Otherw\ ise, they are set to exactly\n ", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertices", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "C", next := 143813, root := ~, start := 143791, stop := 143812 ), rec( content := ". ", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "P", next := 143819, root := ~, start := 143815, stop := 143818 ), rec( content := "\n\n \ Induced subdigraphs, and other operations which create new digraphs from\n \ old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabels(D);\n[ 1 .. 5 ]\ngap> D := Digra\ ph([\"a\", \"b\", \"c\"], [], []);\n\ \ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" ]\ngap> SetDigraphVertex\ Label(D, 2, \"d\");\ngap> DigraphVertexLabels(D);\n[ \"a\", \"d\", \"c\" ]\nga\ p> D := InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"c\" ]\ngap> D := Digraph(IsMutabl\ eDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> SetDigraphVertexLabels(D, [\"h\", \"i\", \"j\"]);\ngap> DigraphVert\ exLabels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"h\", \ \"j\" ]\n", count := [ 5, 1, 10, 10 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 10 ], mathmode := "Text", name := "Example", next := 144782, root := ~, start := 143958, stop := 144781 ), rec( content := "\n ", count := [ 5, 1, 10, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Description", next := 144799, root := ~, start := 142691, stop := 144798 ) ], count := [ 5, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 144813, root := ~, start := 142565, stop := 144812 ), rec( content := "\n\n ", count := [ 5, 1, 10, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i, j", Name := "DigraphEdgeLabe\ l" ), content := 0, count := [ 5, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 144887, root := ~, start := 144834, stop := 144886 ), rec( attributes := rec( Arg := "digraph, i, j, o\ bj", Name := "SetDigraphEdgeLabel" ), content := 0, count := [ 5, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 144951, root := ~, start := 144890, stop := 144950 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 144989, root := ~, start := 144975, stop := 144988 ), rec( content := " is a di\ graph without multiple edges, then the first\n operation returns the label \ of the edge from vertex ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 145106, root := ~, start := 145098, stop := 145105 ), rec( content := " to vert\ ex\n ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 145129, root := ~, start := 145121, stop := 145128 ), rec( content := ". The s\ econd operation can be used to set the label of the edge\n between vertex " , count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 145221, root := ~, start := 145213, stop := 145220 ), rec( content := " and ver\ tex ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 145241, root := ~, start := 145233, stop := 145240 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 145286, root := ~, start := 145276, stop := 145285 ), rec( content := ". ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "P", next := 145292, root := ~, start := 145288, stop := 145291 ), rec( content := "\n\n \ The label of an edge can be changed an arbitrary number of times. If no\n l\ abel has been set for the edge, then the default value is ", count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "A", next := 145441, root := ~, start := 145433, stop := 145440 ), rec( content := ". ", count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "P", next := 145447, root := ~, start := 145443, stop := 145446 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs from\ \n old ones, inherit their edge labels from their parents.\n\n See also \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Ref", next := 145636, root := ~, start := 145605, stop := 145635 ), rec( content := ".\n\n \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabel(D, 3, 1);\n1\ngap> SetDigraphEdgeLa\ bel(D, 2, 5, [42]);\ngap> DigraphEdgeLabel(D, 2, 5);\n[ 42 ]\ngap> D := Induce\ dSubdigraph(D, [2, 5]);\n\ngap> Di\ graphEdgeLabel(D, 1, 2);\n[ 42 ]\ngap> D := ChainDigraph(IsMutableDigraph, 5);\ \n\ngap> DigraphEdgeLabel(D, 2, 3);\ \n1\ngap> SetDigraphEdgeLabel(D, 4, 5, [1729]);\ngap> DigraphEdgeLabel(D, 4, 5\ );\n[ 1729 ]\ngap> InducedSubdigraph(D, [4, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 1729 ]\n", count := [ 5, 1, 11, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "Example", next := 146331, root := ~, start := 145643, stop := 146330 ), rec( content := "\n ", count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 146348, root := ~, start := 144954, stop := 146347 ) ], count := [ 5, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 146362, root := ~, start := 144819, stop := 146361 ), rec( content := "\n\n ", count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdgeLabe\ ls" ), content := 0, count := [ 5, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 146429, root := ~, start := 146383, stop := 146428 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of lists", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 12, 4 ], mathmode := "Text", name := "Oper", next := 146535, root := ~, start := 146432, stop := 146534 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and a function", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 12, 6 ], mathmode := "Text", name := "Oper", next := 146634, root := ~, start := 146538, stop := 146633 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 146672, root := ~, start := 146658, stop := 146671 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeLabels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 146743, root := ~, start := 146719, stop := 146742 ), rec( content := " returns\ a copy of the labels of the edges\n in ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 146807, root := ~, start := 146793, stop := 146806 ), rec( content := " as a li\ st of lists ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 146840, root := ~, start := 146827, stop := 146839 ), rec( content := " such th\ at\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 146874, root := ~, start := 146855, stop := 146873 ), rec( content := " is the \ label on the edge from vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 146920, root := ~, start := 146912, stop := 146919 ), rec( content := "\n to\ vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph)[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 146970, root := ~, start := 146935, stop := 146969 ), rec( content := ".\n\n \ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147004, root := ~, start := 146977, stop := 147003 ), rec( content := " can be \ used to set the labels of the edges in\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 147069, root := ~, start := 147055, stop := 147068 ), rec( content := " without\ multiple edges to the list ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 147118, root := ~, start := 147105, stop := 147117 ), rec( content := " of list\ s of\n arbitrary ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ such that ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147186, root := ~, start := 147169, stop := 147185 ), rec( content := " is the \ label on the edge\n from vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147236, root := ~, start := 147228, stop := 147235 ), rec( content := " to the \ vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph>[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147286, root := ~, start := 147251, stop := 147285 ), rec( content := ".\n\n \ Alternatively ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147334, root := ~, start := 147307, stop := 147333 ), rec( content := " can be \ called with binary function\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 147385, root := ~, start := 147374, stop := 147384 ), rec( content := " that as\ its second argument that when passed two vertices ", count := [ 5, 1, 12, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147452, root := ~, start := 147444, stop := 147451 ), rec( content := "\n an\ d ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147469, root := ~, start := 147461, stop := 147468 ), rec( content := " returns\ the label for the edge between vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147524, root := ~, start := 147516, stop := 147523 ), rec( content := " and ver\ tex\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 147548, root := ~, start := 147540, stop := 147547 ), rec( content := ". ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "P", next := 147554, root := ~, start := 147550, stop := 147553 ), rec( content := "\n\n \ The label of an edge can be changed an arbitrary number of times. If no\n l\ abel has been set for an edge, then the default value is ", count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "C", next := 147702, root := ~, start := 147694, stop := 147701 ), rec( content := ". ", count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 12, 9 ], mathmode := "Text", name := "P", next := 147708, root := ~, start := 147704, stop := 147707 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs\n \ from old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ 1, 1, 1 ], [ 1 ], \ [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 1, \"d\");\ngap> Di\ graphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ]\ ]\ngap> D := InducedSubdigraph(D, [1, 2, 3]);\n\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1 ], [ 1 ] ]\nga\ p> OutNeighbours(D);\n[ [ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := CompleteBipartiteD\ igraph(IsMutableDigraph, 2, 3);\n\n\ gap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ \ 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 4, \"a\");\ngap> DigraphEdgeLabels(D)\ ;\n[ [ 1, 1, 1 ], [ 1, \"a\", 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> Induce\ dSubdigraph(D, [1, 2, 3, 4]);\n\ngap\ > DigraphEdgeLabels(D);\n[ [ 1, 1 ], [ 1, \"a\" ], [ 1, 1 ], [ 1, 1 ] ]\ngap> \ OutNeighbors(D);\n[ [ 3, 4 ], [ 3, 4 ], [ 1, 2 ], [ 1, 2 ] ]\n", count := [ 5, 1, 12, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 10 ], mathmode := "Text", name := "Example", next := 148919, root := ~, start := 147852, stop := 148918 ), rec( content := "\n ", count := [ 5, 1, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "Description", next := 148936, root := ~, start := 146637, stop := 148935 ) ], count := [ 5, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 148950, root := ~, start := 146368, stop := 148949 ), rec( content := "\n\n ", count := [ 5, 1, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphInEdges"\ ), content := 0, count := [ 5, 1, 13, 2 ], mathmode := "Text", name := "Oper", next := 149022, root := ~, start := 148971, stop := 149021 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 149060, root := ~, start := 149025, stop := 149059 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphInEdges", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "C", next := 149102, root := ~, start := 149081, stop := 149101 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "A", next := 149150, root := ~, start := 149136, stop := 149149 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "A", next := 149179, root := ~, start := 149166, stop := 149178 ), rec( content := " as thei\ r range.\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphInEdges(D, 2);\n[ [ 1, 2 ], [ 1, 2 ] ]\ \n", count := [ 5, 1, 13, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 4 ], mathmode := "Text", name := "Example", next := 149385, root := ~, start := 149200, stop := 149384 ), rec( content := "\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 149402, root := ~, start := 149063, stop := 149401 ) ], count := [ 5, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 149416, root := ~, start := 148956, stop := 149415 ), rec( content := "\n\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphOutEdges\ " ), content := 0, count := [ 5, 1, 14, 2 ], mathmode := "Text", name := "Oper", next := 149489, root := ~, start := 149437, stop := 149488 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 149527, root := ~, start := 149492, stop := 149526 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOutEdges", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "C", next := 149570, root := ~, start := 149548, stop := 149569 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "A", next := 149618, root := ~, start := 149604, stop := 149617 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "A", next := 149647, root := ~, start := 149634, stop := 149646 ), rec( content := " as thei\ r source.\n ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] \ ]\n", count := [ 5, 1, 14, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 4 ], mathmode := "Text", name := "Example", next := 149855, root := ~, start := 149669, stop := 149854 ), rec( content := "\n ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 149872, root := ~, start := 149530, stop := 149871 ) ], count := [ 5, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 149886, root := ~, start := 149422, stop := 149885 ), rec( content := "\n\n ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Label := "for digraph an\ d list", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 15, 2 ], mathmode := "Text", name := "Oper", next := 149992, root := ~, start := 149907, stop := 149991 ), rec( attributes := rec( Arg := "digraph, u, v", Label := "for digraph an\ d two pos ints", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 15, 4 ], mathmode := "Text", name := "Oper", next := 150088, root := ~, start := 149995, stop := 150087 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 150111, root := ~, start := 150100, stop := 150110 ), rec( content := " or ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 150127, root := ~, start := 150115, stop := 150126 ), rec( content := ".", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 150138, root := ~, start := 150091, stop := 150137 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, this function returns ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 150211, root := ~, start := 150200, stop := 150210 ), rec( content := " if and \ only if the list\n ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 150251, root := ~, start := 150240, stop := 150250 ), rec( content := " specifi\ es an edge in the digraph ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 150299, root := ~, start := 150285, stop := 150298 ), rec( content := ". Speci\ fically,\n this operation returns ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 150354, root := ~, start := 150343, stop := 150353 ), rec( content := " if ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 150369, root := ~, start := 150358, stop := 150368 ), rec( content := " is a pa\ ir of positive\n integers where ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 150422, root := ~, start := 150411, stop := 150421 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "C", next := 150432, root := ~, start := 150422, stop := 150431 ), rec( content := " is the \ source and ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 150462, root := ~, start := 150451, stop := 150461 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "C", next := 150472, root := ~, start := 150462, stop := 150471 ), rec( content := "\n is\ the range of an edge in ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 150518, root := ~, start := 150504, stop := 150517 ), rec( content := ", and ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 150536, root := ~, start := 150524, stop := 150535 ), rec( content := " otherwi\ se. ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "P", next := 150552, root := ~, start := 150548, stop := 150551 ), rec( content := "\n\n \ The second form simply returns ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "K", next := 150600, root := ~, start := 150589, stop := 150599 ), rec( content := " if ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "u", count := [ 5, 1, 15, 6 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := \ "Text", name := "A", next := 150616, root := ~, start := 150608, stop := 15061\ 5 ), rec( content := ", ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := \ "Text", name := "A", next := 150626, root := ~, start := 150618, stop := 15062\ 5 ), rec( content := "]", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "C", next := 150631, root := ~, start := 150604, stop := 150630 ), rec( content := " is\n \ an edge in ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "A", next := 150664, root := ~, start := 150650, stop := 150663 ), rec( content := ", and ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "K", next := 150682, root := ~, start := 150670, stop := 150681 ), rec( content := " otherwi\ se.\n\n", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [6], [], [3], [], [1]]);\n\ngap> IsDigraphEdge(D, [1, 1]);\nfalse\ngap> IsDigra\ phEdge(D, [1, 2]);\ntrue\ngap> IsDigraphEdge(D, [1, 8]);\nfalse\n", count := [ 5, 1, 15, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "Example", next := 150939, root := ~, start := 150695, stop := 150938 ), rec( content := "\n ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 150956, root := ~, start := 150141, stop := 150955 ) ], count := [ 5, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 150970, root := ~, start := 149892, stop := 150969 ), rec( content := "\n\n ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsMatching" ), content := 0, count := [ 5, 1, 16, 2 ], mathmode := "Text", name := "Oper", next := 151036, root := ~, start := 150991, stop := 151035 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximalMatchi\ ng" ), content := 0, count := [ 5, 1, 16, 4 ], mathmode := "Text", name := "Oper", next := 151091, root := ~, start := 151039, stop := 151090 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximumMatchi\ ng" ), content := 0, count := [ 5, 1, 16, 6 ], mathmode := "Text", name := "Oper", next := 151146, root := ~, start := 151094, stop := 151145 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsPerfectMatchi\ ng" ), content := 0, count := [ 5, 1, 16, 8 ], mathmode := "Text", name := "Oper", next := 151201, root := ~, start := 151149, stop := 151200 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151224, root := ~, start := 151213, stop := 151223 ), rec( content := " or ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151240, root := ~, start := 151228, stop := 151239 ), rec( content := ".", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "Returns", next := 151251, root := ~, start := 151204, stop := 151250 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151289, root := ~, start := 151275, stop := 151288 ), rec( content := " is a di\ graph and ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151318, root := ~, start := 151307, stop := 151317 ), rec( content := " is a li\ st of pairs of\n vertices of ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151371, root := ~, start := 151357, stop := 151370 ), rec( content := ", then " , count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 151395, root := ~, start := 151378, stop := 151394 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151415, root := ~, start := 151404, stop := 151414 ), rec( content := " if\n \ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151434, root := ~, start := 151423, stop := 151433 ), rec( content := " is a ma\ tching of ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151466, root := ~, start := 151452, stop := 151465 ), rec( content := ". The o\ peration\n ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 151511, root := ~, start := 151487, stop := 151510 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151531, root := ~, start := 151520, stop := 151530 ), rec( content := " if ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151546, root := ~, start := 151535, stop := 151545 ), rec( content := " is a ma\ ximal \n matching, ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximumMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 151599, root := ~, start := 151575, stop := 151598 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151619, root := ~, start := 151608, stop := 151618 ), rec( content := " if ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151634, root := ~, start := 151623, stop := 151633 ), rec( content := " is a\n \ maximum matching and ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPerfectMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 151689, root := ~, start := 151665, stop := 151688 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151709, root := ~, start := 151698, stop := 151708 ), rec( content := " if \n \ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151729, root := ~, start := 151718, stop := 151728 ), rec( content := " is a pe\ rfect, matching of ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 151770, root := ~, start := 151756, stop := 151769 ), rec( content := ", respec\ tively. \n Otherwise, each of these operations return ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 151847, root := ~, start := 151835, stop := 151846 ), rec( content := ".\n " , count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "P", next := 151857, root := ~, start := 151853, stop := 151856 ), rec( content := "\n\n \ A ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ atching", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 151880, root := ~, start := 151865, stop := 151879 ), rec( content := " ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 151889, root := ~, start := 151881, stop := 151888 ), rec( content := " of a di\ graph ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "A", next := 151917, root := ~, start := 151903, stop := 151916 ), rec( content := " is a su\ bset of the\n edges of ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "A", next := 151964, root := ~, start := 151950, stop := 151963 ), rec( content := ", i.e. " , count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 1, 16, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := \ "Text", name := "A", next := 152001, root := ~, start := 151987, stop := 15200\ 0 ), rec( content := ")", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 152006, root := ~, start := 151971, stop := 152005 ), rec( content := ", such\n\ that no pair of distinct edges in ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 152059, root := ~, start := 152051, stop := 152058 ), rec( content := " are inc\ ident to the same vertex\n of ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "A", next := 152113, root := ~, start := 152099, stop := 152112 ), rec( content := ". Note \ that this definition allows a matching to contain\n loops. See ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "Ref", next := 152217, root := ~, start := 152187, stop := 152216 ), rec( content := ". The m\ atching ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 152241, root := ~, start := 152233, stop := 152240 ), rec( content := " is\n \ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 152263, root := ~, start := 152249, stop := 152262 ), rec( content := " if it i\ s contained in no larger matching of the digraph,\n is ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximum", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 152342, root := ~, start := 152328, stop := 152341 ), rec( content := " if it h\ as the greatest cardinality among all matchings and\n is ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erfect", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 152423, root := ~, start := 152409, stop := 152422 ), rec( content := " if ever\ y vertex of the digraph is incident to an edge in\n the matching. Every ma\ ximum or perfect matching is maximal. Note, however,\n that not every perfe\ ct matching of digraphs with loops is maximum.\n\n ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [1, 2], [2, 3, 4], [3, 5], [1]]);\n\ngap> IsMatching(D, [[2, 1], [3, 2]]);\nfalse\n\ gap> edges := [[3, 2]];;\ngap> IsMatching(D, edges);\ntrue\ngap> IsMaximalMatc\ hing(D, edges);\nfalse\ngap> edges := [[2, 1], [3, 4]];;\ngap> IsMaximalMatchi\ ng(D, edges);\ntrue\ngap> IsPerfectMatching(D, edges);\nfalse\ngap> edges := [\ [1, 2], [3, 3], [4, 5]];;\ngap> IsPerfectMatching(D, edges);\ntrue\ngap> IsMax\ imumMatching(D, edges);\nfalse\ngap> edges := [[1, 1], [2, 2], [3, 3], [4, 5]]\ ;;\ngap> IsMaximumMatching(D, edges);\ntrue\n", count := [ 5, 1, 16, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 16, 11 ], mathmode := "Text", name := "Example", next := 153237, root := ~, start := 152637, stop := 153236 ), rec( content := "\n ", count := [ 5, 1, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "Description", next := 153254, root := ~, start := 151254, stop := 153253 ) ], count := [ 5, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 153268, root := ~, start := 150976, stop := 153267 ), rec( content := "\n\n ", count := [ 5, 1, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalM\ atching" ), content := 0, count := [ 5, 1, 17, 2 ], mathmode := "Text", name := "Attr", next := 153340, root := ~, start := 153289, stop := 153339 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ pairs of vertices.", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Returns", next := 153390, root := ~, start := 153343, stop := 153389 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a maximal matching of the digraph ", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "A", next := 153481, root := ~, start := 153467, stop := 153480 ), rec( content := ".\n " , count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "P", next := 153491, root := ~, start := 153487, stop := 153490 ), rec( content := "\n\n \ For the definition of a maximal matching, see ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alMatching" ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "Ref", next := 153574, root := ~, start := 153543, stop := 153573 ), rec( content := ". \n\n \ ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".IeAoXCJU@|SHAe?d\");\n\ngap> M := DigraphMaximalMatching(D);; IsMaxim\ alMatching(D, M);\ntrue\ngap> D := RandomDigraph(100);;\ngap> IsMaximalMatchin\ g(D, DigraphMaximalMatching(D));\ntrue\ngap> D := GeneralisedPetersenGraph(IsM\ utableDigraph, 9, 2);\n\ngap> IsMa\ ximalMatching(D, DigraphMaximalMatching(D));\ntrue\n", count := [ 5, 1, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 17, 5 ], mathmode := "Text", name := "Example", next := 154042, root := ~, start := 153582, stop := 154041 ), rec( content := "\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Description", next := 154059, root := ~, start := 153393, stop := 154058 ) ], count := [ 5, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 154073, root := ~, start := 153274, stop := 154072 ), rec( content := "\n\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximumM\ atching" ), content := 0, count := [ 5, 1, 18, 2 ], mathmode := "Text", name := "Attr", next := 154145, root := ~, start := 154094, stop := 154144 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ pairs of vertices.", count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "Returns", next := 154195, root := ~, start := 154148, stop := 154194 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a maximum matching of the digraph ", count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "A", next := 154286, root := ~, start := 154272, stop := 154285 ), rec( content := ".\n " , count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "P", next := 154296, root := ~, start := 154292, stop := 154295 ), rec( content := "\n\n \ For the definition of a maximum matching, see ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ umMatching" ), content := 0, count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "Ref", next := 154379, root := ~, start := 154348, stop := 154378 ), rec( content := ".\n I\ f ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "A", next := 154402, root := ~, start := 154388, stop := 154401 ), rec( content := " is bipa\ rtite (see ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsBipar\ titeDigraph" ), content := 0, count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "Ref", next := 154453, root := ~, start := 154421, stop := 154452 ), rec( content := "), then \ \n the algorithm used has complexity ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m*sqrt(n))", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 154519, root := ~, start := 154500, stop := 154518 ), rec( content := ". Otherw\ ise for general \n graphs the complexity is ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m*n*log(n))", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 154593, root := ~, start := 154573, stop := 154592 ), rec( content := ". Here " , count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 154608, root := ~, start := 154600, stop := 154607 ), rec( content := " is the \ number of vertices \n and ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 154652, root := ~, start := 154644, stop := 154651 ), rec( content := " is the \ number of edges.\n\n ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&I@EA_A?AdDp[_c??OO\");\n\ngap> M := DigraphMaximumMatching(D);; IsMaxi\ malMatching(D, M);\ntrue\ngap> Length(M);\n5\ngap> D := Digraph([[5, 6, 7, 8],\ [6, 7, 8], [7, 8], [8], \n> [], [], [], []]);;\ngap> M := Di\ graphMaximumMatching(D);\n[ [ 1, 5 ], [ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D :\ = GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> M := DigraphMaximumMatching(D);;\ngap> IsMaximalMatc\ hing(D, M);\ntrue\ngap> Length(M);\n9\n", count := [ 5, 1, 18, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 18, 5 ], mathmode := "Text", name := "Example", next := 155279, root := ~, start := 154682, stop := 155278 ), rec( content := "\n ", count := [ 5, 1, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "Description", next := 155296, root := ~, start := 154198, stop := 155295 ) ], count := [ 5, 1, 18, 1 ], mathmode := "Text", name := "ManSection", next := 155310, root := ~, start := 154079, stop := 155309 ), rec( content := "\n\n ", count := [ 5, 1, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 155324, root := ~, start := 133826, stop := 155323 ) , rec( content := "\n\n ", count := [ 5, 1, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Neighbours and de\ gree", count := [ 5, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 155377, root := ~, start := 155337, stop := 155376 ), rec( content := "\n ", count := [ 5, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ " ), content := 0, count := [ 5, 2, 1, 2 ], mathmode := "Text", name := "Attr", next := 155441, root := ~, start := 155397, stop := 155440 ), rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ MutableCopy" ), content := 0, count := [ 5, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 155499, root := ~, start := 155444, stop := 155498 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of non-negative integers.", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 155562, root := ~, start := 155502, stop := 155561 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the adjacency matrix ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 155636, root := ~, start := 155626, stop := 155635 ), rec( content := " of the \ digraph\n ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 155670, root := ~, start := 155656, stop := 155669 ), rec( content := ".\n T\ he value of the matrix entry ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 155722, root := ~, start := 155706, stop := 155721 ), rec( content := " is the \ number of edges\n in ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 155767, root := ~, start := 155753, stop := 155766 ), rec( content := " with so\ urce ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 155788, root := ~, start := 155780, stop := 155787 ), rec( content := " and ran\ ge ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 155807, root := ~, start := 155799, stop := 155806 ), rec( content := ". If ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 155826, root := ~, start := 155812, stop := 155825 ), rec( content := "\n ha\ s no vertices, then the empty list is returned. ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "P", next := 155885, root := ~, start := 155881, stop := 155884 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrix", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 155926, root := ~, start := 155904, stop := 155925 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrixMutableCopy", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 156021, root := ~, start := 155988, stop := 156020 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrix", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 156066, root := ~, start := 156044, stop := 156065 ), rec( content := " that is\ a mutable list of mutable lists. ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 1, 7 ], mathmode := "Text", name := "P", next := 156112, root := ~, start := 156108, stop := 156111 ), rec( content := "\n\n \ ", count := [ 5, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, \ 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4,\ 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> mat := AdjacencyMatrix(gr);;\ngap> Display(mat);\n[ [ 0, \ 3, 0, 0, 0, 0, 0, 0, 0, 0 ],\n [ 1, 0, 1, 0, 0, 1, 0, 1, 1, \ 1 ],\n [ 0, 0, 0, 1, 0, 1, 0, 1, 0, 0 ],\n [ 1, 1, 1, 0, 0, \ 0, 0, 0, 1, 0 ],\n [ 0, 0, 2, 0, 0, 0, 0, 0, 0, 0 ],\n [ 0, \ 0, 1, 0, 1, 1, 0, 0, 0, 1 ],\n [ 1, 1, 0, 0, 0, 0, 1, 0, 0,\ 0 ],\n [ 1, 1, 1, 0, 1, 1, 0, 0, 0, 2 ],\n [ 1, 0, 1, 1, 1,\ 0, 0, 1, 0, 1 ],\n [ 0, 1, 1, 1, 0, 1, 1, 0, 0, 1 ] ]\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(AdjacencyMatrix(D));\n[ [ 0, 1, 0 ],\n [ 0, 0, 1 ],\ \n [ 1, 0, 0 ] ]\n", count := [ 5, 2, 1, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 8 ], mathmode := "Text", name := "Example", next := 157082, root := ~, start := 156118, stop := 157081 ), rec( content := "\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 157099, root := ~, start := 155565, stop := 157098 ) ], count := [ 5, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 157113, root := ~, start := 155382, stop := 157112 ), rec( content := "\n\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "CharacteristicP\ olynomial" ), content := 0, count := [ 5, 2, 2, 2 ], mathmode := "Text", name := "Attr", next := 157187, root := ~, start := 157134, stop := 157186 ), rec( attributes := rec( ), content := [ rec( content := "A polynom\ ial with integer coefficients.", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 157248, root := ~, start := 157190, stop := 157247 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the characteristic polynomial of the digraph\n ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "A", next := 157355, root := ~, start := 157341, stop := 157354 ), rec( content := ". That i\ s it returns the characteristic polynomial\n of the adjacency matrix of t\ he digraph ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "A", next := 157465, root := ~, start := 157451, stop := 157464 ), rec( content := "\n ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3\ , 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, \ 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> CharacteristicPolynomial(D);\nx_1^10-3*x_1^9-7*x_1^8-x_1^7+1\ 4*x_1^6+x_1^5-26*x_1^4+51*x_1^3-10*x_1^2\\\n+18*x_1-30\ngap> D := CompleteDigr\ aph(5);\n\ngap> CharacteristicPoly\ nomial(D);\nx_1^5-10*x_1^3-20*x_1^2-15*x_1-4\ngap> D := CycleDigraph(IsMutable\ Digraph, 3);\n\ngap> CharacteristicP\ olynomial(D);\nx_1^3-1\n", count := [ 5, 2, 2, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 4 ], mathmode := "Text", name := "Example", next := 158131, root := ~, start := 157470, stop := 158130 ), rec( content := "\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 158148, root := ~, start := 157251, stop := 158147 ) ], count := [ 5, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 158162, root := ~, start := 157119, stop := 158161 ), rec( content := "\n\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrix" ), content := 0, count := [ 5, 2, 3, 2 ], mathmode := "Text", name := "Attr", next := 158234, root := ~, start := 158183, stop := 158233 ), rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrixMutableCopy" ), content := 0, count := [ 5, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 158299, root := ~, start := 158237, stop := 158298 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of booleans.", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Returns", next := 158349, root := ~, start := 158302, stop := 158348 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 158387, root := ~, start := 158373, stop := 158386 ), rec( content := " is a di\ graph with a positive number of vertices\n ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158448, root := ~, start := 158440, stop := 158447 ), rec( content := ", then " , count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix(", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158485, root := ~, start := 158455, stop := 158484 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 158499, root := ~, start := 158485, stop := 158498 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158507, root := ~, start := 158499, stop := 158506 ), rec( content := "\n re\ turns the boolean adjacency matrix ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158559, root := ~, start := 158549, stop := 158558 ), rec( content := " of ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 158577, root := ~, start := 158563, stop := 158576 ), rec( content := ". The\n\ value of the matrix entry ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[j][i]", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158630, root := ~, start := 158614, stop := 158629 ), rec( content := " is ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "K", next := 158645, root := ~, start := 158634, stop := 158644 ), rec( content := " if and \ only if\n there exists an edge in ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 158703, root := ~, start := 158689, stop := 158702 ), rec( content := " with so\ urce ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158724, root := ~, start := 158716, stop := 158723 ), rec( content := " and ran\ ge\n ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 158747, root := ~, start := 158739, stop := 158746 ), rec( content := ". If ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 158767, root := ~, start := 158753, stop := 158766 ), rec( content := " has no \ vertices, then the empty list is\n returned. ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 6 ], mathmode := "Text", name := "P", next := 158826, root := ~, start := 158822, stop := 158825 ), rec( content := "\n\n \ Note that the boolean adjacency matrix loses information about multiple\n e\ dges. ", count := [ 5, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "P", next := 158920, root := ~, start := 158916, stop := 158919 ), rec( content := "\n\n \ The attribute ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 158969, root := ~, start := 158940, stop := 158968 ), rec( content := " returns\ an immutable list of\n immutable lists, whereas the function\n ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrixMutableCopy", count := [ 5, 2, 3, 7 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 159085, root := ~, start := 159045, stop := 159084 ), rec( content := " returns\ a copy of the\n ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 159141, root := ~, start := 159112, stop := 159140 ), rec( content := " that is\ a mutable list of mutable lists. ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 8 ], mathmode := "Text", name := "P", next := 159187, root := ~, start := 159183, stop := 159186 ), rec( content := "\n ", count := [ 5, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);\n\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ fal\ se, false, true, true ],\n [ false, true, true, false ],\n [ tr\ ue, true, false, true ],\n [ false, false, false, true ] ]\ngap> gr\ := CycleDigraph(4);;\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ fals\ e, true, false, false ],\n [ false, false, true, false ],\n [ fals\ e, false, false, true ],\n [ true, false, false, false ] ]\ngap> Boo\ leanAdjacencyMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableD\ igraph, 3);\n\ngap> PrintArray(Boole\ anAdjacencyMatrix(D));\n[ [ false, true, false ],\n [ false, false, t\ rue ],\n [ true, false, false ] ]\n", count := [ 5, 2, 3, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 9 ], mathmode := "Text", name := "Example", next := 160022, root := ~, start := 159192, stop := 160021 ), rec( content := "\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Description", next := 160039, root := ~, start := 158352, stop := 160038 ) ], count := [ 5, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 160053, root := ~, start := 158168, stop := 160052 ), rec( content := "\n\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAdjacenc\ yFunction" ), content := 0, count := [ 5, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 160127, root := ~, start := 160074, stop := 160126 ), rec( attributes := rec( ), content := [ rec( content := "A functio\ n.", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 160160, root := ~, start := 160130, stop := 160159 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "A", next := 160198, root := ~, start := 160184, stop := 160197 ), rec( content := " is a di\ graph, then ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAdjacencyFunction", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 160249, root := ~, start := 160218, stop := 160248 ), rec( content := " returns\ \n a function which takes two integer parameters ", count := [ 5, 2, 4, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ , y", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 160319, root := ~, start := 160308, stop := 160318 ), rec( content := " and ret\ urns\n ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "K", next := 160347, root := ~, start := 160336, stop := 160346 ), rec( content := " if ther\ e exists an edge from vertex ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 160392, root := ~, start := 160384, stop := 160391 ), rec( content := " to vert\ ex ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "y\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 160411, root := ~, start := 160403, stop := 160410 ), rec( content := "\n in\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "A", next := 160433, root := ~, start := 160419, stop := 160432 ), rec( content := " and ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "K", next := 160450, root := ~, start := 160438, stop := 160449 ), rec( content := " if not.\ \n\n ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[1, 2], [3], []]);\n\ngap> foo := DigraphAdjacencyFunction(digraph);\nfunction( u, v )\ ... end\ngap> foo(1, 1);\ntrue\ngap> foo(1, 2);\ntrue\ngap> foo(1, 3);\nfalse\ \ngap> foo(3, 1);\nfalse\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ \ngap> foo := DigraphAdjacenc\ yFunction(gr);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(3, 2\ );\nfalse\ngap> foo(3, 1);\nfalse\ngap> D := CycleDigraph(IsMutableDigraph, 3)\ ;\n\ngap> foo := DigraphAdjacencyFun\ ction(D);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(2, 1);\nf\ alse\n", count := [ 5, 2, 4, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 4 ], mathmode := "Text", name := "Example", next := 161231, root := ~, start := 160464, stop := 161230 ), rec( content := "\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 161248, root := ~, start := 160163, stop := 161247 ) ], count := [ 5, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 161262, root := ~, start := 160059, stop := 161261 ), rec( content := "\n\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRange" ) , content := 0, count := [ 5, 2, 5, 2 ], mathmode := "Text", name := "Attr", next := 161324, root := ~, start := 161283, stop := 161323 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSource" ), content := 0, count := [ 5, 2, 5, 4 ], mathmode := "Text", name := "Attr", next := 161369, root := ~, start := 161327, stop := 161368 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers.", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Returns", next := 161419, root := ~, start := 161372, stop := 161418 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphRange", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 161459, root := ~, start := 161440, stop := 161458 ), rec( content := " and ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphSource", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 161484, root := ~, start := 161464, stop := 161483 ), rec( content := " return \ the range and source of\n the digraph ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "A", next := 161546, root := ~, start := 161532, stop := 161545 ), rec( content := ". More p\ recisely, position ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 161581, root := ~, start := 161573, stop := 161580 ), rec( content := " in\n \ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphRange(", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := \ "Text", name := "A", next := 161619, root := ~, start := 161605, stop := 16161\ 8 ), rec( content := ")", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 161624, root := ~, start := 161589, stop := 161623 ), rec( content := " is the \ range of the ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 161653, root := ~, start := 161645, stop := 161652 ), rec( content := "th edge \ of\n ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "A", next := 161682, root := ~, start := 161668, stop := 161681 ), rec( content := ".\n\n \ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 1, 3, 5], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);\ \n\ngap> DigraphRange(gr);\n[ 2, \ 1, 3, 5, 1, 3, 4, 2, 3, 2, 1, 2, 3, 4 ]\ngap> DigraphSource(gr);\n[ 1, 1, 1, 1\ , 2, 2, 2, 3, 3, 4, 5, 5, 5, 5 ]\ngap> DigraphEdges(gr);\n[ [ 1, 2 ], [ 1, 1 ]\ , [ 1, 3 ], [ 1, 5 ], [ 2, 1 ], [ 2, 3 ], \n [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ \ 4, 2 ], [ 5, 1 ], [ 5, 2 ], \n [ 5, 3 ], [ 5, 4 ] ]\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Digraph\ Range(D);\n[ 2, 3, 1 ]\ngap> DigraphSource(D);\n[ 1, 2, 3 ]\n", count := [ 5, 2, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 6 ], mathmode := "Text", name := "Example", next := 162310, root := ~, start := 161689, stop := 162309 ), rec( content := "\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Description", next := 162327, root := ~, start := 161422, stop := 162326 ) ], count := [ 5, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 162341, root := ~, start := 161268, stop := 162340 ), rec( content := "\n\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OutNeighbours" ), content := 0, count := [ 5, 2, 6, 2 ], mathmode := "Text", name := "Attr", next := 162404, root := ~, start := 162362, stop := 162403 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighbors" ) , content := 0, count := [ 5, 2, 6, 4 ], mathmode := "Text", name := "Attr", next := 162448, root := ~, start := 162407, stop := 162447 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighboursMu\ tableCopy" ), content := 0, count := [ 5, 2, 6, 6 ], mathmode := "Text", name := "Oper", next := 162504, root := ~, start := 162451, stop := 162503 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighborsMut\ ableCopy" ), content := 0, count := [ 5, 2, 6, 8 ], mathmode := "Text", name := "Oper", next := 162559, root := ~, start := 162507, stop := 162558 ), rec( attributes := rec( ), content := [ rec( content := "The adjac\ encies of a digraph.", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Returns", next := 162610, root := ~, start := 162562, stop := 162609 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 162672, root := ~, start := 162662, stop := 162671 ), rec( content := " of out-\ neighbours of each vertex\n of the digraph ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "A", next := 162739, root := ~, start := 162725, stop := 162738 ), rec( content := ".\n M\ ore specifically, a vertex ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 162781, root := ~, start := 162773, stop := 162780 ), rec( content := " appears\ in ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 162806, root := ~, start := 162793, stop := 162805 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 162862, root := ~, start := 162854, stop := 162861 ), rec( content := " and ran\ ge ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 162881, root := ~, start := 162873, stop := 162880 ), rec( content := " in\n \ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "A", next := 162903, root := ~, start := 162889, stop := 162902 ), rec( content := ". ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "P", next := 162909, root := ~, start := 162905, stop := 162908 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 162948, root := ~, start := 162928, stop := 162947 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighboursMutableCopy", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 163041, root := ~, start := 163010, stop := 163040 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 163084, root := ~, start := 163064, stop := 163083 ), rec( content := " which i\ s a mutable list of mutable lists. ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 6, 11 ], mathmode := "Text", name := "P", next := 163131, root := ~, start := 163127, stop := 163130 ), rec( content := "\n\n \ ", count := [ 5, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ 1, 3 ], [ ] ]\ngap> \ gr := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> OutNeighbours(gr);\n[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]\ngap> gr :\ = DigraphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n<\ immutable multidigraph with 3 vertices, 7 edges>\ngap> OutNeighbours(gr);\n[ [\ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutNeighboursMutableCopy(gr);\n[ [ 1, 2\ , 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 \ ], [ 1 ] ]\n", count := [ 5, 2, 6, 12 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 12 ], mathmode := "Text", name := "Example", next := 163919, root := ~, start := 163137, stop := 163918 ), rec( content := "\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Description", next := 163936, root := ~, start := 162613, stop := 163935 ) ], count := [ 5, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 163950, root := ~, start := 162347, stop := 163949 ), rec( content := "\n\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "InNeighbours" ) , content := 0, count := [ 5, 2, 7, 2 ], mathmode := "Text", name := "Attr", next := 164012, root := ~, start := 163971, stop := 164011 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighbors" ), content := 0, count := [ 5, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 164055, root := ~, start := 164015, stop := 164054 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighboursMut\ ableCopy" ), content := 0, count := [ 5, 2, 7, 6 ], mathmode := "Text", name := "Oper", next := 164110, root := ~, start := 164058, stop := 164109 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighborsMuta\ bleCopy" ), content := 0, count := [ 5, 2, 7, 8 ], mathmode := "Text", name := "Oper", next := 164164, root := ~, start := 164113, stop := 164163 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Returns", next := 164214, root := ~, start := 164167, stop := 164213 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 164276, root := ~, start := 164266, stop := 164275 ), rec( content := " of in-n\ eighbours of each vertex\n of the digraph ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "A", next := 164342, root := ~, start := 164328, stop := 164341 ), rec( content := ".\n M\ ore specifically, a vertex ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 164384, root := ~, start := 164376, stop := 164383 ), rec( content := " appears\ in ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn[i]", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 164409, root := ~, start := 164396, stop := 164408 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 164465, root := ~, start := 164457, stop := 164464 ), rec( content := " and ran\ ge ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 164484, root := ~, start := 164476, stop := 164483 ), rec( content := " in\n \ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "A", next := 164506, root := ~, start := 164492, stop := 164505 ), rec( content := ". ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "P", next := 164512, root := ~, start := 164508, stop := 164511 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighbours", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 164550, root := ~, start := 164531, stop := 164549 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighboursMutableCopy", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 164642, root := ~, start := 164612, stop := 164641 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighbours", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 164684, root := ~, start := 164665, stop := 164683 ), rec( content := " which i\ s a mutable list of mutable lists. ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "P", next := 164731, root := ~, start := 164727, stop := 164730 ), rec( content := "\n\n \ Note that each entry of ", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "C", next := 164771, root := ~, start := 164761, stop := 164770 ), rec( content := " is sort\ ed into ascending order. ", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 12 ], mathmode := "Text", name := "P", next := 164808, root := ~, start := 164804, stop := 164807 ), rec( content := "\n\n \ ", count := [ 5, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n\ngap> InNeighbours(gr);\n[ [ 2 ], [ 1 ], [ 2 ] ]\ngap> gr \ := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 2 ], [ 1, 3 ] ]\ngap> gr := Di\ graphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2\ ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> InNeighboursMutableCopy(gr);\n[ [ 1, 2 ], [ \ 1, 1, 2 ], [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> InNeighbours(D);\n[ [ 3 ], [ 1 ], [ 2 \ ] ]\n", count := [ 5, 2, 7, 13 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 13 ], mathmode := "Text", name := "Example", next := 165589, root := ~, start := 164814, stop := 165588 ), rec( content := "\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Description", next := 165606, root := ~, start := 164217, stop := 165605 ) ], count := [ 5, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 165620, root := ~, start := 163956, stop := 165619 ), rec( content := "\n\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OutDegrees" ), content := 0, count := [ 5, 2, 8, 2 ], mathmode := "Text", name := "Attr", next := 165680, root := ~, start := 165641, stop := 165679 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSequen\ ce" ), content := 0, count := [ 5, 2, 8, 4 ], mathmode := "Text", name := "Attr", next := 165729, root := ~, start := 165683, stop := 165728 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSet" ) , content := 0, count := [ 5, 2, 8, 6 ], mathmode := "Text", name := "Attr", next := 165773, root := ~, start := 165732, stop := 165772 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ non-negative integers.", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Returns", next := 165827, root := ~, start := 165776, stop := 165826 ), rec( attributes := rec( ), content := [ rec( content := "\n\n G\ iven a digraph ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "A", next := 165879, root := ~, start := 165865, stop := 165878 ), rec( content := " with ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "M", next := 165893, root := ~, start := 165885, stop := 165892 ), rec( content := " vertice\ s, the function\n ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegrees", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 165938, root := ~, start := 165921, stop := 165937 ), rec( content := " returns\ an immutable list ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 165975, root := ~, start := 165965, stop := 165974 ), rec( content := " of leng\ th ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "M", next := 165994, root := ~, start := 165986, stop := 165993 ), rec( content := ", such t\ hat\n for a vertex ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 166031, root := ~, start := 166023, stop := 166030 ), rec( content := " in ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "A", next := 166049, root := ~, start := 166035, stop := 166048 ), rec( content := ", the va\ lue of ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 166077, root := ~, start := 166064, stop := 166076 ), rec( content := " is the\\ n out-degree of vertex ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 166118, root := ~, start := 166110, stop := 166117 ), rec( content := ".\n S\ ee ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Ref", next := 166159, root := ~, start := 166128, stop := 166158 ), rec( content := ". ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "P", next := 166165, root := ~, start := 166161, stop := 166164 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegreeSequence", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "C", next := 166208, root := ~, start := 166184, stop := 166207 ), rec( content := " returns\ the same list,\n after it has been sorted into non-increasing order. ", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "P", next := 166292, root := ~, start := 166288, stop := 166291 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegreeSet", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "C", next := 166330, root := ~, start := 166311, stop := 166329 ), rec( content := " returns\ the same list, sorted into\n increasing order with duplicate entries remov\ ed. ", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 10 ], mathmode := "Text", name := "P", next := 166423, root := ~, start := 166419, stop := 166422 ), rec( content := "\n\n \ ", count := [ 5, 2, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);\n\ngap> OutDegrees(D);\n[ 4, 0, 2, 0 ]\ngap> OutDegreeS\ equence(D);\n[ 4, 2, 0, 0 ]\ngap> OutDegreeSet(D);\n[ 0, 2, 4 ]\ngap> D := Cyc\ leDigraph(IsMutableDigraph, 3);\n\ng\ ap> OutDegrees(D);\n[ 1, 1, 1 ]\n", count := [ 5, 2, 8, 11 ], name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 8, 11 ], mathmode := "Text", name := "Example", next := 166793, root := ~, start := 166429, stop := 166792 ), rec( content := "\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Description", next := 166810, root := ~, start := 165830, stop := 166809 ) ], count := [ 5, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 166824, root := ~, start := 165626, stop := 166823 ), rec( content := "\n\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "InDegrees" ), content := 0, count := [ 5, 2, 9, 2 ], mathmode := "Text", name := "Attr", next := 166883, root := ~, start := 166845, stop := 166882 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSequenc\ e" ), content := 0, count := [ 5, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 166931, root := ~, start := 166886, stop := 166930 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSet" ), content := 0, count := [ 5, 2, 9, 6 ], mathmode := "Text", name := "Attr", next := 166974, root := ~, start := 166934, stop := 166973 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ non-negative integers.", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Returns", next := 167028, root := ~, start := 166977, stop := 167027 ), rec( attributes := rec( ), content := [ rec( content := "\n\n G\ iven a digraph ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "A", next := 167080, root := ~, start := 167066, stop := 167079 ), rec( content := " with ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "M", next := 167094, root := ~, start := 167086, stop := 167093 ), rec( content := " vertice\ s, the function\n ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegrees", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 167138, root := ~, start := 167122, stop := 167137 ), rec( content := " returns\ an immutable list ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 167175, root := ~, start := 167165, stop := 167174 ), rec( content := " of leng\ th ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "M", next := 167194, root := ~, start := 167186, stop := 167193 ), rec( content := ", such t\ hat\n for a vertex ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 167231, root := ~, start := 167223, stop := 167230 ), rec( content := " in ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "A", next := 167249, root := ~, start := 167235, stop := 167248 ), rec( content := ", the va\ lue of ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn[i]", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 167277, root := ~, start := 167264, stop := 167276 ), rec( content := " is\n \ the in-degree of vertex ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 167317, root := ~, start := 167309, stop := 167316 ), rec( content := ".\n S\ ee ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Ref", next := 167357, root := ~, start := 167327, stop := 167356 ), rec( content := ". ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "P", next := 167363, root := ~, start := 167359, stop := 167362 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegreeSequence", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "C", next := 167405, root := ~, start := 167382, stop := 167404 ), rec( content := " returns\ the same list,\n after it has been sorted into non-increasing order. ", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "P", next := 167489, root := ~, start := 167485, stop := 167488 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegreeSet", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "C", next := 167526, root := ~, start := 167508, stop := 167525 ), rec( content := " returns\ the same list, sorted into\n increasing order with duplicate entries remov\ ed. ", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 10 ], mathmode := "Text", name := "P", next := 167619, root := ~, start := 167615, stop := 167618 ), rec( content := "\n\n \ ", count := [ 5, 2, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);\n\ngap> InDegrees(D);\n[ 2, 3, 1, 2 ]\ngap> InDeg\ reeSequence(D);\n[ 3, 2, 2, 1 ]\ngap> InDegreeSet(D);\n[ 1, 2, 3 ]\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> InDegrees(D);\n[ 1, 1, 1 ]\n", count := [ 5, 2, 9, 11 ], name := "PCDAT\ A", root := ~ ) ], count := [ 5, 2, 9, 11 ], mathmode := "Text", name := "Example", next := 167991, root := ~, start := 167625, stop := 167990 ), rec( content := "\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Description", next := 168008, root := ~, start := 167031, stop := 168007 ) ], count := [ 5, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 168022, root := ~, start := 166830, stop := 168021 ), rec( content := "\n\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutDegreeOfVert\ ex" ), content := 0, count := [ 5, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 168097, root := ~, start := 168043, stop := 168096 ), rec( attributes := rec( ), content := [ rec( content := "The non-n\ egative integer.", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 168144, root := ~, start := 168100, stop := 168143 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the out-degree of the vertex ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 168230, root := ~, start := 168217, stop := 168229 ), rec( content := " in the\\ n digraph ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 168264, root := ~, start := 168250, stop := 168263 ), rec( content := ".\n T\ he out-degree of ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 168301, root := ~, start := 168288, stop := 168300 ), rec( content := " is the \ number of edges in ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 168342, root := ~, start := 168328, stop := 168341 ), rec( content := "\n wh\ ose source is ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 168376, root := ~, start := 168363, stop := 168375 ), rec( content := ".\n " , count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 10, 4 ], mathmode := "Text", name := "P", next := 168386, root := ~, start := 168382, stop := 168385 ), rec( content := "\n ", count := [ 5, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> OutDegreeOfVerte\ x(D, 1);\n3\ngap> OutDegreeOfVertex(D, 2);\n2\ngap> OutDegreeOfVertex(D, 3);\n\ 4\ngap> OutDegreeOfVertex(D, 4);\n7\n", count := [ 5, 2, 10, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 10, 5 ], mathmode := "Text", name := "Example", next := 168682, root := ~, start := 168391, stop := 168681 ), rec( content := "\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 168699, root := ~, start := 168147, stop := 168698 ) ], count := [ 5, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 168713, root := ~, start := 168028, stop := 168712 ), rec( content := "\n\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighboursOf\ Vertex" ), content := 0, count := [ 5, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 168792, root := ~, start := 168734, stop := 168791 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighborsOfV\ ertex" ), content := 0, count := [ 5, 2, 11, 4 ], mathmode := "Text", name := "Oper", next := 168852, root := ~, start := 168795, stop := 168851 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Returns", next := 168893, root := ~, start := 168855, stop := 168892 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation returns the list ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 168957, root := ~, start := 168947, stop := 168956 ), rec( content := " of vert\ ices of the digraph\n ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 169003, root := ~, start := 168989, stop := 169002 ), rec( content := ".\n A\ vertex ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 169026, root := ~, start := 169018, stop := 169025 ), rec( content := " appears\ in the list ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 169057, root := ~, start := 169047, stop := 169056 ), rec( content := " each ti\ me there exists an\n edge with source ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 169119, root := ~, start := 169106, stop := 169118 ), rec( content := " and ran\ ge ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 169138, root := ~, start := 169130, stop := 169137 ), rec( content := " in ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 169156, root := ~, start := 169142, stop := 169155 ), rec( content := "; in\n \ particular, this means that ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 169203, root := ~, start := 169193, stop := 169202 ), rec( content := " may con\ tain duplicates.", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 11, 6 ], mathmode := "Text", name := "P", next := 169231, root := ~, start := 169227, stop := 169230 ), rec( content := "\n ", count := [ 5, 2, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> OutNeighboursOfV\ ertex(D, 1);\n[ 2, 2, 3 ]\ngap> OutNeighboursOfVertex(D, 3);\n[ 2, 2, 3 ]\n", count := [ 5, 2, 11, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 11, 7 ], mathmode := "Text", name := "Example", next := 169491, root := ~, start := 169236, stop := 169490 ), rec( content := "\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Description", next := 169508, root := ~, start := 168896, stop := 169507 ) ], count := [ 5, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 169522, root := ~, start := 168719, stop := 169521 ), rec( content := "\n\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "InDegreeOfVerte\ x" ), content := 0, count := [ 5, 2, 12, 2 ], mathmode := "Text", name := "Oper", next := 169596, root := ~, start := 169543, stop := 169595 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer.", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Returns", next := 169641, root := ~, start := 169599, stop := 169640 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the in-degree of the vertex ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 169726, root := ~, start := 169713, stop := 169725 ), rec( content := " in the\\ n digraph ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 169760, root := ~, start := 169746, stop := 169759 ), rec( content := ".\n T\ he in-degree of ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 169796, root := ~, start := 169783, stop := 169795 ), rec( content := " is the \ number of edges in ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 169837, root := ~, start := 169823, stop := 169836 ), rec( content := "\n wh\ ose range is ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 169870, root := ~, start := 169857, stop := 169869 ), rec( content := ".\n " , count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> InDegreeOfVertex\ (D, 1);\n5\ngap> InDegreeOfVertex(D, 2);\n9\ngap> InDegreeOfVertex(D, 3);\n0\n\ gap> InDegreeOfVertex(D, 4);\n2\n", count := [ 5, 2, 12, 4 ], name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 12, 4 ], mathmode := "Text", name := "Example", next := 170163, root := ~, start := 169876, stop := 170162 ), rec( content := "\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 170180, root := ~, start := 169644, stop := 170179 ) ], count := [ 5, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 170194, root := ~, start := 169528, stop := 170193 ), rec( content := "\n\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighboursOfV\ ertex" ), content := 0, count := [ 5, 2, 13, 2 ], mathmode := "Text", name := "Oper", next := 170272, root := ~, start := 170215, stop := 170271 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighborsOfVe\ rtex" ), content := 0, count := [ 5, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 170331, root := ~, start := 170275, stop := 170330 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ postitive vertices.", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Returns", next := 170382, root := ~, start := 170334, stop := 170381 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation returns the list ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 170446, root := ~, start := 170436, stop := 170445 ), rec( content := " of vert\ ices of the digraph\n ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 170492, root := ~, start := 170478, stop := 170491 ), rec( content := ".\n A\ vertex ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 170515, root := ~, start := 170507, stop := 170514 ), rec( content := " appears\ in the list ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 170546, root := ~, start := 170536, stop := 170545 ), rec( content := " each ti\ me there exists an\n edge with source ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 170603, root := ~, start := 170595, stop := 170602 ), rec( content := " and ran\ ge ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 170627, root := ~, start := 170614, stop := 170626 ), rec( content := " in ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 170645, root := ~, start := 170631, stop := 170644 ), rec( content := "; in\n \ particular, this means that ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 170692, root := ~, start := 170682, stop := 170691 ), rec( content := " may con\ tain duplicates. ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 13, 6 ], mathmode := "Text", name := "P", next := 170721, root := ~, start := 170717, stop := 170720 ), rec( content := "\n\n \ ", count := [ 5, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> InNeighboursOfVe\ rtex(D, 1);\n[ 2, 4, 4, 4 ]\ngap> InNeighboursOfVertex(D, 2);\n[ 1, 1, 3, 3, 4\ , 4, 4, 4 ]\n", count := [ 5, 2, 13, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 7 ], mathmode := "Text", name := "Example", next := 170998, root := ~, start := 170727, stop := 170997 ), rec( content := "\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 171015, root := ~, start := 170385, stop := 171014 ) ], count := [ 5, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 171029, root := ~, start := 170200, stop := 171028 ), rec( content := "\n\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLoops" ) , content := 0, count := [ 5, 2, 14, 2 ], mathmode := "Text", name := "Attr", next := 171091, root := ~, start := 171050, stop := 171090 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Returns", next := 171132, root := ~, start := 171094, stop := 171131 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "A", next := 171170, root := ~, start := 171156, stop := 171169 ), rec( content := " is a di\ graph, then ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLoops", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "C", next := 171209, root := ~, start := 171190, stop := 171208 ), rec( content := " returns\ the list\n consisting of the ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 171278, root := ~, start := 171249, stop := 171277 ), rec( content := " of ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "A", next := 171296, root := ~, start := 171282, stop := 171295 ), rec( content := "\n at\ which there is a loop. See ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 171360, root := ~, start := 171331, stop := 171359 ), rec( content := ". ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 14, 4 ], mathmode := "Text", name := "P", next := 171366, root := ~, start := 171362, stop := 171365 ), rec( content := "\n\n \ ", count := [ 5, 2, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], []]);\n\ngap> DigraphHasLoops(D);\nfalse\ngap> DigraphLoops(D);\n[ ]\ngap> D := D\ igraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);\n\ngap> DigraphLoops(D);\n[ 3, 4 ]\ngap> D := Digraph(IsMutableDi\ graph, [[1], [1]]);\n\ngap> DigraphL\ oops(D);\n[ 1 ]\n", count := [ 5, 2, 14, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 5 ], mathmode := "Text", name := "Example", next := 171796, root := ~, start := 171372, stop := 171795 ), rec( content := "\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 171813, root := ~, start := 171135, stop := 171812 ) ], count := [ 5, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 171827, root := ~, start := 171035, stop := 171826 ), rec( content := "\n\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphMeetOfVertices" ), content := 0, count := [ 5, 2, 15, 2 ], mathmode := "Text", name := "Oper", next := 171920, root := ~, start := 171848, stop := 171919 ), rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphJoinOfVertices" ), content := 0, count := [ 5, 2, 15, 4 ], mathmode := "Text", name := "Oper", next := 171995, root := ~, start := 171923, stop := 171994 ), rec( attributes := rec( ), content := [ rec( content := "A positiv\ e integer or ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "K", next := 172040, root := ~, start := 172029, stop := 172039 ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Returns", next := 172050, root := ~, start := 171998, stop := 172049 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the first argument is a partial order digraph ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 172161, root := ~, start := 172122, stop := 172160 ), rec( content := " then th\ ese operations return the meet, or\n the join, of the two input vertices. I\ f the meet (or join) is does not\n exist then ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "K", next := 172305, root := ~, start := 172294, stop := 172304 ), rec( content := " is retu\ rned. The meet (or join) is guaranteed to\n exist when the first argument s\ atisfies\n ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 172441, root := ~, start := 172403, stop := 172440 ), rec( content := " (or\n \ ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 172489, root := ~, start := 172451, stop := 172488 ), rec( content := ") - see \ the documentation for these\n properties for the definition of the meet (or\ the join).\n\n ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [1, 2], [1, 3], [1, 2, 3, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 2, 3);\n4\ \ngap> PartialOrderDigraphJoinOfVertices(D, 2, 3);\n1\ngap> PartialOrderDigrap\ hMeetOfVertices(D, 1, 2);\n2\ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\ \n3\ngap> D := Digraph([[1], [2], [1, 2, 3], [1, 2, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 3, 4);\nf\ ail\ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\nfail\ngap> PartialOrder\ DigraphMeetOfVertices(D, 1, 2);\nfail\ngap> PartialOrderDigraphJoinOfVertices(\ D, 1, 2);\nfail\n", count := [ 5, 2, 15, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 6 ], mathmode := "Text", name := "Example", next := 173242, root := ~, start := 172591, stop := 173241 ), rec( content := "\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Description", next := 173259, root := ~, start := 172055, stop := 173258 ) ], count := [ 5, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 173273, root := ~, start := 171833, stop := 173272 ), rec( content := "\n\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DegreeMatrix" ) , content := 0, count := [ 5, 2, 16, 2 ], mathmode := "Text", name := "Attr", next := 173335, root := ~, start := 173294, stop := 173334 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of non-negative integers.", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 173398, root := ~, start := 173338, stop := 173397 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the outdegree matrix ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 173472, root := ~, start := 173462, stop := 173471 ), rec( content := " of the \ digraph\n ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 173506, root := ~, start := 173492, stop := 173505 ), rec( content := ". The va\ lue of the ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 173533, root := ~, start := 173525, stop := 173532 ), rec( content := "th diago\ nal matrix entry is the\n outdegree of the vertex ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 173601, root := ~, start := 173593, stop := 173600 ), rec( content := " in ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 173619, root := ~, start := 173605, stop := 173618 ), rec( content := ". All of\ f-diagonal entries \n are ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 173663, root := ~, start := 173655, stop := 173662 ), rec( content := ".\n I\ f ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 173686, root := ~, start := 173672, stop := 173685 ), rec( content := " has no \ vertices, then the empty list is returned. ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "P", next := 173741, root := ~, start := 173737, stop := 173740 ), rec( content := "\n\n \ See ", count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ ees" ), content := 0, count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "Ref", next := 173775, root := ~, start := 173751, stop := 173774 ), rec( content := " for mor\ e information.\n ", count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);\n\ngap> PrintArray(DegreeMatrix(D));\n[ [ 3, 0, 0, 0 ],\ \n [ 0, 1, 0, 0 ],\n [ 0, 0, 3, 0 ],\n [ 0, 0, 0, 0 ] ]\ngap> D\ := CycleDigraph(5);;\ngap> PrintArray(DegreeMatrix(D));\n[ [ 1, 0, 0, 0, \ 0 ],\n [ 0, 1, 0, 0, 0 ],\n [ 0, 0, 1, 0, 0 ],\n [ 0, 0, 0, \ 1, 0 ],\n [ 0, 0, 0, 0, 1 ] ]\ngap> DegreeMatrix(EmptyDigraph(0));\n[ \ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(DegreeMatrix(D));\n[ [ 1, 0, 0 ],\n [ 0, 1, \ 0 ],\n [ 0, 0, 1 ] ]\n", count := [ 5, 2, 16, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 5 ], mathmode := "Text", name := "Example", next := 174462, root := ~, start := 173802, stop := 174461 ), rec( content := "\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 174479, root := ~, start := 173401, stop := 174478 ) ], count := [ 5, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 174493, root := ~, start := 173279, stop := 174492 ), rec( content := "\n\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LaplacianMatrix\ " ), content := 0, count := [ 5, 2, 17, 2 ], mathmode := "Text", name := "Attr", next := 174558, root := ~, start := 174514, stop := 174557 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of integers.", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 174608, root := ~, start := 174561, stop := 174607 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the outdegree Laplacian matrix ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "C", next := 174692, root := ~, start := 174682, stop := 174691 ), rec( content := " of the\\ n digraph ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "A", next := 174726, root := ~, start := 174712, stop := 174725 ), rec( content := ". The ou\ tdegree Laplacian matrix is defined as\n ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ egreeMatrix(digraph) - AdjacencyMatrix(digraph)", count := [ 5, 2, 17, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "C", next := 174832, root := ~, start := 174777, stop := 174831 ), rec( content := ". If\n \ ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "A", next := 174856, root := ~, start := 174842, stop := 174855 ), rec( content := " has no \ vertices, then the empty list is returned. ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "P", next := 174911, root := ~, start := 174907, stop := 174910 ), rec( content := "\n\n \ See ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DegreeM\ atrix" ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "Ref", next := 174947, root := ~, start := 174921, stop := 174946 ), rec( content := " and ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "Ref", next := 174981, root := ~, start := 174952, stop := 174980 ), rec( content := " for mor\ e\n information.\n ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);\n\ngap> PrintArray(LaplacianMatrix(gr));\n[ [ 2, -1, -\ 1, 0 ],\n [ 0, 1, 0, -1 ],\n [ -1, 0, 2, -1 ],\n [ 0, 0,\ 0, 0 ] ]\ngap> LaplacianMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDi\ graph(IsMutableDigraph, 3);\n\ngap> \ Display(LaplacianMatrix(D));\n[ [ 1, -1, 0 ],\n [ 0, 1, -1 ],\n [ \ -1, 0, 1 ] ]\n", count := [ 5, 2, 17, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 5 ], mathmode := "Text", name := "Example", next := 175515, root := ~, start := 175012, stop := 175514 ), rec( content := "\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 175532, root := ~, start := 174611, stop := 175531 ) ], count := [ 5, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 175546, root := ~, start := 174499, stop := 175545 ), rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 175560, root := ~, start := 155328, stop := 175559 ) , rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Reachability and \ connectivity", count := [ 5, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 175621, root := ~, start := 175573, stop := 175620 ), rec( content := "\n ", count := [ 5, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDiameter\ " ), content := 0, count := [ 5, 3, 1, 2 ], mathmode := "Text", name := "Attr", next := 175685, root := ~, start := 175641, stop := 175684 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "C", next := 175722, root := ~, start := 175711, stop := 175721 ), rec( content := ".", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 175733, root := ~, start := 175688, stop := 175732 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the diameter of the digraph ", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "A", next := 175818, root := ~, start := 175804, stop := 175817 ), rec( content := ".\n " , count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "P", next := 175828, root := ~, start := 175824, stop := 175827 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "A", next := 175861, root := ~, start := 175847, stop := 175860 ), rec( content := " is stro\ ngly connected and has at least 1\n vertex, then the ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ iameter", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "E", next := 175939, root := ~, start := 175924, stop := 175938 ), rec( content := " is the \ maximum shortest distance between\n any pair of distinct vertices. Otherwis\ e then the diameter of\n ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "A", next := 176065, root := ~, start := 176051, stop := 176064 ), rec( content := " is unde\ fined, and this function returns the value\n ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "C", next := 176131, root := ~, start := 176120, stop := 176130 ), rec( content := ". ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "P", next := 176137, root := ~, start := 176133, stop := 176136 ), rec( content := "\n\n \ See ", count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ShortestDistances" ), content := 0, count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "Ref", next := 176185, root := ~, start := 176147, stop := 176184 ), rec( content := ". ", count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 6 ], mathmode := "Text", name := "P", next := 176191, root := ~, start := 176187, stop := 176190 ), rec( content := "\n ", count := [ 5, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [4, 5], [5], [1, 2, 3, 4, 5]]);\n\ngap> DigraphDiameter(D);\n3\ngap> D := ChainDi\ graph(2);\n\ngap> DigraphDiameter(D);\ \nfail\ngap> IsStronglyConnectedDigraph(D);\nfalse\ngap> D := GeneralisedPeter\ senGraph(IsMutableDigraph, 6, 2);\n\ngap> DigraphDiameter(D);\n4\n", count := [ 5, 3, 1, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 7 ], mathmode := "Text", name := "Example", next := 176635, root := ~, start := 176196, stop := 176634 ), rec( content := "\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 176652, root := ~, start := 175736, stop := 176651 ) ], count := [ 5, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 176666, root := ~, start := 175626, stop := 176665 ), rec( content := "\n\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Label := "for a digraph \ and two vertices", Name := "DigraphShortestDistance" ), content := 0, count := [ 5, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 176792, root := ~, start := 176687, stop := 176791 ), rec( attributes := rec( Arg := "digraph, list", Label := "for a digraph \ and a list", Name := "DigraphShortestDistance" ), content := 0, count := [ 5, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 176894, root := ~, start := 176795, stop := 176893 ), rec( attributes := rec( Arg := "digraph, list1, \ list2", Label := "for a digraph, a list, and a list", Name := "DigraphShortest\ Distance" ), content := 0, count := [ 5, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 177013, root := ~, start := 176897, stop := 177012 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "K", next := 177050, root := ~, start := 177039, stop := 177049 ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Returns", next := 177060, root := ~, start := 177016, stop := 177059 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ there is a directed path in the digraph ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 177138, root := ~, start := 177124, stop := 177137 ), rec( content := " between\ vertex\n ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 177166, root := ~, start := 177158, stop := 177165 ), rec( content := " and ver\ tex ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 177186, root := ~, start := 177178, stop := 177185 ), rec( content := ", then t\ his operation returns the length of the\n shortest such directed path. If \ no such directed path exists, then this\n operation returns ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "K", next := 177344, root := ~, start := 177333, stop := 177343 ), rec( content := ". See Se\ ction ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Ref", next := 177408, root := ~, start := 177358, stop := 177407 ), rec( content := " for the\ definition of a directed path. ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "P", next := 177452, root := ~, start := 177448, stop := 177451 ), rec( content := "\n\n \ If the second form is used, then ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 177502, root := ~, start := 177491, stop := 177501 ), rec( content := " should \ be a list of length two,\n containing two positive integers which correspon\ d to the vertices ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 177613, root := ~, start := 177605, stop := 177612 ), rec( content := "\n an\ d ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 177630, root := ~, start := 177622, stop := 177629 ), rec( content := ". ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 9 ], mathmode := "Text", name := "P", next := 177636, root := ~, start := 177632, stop := 177635 ), rec( content := "\n\n \ Note that as usual, a vertex is considered to be at distance 0 from\n itsel\ f .", count := [ 5, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "P", next := 177726, root := ~, start := 177722, stop := 177725 ), rec( content := "\n\n \ If the third form is used, then ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 177776, root := ~, start := 177764, stop := 177775 ), rec( content := " and ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 177793, root := ~, start := 177781, stop := 177792 ), rec( content := " are bot\ h lists\n of vertices. The shortest directed path between ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 177873, root := ~, start := 177861, stop := 177872 ), rec( content := " and\n \ ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 177894, root := ~, start := 177882, stop := 177893 ), rec( content := " is then\ the length of the shortest directed path which starts\n with a vertex in " , count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 177990, root := ~, start := 177978, stop := 177989 ), rec( content := " and ter\ minates at a vertex in ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 178033, root := ~, start := 178021, stop := 178032 ), rec( content := ", if\n \ such directed path exists. If ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 178084, root := ~, start := 178072, stop := 178083 ), rec( content := " and ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 178101, root := ~, start := 178089, stop := 178100 ), rec( content := " have no\ n-empty\n intersection, the operation returns ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "C", next := 178165, root := ~, start := 178157, stop := 178164 ), rec( content := ".", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 11 ], mathmode := "Text", name := "P", next := 178170, root := ~, start := 178166, stop := 178169 ), rec( content := "\n ", count := [ 5, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [1, 4], [1, 3], [5]]);\n\ngap> DigraphShortestDistance(D, 1, 3);\n2\ngap> DigraphS\ hortestDistance(D, [3, 3]);\n0\ngap> DigraphShortestDistance(D, 5, 2);\nfail\n\ gap> DigraphShortestDistance(D, [1, 2], [4, 5]);\n2\ngap> DigraphShortestDista\ nce(D, [1, 3], [3, 5]);\n0\n", count := [ 5, 3, 2, 12 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 12 ], mathmode := "Text", name := "Example", next := 178534, root := ~, start := 178175, stop := 178533 ), rec( content := "\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Description", next := 178551, root := ~, start := 177063, stop := 178550 ) ], count := [ 5, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 178565, root := ~, start := 176672, stop := 178564 ), rec( content := "\n\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphShortest\ Distances" ), content := 0, count := [ 5, 3, 3, 2 ], mathmode := "Text", name := "Attr", next := 178639, root := ~, start := 178586, stop := 178638 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix.", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 178677, root := ~, start := 178642, stop := 178676 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "A", next := 178715, root := ~, start := 178701, stop := 178714 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 178742, root := ~, start := 178734, stop := 178741 ), rec( content := " vertice\ s, then this\n function returns an ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 178804, root := ~, start := 178787, stop := 178803 ), rec( content := " matrix \ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "C", next := 178822, root := ~, start := 178812, stop := 178821 ), rec( content := ", where \ each entry is\n either a non-negative integer, or ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "K", next := 178893, root := ~, start := 178882, stop := 178892 ), rec( content := ". If ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ = 0", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 178911, root := ~, start := 178899, stop := 178910 ), rec( content := ", then a\ n\n empty list is returned. ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "P", next := 178953, root := ~, start := 178949, stop := 178952 ), rec( content := "\n\n \ If there is a directed path from vertex ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179007, root := ~, start := 178999, stop := 179006 ), rec( content := " to vert\ ex ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179026, root := ~, start := 179018, stop := 179025 ), rec( content := ", then\n\ the value of ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179066, root := ~, start := 179050, stop := 179065 ), rec( content := " is the \ length of the shortest such directed\n path. If no such directed path exist\ s, then the value of ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179188, root := ~, start := 179172, stop := 179187 ), rec( content := " is\n \ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179207, root := ~, start := 179196, stop := 179206 ), rec( content := ". We us\ e the convention that the distance from every vertex to\n itself is ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179293, root := ~, start := 179285, stop := 179292 ), rec( content := ", i.e. " , count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][i] = 0", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179320, root := ~, start := 179300, stop := 179319 ), rec( content := " for all\ vertices ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 179346, root := ~, start := 179338, stop := 179345 ), rec( content := ".\n " , count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "P", next := 179356, root := ~, start := 179352, stop := 179355 ), rec( content := "\n\n \ The method used in this function is a version of the Floyd-Warshall\n algor\ ithm, and has complexity ", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "M", next := 179477, root := ~, start := 179464, stop := 179476 ), rec( content := ".\n\n \ ", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [1, 2], [4]]);\n\ngap> mat := DigraphShortestDistances(D);;\ngap> PrintArray(ma\ t);\n[ [ 0, 1, 2, fail ],\n [ 2, 0, 1, fail ],\n [\ 1, 1, 0, fail ],\n [ fail, fail, fail, 0 ] ]\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> DigraphShortestDistances(D);\n[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]\ \n", count := [ 5, 3, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 6 ], mathmode := "Text", name := "Example", next := 179973, root := ~, start := 179484, stop := 179972 ), rec( content := "\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 179990, root := ~, start := 178680, stop := 179989 ) ], count := [ 5, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 180004, root := ~, start := 178571, stop := 180003 ), rec( content := "\n\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphLongestD\ istanceFromVertex" ), content := 0, count := [ 5, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 180089, root := ~, start := 180025, stop := 180088 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "K", next := 180131, root := ~, start := 180116, stop := 180130 ), rec( content := ".", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 180142, root := ~, start := 180092, stop := 180141 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 180180, root := ~, start := 180166, stop := 180179 ), rec( content := " is a di\ graph and ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 180206, root := ~, start := 180198, stop := 180205 ), rec( content := " is a ve\ rtex in ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 180236, root := ~, start := 180222, stop := 180235 ), rec( content := ",\n t\ hen this operation returns the length of the longest directed walk in\n ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 180331, root := ~, start := 180317, stop := 180330 ), rec( content := " which b\ egins at vertex ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 180363, root := ~, start := 180355, stop := 180362 ), rec( content := ". See S\ ection ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 180428, root := ~, start := 180378, stop := 180427 ), rec( content := " for the\ definitions of directed\n walk, directed cycle, and loop.\n ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 4, 4 ], mathmode := "Text", name := "P", next := 180505, root := ~, start := 180501, stop := 180504 ), rec( content := "\n\n \ ", count := [ 5, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n If there exists a directed w\ alk starting at vertex ", count := [ 5, 3, 4, 6 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "v", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "A", next := 1805\ 98, root := ~, start := 180590, stop := 180597 ), rec( content := "\n w\ hich traverses a loop or a directed cycle,\n then we consider there to \ be a walk of infinite length from ", count := [ 5, 3, 4, 6 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "v", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "A", next\ := 180727, root := ~, start := 180719, stop := 180726 ), rec( content := "\n \ (realised by repeatedly traversing the loop/directed cycle),\n a\ nd so the result is ", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "infinity", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "K", next := \ 180841, root := ~, start := 180826, stop := 180840 ), rec( content := ".\n \ To disallow walks using loops, try using\n ", count := [ 5, 3, 4, 6\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ Oper := "DigraphRemoveLoops" ), content := 0, count := [ 5, 3, 4, 6 ], mathmo\ de := "Text", name := "Ref", next := 180932, root := ~, start := 180900, stop \ := 180931 ), rec( content := ":", count := [ 5, 3, 4, 6 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 5, 3, 4, 7 ], mathmode := "Text", name := "P", next := 180937, root := ~, star\ t := 180933, stop := 180936 ), rec( content := "\n\n ", count := [ 5, 3\ , 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "DigraphLongestDistanceFromVertex(Digr\ aphRemoveLoops(", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digr\ aph", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "A", next := 181016\ , root := ~, start := 181002, stop := 181015 ), rec( content := ",", count := \ [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "v", count := [ 5, 3, 4, 7 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], ma\ thmode := "Text", name := "A", next := 181025, root := ~, start := 181017, sto\ p := 181024 ), rec( content := "))", count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "C", next := 181031, root := ~, start := 180947, stop := 181030 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 3, 4, 6 ], name := "Item", next := 181046, root := ~, start := 180\ 524, stop := 181045 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n Otherwise, if all directed walks\n starting at vertex ", coun\ t := [ 5, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v", count := [ 5, 3, 4, 9 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 9 ], math\ mode := "Text", name := "A", next := 181137, root := ~, start := 181129, stop \ := 181136 ), rec( content := " have finite length,\n then the length of\ the longest such walk is returned.\n ", count := [ 5, 3, 4, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 9 ], name := "Ite\ m", next := 181233, root := ~, start := 181054, stop := 181232 ) ], count := [ 5, 3, 4, 5 ], mathmode := "Text", name := "List", next := 181245, root := ~, start := 180511, stop := 181244 ), rec( content := "\n\n \ Note that the result is ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "C", next := 181283, root := ~, start := 181275, stop := 181282 ), rec( content := " if and \ only if ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "A", next := 181307, root := ~, start := 181299, stop := 181306 ), rec( content := " is a si\ nk of\n ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "A", next := 181339, root := ~, start := 181325, stop := 181338 ), rec( content := ". See " , count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sinks" ), content := 0, count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "Ref", next := 181372, root := ~, start := 181346, stop := 181371 ), rec( content := ".\n\n", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3, 4], [], [5], [], [6]]);\n\ngap> DigraphLongestDistanceFromVertex(D, 1);\n3\ngap> Di\ graphLongestDistanceFromVertex(D, 3);\n0\ngap> 3 in DigraphSinks(D);\ntrue\nga\ p> DigraphLongestDistanceFromVertex(D, 6);\ninfinity\n", count := [ 5, 3, 4, 12 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 4, 12 ], mathmode := "Text", name := "Example", next := 181684, root := ~, start := 181375, stop := 181683 ), rec( content := "\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 181701, root := ~, start := 180145, stop := 181700 ) ], count := [ 5, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 181715, root := ~, start := 180010, stop := 181714 ), rec( content := "\n\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex,\ distance", Label := "for a digraph, a pos int, and an int", Name := "DigraphDistance\ Set" ), content := 0, count := [ 5, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 181846, root := ~, start := 181736, stop := 181845 ), rec( attributes := rec( Arg := "digraph, vertex,\ distances", Label := "for a digraph, a pos int, and a list", Name := "DigraphDistance\ Set" ), content := 0, count := [ 5, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 181960, root := ~, start := 181849, stop := 181959 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Returns", next := 182000, root := ~, start := 181963, stop := 181999 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the list of all vertices in digraph ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 182094, root := ~, start := 182080, stop := 182093 ), rec( content := "\n su\ ch that the shortest distance to a vertex ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 182156, root := ~, start := 182143, stop := 182155 ), rec( content := " is\n \ ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ istance", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 182179, root := ~, start := 182164, stop := 182178 ), rec( content := " or is i\ n the list ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istances", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 182214, root := ~, start := 182198, stop := 182213 ), rec( content := ". ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "P", next := 182220, root := ~, start := 182216, stop := 182219 ), rec( content := "\n\n \ ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 182240, root := ~, start := 182226, stop := 182239 ), rec( content := " should \ be a digraph, ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 182275, root := ~, start := 182262, stop := 182274 ), rec( content := " should \ be a positive\n integer, ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istance", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 182325, root := ~, start := 182310, stop := 182324 ), rec( content := " should \ be a non-negative integer, and\n ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istances", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 182384, root := ~, start := 182368, stop := 182383 ), rec( content := " should \ be a list of non-negative integers.\n\n ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [1, 4], [1, 3]]);\n\ngap> DigraphDistanceSet(D, 2, [1, 2]);\n[ 3, 1, 4 ]\ngap> Dig\ raphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ngap> DigraphDistanceSet(D, 2, 0);\n[ 2 ]\ \n", count := [ 5, 3, 5, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 7 ], mathmode := "Text", name := "Example", next := 182691, root := ~, start := 182433, stop := 182690 ), rec( content := "\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 182708, root := ~, start := 182003, stop := 182707 ) ], count := [ 5, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 182722, root := ~, start := 181721, stop := 182721 ), rec( content := "\n\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphGirth" ) , content := 0, count := [ 5, 3, 6, 2 ], mathmode := "Text", name := "Attr", next := 182784, root := ~, start := 182743, stop := 182783 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "K", next := 182826, root := ~, start := 182811, stop := 182825 ), rec( content := ".", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 182837, root := ~, start := 182787, stop := 182836 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s attribute returns the ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "E", next := 182897, root := ~, start := 182885, stop := 182896 ), rec( content := " of the \ digraph ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "A", next := 182927, root := ~, start := 182913, stop := 182926 ), rec( content := ".\n T\ he ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "E", next := 182949, root := ~, start := 182937, stop := 182948 ), rec( content := " of a di\ graph is the length of its shortest simple circuit.\n See Section ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 183069, root := ~, start := 183025, stop := 183068 ), rec( content := " for the\ definitions\n of simple circuit, directed cycle, and loop.\n ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "P", next := 183147, root := ~, start := 183143, stop := 183146 ), rec( content := "\n\n \ If ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "A", next := 183170, root := ~, start := 183156, stop := 183169 ), rec( content := " has no \ directed cycles, then this function will return\n ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "K", next := 183245, root := ~, start := 183230, stop := 183244 ), rec( content := ". If ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "A", next := 183265, root := ~, start := 183251, stop := 183264 ), rec( content := " contain\ s a loop, then this function will\n return ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "C", next := 183326, root := ~, start := 183318, stop := 183325 ), rec( content := ".\n " , count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "P", next := 183336, root := ~, start := 183332, stop := 183335 ), rec( content := "\n\n \ In the worst case, the method used in this function is a version of the\n F\ loyd-Warshall algorithm, and has complexity ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 5, 3, 6, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := \ "Text", name := "A", next := 183476, root := ~, start := 183468, stop := 18347\ 5 ), rec( content := " ^ 3)", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "C", next := 183485, root := ~, start := 183463, stop := 183484 ), rec( content := ", where\\ n ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "A", next := 183505, root := ~, start := 183497, stop := 183504 ), rec( content := " is the \ number of vertices in ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "A", next := 183549, root := ~, start := 183535, stop := 183548 ), rec( content := ". If th\ e digraph has\n known automorphisms [see ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "Ref", next := 183626, root := ~, start := 183600, stop := 183625 ), rec( content := "], then \ the performance\n is likely to be better.\n ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "P", next := 183686, root := ~, start := 183682, stop := 183685 ), rec( content := "\n\n \ For symmetric digraphs, see also ", count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ UndirectedGirth" ), content := 0, count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "Ref", next := 183761, root := ~, start := 183725, stop := 183760 ), rec( content := ". ", count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 7 ], mathmode := "Text", name := "P", next := 183767, root := ~, start := 183763, stop := 183766 ), rec( content := "\n ", count := [ 5, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [1]]);\n\ \ngap> DigraphGirth(D);\n1\ngap> D := Digraph([[2, 3], [3], [4], []]);\n\ngap> DigraphGirth(D);\ninfinity\ngap> \ D := Digraph([[2, 3], [3], [4], [1]]);\n\ngap> DigraphGirth(D);\n3\ngap> D := GeneralisedPetersenGraph(IsMutable\ Digraph, 6, 2);\n\ngap> DigraphGir\ th(D);\n2\n", count := [ 5, 3, 6, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 8 ], mathmode := "Text", name := "Example", next := 184265, root := ~, start := 183772, stop := 184264 ), rec( content := "\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 184282, root := ~, start := 182840, stop := 184281 ) ], count := [ 5, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 184296, root := ~, start := 182728, stop := 184295 ), rec( content := "\n\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOddGirth\ " ), content := 0, count := [ 5, 3, 7, 2 ], mathmode := "Text", name := "Attr", next := 184361, root := ~, start := 184317, stop := 184360 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "K", next := 184403, root := ~, start := 184388, stop := 184402 ), rec( content := ".", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 184414, root := ~, start := 184364, stop := 184413 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s attribute returns the ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ dd girth", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "E", next := 184478, root := ~, start := 184462, stop := 184477 ), rec( content := " of the \ digraph ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "A", next := 184508, root := ~, start := 184494, stop := 184507 ), rec( content := ".\n T\ he ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ dd girth", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "E", next := 184534, root := ~, start := 184518, stop := 184533 ), rec( content := " of a di\ graph is the length of its shortest simple\n circuit of odd length. See Se\ ction ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Ref", next := 184673, root := ~, start := 184625, stop := 184672 ), rec( content := " for the\ definitions of simple circuit, directed\n cycle, and loop.\n ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "P", next := 184751, root := ~, start := 184747, stop := 184750 ), rec( content := "\n\n \ If ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 184774, root := ~, start := 184760, stop := 184773 ), rec( content := " has no \ directed cycles of odd length, then this function\n will return ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "K", next := 184863, root := ~, start := 184848, stop := 184862 ), rec( content := ", even i\ f ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 184887, root := ~, start := 184873, stop := 184886 ), rec( content := " has a d\ irected cycle\n of even length. If ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 184947, root := ~, start := 184933, stop := 184946 ), rec( content := " contain\ s a loop, then this function\n will return ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "C", next := 185008, root := ~, start := 185000, stop := 185007 ), rec( content := ".\n " , count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "P", next := 185018, root := ~, start := 185014, stop := 185017 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Girth" ), content := 0, count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 185059, root := ~, start := 185033, stop := 185058 ), rec( content := ". ", count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 6 ], mathmode := "Text", name := "P", next := 185065, root := ~, start := 185061, stop := 185064 ), rec( content := "\n ", count := [ 5, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3, 1], [1]]);\n\ngap> DigraphOddGirth(D);\n3\ngap> D := CycleDigraph(4);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := D\ igraph([[2], [3], [], [3], [4]]);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := CycleDigraph(IsMutableDigraph\ , 4);\n\ngap> DigraphDisjointUnion(D\ , CycleDigraph(5));\n\ngap> DigraphO\ ddGirth(D);\n5\n", count := [ 5, 3, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 7 ], mathmode := "Text", name := "Example", next := 185645, root := ~, start := 185070, stop := 185644 ), rec( content := "\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 185662, root := ~, start := 184417, stop := 185661 ) ], count := [ 5, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 185676, root := ~, start := 184302, stop := 185675 ), rec( content := "\n\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphUndirect\ edGirth" ), content := 0, count := [ 5, 3, 8, 2 ], mathmode := "Text", name := "Attr", next := 185748, root := ~, start := 185697, stop := 185747 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "K", next := 185789, root := ~, start := 185774, stop := 185788 ), rec( content := ".", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 185800, root := ~, start := 185751, stop := 185799 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "A", next := 185838, root := ~, start := 185824, stop := 185837 ), rec( content := " is a sy\ mmetric digraph, then this function returns the\n girth of ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "A", next := 185921, root := ~, start := 185907, stop := 185920 ), rec( content := " when tr\ eated as an undirected graph (i.e. each pair\n of edges ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 186000, root := ~, start := 185987, stop := 185999 ), rec( content := " and ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, i]", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 186018, root := ~, start := 186005, stop := 186017 ), rec( content := " is trea\ ted as a single edge between\n ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 186067, root := ~, start := 186059, stop := 186066 ), rec( content := " and ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 186080, root := ~, start := 186072, stop := 186079 ), rec( content := "). ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "P", next := 186087, root := ~, start := 186083, stop := 186086 ), rec( content := "\n\n \ The ", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "E", next := 186109, root := ~, start := 186097, stop := 186108 ), rec( content := " of an u\ ndirected graph is the length of its shortest simple\n cycle, i.e. the shor\ test non-trivial path starting and ending at the same\n vertex and passing \ through no vertex or edge more than once. ", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "P", next := 186317, root := ~, start := 186313, stop := 186316 ), rec( content := "\n\n \ If ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "A", next := 186340, root := ~, start := 186326, stop := 186339 ), rec( content := " has no \ cycles, then this function will return\n ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "K", next := 186406, root := ~, start := 186391, stop := 186405 ), rec( content := ". ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 6 ], mathmode := "Text", name := "P", next := 186412, root := ~, start := 186408, stop := 186411 ), rec( content := "\n ", count := [ 5, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);\n\ngap> DigraphUndirectedGirth(D);\n4\ngap> D := Digraph([\ [2], [1, 3], [2]]);\n\ngap> Digrap\ hUndirectedGirth(D);\ninfinity\ngap> D := Digraph([[1], [], [4], [3]]);\n\ngap> DigraphUndirectedGirth(D);\n1\ng\ ap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> DigraphUndirectedGirth(D);\n5\n", count := [ 5, 3, 8, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 8, 7 ], mathmode := "Text", name := "Example", next := 186964, root := ~, start := 186417, stop := 186963 ), rec( content := "\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 186981, root := ~, start := 185803, stop := 186980 ) ], count := [ 5, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 186995, root := ~, start := 185682, stop := 186994 ), rec( content := "\n\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphConnecte\ dComponents" ), content := 0, count := [ 5, 3, 9, 2 ], mathmode := "Text", name := "Attr", next := 187071, root := ~, start := 187016, stop := 187070 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrConnec\ tedComponents" ), content := 0, count := [ 5, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 187131, root := ~, start := 187074, stop := 187130 ), rec( attributes := rec( ), content := [ rec( content := "A record.\ ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 187162, root := ~, start := 187134, stop := 187161 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the record ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ cc", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "C", next := 187226, root := ~, start := 187216, stop := 187225 ), rec( content := " corresp\ onding to the weakly\n connected components of the digraph ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 187309, root := ~, start := 187295, stop := 187308 ), rec( content := ". Two v\ ertices of\n ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 187346, root := ~, start := 187332, stop := 187345 ), rec( content := " are in \ the same weakly connected component whenever they are\n equal, or there exi\ sts a directed path (ignoring the orientation of edges)\n between them. Mo\ re formally, two vertices are in the same weakly connected\n component of " , count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 187598, root := ~, start := 187584, stop := 187597 ), rec( content := " if and \ only if they are in the same strongly\n connected component (see ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ StronglyConnectedComponents" ), content := 0, count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 187721, root := ~, start := 187673, stop := 187720 ), rec( content := ")\n o\ f the ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 187771, root := ~, start := 187734, stop := 187770 ), rec( content := " of ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 187789, root := ~, start := 187775, stop := 187788 ), rec( content := ". ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "P", next := 187796, root := ~, start := 187792, stop := 187795 ), rec( content := "\n\n \ The set of weakly connected components is a partition of\n the vertex set o\ f ", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "A", next := 187895, root := ~, start := 187881, stop := 187894 ), rec( content := ". ", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "P", next := 187901, root := ~, start := 187897, stop := 187900 ), rec( content := "\n\n \ The record ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ cc", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 187928, root := ~, start := 187918, stop := 187927 ), rec( content := " has 2 c\ omponents: ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 187959, root := ~, start := 187947, stop := 187958 ), rec( content := " and ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 187973, root := ~, start := 187964, stop := 187972 ), rec( content := ".\n T\ he component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 188005, root := ~, start := 187993, stop := 188004 ), rec( content := " is a li\ st of the weakly connected components\n of ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "A", next := 188073, root := ~, start := 188059, stop := 188072 ), rec( content := " (each o\ f which is a list of vertices).\n The component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 188140, root := ~, start := 188131, stop := 188139 ), rec( content := " is a li\ st such that the vertex ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 188180, root := ~, start := 188172, stop := 188179 ), rec( content := " is an\n\ element of the weakly connected component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps[id[i]]", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 188252, root := ~, start := 188233, stop := 188251 ), rec( content := ". ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "P", next := 188258, root := ~, start := 188254, stop := 188257 ), rec( content := "\n\n \ The method used in this function has complexity ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 188326, root := ~, start := 188313, stop := 188325 ), rec( content := ", where\\ n ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 188346, root := ~, start := 188338, stop := 188345 ), rec( content := " is the \ number of edges and\n ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 188386, root := ~, start := 188378, stop := 188385 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "P", next := 188433, root := ~, start := 188429, stop := 188432 ), rec( content := "\n\n \ ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrConnectedComponents(", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "A", next := 1884\ 85, root := ~, start := 188471, stop := 188484 ), rec( content := ")", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 188490, root := ~, start := 188439, stop := 188489 ), rec( content := " is simp\ ly a shortcut\n for ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(DigraphConnectedComponents(", count := [ 5, 3, 9, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "digraph", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "A", next := 1885\ 71, root := ~, start := 188557, stop := 188570 ), rec( content := ").comps)", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 188583, root := ~, start := 188520, stop := 188582 ), rec( content := ",\n a\ nd is no more efficient.\n\n ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ [ 1, 2, 3 ] ], \ id := [ 1, 1, 1 ] )\ngap> gr := Digraph([[1], [1, 2], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps \ := [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )\ngap> DigraphNrConnectedComponents\ (gr);\n2\ngap> gr := EmptyDigraph(0);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ ], id := [ ] )\nga\ p> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] \ ], id := [ 1, 1, 2, 2, 2 ] )\n", count := [ 5, 3, 9, 10 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 10 ], mathmode := "Text", name := "Example", next := 189517, root := ~, start := 188620, stop := 189516 ), rec( content := "\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 189534, root := ~, start := 187165, stop := 189533 ) ], count := [ 5, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 189548, root := ~, start := 187001, stop := 189547 ), rec( content := "\n\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphConnecte\ dComponent" ), content := 0, count := [ 5, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 189631, root := ~, start := 189569, stop := 189630 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 189672, root := ~, start := 189634, stop := 189671 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 189709, root := ~, start := 189696, stop := 189708 ), rec( content := " is a ve\ rtex in the digraph ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 189751, root := ~, start := 189737, stop := 189750 ), rec( content := ", then\n\ this operation returns the connected component of ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 189825, root := ~, start := 189812, stop := 189824 ), rec( content := "\n in\ ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 189847, root := ~, start := 189833, stop := 189846 ), rec( content := ".\n S\ ee ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Ref", next := 189897, root := ~, start := 189857, stop := 189896 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [2], [1, 2], [4]]);\n\ngap> DigraphConnectedComponent(D, 3);\n[ 1, 2, 3 ]\ngap> Digraph\ ConnectedComponent(D, 2);\n[ 1, 2, 3 ]\ngap> DigraphConnectedComponent(D, 4);\ \n[ 4 ]\n", count := [ 5, 3, 10, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 4 ], mathmode := "Text", name := "Example", next := 190189, root := ~, start := 189924, stop := 190188 ), rec( content := "\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 190206, root := ~, start := 189675, stop := 190205 ) ], count := [ 5, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 190220, root := ~, start := 189554, stop := 190219 ), rec( content := "\n\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphStrongly\ ConnectedComponents" ), content := 0, count := [ 5, 3, 11, 2 ], mathmode := "Text", name := "Attr", next := 190304, root := ~, start := 190241, stop := 190303 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrStrong\ lyConnectedComponents" ), content := 0, count := [ 5, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 190372, root := ~, start := 190307, stop := 190371 ), rec( attributes := rec( ), content := [ rec( content := "A record.\ ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 190403, root := ~, start := 190375, stop := 190402 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the record ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ cc", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "C", next := 190467, root := ~, start := 190457, stop := 190466 ), rec( content := " corresp\ onding to the strongly\n connected components of the digraph ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 190552, root := ~, start := 190538, stop := 190551 ), rec( content := ". Two v\ ertices of\n ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 190589, root := ~, start := 190575, stop := 190588 ), rec( content := " are in \ the same strongly connected component whenever they\n are equal, or there i\ s a directed path from each vertex to the other. The\n set of strongly con\ nected components is a partition of the vertex set of\n ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 190823, root := ~, start := 190809, stop := 190822 ), rec( content := ".\n " , count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "P", next := 190833, root := ~, start := 190829, stop := 190832 ), rec( content := "\n\n \ The record ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ cc", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 190860, root := ~, start := 190850, stop := 190859 ), rec( content := " has 2 c\ omponents: ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 190891, root := ~, start := 190879, stop := 190890 ), rec( content := " and ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 190905, root := ~, start := 190896, stop := 190904 ), rec( content := ".\n T\ he component ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 190937, root := ~, start := 190925, stop := 190936 ), rec( content := " is a li\ st of the strongly connected components\n of ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "A", next := 191006, root := ~, start := 190992, stop := 191005 ), rec( content := " (each o\ f which is a list of vertices).\n The component ", count := [ 5, 3, 11, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 191073, root := ~, start := 191064, stop := 191072 ), rec( content := " is a li\ st such that the vertex ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 191113, root := ~, start := 191105, stop := 191112 ), rec( content := " is an\n\ element of the strongly connected component ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps[id[i]]", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 191187, root := ~, start := 191168, stop := 191186 ), rec( content := ". ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "P", next := 191193, root := ~, start := 191189, stop := 191192 ), rec( content := "\n\n \ The method used in this function is a non-recursive version of Gabow's\n Al\ gorithm ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "Cite", next := 191303, root := ~, start := 191284, stop := 191302 ), rec( content := " and has\ complexity ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 191336, root := ~, start := 191323, stop := 191335 ), rec( content := " where\n\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 191355, root := ~, start := 191347, stop := 191354 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 191428, root := ~, start := 191420, stop := 191427 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "P", next := 191475, root := ~, start := 191471, stop := 191474 ), rec( content := "\n\n \ ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrStronglyConnectedComponents(", count := [ 5, 3, 11, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "A", next := 1915\ 35, root := ~, start := 191521, stop := 191534 ), rec( content := ")", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 191540, root := ~, start := 191481, stop := 191539 ), rec( content := " is simp\ ly a\n shortcut for\n ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(DigraphStronglyConnectedComponents(", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "A", next := 1916\ 33, root := ~, start := 191619, stop := 191632 ), rec( content := ").comps)", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 191645, root := ~, start := 191574, stop := 191644 ), rec( content := ",\n a\ nd is no more efficient.\n\n ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphStronglyConnectedComponents(gr);\nrec( comps := [ [ 3 ], \ [ 1, 2 ] ], id := [ 2, 2, 1 ] )\ngap> DigraphNrStronglyConnectedComponents(gr)\ ;\n2\ngap> D := DigraphDisjointUnion(CycleDigraph(4), CycleDigraph(5));\n\ngap> DigraphStronglyConnectedComponen\ ts(D);\nrec( comps := [ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8, 9 ] ], \n id := [ 1, 1,\ 1, 1, 2, 2, 2, 2, 2 ] )\ngap> DigraphNrStronglyConnectedComponents(D);\n2\nga\ p> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphStronglyConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3,\ 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n", count := [ 5, 3, 11, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 11, 9 ], mathmode := "Text", name := "Example", next := 192588, root := ~, start := 191682, stop := 192587 ), rec( content := "\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 192605, root := ~, start := 190406, stop := 192604 ) ], count := [ 5, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 192619, root := ~, start := 190226, stop := 192618 ), rec( content := "\n\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphStrongly\ ConnectedComponent" ), content := 0, count := [ 5, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 192710, root := ~, start := 192640, stop := 192709 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 192751, root := ~, start := 192713, stop := 192750 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 192788, root := ~, start := 192775, stop := 192787 ), rec( content := " is a ve\ rtex in the digraph ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 192830, root := ~, start := 192816, stop := 192829 ), rec( content := ", then\n\ this operation returns the strongly connected component of ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 192913, root := ~, start := 192900, stop := 192912 ), rec( content := "\n in\ ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 192935, root := ~, start := 192921, stop := 192934 ), rec( content := ".\n S\ ee ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ StronglyConnectedComponents" ), content := 0, count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Ref", next := 192993, root := ~, start := 192945, stop := 192992 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [2], [1, 2], [3]]);\n\ngap> DigraphStronglyConnectedComponent(D, 3);\n[ 1, 3 ]\ngap> Di\ graphStronglyConnectedComponent(D, 2);\n[ 2 ]\ngap> DigraphStronglyConnectedCo\ mponent(D, 4);\n[ 4 ]\n", count := [ 5, 3, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 4 ], mathmode := "Text", name := "Example", next := 193300, root := ~, start := 193020, stop := 193299 ), rec( content := "\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 193317, root := ~, start := 192754, stop := 193316 ) ], count := [ 5, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 193331, root := ~, start := 192625, stop := 193330 ), rec( content := "\n\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphBicompon\ ents" ), content := 0, count := [ 5, 3, 13, 2 ], mathmode := "Text", name := "Attr", next := 193400, root := ~, start := 193352, stop := 193399 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists of vertices, or ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "K", next := 193455, root := ~, start := 193444, stop := 193454 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Returns", next := 193466, root := ~, start := 193403, stop := 193465 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "A", next := 193504, root := ~, start := 193490, stop := 193503 ), rec( content := " is a bi\ partite digraph, i.e. if it satisfies ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Ref", next := 193588, root := ~, start := 193550, stop := 193587 ), rec( content := ", then " , count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphBicomponents", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "C", next := 193621, root := ~, start := 193595, stop := 193620 ), rec( content := " returns\ a\n pair of bicomponents of ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "A", next := 193674, root := ~, start := 193660, stop := 193673 ), rec( content := ". Otherw\ ise,\n ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphBicomponents", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "C", next := 193717, root := ~, start := 193691, stop := 193716 ), rec( content := " returns\ ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "K", next := 193737, root := ~, start := 193726, stop := 193736 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "P", next := 193742, root := ~, start := 193738, stop := 193741 ), rec( content := "\n\n \ For a bipartite digraph, the vertices can be partitioned into two non-empty\n \ sets such that the source and range of any edge are in distinct sets. The\n\ parts of this partition are called ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ icomponents", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "E", next := 193960, root := ~, start := 193941, stop := 193959 ), rec( content := " of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 193978, root := ~, start := 193964, stop := 193977 ), rec( content := ".\n E\ quivalently, a pair of bicomponents of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 194038, root := ~, start := 194024, stop := 194037 ), rec( content := " consist\ s of the\n color-classes of a 2-coloring of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 194106, root := ~, start := 194092, stop := 194105 ), rec( content := ". ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "P", next := 194112, root := ~, start := 194108, stop := 194111 ), rec( content := "\n\n \ For a bipartite digraph with at least 3 vertices, there is a unique pair of\n \ bicomponents of bipartite if and only if the digraph is connected. See ", count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 194307, root := ~, start := 194269, stop := 194306 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(3);\n\ngap> D\ igraphBicomponents(D);\nfail\ngap> D := ChainDigraph(5);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 3, 5 ], [ 2, 4 ] ]\ \ngap> D := Digraph([[5], [1, 4], [5], [5], []]);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 3, 4 ], [ 2, 5 ] ]\nga\ p> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 2 ], [ 3, 4, 5 \ ] ]\n", count := [ 5, 3, 13, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 6 ], mathmode := "Text", name := "Example", next := 194900, root := ~, start := 194334, stop := 194899 ), rec( content := "\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Description", next := 194917, root := ~, start := 193469, stop := 194916 ) ], count := [ 5, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 194931, root := ~, start := 193337, stop := 194930 ), rec( content := "\n\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ArticulationPoi\ nts" ), content := 0, count := [ 5, 3, 14, 2 ], mathmode := "Text", name := "Attr", next := 194999, root := ~, start := 194952, stop := 194998 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 195040, root := ~, start := 195002, stop := 195039 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ iconnected", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "E", next := 195102, root := ~, start := 195084, stop := 195101 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 195181, root := ~, start := 195149, stop := 195180 ), rec( content := ") when a\ ny vertex is removed. \n If the digraph ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "A", next := 195245, root := ~, start := 195231, stop := 195244 ), rec( content := " is not \ biconnected but is connected, then any\n vertex ", count := [ 5, 3, 14, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "C", next := 195311, root := ~, start := 195303, stop := 195310 ), rec( content := " of ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "A", next := 195329, root := ~, start := 195315, stop := 195328 ), rec( content := " whose r\ emoval makes the resulting digraph\n disconnected is called an ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ rticulation point", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "E", next := 195427, root := ~, start := 195402, stop := 195426 ), rec( content := ".", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "P", next := 195432, root := ~, start := 195428, stop := 195431 ), rec( content := "\n\n \ ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "A\ rticulationPoints", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "C", next := 195463, root := ~, start := 195438, stop := 195462 ), rec( content := " returns\ a list of the articulation points of\n ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "A", next := 195527, root := ~, start := 195513, stop := 195526 ), rec( content := ", if any\ , and, in particular, returns the empty list if\n ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "A", next := 195601, root := ~, start := 195587, stop := 195600 ), rec( content := " is not \ connected. ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 5 ], mathmode := "Text", name := "P", next := 195624, root := ~, start := 195620, stop := 195623 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 5, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "P", next := 195677, root := ~, start := 195673, stop := 195676 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 195745, root := ~, start := 195732, stop := 195744 ), rec( content := " where\n\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 195764, root := ~, start := 195756, stop := 195763 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 195800, root := ~, start := 195792, stop := 195799 ), rec( content := " is the \ number of vertices in\n the digraph.", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "P", next := 195850, root := ~, start := 195846, stop := 195849 ), rec( content := "\n \ \n If ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "A", next := 195873, root := ~, start := 195865, stop := 195872 ), rec( content := " has a b\ ridge (see ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "Ref", next := 195913, root := ~, start := 195892, stop := 195912 ), rec( content := "), then \ a node incident\n to the bridge is an articulation point if and only if it \ has degree at\n least ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "M", next := 196030, root := ~, start := 196022, stop := 196029 ), rec( content := ". It fo\ llows that if ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "A", next := 196060, root := ~, start := 196052, stop := 196059 ), rec( content := " has a b\ ridge and at least\n ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "M", next := 196099, root := ~, start := 196091, stop := 196098 ), rec( content := " nodes, \ then at least one of the nodes incident to the bridge is\n an articulation \ point. The converse does not hold, there are digraphs\n with articulation \ points, but no bridges.\n ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "P", next := 196293, root := ~, start := 196289, stop := 196292 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "Ref", next := 196342, root := ~, start := 196308, stop := 196341 ), rec( content := " and \n \ ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "Ref", next := 196385, root := ~, start := 196352, stop := 196384 ), rec( content := ".\n", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ArticulationPoints(CycleDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [\ 3, 5], [4], [2], [6], [1], []]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap>\ ArticulationPoints(ChainDigraph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(Nu\ llDigraph(5));\n[ ]\ngap> D := ChainDigraph(IsMutableDigraph, 4);\n\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n", count := [ 5, 3, 14, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 9 ], mathmode := "Text", name := "Example", next := 196790, root := ~, start := 196387, stop := 196789 ), rec( content := "\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 196807, root := ~, start := 195043, stop := 196806 ) ], count := [ 5, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 196821, root := ~, start := 194937, stop := 196820 ), rec( content := "\n\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "Bridges" ), content := 0, count := [ 5, 3, 15, 2 ], mathmode := "Text", name := "Attr", next := 196872, root := ~, start := 196842, stop := 196871 ), rec( attributes := rec( ), content := [ rec( content := "A (possib\ ly empty) list of edges.", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Returns", next := 196927, root := ~, start := 196875, stop := 196926 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -edge-connected", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "E", next := 196994, root := ~, start := 196971, stop := 196993 ), rec( content := " if it i\ s still connected (in\n the sense of ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Ref", next := 197073, root := ~, start := 197041, stop := 197072 ), rec( content := ") when a\ ny edge is\n removed. If the digraph ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 197129, root := ~, start := 197121, stop := 197128 ), rec( content := " is not \ 2-edge-connected but is\n connected, then any edge ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "C", next := 197203, root := ~, start := 197190, stop := 197202 ), rec( content := " of ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 197215, root := ~, start := 197207, stop := 197214 ), rec( content := " whose r\ emoval\n makes the resulting digraph disconnected is called a ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ ridge", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "E", next := 197300, root := ~, start := 197287, stop := 197299 ), rec( content := ".", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "P", next := 197305, root := ~, start := 197301, stop := 197304 ), rec( content := "\n\n \ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ ridges", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "C", next := 197325, root := ~, start := 197311, stop := 197324 ), rec( content := " returns\ a list of the bridges of ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 197367, root := ~, start := 197359, stop := 197366 ), rec( content := ", if any\ , and, in\n particular, returns the empty list if ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 197435, root := ~, start := 197427, stop := 197434 ), rec( content := " is not \ connected. ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "P", next := 197458, root := ~, start := 197454, stop := 197457 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "P", next := 197511, root := ~, start := 197507, stop := 197510 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 197579, root := ~, start := 197566, stop := 197578 ), rec( content := " where\n\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 197598, root := ~, start := 197590, stop := 197597 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 197634, root := ~, start := 197626, stop := 197633 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "P", next := 197689, root := ~, start := 197685, stop := 197688 ), rec( content := "\n\n \ If ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "A", next := 197706, root := ~, start := 197698, stop := 197705 ), rec( content := " has a b\ ridge, then a node incident to the bridge is an\n articulation point (see " , count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 197822, root := ~, start := 197790, stop := 197821 ), rec( content := ") if and\ only if\n it has degree at least ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "M", next := 197874, root := ~, start := 197866, stop := 197873 ), rec( content := ". It fo\ llows that if ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "A", next := 197904, root := ~, start := 197896, stop := 197903 ), rec( content := " has a\n\ bridge and at least ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "M", next := 197943, root := ~, start := 197935, stop := 197942 ), rec( content := " nodes, \ then at least one of the nodes\n incident to the bridge is an articulation \ point. The converse does not\n hold, there are digraphs with articulation p\ oints, but no bridges.\n ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "P", next := 198136, root := ~, start := 198132, stop := 198135 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "Ref", next := 198185, root := ~, start := 198151, stop := 198184 ), rec( content := " and \n \ ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "Ref", next := 198228, root := ~, start := 198195, stop := 198227 ), rec( content := ".\n", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> Bridges(D);\n[ ]\ngap> D := Digrap\ h([[2], [3], [4], [2]]);\n\ngap> B\ ridges(D);\n[ [ 1, 2 ] ]\ngap> Bridges(ChainDigraph(2));\n[ [ 1, 2 ] ]\ngap> A\ rticulationPoints(ChainDigraph(2));\n[ ]\n", count := [ 5, 3, 15, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 15, 9 ], mathmode := "Text", name := "Example", next := 198604, root := ~, start := 198230, stop := 198603 ), rec( content := "\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Description", next := 198621, root := ~, start := 196930, stop := 198620 ) ], count := [ 5, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 198635, root := ~, start := 196827, stop := 198634 ), rec( content := "\n\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ on" ), content := 0, count := [ 5, 3, 16, 2 ], mathmode := "Text", name := "Oper", next := 198696, root := ~, start := 198656, stop := 198695 ), rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ onAttr" ), content := 0, count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "Attr", next := 198743, root := ~, start := 198699, stop := 198742 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ or ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "K", next := 198779, root := ~, start := 198768, stop := 198778 ), rec( content := ".", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 198790, root := ~, start := 198746, stop := 198789 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trong orientation", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "E", next := 198838, root := ~, start := 198813, stop := 198837 ), rec( content := " of a co\ nnected symmetric digraph ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 198880, root := ~, start := 198872, stop := 198879 ), rec( content := " (if\n \ it exists) is a strongly connected subdigraph ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 198943, root := ~, start := 198935, stop := 198942 ), rec( content := " of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 198955, root := ~, start := 198947, stop := 198954 ), rec( content := " such\n \ that for every edge ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 198998, root := ~, start := 198985, stop := 198997 ), rec( content := " of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 199010, root := ~, start := 199002, stop := 199009 ), rec( content := " either \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 199031, root := ~, start := 199018, stop := 199030 ), rec( content := " or\n \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, u]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 199052, root := ~, start := 199039, stop := 199051 ), rec( content := " is an e\ dge of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 199075, root := ~, start := 199067, stop := 199074 ), rec( content := " but not\ both. Robbin's Theorem states\n that a digraph admits a strong orientatio\ n if and only if it is\n bridgeless (see ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Ref", next := 199236, root := ~, start := 199203, stop := 199235 ), rec( content := "). \n \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "P", next := 199248, root := ~, start := 199244, stop := 199247 ), rec( content := "\n\n \ This operation returns a strong orientation of the digraph ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 199321, root := ~, start := 199313, stop := 199320 ), rec( content := " if \n \ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 199338, root := ~, start := 199330, stop := 199337 ), rec( content := " is symm\ etric and ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 199364, root := ~, start := 199356, stop := 199363 ), rec( content := " admits \ a strong orientation. If ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 199405, root := ~, start := 199397, stop := 199404 ), rec( content := "\n is\ symmetric but does not admit a strong orientation, then ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "K", next := 199480, root := ~, start := 199469, stop := 199479 ), rec( content := " is\n \ returned. If ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 199509, root := ~, start := 199501, stop := 199508 ), rec( content := " is not \ symmetric, then an error is given. \n ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "P", next := 199561, root := ~, start := 199557, stop := 199560 ), rec( content := "\n\n \ If ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "A", next := 199578, root := ~, start := 199570, stop := 199577 ), rec( content := " is immu\ table, ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ trongOrientation(", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := \ "Text", name := "A", next := 199622, root := ~, start := 199614, stop := 19962\ 1 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 199627, root := ~, start := 199593, stop := 199626 ), rec( content := " returns\ an\n immutable digraph, and if ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "A", next := 199677, root := ~, start := 199669, stop := 199676 ), rec( content := " is muta\ ble, then\n ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ trongOrientation(", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := \ "Text", name := "A", next := 199728, root := ~, start := 199720, stop := 19972\ 7 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 199733, root := ~, start := 199699, stop := 199732 ), rec( content := " returns\ a mutable digraph. ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "P", next := 199765, root := ~, start := 199761, stop := 199764 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 199833, root := ~, start := 199820, stop := 199832 ), rec( content := " where\n\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 199852, root := ~, start := 199844, stop := 199851 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 199888, root := ~, start := 199880, stop := 199887 ), rec( content := " is the \ number of vertices in\n the digraph.\n", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> StrongOrientation(DigraphSymmetricClosure(CycleDigraph(5))) \n> = CycleD\ igraph(5);\ntrue\ngap> D := DigraphSymmetricClosure(Digraph(\n> [[2, 7], [3, 5\ ], [4], [2], [6], [1], []]));;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> Stro\ ngOrientation(D);\nfail\ngap> StrongOrientation(NullDigraph(0));\n\ngap> StrongOrientation(DigraphDisjointUnion(Com\ pleteDigraph(3), \n> CompleteDigraph\ (3)));\nfail\n", count := [ 5, 3, 16, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 9 ], mathmode := "Text", name := "Example", next := 200434, root := ~, start := 199935, stop := 200433 ), rec( content := "\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 200451, root := ~, start := 198793, stop := 200450 ) ], count := [ 5, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 200465, root := ~, start := 198641, stop := 200464 ), rec( content := "\n\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphPeriod" ), content := 0, count := [ 5, 3, 17, 2 ], mathmode := "Text", name := "Attr", next := 200528, root := ~, start := 200486, stop := 200527 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 200561, root := ~, start := 200531, stop := 200560 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the period of the digraph ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "A", next := 200644, root := ~, start := 200630, stop := 200643 ), rec( content := ".", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "P", next := 200649, root := ~, start := 200645, stop := 200648 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 200682, root := ~, start := 200668, stop := 200681 ), rec( content := " has at \ least one directed cycle, then the period\n is the greatest positive intege\ r which divides the lengths of all directed\n cycles of ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 200839, root := ~, start := 200825, stop := 200838 ), rec( content := ". If ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 200859, root := ~, start := 200845, stop := 200858 ), rec( content := " has no \ directed cycles, then\n this function returns ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "M", next := 200923, root := ~, start := 200915, stop := 200922 ), rec( content := ". See S\ ection ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 200988, root := ~, start := 200938, stop := 200987 ), rec( content := " for the\ definition of a directed cycle. ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "P", next := 201033, root := ~, start := 201029, stop := 201032 ), rec( content := "\n\n \ A digraph with a period of ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "M", next := 201074, root := ~, start := 201066, stop := 201073 ), rec( content := " is said\ to be ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ periodic", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "E", next := 201105, root := ~, start := 201089, stop := 201104 ), rec( content := ". See\n\ ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAperi\ odicDigraph" ), content := 0, count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "Ref", next := 201148, root := ~, start := 201116, stop := 201147 ), rec( content := ". ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "P", next := 201154, root := ~, start := 201150, stop := 201153 ), rec( content := "\n ", count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> DigraphPeriod(D);\n6\ngap> D := Digraph([[2]\ , [3, 5], [4], [5], [1, 2]]);\n\ng\ ap> DigraphPeriod(D);\n1\ngap> D := ChainDigraph(2);\n\ngap> DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\ \ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> DigraphPeriod(D);\n1\n", count := [ 5, 3, 17, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 17, 7 ], mathmode := "Text", name := "Example", next := 201697, root := ~, start := 201159, stop := 201696 ), rec( content := "\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 201714, root := ~, start := 200564, stop := 201713 ) ], count := [ 5, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 201728, root := ~, start := 200471, stop := 201727 ), rec( content := "\n\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, func, n\ opath, edge", Name := "DigraphFloydWarshall" ), content := 0, count := [ 5, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 201818, root := ~, start := 201749, stop := 201817 ), rec( attributes := rec( ), content := [ rec( content := "A matrix.\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 201849, root := ~, start := 201821, stop := 201848 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 201887, root := ~, start := 201873, stop := 201886 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "M", next := 201914, root := ~, start := 201906, stop := 201913 ), rec( content := " vertice\ s, then\n this operation returns an ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "M", next := 201977, root := ~, start := 201960, stop := 201976 ), rec( content := " matrix \ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "C", next := 201995, root := ~, start := 201985, stop := 201994 ), rec( content := " contain\ ing\n the output of a generalised version of the Floyd-Warshall algorithm,\ \n applied to ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 202109, root := ~, start := 202095, stop := 202108 ), rec( content := ". ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "P", next := 202115, root := ~, start := 202111, stop := 202114 ), rec( content := "\n\n \ The operation ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 202162, root := ~, start := 202135, stop := 202161 ), rec( content := " is cust\ omised by the arguments\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202209, root := ~, start := 202198, stop := 202208 ), rec( content := ", ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202224, root := ~, start := 202211, stop := 202223 ), rec( content := ", and ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202241, root := ~, start := 202230, stop := 202240 ), rec( content := ".\n T\ he arguments ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202274, root := ~, start := 202261, stop := 202273 ), rec( content := " and ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202290, root := ~, start := 202279, stop := 202289 ), rec( content := " can be \ arbitrary ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ .\n The argument ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202351, root := ~, start := 202340, stop := 202350 ), rec( content := " must be\ a function which accepts 4 arguments:\n the matrix ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 202423, root := ~, start := 202413, stop := 202422 ), rec( content := ", follow\ ed by 3 postive integers. The function\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 202485, root := ~, start := 202474, stop := 202484 ), rec( content := " is wher\ e the work to calculate the desired outcome must be\n performed. ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "P", next := 202564, root := ~, start := 202560, stop := 202563 ), rec( content := "\n\n \ This method initialises the matrix ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 202615, root := ~, start := 202605, stop := 202614 ), rec( content := " by sett\ ing entry\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 202653, root := ~, start := 202637, stop := 202652 ), rec( content := " to equa\ l ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 202674, root := ~, start := 202663, stop := 202673 ), rec( content := " if ther\ e is an edge with source\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 202719, root := ~, start := 202711, stop := 202718 ), rec( content := " and ran\ ge ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 202738, root := ~, start := 202730, stop := 202737 ), rec( content := ", and by\ setting entry ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 202777, root := ~, start := 202761, stop := 202776 ), rec( content := " to equa\ l\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 202804, root := ~, start := 202791, stop := 202803 ), rec( content := " otherwi\ se.\n The final part of ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 202865, root := ~, start := 202838, stop := 202864 ), rec( content := " then ca\ lls the function\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 202905, root := ~, start := 202894, stop := 202904 ), rec( content := " inside \ three nested for loops, over the vertices of\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 202976, root := ~, start := 202962, stop := 202975 ), rec( content := ": ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "P", next := 202982, root := ~, start := 202978, stop := 202981 ), rec( content := "\n\n", count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ nfor i in DigraphsVertices(digraph) do\n for j in DigraphsVertices(digraph) d\ o\n for k in DigraphsVertices(digraph) do\n func(mat, i, j, k);\n o\ d;\n od;\nod;\n", count := [ 5, 3, 18, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 7 ], mathmode := "Text", name := "Listing", next := 203180, root := ~, start := 202984, stop := 203179 ), rec( content := "\n\n \ The matrix ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 203207, root := ~, start := 203197, stop := 203206 ), rec( content := " is then\ returned as the result. An example of using\n ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 203291, root := ~, start := 203264, stop := 203290 ), rec( content := " to calc\ ulate the shortest (non-zero) distances\n between the vertices of a digraph\ is shown below: ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 9 ], mathmode := "Text", name := "P", next := 203397, root := ~, start := 203393, stop := 203396 ), rec( content := "\n\n \ ", count := [ 5, 3, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&EAHQeDB\");\n\ngap> func := function(mat, i, j, k)\n> if mat[i][k] <>\ -1 and mat[k][j] <> -1 then\n> if (mat[i][j] = -1) or (mat[i][j] > mat[i]\ [k] + mat[k][j]) then\n> mat[i][j] := mat[i][k] + mat[k][j];\n> fi;\ \n> fi;\n> end;\nfunction( mat, i, j, k ) ... end\ngap> shortest_distances :\ = DigraphFloydWarshall(D, func, -1, 1);;\ngap> Display(shortest_distances);\n[\ [ 3, -1, -1, 2, 1, 2 ],\n [ 4, 2, 1, 3, 2, 1 ],\n [ \ 3, 1, 2, 2, 1, 2 ],\n [ 1, -1, -1, 1, 1, 2 ],\n [ 2, \ -1, -1, 1, 2, 1 ],\n [ 3, -1, -1, 2, 1, 1 ] ]\n", count := [ 5, 3, 18, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "Example", next := 204103, root := ~, start := 203403, stop := 204102 ), rec( content := "\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 204120, root := ~, start := 201852, stop := 204119 ) ], count := [ 5, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 204134, root := ~, start := 201734, stop := 204133 ), rec( content := "\n\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IsReachable" ), content := 0, count := [ 5, 3, 19, 2 ], mathmode := "Text", name := "Oper", next := 204201, root := ~, start := 204155, stop := 204200 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 204224, root := ~, start := 204213, stop := 204223 ), rec( content := " or ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 204240, root := ~, start := 204228, stop := 204239 ), rec( content := ".", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 204251, root := ~, start := 204204, stop := 204250 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 204306, root := ~, start := 204295, stop := 204305 ), rec( content := " if ther\ e exists a non-trivial directed\n walk from vertex ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 204375, root := ~, start := 204367, stop := 204374 ), rec( content := " to vert\ ex ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 204394, root := ~, start := 204386, stop := 204393 ), rec( content := " in the \ digraph ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 204424, root := ~, start := 204410, stop := 204423 ), rec( content := ",\n a\ nd ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 204446, root := ~, start := 204434, stop := 204445 ), rec( content := " if ther\ e does not exist such a directed walk. See Section\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Ref", next := 204554, root := ~, start := 204510, stop := 204553 ), rec( content := " for the\ definition of a\n non-trivial directed walk.\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "P", next := 204618, root := ~, start := 204614, stop := 204617 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sReachable", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "C", next := 204657, root := ~, start := 204639, stop := 204656 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 204708, root := ~, start := 204687, stop := 204707 ), rec( content := " where " , count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 204723, root := ~, start := 204715, stop := 204722 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 204759, root := ~, start := 204751, stop := 204758 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "A", next := 204804, root := ~, start := 204790, stop := 204803 ), rec( content := ".\n\n", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [2, 3]]);\n\ngap> IsReachable(D, 1, 3);\ntrue\ngap> IsReachable(D, 2, 1);\nfalse\n\ gap> IsReachable(D, 3, 3);\ntrue\ngap> IsReachable(D, 1, 1);\nfalse\n", count := [ 5, 3, 19, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 19, 5 ], mathmode := "Text", name := "Example", next := 205053, root := ~, start := 204807, stop := 205052 ), rec( content := "\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 205070, root := ~, start := 204254, stop := 205069 ) ], count := [ 5, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 205084, root := ~, start := 204140, stop := 205083 ), rec( content := "\n\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, root", Name := "VerticesReachab\ leFrom" ), content := 0, count := [ 5, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 205161, root := ~, start := 205105, stop := 205160 ), rec( attributes := rec( ), content := [ rec( content := "A list.", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 205190, root := ~, start := 205164, stop := 205189 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a list of the vertices ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 205266, root := ~, start := 205258, stop := 205265 ), rec( content := ", for wh\ ich there exists\n a non-trivial directed walk from vertex ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ oot", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 205346, root := ~, start := 205335, stop := 205345 ), rec( content := " to vert\ ex ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 205365, root := ~, start := 205357, stop := 205364 ), rec( content := " in the \ digraph\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 205399, root := ~, start := 205385, stop := 205398 ), rec( content := ". See Se\ ction ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Ref", next := 205457, root := ~, start := 205413, stop := 205456 ), rec( content := "\n fo\ r the definition of a non-trivial directed walk.\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "P", next := 205521, root := ~, start := 205517, stop := 205520 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ erticesReachableFrom", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "C", next := 205570, root := ~, start := 205542, stop := 205569 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 205621, root := ~, start := 205600, stop := 205620 ), rec( content := " where " , count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 205636, root := ~, start := 205628, stop := 205635 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 205672, root := ~, start := 205664, stop := 205671 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "A", next := 205717, root := ~, start := 205703, stop := 205716 ), rec( content := ".\n\n", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> VerticesReachableFrom(D, 1);\n[ 2, 1, 3, 4, 5 ]\ngap> VerticesReachableFr\ om(D, 3);\n[ 1, 2, 3, 4, 5 ]\ngap> D := EmptyDigraph(5);\n\ngap> VerticesReachableFrom(D, 1);\n[ ]\ngap> VerticesR\ eachableFrom(D, 3);\n[ ]\ngap> D := CycleDigraph(4);\n\ngap> VerticesReachableFrom(D, 1);\n[ 2, 3, 4, 1 ]\ngap> Ve\ rticesReachableFrom(D, 3);\n[ 4, 1, 2, 3 ]\ngap> D := ChainDigraph(5);\n\ngap> VerticesReachableFrom(D, 1);\n[ 2, 3\ , 4, 5 ]\ngap> VerticesReachableFrom(D, 3);\n[ 4, 5 ]\ngap> VerticesReachableF\ rom(D, 5);\n[ ]\n", count := [ 5, 3, 20, 5 ], name := "PCDATA", root := ~ ) ] , count := [ 5, 3, 20, 5 ], mathmode := "Text", name := "Example", next := 206445, root := ~, start := 205720, stop := 206444 ), rec( content := "\n ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 206462, root := ~, start := 205193, stop := 206461 ) ], count := [ 5, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 206476, root := ~, start := 205090, stop := 206475 ), rec( content := "\n\n ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphPath" ), content := 0, count := [ 5, 3, 21, 2 ], mathmode := "Text", name := "Oper", next := 206543, root := ~, start := 206497, stop := 206542 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "K", next := 206586, root := ~, start := 206575, stop := 206585 ), rec( content := ".", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Returns", next := 206597, root := ~, start := 206546, stop := 206596 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ there exists a non-trivial directed path (or a non-trivial cycle, in the\n \ case that ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 206716, root := ~, start := 206708, stop := 206715 ), rec( content := " ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "C", next := 206725, root := ~, start := 206717, stop := 206724 ), rec( content := " ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 206734, root := ~, start := 206726, stop := 206733 ), rec( content := ") from v\ ertex ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 206756, root := ~, start := 206748, stop := 206755 ), rec( content := " to vert\ ex\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 206779, root := ~, start := 206771, stop := 206778 ), rec( content := " in the \ digraph ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 206809, root := ~, start := 206795, stop := 206808 ), rec( content := ", then t\ his operation returns such a\n directed path (or directed cycle). Otherwise\ , this operation returns\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "K", next := 206934, root := ~, start := 206923, stop := 206933 ), rec( content := ". See S\ ection ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Ref", next := 206990, root := ~, start := 206949, stop := 206989 ), rec( content := " for the\ \n definition of a directed path and a directed cycle.\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "P", next := 207063, root := ~, start := 207059, stop := 207062 ), rec( content := "\n\n \ A directed path (or directed cycle) of non-zero length ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "C", next := 207134, root := ~, start := 207124, stop := 207133 ), rec( content := ",\n " , count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)", count := [ 5, 3, 21, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "M", next := 207186, root := ~, start := 207140, stop := 207185 ), rec( content := ",\n i\ s represented by a pair of lists ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,a]", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "C", next := 207238, root := ~, start := 207226, stop := 207237 ), rec( content := " as foll\ ows:\n\n ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 5, 3, 21, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "v", count := [ 5, 3, 21, 6 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name\ := "C", next := 207292, root := ~, start := 207284, stop := 207291 ), rec( cont\ ent := " is the list ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "[v_1, v_2, ..., v_n]", count := [ 5, 3, 21, 6 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name \ := "M", next := 207332, root := ~, start := 207305, stop := 207331 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], name := "Item", next := 207347, root := ~, start := 207\ 269, stop := 207346 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "a", count :\ = [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 21, 8 ], mathmode := "Text", name := "C", next := 207377, root := ~ , star\ t := 207369, stop := 207376 ), rec( content := " is the list of positive integ\ ers ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "[a_1, a_2, ...\ , a_{n-1}]", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M", next \ := 207442, root := ~, start := 207411, stop := 207441 ), rec( content := "\n \ where for each each ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i < n", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M", next\ := 207486, root := ~, start := 207471, stop := 207485 ), rec( content := ", " , coun\ t := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "a_i", count := [ 5, 3, 21, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8\ ], mathmode := "Text", name := "M", next := 207498, root := ~, start := 20748\ 8, stop := 207497 ), rec( content := " is the position of\n ", count :=\ [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 5, 3, 21\ , 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2\ 1, 8 ], mathmode := "Text", name := "M", next := 207540, root := ~, start := 2\ 07526, stop := 207539 ), rec( content := " in\n ", count := [ 5, 3, 21,\ 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "OutNeighboursOfVertex(", count := [ 5, 3,\ 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3\ , 21, 8 ], mathmode := "Text", name := "C", next := 207581, root := ~, start :\ = 207552, stop := 207580 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "A", next\ := 207595, root := ~, start := 207581, stop := 207594 ), rec( attributes := re\ c( ), content := [ rec( content := ",", count := [ 5, 3, 21, 8 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode :\ = "Text", name := "C", next := 207603, root := ~, start := 207595, stop := 207\ 602 ), rec( attributes := rec( ), content := [ rec( content := "v_i", count :\ = [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M", next := 207613, root := ~ , star\ t := 207603, stop := 207612 ), rec( attributes := rec( ), content := [ rec( c\ ontent := ")", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "C", next\ := 207621, root := ~, start := 207613, stop := 207620 ), rec( content := "\n \ corresponding to the edge ", count := [ 5, 3, 21, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "e_i", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M\ ", next := 207666, root := ~, start := 207656, stop := 207665 ), rec( content \ := ". This can be useful if the\n position of a vertex in a list of ou\ t-neighours is significant, for\n example in orbit digraphs.\n ", coun\ t := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], name := "Ite\ m", next := 207819, root := ~, start := 207354, stop := 207818 ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "List", next := 207831, root := ~, start := 207256, stop := 207830 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPath", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "C", next := 207870, root := ~, start := 207852, stop := 207869 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "M", next := 207921, root := ~, start := 207900, stop := 207920 ), rec( content := " where " , count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "M", next := 207936, root := ~, start := 207928, stop := 207935 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "M", next := 207972, root := ~, start := 207964, stop := 207971 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "A", next := 208017, root := ~, start := 208003, stop := 208016 ), rec( content := ".\n\n", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [2, 3]]);\n\ngap> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ngap> DigraphPa\ th(D, 2, 1);\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ 2 ] ]\ngap> Digr\ aphPath(D, 1, 1);\nfail\n", count := [ 5, 3, 21, 11 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 11 ], mathmode := "Text", name := "Example", next := 208300, root := ~, start := 208020, stop := 208299 ), rec( content := "\n ", count := [ 5, 3, 21, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Description", next := 208317, root := ~, start := 206600, stop := 208316 ) ], count := [ 5, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 208331, root := ~, start := 206482, stop := 208330 ), rec( content := "\n\n ", count := [ 5, 3, 21, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphShortest\ Path" ), content := 0, count := [ 5, 3, 22, 2 ], mathmode := "Text", name := "Oper", next := 208406, root := ~, start := 208352, stop := 208405 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "K", next := 208449, root := ~, start := 208438, stop := 208448 ), rec( content := ".", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 208460, root := ~, start := 208409, stop := 208459 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the shortest directed path in the digraph digraph from the vertex\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 208567, root := ~, start := 208559, stop := 208566 ), rec( content := " to the \ vertex ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 208590, root := ~, start := 208582, stop := 208589 ), rec( content := ", if suc\ h a path exists. If ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "u", count := [ 5, 3, 22, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := \ "Text", name := "A", next := 208629, root := ~, start := 208621, stop := 20862\ 8 ), rec( content := " =\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := \ "Text", name := "A", next := 208644, root := ~, start := 208636, stop := 20864\ 3 ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "C", next := 208648, root := ~, start := 208618, stop := 208647 ), rec( content := ", then t\ he shortest non-trivial cycle is returned, again, if it\n exists. Otherwise\ , this operation returns ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "K", next := 208769, root := ~, start := 208758, stop := 208768 ), rec( content := ". See S\ ection\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 208829, root := ~, start := 208788, stop := 208828 ), rec( content := " for the\ definition of a directed\n path and a directed cycle.\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "P", next := 208902, root := ~, start := 208898, stop := 208901 ), rec( content := "\n\n \ See ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "Ref", next := 208937, root := ~, start := 208912, stop := 208936 ), rec( content := " for det\ ails on the output.\n\n The method for ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphShortestPath", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "C", next := 209011, root := ~, start := 208985, stop := 209010 ), rec( content := " has wor\ st case complexity of\n ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n)", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "M", next := 209060, root := ~, start := 209045, stop := 209059 ), rec( content := " where " , count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "M", next := 209075, root := ~, start := 209067, stop := 209074 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "M", next := 209111, root := ~, start := 209103, stop := 209110 ), rec( content := " the\n \ number of vertices in ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "A", next := 209156, root := ~, start := 209142, stop := 209155 ), rec( content := ".\n\n", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [2, 4], [1], [2, 4]]);\n\ngap> DigraphShortestPath(D, 5, 1);\n[ [ 5, 4, 1 ], [ \ 2, 1 ] ]\ngap> DigraphShortestPath(D, 3, 3);\n[ [ 3, 2, 3 ], [ 1, 1 ] ]\ngap> \ DigraphShortestPath(D, 5, 5);\nfail\ngap> DigraphShortestPath(D, 1, 1);\n[ [ 1\ , 1 ], [ 1 ] ]\n", count := [ 5, 3, 22, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "Example", next := 209508, root := ~, start := 209159, stop := 209507 ), rec( content := "\n ", count := [ 5, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 209525, root := ~, start := 208463, stop := 209524 ) ], count := [ 5, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 209539, root := ~, start := 208337, stop := 209538 ), rec( content := "\n\n ", count := [ 5, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IteratorOfPaths\ " ), content := 0, count := [ 5, 3, 23, 2 ], mathmode := "Text", name := "Oper", next := 209610, root := ~, start := 209560, stop := 209609 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 209644, root := ~, start := 209613, stop := 209643 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 209682, root := ~, start := 209668, stop := 209681 ), rec( content := " is a di\ graph or a list of adjacencies which defines a\n digraph - see ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Ref", next := 209782, root := ~, start := 209755, stop := 209781 ), rec( content := " - then \ this operation returns an\n iterator of the non-trivial directed paths (or \ directed cycles, in the case\n that ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 209913, root := ~, start := 209905, stop := 209912 ), rec( content := " ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "C", next := 209922, root := ~, start := 209914, stop := 209921 ), rec( content := " ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 209931, root := ~, start := 209923, stop := 209930 ), rec( content := ") in ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 209950, root := ~, start := 209936, stop := 209949 ), rec( content := " from th\ e vertex ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 209975, root := ~, start := 209967, stop := 209974 ), rec( content := "\n to\ the vertex ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 210002, root := ~, start := 209994, stop := 210001 ), rec( content := ".\n " , count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "P", next := 210012, root := ~, start := 210008, stop := 210011 ), rec( content := "\n\n \ See ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 210047, root := ~, start := 210022, stop := 210046 ), rec( content := " for mor\ e information about the repesentation\n of a directed path or directed cycl\ e which is used, and see ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Sect := "Iterators" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 210201, root := ~, start := 210157, stop := 210200 ), rec( content := " for mor\ e information about iterators.\n See Section ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 210297, root := ~, start := 210256, stop := 210296 ), rec( content := " for the\ definition of\n a directed path and a directed cycle.\n ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "P", next := 210370, root := ~, start := 210366, stop := 210369 ), rec( content := "\n\n", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 4, 4, 2], [3, 5], [2, 3], [1, 2], [4]]);\n\ngap> iter := IteratorOfPaths(D, 1, 4);\ \n\ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 2 ] ]\ngap> NextIterator\ (iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> NextIterator(iter);\n[ [ 1, 2, 5, 4 ], [ 4,\ 2, 1 ] ]\ngap> IsDoneIterator(iter);\ntrue\ngap> iter := IteratorOfPaths(D, 4\ , 3);\n\ngap> NextIterator(iter);\n[ [ 4, 1, 2, 3 ], [ 1, 4, 1 ] ]\n\ ", count := [ 5, 3, 23, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "Example", next := 210855, root := ~, start := 210372, stop := 210854 ), rec( content := "\n ", count := [ 5, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 210872, root := ~, start := 209647, stop := 210871 ) ], count := [ 5, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 210886, root := ~, start := 209545, stop := 210885 ), rec( content := "\n\n ", count := [ 5, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAllSimpl\ eCircuits" ), content := 0, count := [ 5, 3, 24, 2 ], mathmode := "Text", name := "Attr", next := 210960, root := ~, start := 210907, stop := 210959 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Returns", next := 211010, root := ~, start := 210963, stop := 211009 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "A", next := 211048, root := ~, start := 211034, stop := 211047 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "C", next := 211099, root := ~, start := 211068, stop := 211098 ), rec( content := "\n re\ turns a list of the ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuits", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "E", next := 211148, root := ~, start := 211126, stop := 211147 ), rec( content := " in ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "A", next := 211166, root := ~, start := 211152, stop := 211165 ), rec( content := ". ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "P", next := 211172, root := ~, start := 211168, stop := 211171 ), rec( content := "\n\n \ See Section ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "Ref", next := 211234, root := ~, start := 211190, stop := 211233 ), rec( content := " for the\ definition\n of a simple circuit, and related notions. Note that a loop is\ a simple\n circuit. ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "P", next := 211346, root := ~, start := 211342, stop := 211345 ), rec( content := "\n\n \ For a digraph without multiple edges, a simple circuit is uniquely\n determ\ ined by its subsequence of vertices. However this is not the case for\n a m\ ultidigraph. The attribute ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "C", next := 211565, root := ~, start := 211534, stop := 211564 ), rec( content := " ignores\ \n multiple edges, and identifies a simple circuit using only its subsequen\ ce\n of vertices. For example, although the simple circuits ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e, v)", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 211728, root := ~, start := 211712, stop := 211727 ), rec( content := " and\n \ ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e', v)", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 211754, root := ~, start := 211737, stop := 211753 ), rec( content := " (for di\ stinct edges ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 211783, root := ~, start := 211775, stop := 211782 ), rec( content := " and ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ '", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 211797, root := ~, start := 211788, stop := 211796 ), rec( content := ") are\n \ mathematically distinct, ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "C", next := 211863, root := ~, start := 211832, stop := 211862 ), rec( content := " conside\ rs them to\n be the same. ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "P", next := 211903, root := ~, start := 211899, stop := 211902 ), rec( content := "\n\n \ With this approach, a directed circuit of length ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "C", next := 211966, root := ~, start := 211958, stop := 211965 ), rec( content := " can be \ determined\n by a list of its first ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "C", next := 212020, root := ~, start := 212012, stop := 212019 ), rec( content := " vertice\ s. Thus a simple circuit ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v_1,\n e_1, v_2, e_2, ..., e_{n-1}, v_n, e_{n+1}, v_1)", count := [ 5, 3, 24, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "M", next := 212119, root := ~, start := 212053, stop := 212118 ), rec( content := " can be \ represented as\n the list ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v_1, \\ldots, v_n]", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "M", next := 212180, root := ~, start := 212155, stop := 212179 ), rec( content := ", or any\ cyclic permutation thereof. For\n each simple circuit of ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "A", next := 212263, root := ~, start := 212249, stop := 212262 ), rec( content := ",\n " , count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "A", next := 2123\ 11, root := ~, start := 212297, stop := 212310 ), rec( content := ")", count := [ 5, 3, 24, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "C", next := 212316, root := ~, start := 212269, stop := 212315 ), rec( content := " include\ s precisely one such\n list to represent the circuit. ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "P", next := 212385, root := ~, start := 212381, stop := 212384 ), rec( content := "\n\n \ ", count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 4 ], [ 4, 5 ], [ 2\ , 3 ] ]\ngap> D := ChainDigraph(10);;\ngap> DigraphAllSimpleCircuits(D);\n[ ]\ \ngap> D := Digraph([[3], [1], [1]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D := Digraph([[1,\ 1]]);\n\ngap> DigraphAllSimple\ Circuits(D);\n[ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1,\ 2, 3 ] ]\n", count := [ 5, 3, 24, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 8 ], mathmode := "Text", name := "Example", next := 213038, root := ~, start := 212391, stop := 213037 ), rec( content := "\n ", count := [ 5, 3, 24, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Description", next := 213055, root := ~, start := 211013, stop := 213054 ) ], count := [ 5, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 213069, root := ~, start := 210892, stop := 213068 ), rec( content := "\n\n ", count := [ 5, 3, 24, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLongestS\ impleCircuit" ), content := 0, count := [ 5, 3, 25, 2 ], mathmode := "Text", name := "Attr", next := 213146, root := ~, start := 213090, stop := 213145 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices, or ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "K", next := 213192, root := ~, start := 213181, stop := 213191 ), rec( content := ".", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Returns", next := 213203, root := ~, start := 213149, stop := 213202 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 213241, root := ~, start := 213227, stop := 213240 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLongestSimpleCircuit", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "C", next := 213295, root := ~, start := 213261, stop := 213294 ), rec( content := "\n re\ turns the longest ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuit", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "E", next := 213341, root := ~, start := 213320, stop := 213340 ), rec( content := " in ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 213359, root := ~, start := 213345, stop := 213358 ), rec( content := ". See Se\ ction\n ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Ref", next := 213421, root := ~, start := 213377, stop := 213420 ), rec( content := " for the\ definition of simple\n circuit, and the definition of length for a simple \ circuit.", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "P", next := 213518, root := ~, start := 213514, stop := 213517 ), rec( content := "\n\n \ This attribute computes\n ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 213584, root := ~, start := 213552, stop := 213583 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "A", next := 213598, root := ~, start := 213584, stop := 213597 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 213606, root := ~, start := 213598, stop := 213605 ), rec( content := " to find\ all the\n simple circuits of ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "A", next := 213660, root := ~, start := 213646, stop := 213659 ), rec( content := ", and re\ turns one of maximal length. A\n simple circuit is represented as a list o\ f vertices, in the same way as\n described in ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "Ref", next := 213831, root := ~, start := 213793, stop := 213830 ), rec( content := ".", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "P", next := 213836, root := ~, start := 213832, stop := 213835 ), rec( content := "\n\n \ If ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "A", next := 213859, root := ~, start := 213845, stop := 213858 ), rec( content := " has no \ simple circuits, then this attribute returns\n ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "K", next := 213927, root := ~, start := 213916, stop := 213926 ), rec( content := ". If ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "A", next := 213947, root := ~, start := 213933, stop := 213946 ), rec( content := " has mul\ tiple simple circuits of maximal\n length, then this attribute returns one \ of them.", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 25, 6 ], mathmode := "Text", name := "P", next := 214044, root := ~, start := 214040, stop := 214043 ), rec( content := "\n\n \ ", count := [ 5, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSimpl\ eCircuit(D);\n[ 4, 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSimpl\ eCircuit(D);\nfail\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> Digr\ aphLongestSimpleCircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGraph(I\ sMutableDigraph, 4, 1);\n\ngap> Dig\ raphLongestSimpleCircuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n", count := [ 5, 3, 25, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 7 ], mathmode := "Text", name := "Example", next := 214516, root := ~, start := 214050, stop := 214515 ), rec( content := "\n ", count := [ 5, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Description", next := 214533, root := ~, start := 213206, stop := 214532 ) ], count := [ 5, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 214547, root := ~, start := 213075, stop := 214546 ), rec( content := "\n\n ", count := [ 5, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphLayers" ), content := 0, count := [ 5, 3, 26, 2 ], mathmode := "Text", name := "Oper", next := 214618, root := ~, start := 214568, stop := 214617 ), rec( attributes := rec( ), content := [ rec( content := "A list.", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Returns", next := 214647, root := ~, start := 214621, stop := 214646 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a list ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 214709, root := ~, start := 214698, stop := 214708 ), rec( content := " such th\ at ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i]", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 214734, root := ~, start := 214720, stop := 214733 ), rec( content := " is the\\ n list of vertices whose minimum distance from the vertex ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 214815, root := ~, start := 214802, stop := 214814 ), rec( content := " in\n \ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 214837, root := ~, start := 214823, stop := 214836 ), rec( content := " is ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 214853, root := ~, start := 214841, stop := 214852 ), rec( content := ". Vertex\ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 214875, root := ~, start := 214862, stop := 214874 ), rec( content := " is\n \ assumed to be at distance ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 214917, root := ~, start := 214909, stop := 214916 ), rec( content := " from it\ self.\n ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4\ ] ]\n", count := [ 5, 3, 26, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "Example", next := 215047, root := ~, start := 214935, stop := 215046 ), rec( content := "\n ", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Description", next := 215064, root := ~, start := 214650, stop := 215063 ) ], count := [ 5, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 215078, root := ~, start := 214553, stop := 215077 ), rec( content := "\n\n ", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cy" ), content := 0, count := [ 5, 3, 27, 2 ], mathmode := "Text", name := "Attr", next := 215145, root := ~, start := 215099, stop := 215144 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer, or ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "K", next := 215195, root := ~, start := 215184, stop := 215194 ), rec( content := ".", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Returns", next := 215206, root := ~, start := 215148, stop := 215205 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 215244, root := ~, start := 215230, stop := 215243 ), rec( content := " is a sy\ mmetric digraph without multiple edges - see\n ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Ref", next := 215333, root := ~, start := 215301, stop := 215332 ), rec( content := " and ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Ref", next := 215366, root := ~, start := 215338, stop := 215365 ), rec( content := " - then\\ n this attribute returns the degeneracy of ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 215433, root := ~, start := 215419, stop := 215432 ), rec( content := ". ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "P", next := 215439, root := ~, start := 215435, stop := 215438 ), rec( content := "\n\n \ The degeneracy of a digraph is the least integer ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 215502, root := ~, start := 215494, stop := 215501 ), rec( content := " such\n \ that every induced of ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "A", next := 215548, root := ~, start := 215534, stop := 215547 ), rec( content := " contain\ s a vertex whose number of\n neighbours (excluding itself) is at most ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 215636, root := ~, start := 215628, stop := 215635 ), rec( content := ". Note t\ hat this means\n that loops are ignored.", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "P", next := 215690, root := ~, start := 215686, stop := 215689 ), rec( content := "\n\n \ If ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "A", next := 215713, root := ~, start := 215699, stop := 215712 ), rec( content := " is not \ symmetric or has multiple edges then this\n attribute returns ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "K", next := 215796, root := ~, start := 215785, stop := 215795 ), rec( content := ".\n " , count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracy(\ D);\n1\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracy(D);\n4\ngap> D \ := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n\ngap> DigraphDegeneracy(D);\n1\ngap> D := Generalis\ edPetersenGraph(IsMutableDigraph, 10, 3);\n\ngap> DigraphDegeneracy(D);\n3\n", count := [ 5, 3, 27, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 27, 6 ], mathmode := "Text", name := "Example", next := 216251, root := ~, start := 215802, stop := 216250 ), rec( content := "\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Description", next := 216268, root := ~, start := 215209, stop := 216267 ) ], count := [ 5, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 216282, root := ~, start := 215084, stop := 216281 ), rec( content := "\n\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cyOrdering" ), content := 0, count := [ 5, 3, 28, 2 ], mathmode := "Text", name := "Attr", next := 216357, root := ~, start := 216303, stop := 216356 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers, or ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "K", next := 216403, root := ~, start := 216392, stop := 216402 ), rec( content := ".", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Returns", next := 216414, root := ~, start := 216360, stop := 216413 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "A", next := 216452, root := ~, start := 216438, stop := 216451 ), rec( content := " is a di\ graph for which\n ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "C", next := 216505, root := ~, start := 216480, stop := 216504 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "A", next := 216519, root := ~, start := 216505, stop := 216518 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "C", next := 216527, root := ~, start := 216519, stop := 216526 ), rec( content := " is a no\ n-negative integer\n ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "C", next := 216566, root := ~, start := 216558, stop := 216565 ), rec( content := " - see " , count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Degeneracy" ), content := 0, count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Ref", next := 216604, root := ~, start := 216573, stop := 216603 ), rec( content := " - then \ this attribute\n returns a degeneracy ordering of the vertices of the verti\ ces of\n ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "A", next := 216714, root := ~, start := 216700, stop := 216713 ), rec( content := ".", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "P", next := 216719, root := ~, start := 216715, stop := 216718 ), rec( content := "\n\n \ A degeneracy ordering of ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "A", next := 216764, root := ~, start := 216750, stop := 216763 ), rec( content := " is a li\ st ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 216790, root := ~, start := 216775, stop := 216789 ), rec( content := " of the\\ n vertices of ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "A", next := 216828, root := ~, start := 216814, stop := 216827 ), rec( content := " ordered\ such that for any\n position ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 216876, root := ~, start := 216868, stop := 216875 ), rec( content := " of the \ list, the vertex ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering[i]", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 216919, root := ~, start := 216901, stop := 216918 ), rec( content := " has at \ most\n ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 216944, root := ~, start := 216936, stop := 216943 ), rec( content := " neighbo\ urs in later position of the list.", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "P", next := 216990, root := ~, start := 216986, stop := 216989 ), rec( content := "\n\n \ If ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "C", next := 217024, root := ~, start := 216999, stop := 217023 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "A", next := 217038, root := ~, start := 217024, stop := 217037 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "C", next := 217046, root := ~, start := 217038, stop := 217045 ), rec( content := " returns\ ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "K", next := 217066, root := ~, start := 217055, stop := 217065 ), rec( content := ",\n t\ hen this attribute returns ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "K", next := 217111, root := ~, start := 217100, stop := 217110 ), rec( content := ".\n " , count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracyO\ rdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := CompleteDigraph(5);;\ngap> DigraphDe\ generacyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := Digraph([[1], [2, 4, 5], [3\ , 4], [2, 3, 4], [2], []]);\n\nga\ p> DigraphDegeneracyOrdering(D);\n[ 1, 6, 5, 2, 4, 3 ]\ngap> D := GeneralisedP\ etersenGraph(IsMutableDigraph, 3, 1);\n\ngap> DigraphDegeneracyOrdering(D);\n[ 6, 5, 4, 1, 3, 2 ]\n", count := [ 5, 3, 28, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 6 ], mathmode := "Text", name := "Example", next := 217666, root := ~, start := 217117, stop := 217665 ), rec( content := "\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Description", next := 217683, root := ~, start := 216417, stop := 217682 ) ], count := [ 5, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 217697, root := ~, start := 216288, stop := 217696 ), rec( content := "\n\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "HamiltonianPath\ " ), content := 0, count := [ 5, 3, 29, 2 ], mathmode := "Text", name := "Attr", next := 217762, root := ~, start := 217718, stop := 217761 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "K", next := 217795, root := ~, start := 217784, stop := 217794 ), rec( content := ".", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Returns", next := 217806, root := ~, start := 217765, stop := 217805 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a Hamiltonian path if one exists, ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "K", next := 217880, root := ~, start := 217869, stop := 217879 ), rec( content := " if not.\ ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "P", next := 217892, root := ~, start := 217888, stop := 217891 ), rec( content := "\n\n \ A ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian path", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "E", next := 217923, root := ~, start := 217900, stop := 217922 ), rec( content := " of a di\ graph with ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 217950, root := ~, start := 217942, stop := 217949 ), rec( content := " vertice\ s is directed\n cycle of length ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 218000, root := ~, start := 217992, stop := 217999 ), rec( content := ". If ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "A", next := 218019, root := ~, start := 218005, stop := 218018 ), rec( content := " is a di\ graph that contains a\n Hamiltonian path, then this function returns one, d\ escribed in the form\n used by ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "Ref", next := 218175, root := ~, start := 218137, stop := 218174 ), rec( content := ". Note i\ f ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "A", next := 218199, root := ~, start := 218185, stop := 218198 ), rec( content := " has\n \ 0 or 1 vertices, then ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonianPath", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 218252, root := ~, start := 218230, stop := 218251 ), rec( content := " returns\ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ]", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 218270, root := ~, start := 218261, stop := 218269 ), rec( content := " or\n \ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 218288, root := ~, start := 218278, stop := 218287 ), rec( content := ", respec\ tively.", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "P", next := 218307, root := ~, start := 218303, stop := 218306 ), rec( content := "\n\n \ The method used in this attribute has the same worst case complexity as\n " , count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "Ref", next := 218422, root := ~, start := 218389, stop := 218421 ), rec( content := ".\n\n \ ", count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[]]);\n\ngap> Hamil\ tonianPath(D);\n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D := Digraph([\ [3], [], [2]]);\n\ngap> Hamiltonia\ nPath(D);\nfail\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := General\ isedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> HamiltonianPath(D);\nfail\n", count := [ 5, 3, 29, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 29, 6 ], mathmode := "Text", name := "Example", next := 219030, root := ~, start := 218429, stop := 219029 ), rec( content := "\n ", count := [ 5, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Description", next := 219047, root := ~, start := 217809, stop := 219046 ) ], count := [ 5, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 219061, root := ~, start := 217703, stop := 219060 ), rec( content := "\n\n ", count := [ 5, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "NrSpanningTrees\ " ), content := 0, count := [ 5, 3, 30, 2 ], mathmode := "Text", name := "Attr", next := 219126, root := ~, start := 219082, stop := 219125 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "Returns", next := 219159, root := ~, start := 219129, stop := 219158 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of spanning trees of the symmetric digraph\n ", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "A", next := 219260, root := ~, start := 219246, stop := 219259 ), rec( content := ".\n " , count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ rSpanningTrees", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "C", next := 219288, root := ~, start := 219266, stop := 219287 ), rec( content := " will re\ turn an error if ", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "A", next := 219327, root := ~, start := 219313, stop := 219326 ), rec( content := " is not \ a\n symmetric digraph. ", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "P", next := 219364, root := ~, start := 219360, stop := 219363 ), rec( content := " \n\n \ See ", count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "Ref", next := 219407, root := ~, start := 219375, stop := 219406 ), rec( content := " and\n \ ", count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "Ref", next := 219455, root := ~, start := 219416, stop := 219454 ), rec( content := " for mor\ e information. \n\n ", count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> NrSpanningTrees(D);\n125\ngap> D := DigraphSymmetricClosure(CycleDigraph(\ 24));;\ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(EmptyDigraph(0));\n\ 0\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> NrSpanningTrees(D);\n1134225\n", count := [ 5, 3, 30, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "Example", next := 219881, root := ~, start := 219484, stop := 219880 ), rec( content := "\n ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "Description", next := 219898, root := ~, start := 219162, stop := 219897 ) ], count := [ 5, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 219912, root := ~, start := 219067, stop := 219911 ), rec( content := "\n\n ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, source,\ target", Label := "for a source and target", Name := "DigraphDijkstra" ), content := 0, count := [ 5, 3, 31, 2 ], mathmode := "Text", name := "Oper", next := 220025, root := ~, start := 219933, stop := 220024 ), rec( attributes := rec( Arg := "digraph, source" , Label := "for a source", Name := "DigraphDijkstra\ " ), content := 0, count := [ 5, 3, 31, 4 ], mathmode := "Text", name := "Oper", next := 220101, root := ~, start := 220028, stop := 220100 ), rec( attributes := rec( ), content := [ rec( content := "Two lists\ .", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "Returns", next := 220133, root := ~, start := 220104, stop := 220132 ), rec( attributes := rec( ), content := [ rec( content := "\n If ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220169, root := ~, start := 220155, stop := 220168 ), rec( content := " is a di\ graph and ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220200, root := ~, start := 220187, stop := 220199 ), rec( content := " and ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220218, root := ~, start := 220205, stop := 220217 ), rec( content := " are\n \ vertices of ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220252, root := ~, start := 220238, stop := 220251 ), rec( content := ", then " , count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDijkstra", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "C", next := 220281, root := ~, start := 220259, stop := 220280 ), rec( content := " calcula\ tes the\n length of the shortest path from ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220345, root := ~, start := 220332, stop := 220344 ), rec( content := " to ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220362, root := ~, start := 220349, stop := 220361 ), rec( content := " and ret\ urns\n two lists. Each element of the first list is the distance of the\n co\ rresponding element from ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220484, root := ~, start := 220471, stop := 220483 ), rec( content := ". If a \ vertex was not visited in\n the process of calculating the shortest distance \ to ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220585, root := ~, start := 220572, stop := 220584 ), rec( content := " or if\n\ there is no path connecting that vertex with ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220652, root := ~, start := 220639, stop := 220651 ), rec( content := ", then\n\ the corresponding distance is ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "K", next := 220706, root := ~, start := 220691, stop := 220705 ), rec( content := ". Each \ element of the\n second list gives the previous vertex in the shortest path\n\ from ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220810, root := ~, start := 220797, stop := 220809 ), rec( content := " to the \ corresponding vertex. For\n ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 220860, root := ~, start := 220847, stop := 220859 ), rec( content := " and for\ any vertices that remained unvisited this\n will be ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "C", next := 220932, root := ~, start := 220923, stop := 220931 ), rec( content := ".", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "P", next := 220937, root := ~, start := 220933, stop := 220936 ), rec( content := "\n\n \ If the optional second argument ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "A", next := 220988, root := ~, start := 220975, stop := 220987 ), rec( content := " is not \ present, then\n ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDijkstra", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "C", next := 221036, root := ~, start := 221014, stop := 221035 ), rec( content := " returns\ the shortest path from ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "A", next := 221081, root := ~, start := 221068, stop := 221080 ), rec( content := " to\n \ every vertex that is reachable from ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "A", next := 221138, root := ~, start := 221125, stop := 221137 ), rec( content := ".\n\n " , count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> mat := [[0, 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1 ], [ 0, 0, 1 ], [\ 0, 0, 0 ] ]\ngap> D := DigraphByAdjacencyMatrix(mat);\n\ngap> DigraphDijkstra(D, 2, 3);\n[ [ infinity, 0, 1 ], \ [ -1, -1, 2 ] ]\ngap> DigraphDijkstra(D, 1, 3);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\ \ngap> DigraphDijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n", count := [ 5, 3, 31, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 31, 7 ], mathmode := "Text", name := "Example", next := 221541, root := ~, start := 221143, stop := 221540 ), rec( content := "\n", count := [ 5, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 221556, root := ~, start := 220136, stop := 221555 ) ], count := [ 5, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 221570, root := ~, start := 219918, stop := 221569 ), rec( content := "\n\n ", count := [ 5, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 221584, root := ~, start := 175564, stop := 221583 ) , rec( content := "\n\n ", count := [ 5, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Cayley graphs of \ groups", count := [ 5, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 221639, root := ~, start := 221597, stop := 221638 ), rec( content := "\n ", count := [ 5, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "GroupOfCayleyDi\ graph" ), content := 0, count := [ 5, 4, 1, 2 ], mathmode := "Text", name := "Attr", next := 221708, root := ~, start := 221659, stop := 221707 ), rec( attributes := rec( Arg := "digraph", Name := "SemigroupOfCayl\ eyDigraph" ), content := 0, count := [ 5, 4, 1, 4 ], mathmode := "Text", name := "Attr", next := 221764, root := ~, start := 221711, stop := 221763 ), rec( attributes := rec( ), content := [ rec( content := "A group o\ r semigroup.", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Returns", next := 221807, root := ~, start := 221767, stop := 221806 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "A", next := 221845, root := ~, start := 221831, stop := 221844 ), rec( content := " is an i\ mmutable Cayley graph of a group ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 221894, root := ~, start := 221886, stop := 221893 ), rec( content := " and\n \ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "A", next := 221917, root := ~, start := 221903, stop := 221916 ), rec( content := " belongs\ to the category ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Ref", next := 221971, root := ~, start := 221942, stop := 221970 ), rec( content := ", then\n\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ roupOfCayleyDigraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 222009, root := ~, start := 221982, stop := 222008 ), rec( content := " returns\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 222026, root := ~, start := 222018, stop := 222025 ), rec( content := ".\n " , count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "P", next := 222036, root := ~, start := 222032, stop := 222035 ), rec( content := "\n\n \ If ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "A", next := 222059, root := ~, start := 222045, stop := 222058 ), rec( content := " is a Ca\ yley graph of a semigroup ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 222101, root := ~, start := 222093, stop := 222100 ), rec( content := " and\n \ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "A", next := 222124, root := ~, start := 222110, stop := 222123 ), rec( content := " belongs\ to the category ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "Ref", next := 222178, root := ~, start := 222149, stop := 222177 ), rec( content := ", then\n\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ emigroupOfCayleyDigraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 222220, root := ~, start := 222189, stop := 222219 ), rec( content := " returns\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "S\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 222237, root := ~, start := 222229, stop := 222236 ), rec( content := ".\n " , count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "P", next := 222247, root := ~, start := 222243, stop := 222246 ), rec( content := "\n\n \ See also ", count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfCayleyDigraph" ), content := 0, count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "Ref", next := 222301, root := ~, start := 222262, stop := 222300 ), rec( content := ".\n " , count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> d\ igraph := CayleyDigraph(G);\n\nga\ p> GroupOfCayleyDigraph(digraph) = G;\ntrue\n", count := [ 5, 4, 1, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 8 ], mathmode := "Text", name := "Example", next := 222533, root := ~, start := 222307, stop := 222532 ), rec( content := "\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Description", next := 222550, root := ~, start := 221810, stop := 222549 ) ], count := [ 5, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 222564, root := ~, start := 221644, stop := 222563 ), rec( content := "\n\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfCay\ leyDigraph" ), content := 0, count := [ 5, 4, 2, 2 ], mathmode := "Text", name := "Attr", next := 222639, root := ~, start := 222585, stop := 222638 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ generators.", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 222682, root := ~, start := 222642, stop := 222681 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 222720, root := ~, start := 222706, stop := 222719 ), rec( content := " is an i\ mmutable Cayley graph of a group or semigroup with\n respect to a set of ge\ nerators ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 222825, root := ~, start := 222814, stop := 222824 ), rec( content := " and ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 222844, root := ~, start := 222830, stop := 222843 ), rec( content := " belongs\ to\n the category ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Ref", next := 222902, root := ~, start := 222873, stop := 222901 ), rec( content := ", then\n\ ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfCayleyDigraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 222945, root := ~, start := 222913, stop := 222944 ), rec( content := " return \ the list of generators ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 222987, root := ~, start := 222976, stop := 222986 ), rec( content := "\n ov\ er which ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 223017, root := ~, start := 223003, stop := 223016 ), rec( content := " is defi\ ned.\n ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "P", next := 223038, root := ~, start := 223034, stop := 223037 ), rec( content := "\n\n \ See also ", count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "GroupOf\ CayleyDigraph" ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "Ref", next := 223087, root := ~, start := 223053, stop := 223086 ), rec( content := " \n o\ r ", count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Semigro\ upOfCayleyDigraph" ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "Ref", next := 223134, root := ~, start := 223096, stop := 223133 ), rec( content := ".\n " , count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> d\ igraph := CayleyDigraph(G);\n\nga\ p> GeneratorsOfCayleyDigraph(digraph) = GeneratorsOfGroup(G);\ntrue\ngap> digr\ aph := CayleyDigraph(G, [()]);\n\n\ gap> GeneratorsOfCayleyDigraph(digraph) = [()];\ntrue", count := [ 5, 4, 2, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 2, 5 ], mathmode := "Text", name := "Example", next := 223527, root := ~, start := 223140, stop := 223526 ), rec( content := "\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 223544, root := ~, start := 222685, stop := 223543 ) ], count := [ 5, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 223558, root := ~, start := 222570, stop := 223557 ), rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 223572, root := ~, start := 221588, stop := 223571 ) , rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Associated semigr\ oups", count := [ 5, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 223625, root := ~, start := 223585, stop := 223624 ), rec( content := "\n ", count := [ 5, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filt, digraph", Label := "for a filter a\ nd a digraph", Name := "AsSemigroup" ), content := 0, count := [ 5, 5, 1, 2 ], mathmode := "Text", name := "Oper", next := 223736, root := ~, start := 223649, stop := 223735 ), rec( attributes := rec( Arg := "filt, digraph", Name := "AsMonoid" ), content := 0, count := [ 5, 5, 1, 4 ], mathmode := "Text", name := "Oper", next := 223784, root := ~, start := 223741, stop := 223783 ), rec( attributes := rec( ), content := [ rec( content := "A semilat\ tice of partial perms.", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Returns", next := 223839, root := ~, start := 223789, stop := 223838 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 223896, root := ~, start := 223878, stop := 223895 ), rec( content := " require\ s that ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 223922, root := ~, start := 223911, stop := 223921 ), rec( content := " be equa\ l to\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermSemigroup" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 223992, root := ~, start := 223941, stop := 223991 ), rec( content := ".\n \ If ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 224017, root := ~, start := 224003, stop := 224016 ), rec( content := " is a\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 224067, root := ~, start := 224029, stop := 224066 ), rec( content := " or ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 224101, root := ~, start := 224071, stop := 224100 ), rec( content := "\n \ then ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 224131, root := ~, start := 224113, stop := 224130 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the semigroup whos\ e elements are the vertices of\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 224267, root := ~, start := 224253, stop := 224266 ), rec( content := " with th\ e binary operation\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Partial\ OrderDigraphJoinOfVertices" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 224347, root := ~, start := 224300, stop := 224346 ), rec( content := ".\n \ If ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 224372, root := ~, start := 224358, stop := 224371 ), rec( content := "\n \ satisfies ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 224428, root := ~, start := 224390, stop := 224427 ), rec( content := " but not\ \n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 224481, root := ~, start := 224443, stop := 224480 ), rec( content := " then\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 224511, root := ~, start := 224493, stop := 224510 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the semigroup whos\ e elements are the vertices of\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 224647, root := ~, start := 224633, stop := 224646 ), rec( content := " with th\ e binary operation\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Partial\ OrderDigraphMeetOfVertices" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 224727, root := ~, start := 224680, stop := 224726 ), rec( content := ".\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "P", next := 224739, root := ~, start := 224735, stop := 224738 ), rec( content := "\n\n \ The operation ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sMonoid", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "C", next := 224776, root := ~, start := 224761, stop := 224775 ), rec( content := " behaves\ similarly to ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "C", next := 224816, root := ~, start := 224798, stop := 224815 ), rec( content := "\n \ except that ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "A", next := 224846, root := ~, start := 224835, stop := 224845 ), rec( content := " may als\ o be equal to\n ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermMonoid" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 224922, root := ~, start := 224874, stop := 224921 ), rec( content := ",\n \ ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "A", next := 224944, root := ~, start := 224930, stop := 224943 ), rec( content := " must sa\ tisfy ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 224988, root := ~, start := 224958, stop := 224987 ), rec( content := ", and th\ e\n output satisfies ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsMonoid" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 225058, root := ~, start := 225021, stop := 225057 ), rec( content := ". ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "P", next := 225064, root := ~, start := 225060, stop := 225063 ), rec( content := "\n\n \ The output of both of these operations is guaranteed to be of minimal\n \ degree (see ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DegreeO\ fPartialPermSemigroup", BookName := "ref" ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "Ref", next := 225217, root := ~, start := 225160, stop := 225216 ), rec( content := ").\n \ Furthermore the ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfSemigroup", BookName := "ref" ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "Ref", next := 225292, root := ~, start := 225242, stop := 225291 ), rec( content := " of\n \ the output is guaranteed to be the unique generating set of minimal size.\n\ ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> di := Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 2, 3, 5]]);\n\ngap> S := AsSemigroup(IsPartialPermSemigro\ up, di);\n\ngap> ForAll(El\ ements(S), IsIdempotent);\ntrue\ngap> IsInverseSemigroup(S);\ntrue\ngap> Size(\ S);\n5\ngap> di := Digraph([[1], [1, 2], [1, 2, 3]]);\n\ngap> M := AsMonoid(IsPartialPermMonoid, di);\n\ngap> Size(M);\n3", count := [ 5, 5, 1, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 5, 1, 8 ], mathmode := "Text", name := "Example", next := 225924, root := ~, start := 225382, stop := 225923 ), rec( content := "\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Description", next := 225943, root := ~, start := 223844, stop := 225942 ) ], count := [ 5, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 225959, root := ~, start := 223632, stop := 225958 ), rec( content := "\n\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filt, Y, gps, ho\ ms", Label := "for a filter, semilattice digraph, and two lists", Name := "AsSemigroup" ), content := 0, count := [ 5, 5, 2, 2 ], mathmode := "Text", name := "Oper", next := 226098, root := ~, start := 225984, stop := 226097 ), rec( attributes := rec( ), content := [ rec( content := " A Cliffo\ rd semigroup of partial perms. ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 226162, root := ~, start := 226103, stop := 226161 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226219, root := ~, start := 226201, stop := 226218 ), rec( content := " require\ s that ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 226245, root := ~, start := 226234, stop := 226244 ), rec( content := " be equa\ l to\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermSemigroup" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 226315, root := ~, start := 226264, stop := 226314 ), rec( content := ".\n \ If ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Y\ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 226334, root := ~, start := 226326, stop := 226333 ), rec( content := " is a ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 226378, root := ~, start := 226340, stop := 226377 ), rec( content := " or\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 226426, root := ~, start := 226388, stop := 226425 ), rec( content := ",\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "g\ ps", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 226444, root := ~, start := 226434, stop := 226443 ), rec( content := " is a li\ st of groups corresponding to each vertex, and\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ oms", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 226516, root := ~, start := 226505, stop := 226515 ), rec( content := " is a li\ st containing for each edge ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ i, j)", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226565, root := ~, start := 226552, stop := 226564 ), rec( content := " in\n \ the transitive reduction of ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 226617, root := ~, start := 226603, stop := 226616 ), rec( content := " a tripl\ e ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j, hom]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226645, root := ~, start := 226627, stop := 226644 ), rec( content := "\n \ where ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ om", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226668, root := ~, start := 226658, stop := 226667 ), rec( content := " is a gr\ oup homomorphism from ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ps[i]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226711, root := ~, start := 226698, stop := 226710 ), rec( content := " to\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ps[j]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226734, root := ~, start := 226721, stop := 226733 ), rec( content := ", and th\ e diagram of homomorphisms commutes, then\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 226808, root := ~, start := 226790, stop := 226807 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the strong semilat\ tice of groups ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ [Y; gps; homs]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "M", next := 226930, root := ~, start := 226908, stop := 226929 ), rec( content := ".\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G1 := AlternatingGroup(4);;\ngap> G2 := SymmetricGroup(2);;\ngap> G3 := \ SymmetricGroup(3);;\ngap> gr := Digraph([[1, 3], [2, 3], [3]]);;\ngap> sgn := \ function(x)\n> if SignPerm(x) = 1 then\n> return ();\n> fi;\n> return (1, 2);\ \n> end;;\ngap> hom13 := GroupHomomorphismByFunction(G1, G3, sgn);;\ngap> hom2\ 3 := GroupHomomorphismByFunction(G2, G3, sgn);;\ngap> T := AsSemigroup(IsParti\ alPermSemigroup,\n> gr,\n> [G1, G2, G3], [[1, 3, hom13], [2, 3, hom23]]);;\nga\ p> Size(T);\n20\ngap> D := GreensDClasses(T);;\ngap> List(D, x -> Size(x));\n[\ 6, 12, 2 ]\n", count := [ 5, 5, 2, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 4 ], mathmode := "Text", name := "Example", next := 227507, root := ~, start := 226938, stop := 227506 ), rec( content := "\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 227526, root := ~, start := 226167, stop := 227525 ) ], count := [ 5, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 227542, root := ~, start := 225967, stop := 227541 ), rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 227556, root := ~, start := 223576, stop := 227555 ) , rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Planarity", count := [ 5, 6, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 2 ], mathmode := "Text", name := "Heading", next := 227597, root := ~, start := 227569, stop := 227596 ), rec( content := "\n ", count := [ 5, 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "KuratowskiPlana\ rSubdigraph" ), content := 0, count := [ 5, 6, 1, 2 ], mathmode := "Text", name := "Attr", next := 227672, root := ~, start := 227617, stop := 227671 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "K", next := 227705, root := ~, start := 227694, stop := 227704 ), rec( content := ".", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Returns", next := 227716, root := ~, start := 227675, stop := 227715 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiPlanarSubdigraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 227770, root := ~, start := 227737, stop := 227769 ), rec( content := " returns\ the immutable list of\n lists of out-neighbours of a (not necessarily indu\ ced) subdigraph of the\n digraph ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 227904, root := ~, start := 227890, stop := 227903 ), rec( content := " that wi\ tnesses the fact that ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 227948, root := ~, start := 227934, stop := 227947 ), rec( content := " is not\\ n planar, or ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "K", next := 227982, root := ~, start := 227971, stop := 227981 ), rec( content := " if ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 228000, root := ~, start := 227986, stop := 227999 ), rec( content := " is plan\ ar. In other words,\n ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiPlanarSubdigraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 228066, root := ~, start := 228033, stop := 228065 ), rec( content := " returns\ the out-neighbours of a\n subdigraph of ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 228131, root := ~, start := 228117, stop := 228130 ), rec( content := " that is\ homeomorphic to the complete graph\n with ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "5\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 228192, root := ~, start := 228184, stop := 228191 ), rec( content := " vertice\ s, or to the complete bipartite graph with vertex sets\n of sizes ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 228276, root := ~, start := 228268, stop := 228275 ), rec( content := " and ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 228289, root := ~, start := 228281, stop := 228288 ), rec( content := ". ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "P", next := 228295, root := ~, start := 228291, stop := 228294 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "A", next := 228369, root := ~, start := 228355, stop := 228368 ), rec( content := " are\n \ ignored when considering whether or not ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "A", next := 228432, root := ~, start := 228418, stop := 228431 ), rec( content := " is plan\ ar. ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "P", next := 228448, root := ~, start := 228444, stop := 228447 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "Ref", next := 228492, root := ~, start := 228463, stop := 228491 ), rec( content := "\n an\ d ", count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK33" ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "Ref", next := 228542, root := ~, start := 228501, stop := 228541 ), rec( content := ".\n " , count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "P", next := 228552, root := ~, start := 228548, stop := 228551 ), rec( content := "\n \n\ This method uses the reference implementation in\n ", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 1, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "Cite", next := 228714, root := ~, start := 228696, stop := 228713 ), rec( content := ".\n\n", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := Digra\ ph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3\ , 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, \ 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsPlanarDigraph(D);\nfalse\ngap> KuratowskiPlanarSubdigraph(D);\n[ [\ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ],\ [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\ \n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D)\ ;\nfail\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6\ , 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9,\ 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ \ngap> IsPlanarDigraph(D);\nfalse\ \ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], \ [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\n", count := [ 5, 6, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 7 ], mathmode := "Text", name := "Example", next := 230016, root := ~, start := 228717, stop := 230015 ), rec( content := "\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Description", next := 230033, root := ~, start := 227719, stop := 230032 ) ], count := [ 5, 6, 1, 1 ], mathmode := "Text", name := "ManSection", next := 230047, root := ~, start := 227602, stop := 230046 ), rec( content := "\n\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "KuratowskiOuter\ PlanarSubdigraph" ), content := 0, count := [ 5, 6, 2, 2 ], mathmode := "Text", name := "Attr", next := 230128, root := ~, start := 230068, stop := 230127 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "K", next := 230161, root := ~, start := 230150, stop := 230160 ), rec( content := ".", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Returns", next := 230172, root := ~, start := 230131, stop := 230171 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiOuterPlanarSubdigraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 230231, root := ~, start := 230193, stop := 230230 ), rec( content := " returns\ the immutable list of\n immutable lists of out-neighbours of a (not necess\ arily induced)\n subdigraph of the digraph ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 230375, root := ~, start := 230361, stop := 230374 ), rec( content := " that wi\ tnesses the fact\n that ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 230423, root := ~, start := 230409, stop := 230422 ), rec( content := " is not \ outer planar, or ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "K", next := 230459, root := ~, start := 230448, stop := 230458 ), rec( content := " if\n \ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 230481, root := ~, start := 230467, stop := 230480 ), rec( content := " is oute\ r planar. In other words,\n ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiOuterPlanarSubdigraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 230558, root := ~, start := 230520, stop := 230557 ), rec( content := " returns\ the out-neighbours of a\n subdigraph of ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 230623, root := ~, start := 230609, stop := 230622 ), rec( content := " that is\ homeomorphic to the complete graph\n with ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 230684, root := ~, start := 230676, stop := 230683 ), rec( content := " vertice\ s, or to the complete bipartite graph with vertex sets\n of sizes ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 230768, root := ~, start := 230760, stop := 230767 ), rec( content := " and ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 230781, root := ~, start := 230773, stop := 230780 ), rec( content := ". ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "P", next := 230787, root := ~, start := 230783, stop := 230786 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "A", next := 230861, root := ~, start := 230847, stop := 230860 ), rec( content := " are\n \ ignored when considering whether or not ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "A", next := 230924, root := ~, start := 230910, stop := 230923 ), rec( content := " is oute\ r planar. \n ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "P", next := 230951, root := ~, start := 230947, stop := 230950 ), rec( content := "\n\n \ See also \n ", count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 231005, root := ~, start := 230971, stop := 231004 ), rec( content := ",\n " , count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK4" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 231051, root := ~, start := 231011, stop := 231050 ), rec( content := ", and \n\ ", count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK23" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 231103, root := ~, start := 231062, stop := 231102 ), rec( content := ".\n " , count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "P", next := 231113, root := ~, start := 231109, stop := 231112 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 2, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "Cite", next := 231271, root := ~, start := 231253, stop := 231270 ), rec( content := ".\n\n", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], \ [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ]\ ]\ngap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2,\ 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9,\ 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiO\ uterPlanarSubdigraph(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], \ [ 7, 9 ], [ ], \n [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [\ 8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1,\ 2], [5, 9]]);\n\ngap> KuratowskiO\ uterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], \ [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigra\ ph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2\ , 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7\ , 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigrap\ h(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n \ [ ] ]", count := [ 5, 6, 2, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 7 ], mathmode := "Text", name := "Example", next := 232764, root := ~, start := 231274, stop := 232763 ), rec( content := "\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Description", next := 232781, root := ~, start := 230175, stop := 232780 ) ], count := [ 5, 6, 2, 1 ], mathmode := "Text", name := "ManSection", next := 232795, root := ~, start := 230053, stop := 232794 ), rec( content := "\n\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlanarEmbedding\ " ), content := 0, count := [ 5, 6, 3, 2 ], mathmode := "Text", name := "Attr", next := 232860, root := ~, start := 232816, stop := 232859 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "K", next := 232893, root := ~, start := 232882, stop := 232892 ), rec( content := ".", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Returns", next := 232904, root := ~, start := 232863, stop := 232903 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 232942, root := ~, start := 232928, stop := 232941 ), rec( content := " is a pl\ anar digraph, then \n ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lanarEmbedding", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "C", next := 232996, root := ~, start := 232974, stop := 232995 ), rec( content := " returns\ the immutable list of lists of\n out-neighbours of a subdigraph of ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 233088, root := ~, start := 233074, stop := 233087 ), rec( content := " such th\ at each vertex's\n neighbours are given in clockwise order. If ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 233175, root := ~, start := 233161, stop := 233174 ), rec( content := " is not \ planar,\n then ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "K", next := 233211, root := ~, start := 233200, stop := 233210 ), rec( content := " is retu\ rned. ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "P", next := 233229, root := ~, start := 233225, stop := 233228 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "A", next := 233303, root := ~, start := 233289, stop := 233302 ), rec( content := " are\n \ ignored by ", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lanarEmbedding", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "C", next := 233345, root := ~, start := 233323, stop := 233344 ), rec( content := ".\n " , count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "P", next := 233355, root := ~, start := 233351, stop := 233354 ), rec( content := "\n\n \ See also \n ", count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "Ref", next := 233404, root := ~, start := 233375, stop := 233403 ), rec( content := ".\n " , count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "P", next := 233414, root := ~, start := 233410, stop := 233413 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 3, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "Cite", next := 233572, root := ~, start := 233554, stop := 233571 ), rec( content := ".\n\n", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ \ 4 ], [ 6 ], [ 11, 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D :=\ Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1\ , 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, \ 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigraph,\ [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7\ ], [2, 11], [1, 2], [5, 9]]);\n\ng\ ap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11, 7 ]\ , [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDig\ raph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1,\ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5,\ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\n", count := [ 5, 6, 3, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 3, 7 ], mathmode := "Text", name := "Example", next := 234790, root := ~, start := 233575, stop := 234789 ), rec( content := "\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Description", next := 234807, root := ~, start := 232907, stop := 234806 ) ], count := [ 5, 6, 3, 1 ], mathmode := "Text", name := "ManSection", next := 234821, root := ~, start := 232801, stop := 234820 ), rec( content := "\n\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OuterPlanarEmbe\ dding" ), content := 0, count := [ 5, 6, 4, 2 ], mathmode := "Text", name := "Attr", next := 234891, root := ~, start := 234842, stop := 234890 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "K", next := 234924, root := ~, start := 234913, stop := 234923 ), rec( content := ".", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Returns", next := 234935, root := ~, start := 234894, stop := 234934 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 234973, root := ~, start := 234959, stop := 234972 ), rec( content := " is an o\ uter planar digraph, then\n ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ uterPlanarEmbedding", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "C", next := 235038, root := ~, start := 235011, stop := 235037 ), rec( content := " returns\ the immutable list of lists\n of out-neighbours of a subdigraph of ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 235130, root := ~, start := 235116, stop := 235129 ), rec( content := " such th\ at each\n vertex's neighbours are given in clockwise order. If ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 235217, root := ~, start := 235203, stop := 235216 ), rec( content := " is not\\ n outer planar, then ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "K", next := 235259, root := ~, start := 235248, stop := 235258 ), rec( content := " is retu\ rned. ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "P", next := 235277, root := ~, start := 235273, stop := 235276 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "A", next := 235351, root := ~, start := 235337, stop := 235350 ), rec( content := " are\n \ ignored by ", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ uterPlanarEmbedding", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "C", next := 235398, root := ~, start := 235371, stop := 235397 ), rec( content := ".\n " , count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "P", next := 235408, root := ~, start := 235404, stop := 235407 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "Ref", next := 235457, root := ~, start := 235423, stop := 235456 ), rec( content := ".\n " , count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "P", next := 235467, root := ~, start := 235463, stop := 235466 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 4, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 4, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "Cite", next := 235625, root := ~, start := 235607, stop := 235624 ), rec( content := ".\n\n", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph([[2\ , 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6\ , 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2\ , 3, 5, 6, 7, 8], [3, 5]]);\n\ng\ ap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteBipartit\ eDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ngap> D := Digraph(IsMuta\ bleDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6\ , 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigr\ aph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, \ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, \ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteB\ ipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\n", count := [ 5, 6, 4, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 7 ], mathmode := "Text", name := "Example", next := 236863, root := ~, start := 235628, stop := 236862 ), rec( content := "\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Description", next := 236880, root := ~, start := 234938, stop := 236879 ) ], count := [ 5, 6, 4, 1 ], mathmode := "Text", name := "ManSection", next := 236894, root := ~, start := 234827, stop := 236893 ), rec( content := "\n\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK23" ), content := 0, count := [ 5, 6, 5, 2 ], mathmode := "Text", name := "Attr", next := 236971, root := ~, start := 236915, stop := 236970 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK33" ), content := 0, count := [ 5, 6, 5, 4 ], mathmode := "Text", name := "Attr", next := 237030, root := ~, start := 236974, stop := 237029 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK4" ), content := 0, count := [ 5, 6, 5, 6 ], mathmode := "Text", name := "Attr", next := 237088, root := ~, start := 237033, stop := 237087 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "K", next := 237121, root := ~, start := 237110, stop := 237120 ), rec( content := ".", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Returns", next := 237132, root := ~, start := 237091, stop := 237131 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se attributes return the immutable list of lists of\n out-neighbours of a s\ ubdigraph of the digraph ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "A", next := 237272, root := ~, start := 237258, stop := 237271 ), rec( content := " which i\ s\n homeomorphic to one of the following:\n the complete bipartite graph\ with vertex sets of sizes ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 237391, root := ~, start := 237383, stop := 237390 ), rec( content := " and\n \ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 237408, root := ~, start := 237400, stop := 237407 ), rec( content := "; the co\ mplete bipartite graph with vertex sets of sizes ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 237473, root := ~, start := 237465, stop := 237472 ), rec( content := "\n an\ d ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 237490, root := ~, start := 237482, stop := 237489 ), rec( content := "; or the\ complete graph with ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 237527, root := ~, start := 237519, stop := 237526 ), rec( content := " vertice\ s. If\n ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "A", next := 237559, root := ~, start := 237545, stop := 237558 ), rec( content := " has no \ such subdigraphs, then ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "K", next := 237601, root := ~, start := 237590, stop := 237600 ), rec( content := " is retu\ rned. \n ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "P", next := 237624, root := ~, start := 237620, stop := 237623 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "Ref", next := 237668, root := ~, start := 237639, stop := 237667 ), rec( content := " and ", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "Ref", next := 237713, root := ~, start := 237673, stop := 237712 ), rec( content := " for mor\ e details.", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "P", next := 237735, root := ~, start := 237731, stop := 237734 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 9 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 5, 9 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 5, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 9 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "Cite", next := 237893, root := ~, start := 237875, stop := 237892 ), rec( content := ".\n\n", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 7, 11], \n> [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, \ 8, 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ \ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [\ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(\ [[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 11], \n> [4, 7], [6, 8], [2, 7],\ [2, 11], [1, 2], [5, 9]]);\n\ng\ ap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23(D);\ \n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], \ [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> Subdigr\ aphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK33(\ CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipartite\ Digraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ] ]\n\ gap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\ngap> D := Digraph(\ IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, \ 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, 8, 1\ 0 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\nga\ p> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ]\ , [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMu\ tableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 11], [4, 7], [6,\ 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphi\ cToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \ \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\nga\ p> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomor\ phicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(Complet\ eBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ]\ , [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\n", count := [ 5, 6, 5, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 5, 10 ], mathmode := "Text", name := "Example", next := 240205, root := ~, start := 237896, stop := 240204 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Description", next := 240222, root := ~, start := 237135, stop := 240221 ) ], count := [ 5, 6, 5, 1 ], mathmode := "Text", name := "ManSection", next := 240236, root := ~, start := 236900, stop := 240235 ), rec( content := "\n\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 1 ], mathmode := "Text", name := "Section", next := 240250, root := ~, start := 227560, stop := 240249 ) , rec( content := "\n\n", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 240262, root := ~, start := 133736, stop := 240261 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " attributes, operations ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Properties of digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Properties of digraphs", count := [ 6, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 240379, root := ~, start := 240338, stop := 240378 ) , rec( content := "\n\n ", count := [ 6, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Edge properties", count := [ 6, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 240426, root := ~, start := 240392, stop := 240425 ), rec( content := "\n ", count := [ 6, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphHasLoops\ " ), content := 0, count := [ 6, 1, 1, 2 ], mathmode := "Text", name := "Prop", next := 240490, root := ~, start := 240446, stop := 240489 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 240513, root := ~, start := 240502, stop := 240512 ), rec( content := " or ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 240529, root := ~, start := 240517, stop := 240528 ), rec( content := ".", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 240540, root := ~, start := 240493, stop := 240539 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 240580, root := ~, start := 240569, stop := 240579 ), rec( content := " if the \ digraph ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "A", next := 240610, root := ~, start := 240596, stop := 240609 ), rec( content := " has loo\ ps, and\n ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 240642, root := ~, start := 240630, stop := 240641 ), rec( content := " if it d\ oes not. A loop is an edge with equal source and range.\n ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "P", next := 240714, root := ~, start := 240710, stop := 240713 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "A", next := 240756, root := ~, start := 240742, stop := 240755 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ngap> DigraphHasLo\ ops(D);\ntrue\ngap> D := Digraph([[2, 3], [1], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], \ [ 3, 2 ] ]\ngap> DigraphHasLoops(D);\nfalse\ngap> D := CompleteDigraph(IsMutab\ leDigraph, 4);\n\ngap> DigraphHasLo\ ops(D);\nfalse", count := [ 6, 1, 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 5 ], mathmode := "Text", name := "Example", next := 241351, root := ~, start := 240853, stop := 241350 ), rec( content := "\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 241368, root := ~, start := 240543, stop := 241367 ) ], count := [ 6, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 241382, root := ~, start := 240431, stop := 241381 ), rec( content := "\n\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAntiSymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 2 ], mathmode := "Text", name := "Prop", next := 241454, root := ~, start := 241403, stop := 241453 ), rec( attributes := rec( Arg := "digraph", Name := "IsAntisymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 4 ], mathmode := "Text", name := "Prop", next := 241508, root := ~, start := 241457, stop := 241507 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 241531, root := ~, start := 241520, stop := 241530 ), rec( content := " or ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 241547, root := ~, start := 241535, stop := 241546 ), rec( content := ".", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 241558, root := ~, start := 241511, stop := 241557 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 241607, root := ~, start := 241596, stop := 241606 ), rec( content := " if the \ digraph ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "A", next := 241637, root := ~, start := 241623, stop := 241636 ), rec( content := "\n is\ antisymmetric, and ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 241676, root := ~, start := 241664, stop := 241675 ), rec( content := " if it i\ s not.\n ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "P", next := 241699, root := ~, start := 241695, stop := 241698 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ntisymmetric", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "E", next := 241738, root := ~, start := 241718, stop := 241737 ), rec( content := " if when\ ever there is an edge with source\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 241792, root := ~, start := 241784, stop := 241791 ), rec( content := " and ran\ ge ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 241811, root := ~, start := 241803, stop := 241810 ), rec( content := ", and an\ edge with source ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 241845, root := ~, start := 241837, stop := 241844 ), rec( content := " and ran\ ge\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 241868, root := ~, start := 241860, stop := 241867 ), rec( content := ", then t\ he vertices ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 241896, root := ~, start := 241888, stop := 241895 ), rec( content := " and ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 241909, root := ~, start := 241901, stop := 241908 ), rec( content := " are equ\ al.\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "P", next := 241929, root := ~, start := 241925, stop := 241928 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "A", next := 241971, root := ~, start := 241957, stop := 241970 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsAntisymmetricDigraph(gr1);\nfalse\ngap> DigraphEdges(gr1)\ {[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2 := Digraph([[1, 2], [3, 3], [1]]);\ \n\ngap> IsAntisymmetricDigra\ ph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2, \ 3 ], [ 3, 1 ] ]\n", count := [ 6, 1, 2, 8 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 8 ], mathmode := "Text", name := "Example", next := 242498, root := ~, start := 242072, stop := 242497 ), rec( content := "\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 242515, root := ~, start := 241561, stop := 242514 ) ], count := [ 6, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 242529, root := ~, start := 241388, stop := 242528 ), rec( content := "\n\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsBipartiteDigr\ aph" ), content := 0, count := [ 6, 1, 3, 2 ], mathmode := "Text", name := "Prop", next := 242597, root := ~, start := 242550, stop := 242596 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 242620, root := ~, start := 242609, stop := 242619 ), rec( content := " or ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 242636, root := ~, start := 242624, stop := 242635 ), rec( content := ".", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 242647, root := ~, start := 242600, stop := 242646 ), rec( attributes := rec( ), content := [ rec( content := "\n This \ property is ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 242694, root := ~, start := 242683, stop := 242693 ), rec( content := " if the \ digraph ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 242724, root := ~, start := 242710, stop := 242723 ), rec( content := " is bipa\ rtite, and\n ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 242757, root := ~, start := 242745, stop := 242756 ), rec( content := " if it i\ s not. A digraph is bipartite if and only if the\n vertices of ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 242843, root := ~, start := 242829, stop := 242842 ), rec( content := " can be \ partitioned into two non-empty sets such\n that the source and range of any e\ dge of ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 242949, root := ~, start := 242935, stop := 242948 ), rec( content := " lie in \ distinct sets.\n Equivalently, a digraph is bipartite if and only if it is 2-\ colorable; see\n ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 243109, root := ~, start := 243051, stop := 243108 ), rec( content := ". ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "P", next := 243115, root := ~, start := 243111, stop := 243114 ), rec( content := "\n\n Se\ e also ", count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "Ref", next := 243161, root := ~, start := 243128, stop := 243160 ), rec( content := ".\n " , count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "P", next := 243171, root := ~, start := 243167, stop := 243170 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "A", next := 243213, root := ~, start := 243199, stop := 243212 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(4);\n\ngap> I\ sBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(3);\n\ngap> IsBipartiteDigraph(D);\nfalse\ngap> D := CompleteBi\ partiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBipartiteDigraph(D);\ntrue", count := [ 6, 1, 3, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 3, 6 ], mathmode := "Text", name := "Example", next := 243686, root := ~, start := 243313, stop := 243685 ), rec( content := "\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 243703, root := ~, start := 242650, stop := 243702 ) ], count := [ 6, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 243717, root := ~, start := 242535, stop := 243716 ), rec( content := "\n\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteBipar\ titeDigraph" ), content := 0, count := [ 6, 1, 4, 2 ], mathmode := "Text", name := "Prop", next := 243793, root := ~, start := 243738, stop := 243792 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 243816, root := ~, start := 243805, stop := 243815 ), rec( content := " or ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 243832, root := ~, start := 243820, stop := 243831 ), rec( content := ".", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 243843, root := ~, start := 243796, stop := 243842 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 243883, root := ~, start := 243872, stop := 243882 ), rec( content := " if the \ digraph ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "A", next := 243913, root := ~, start := 243899, stop := 243912 ), rec( content := " is a co\ mplete bipartite\n digraph, and ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 243967, root := ~, start := 243955, stop := 243966 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "P", next := 243986, root := ~, start := 243982, stop := 243985 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete bipartite digraph", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "E", next := 244040, root := ~, start := 244007, stop := 244039 ), rec( content := " if it i\ s bipartite, see\n ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 244101, root := ~, start := 244069, stop := 244100 ), rec( content := ", and th\ ere exists a unique edge with\n source ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 244158, root := ~, start := 244150, stop := 244157 ), rec( content := " and ran\ ge ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 244177, root := ~, start := 244169, stop := 244176 ), rec( content := " if and \ only if ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 244201, root := ~, start := 244193, stop := 244200 ), rec( content := " and ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 244214, root := ~, start := 244206, stop := 244213 ), rec( content := " lie\n \ in different bicomponents of ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "A", next := 244266, root := ~, start := 244252, stop := 244265 ), rec( content := ", see ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 244311, root := ~, start := 244272, stop := 244310 ), rec( content := ". ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "P", next := 244317, root := ~, start := 244313, stop := 244316 ), rec( content := "\n\n \ Equivalently, a bipartite digraph with bicomponents of size ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 244391, root := ~, start := 244383, stop := 244390 ), rec( content := " and\n \ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 244408, root := ~, start := 244400, stop := 244407 ), rec( content := " is comp\ lete precisely when it has ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ mn", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 244453, root := ~, start := 244443, stop := 244452 ), rec( content := " edges, \ none of which\n are multiple edges. ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "P", next := 244503, root := ~, start := 244499, stop := 244502 ), rec( content := "\n\n \ See also ", count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eBipartiteDigraph" ), content := 0, count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "Ref", next := 244556, root := ~, start := 244518, stop := 244555 ), rec( content := ".\n " , count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "P", next := 244566, root := ~, start := 244562, stop := 244565 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "A", next := 244608, root := ~, start := 244594, stop := 244607 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(2);\n\ngap> I\ sCompleteBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(4);\n\ngap> IsBipartiteDigraph(D);\ntrue\ngap> IsComple\ teBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDig\ raph, 5, 4);\n\ngap> IsCompleteBipa\ rtiteDigraph(D);\ntrue", count := [ 6, 1, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 8 ], mathmode := "Text", name := "Example", next := 245139, root := ~, start := 244709, stop := 245138 ), rec( content := "\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 245156, root := ~, start := 243846, stop := 245155 ) ], count := [ 6, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 245170, root := ~, start := 243723, stop := 245169 ), rec( content := "\n\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteDigra\ ph" ), content := 0, count := [ 6, 1, 5, 2 ], mathmode := "Text", name := "Prop", next := 245237, root := ~, start := 245191, stop := 245236 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 245260, root := ~, start := 245249, stop := 245259 ), rec( content := " or ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 245276, root := ~, start := 245264, stop := 245275 ), rec( content := ".", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 245287, root := ~, start := 245240, stop := 245286 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 245327, root := ~, start := 245316, stop := 245326 ), rec( content := " if the \ digraph ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "A", next := 245357, root := ~, start := 245343, stop := 245356 ), rec( content := " is comp\ lete, and\n ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 245391, root := ~, start := 245379, stop := 245390 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "P", next := 245410, root := ~, start := 245406, stop := 245409 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "E", next := 245444, root := ~, start := 245429, stop := 245443 ), rec( content := " if it h\ as no loops, and for all\n ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istinct", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "E", next := 245496, root := ~, start := 245481, stop := 245495 ), rec( content := " vertice\ s ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 245514, root := ~, start := 245506, stop := 245513 ), rec( content := " and ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 245527, root := ~, start := 245519, stop := 245526 ), rec( content := ",\n t\ here is exactly one edge with source ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 245579, root := ~, start := 245571, stop := 245578 ), rec( content := " and ran\ ge ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 245598, root := ~, start := 245590, stop := 245597 ), rec( content := ".\n\n \ Equivalently, a digraph with ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "M", next := 245642, root := ~, start := 245634, stop := 245641 ), rec( content := " vertice\ s is complete precisely when\n it has ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ (n - 1)", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "M", next := 245705, root := ~, start := 245690, stop := 245704 ), rec( content := " edges, \ no loops, and no multiple edges.\n ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "P", next := 245754, root := ~, start := 245750, stop := 245753 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "A", next := 245796, root := ~, start := 245782, stop := 245795 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [1, 3], [1, 2]]);\n\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Digraph([[2, 2], [1\ ]]);\n\ngap> IsCompleteDigrap\ h(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteDigraph(D);\nfalse", count := [ 6, 1, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 5, 6 ], mathmode := "Text", name := "Example", next := 246304, root := ~, start := 245897, stop := 246303 ), rec( content := "\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 246321, root := ~, start := 245290, stop := 246320 ) ], count := [ 6, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 246335, root := ~, start := 245176, stop := 246334 ), rec( content := "\n\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteMulti\ partiteDigraph" ), content := 0, count := [ 6, 1, 6, 2 ], mathmode := "Text", name := "Prop", next := 246414, root := ~, start := 246356, stop := 246413 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 246437, root := ~, start := 246426, stop := 246436 ), rec( content := " or ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 246453, root := ~, start := 246441, stop := 246452 ), rec( content := ".", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 246464, root := ~, start := 246417, stop := 246463 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 246518, root := ~, start := 246507, stop := 246517 ), rec( content := " if ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "A", next := 246536, root := ~, start := 246522, stop := 246535 ), rec( content := " is a co\ mplete\n multipartite digraph, and ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 246593, root := ~, start := 246581, stop := 246592 ), rec( content := " if not.\ ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "P", next := 246606, root := ~, start := 246602, stop := 246605 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete multipartite digraph", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "E", next := 246663, root := ~, start := 246627, stop := 246662 ), rec( content := " if and \ only if\n its vertices can be partitioned into at least two maximal indepen\ dent sets,\n where every possible edge between these independent sets occur\ s in the\n digraph exactly once.\n ", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "P", next := 246868, root := ~, start := 246864, stop := 246867 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "A", next := 246910, root := ~, start := 246896, stop := 246909 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteMultipartiteDigraph([2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph\ (D);\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteMultipartiteDigraph(D);\ \ntrue", count := [ 6, 1, 6, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 6 ], mathmode := "Text", name := "Example", next := 247353, root := ~, start := 247011, stop := 247352 ), rec( content := "\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 247370, root := ~, start := 246467, stop := 247369 ) ], count := [ 6, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 247384, root := ~, start := 246341, stop := 247383 ), rec( content := "\n\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEmptyDigraph"\ ), content := 0, count := [ 6, 1, 7, 2 ], mathmode := "Text", name := "Prop", next := 247448, root := ~, start := 247405, stop := 247447 ), rec( attributes := rec( Arg := "digraph", Name := "IsNullDigraph" ), content := 0, count := [ 6, 1, 7, 4 ], mathmode := "Text", name := "Prop", next := 247493, root := ~, start := 247451, stop := 247492 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 247516, root := ~, start := 247505, stop := 247515 ), rec( content := " or ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 247532, root := ~, start := 247520, stop := 247531 ), rec( content := ".", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Returns", next := 247543, root := ~, start := 247496, stop := 247542 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 247583, root := ~, start := 247572, stop := 247582 ), rec( content := " if the \ digraph ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "A", next := 247613, root := ~, start := 247599, stop := 247612 ), rec( content := " is empt\ y, and\n ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 247644, root := ~, start := 247632, stop := 247643 ), rec( content := " if it i\ s not. A digraph is ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mpty", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "E", next := 247684, root := ~, start := 247672, stop := 247683 ), rec( content := " if it h\ as no\n edges.", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "P", next := 247712, root := ~, start := 247708, stop := 247711 ), rec( content := "\n\n \ ", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sNullDigraph", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "C", next := 247738, root := ~, start := 247718, stop := 247737 ), rec( content := " is a sy\ nonym for ", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEmptyDigraph", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "C", next := 247777, root := ~, start := 247756, stop := 247776 ), rec( content := ".\n " , count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "P", next := 247787, root := ~, start := 247783, stop := 247786 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "A", next := 247829, root := ~, start := 247815, stop := 247828 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], []]);\n\ngap>\ IsEmptyDigraph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D := Digraph([[]\ , [1]]);\n\ngap> IsEmptyDigraph(D);\ \nfalse\ngap> IsNullDigraph(D);\nfalse", count := [ 6, 1, 7, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 7, 8 ], mathmode := "Text", name := "Example", next := 248222, root := ~, start := 247930, stop := 248221 ), rec( content := "\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Description", next := 248239, root := ~, start := 247546, stop := 248238 ) ], count := [ 6, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 248253, root := ~, start := 247390, stop := 248252 ), rec( content := "\n\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEquivalenceDi\ graph" ), content := 0, count := [ 6, 1, 8, 2 ], mathmode := "Text", name := "Prop", next := 248323, root := ~, start := 248274, stop := 248322 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "K", next := 248346, root := ~, start := 248335, stop := 248345 ), rec( content := " or ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "K", next := 248362, root := ~, start := 248350, stop := 248361 ), rec( content := ".", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 248373, root := ~, start := 248326, stop := 248372 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is an equivalence digraph if and only if the digraph satisifies\n al\ l of ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 248511, root := ~, start := 248479, stop := 248510 ), rec( content := ",\n " , count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 248549, root := ~, start := 248517, stop := 248548 ), rec( content := " and ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 248587, root := ~, start := 248554, stop := 248586 ), rec( content := ".\n A\ partial order ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "A", next := 248623, root := ~, start := 248609, stop := 248622 ), rec( content := " corresp\ onds to an equivalence relation.\n ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "P", next := 248672, root := ~, start := 248668, stop := 248671 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "A", next := 248714, root := ~, start := 248700, stop := 248713 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2], [1, 3]]);\n\ngap> IsEquivalenceDigraph(D);\ntrue\n", count := [ 6, 1, 8, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 8, 5 ], mathmode := "Text", name := "Example", next := 248965, root := ~, start := 248811, stop := 248964 ), rec( content := "\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 248982, root := ~, start := 248376, stop := 248981 ) ], count := [ 6, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 248996, root := ~, start := 248259, stop := 248995 ), rec( content := "\n\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsFunctionalDig\ raph" ), content := 0, count := [ 6, 1, 9, 2 ], mathmode := "Text", name := "Prop", next := 249065, root := ~, start := 249017, stop := 249064 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 249088, root := ~, start := 249077, stop := 249087 ), rec( content := " or ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 249104, root := ~, start := 249092, stop := 249103 ), rec( content := ".", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 249115, root := ~, start := 249068, stop := 249114 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 249164, root := ~, start := 249153, stop := 249163 ), rec( content := " if the \ digraph ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "A", next := 249194, root := ~, start := 249180, stop := 249193 ), rec( content := " is\n \ functional. ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "P", next := 249218, root := ~, start := 249214, stop := 249217 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unctional", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "E", next := 249254, root := ~, start := 249237, stop := 249253 ), rec( content := " if ever\ y vertex is the source of a\n unique edge.\n ", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "P", next := 249315, root := ~, start := 249311, stop := 249314 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "A", next := 249357, root := ~, start := 249343, stop := 249356 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[3], [2], [2], [1], [6], [5]]);\n\ngap> IsFunctionalDigraph(gr1);\ntrue\ngap> gr2 := Digra\ ph([[1, 2], [1]]);\n\ngap> IsFunct\ ionalDigraph(gr2);\nfalse\ngap> gr3 := Digraph(3, [1, 2, 3], [2, 3, 1]);\n\ngap> IsFunctionalDigraph(gr3);\ntrue\ \n", count := [ 6, 1, 9, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 6 ], mathmode := "Text", name := "Example", next := 249868, root := ~, start := 249457, stop := 249867 ), rec( content := "\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 249885, root := ~, start := 249118, stop := 249884 ) ], count := [ 6, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 249899, root := ~, start := 249002, stop := 249898 ), rec( content := "\n\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsMultiDigraph"\ ), content := 0, count := [ 6, 1, 10, 2 ], mathmode := "Text", name := "Prop", next := 249963, root := ~, start := 249920, stop := 249962 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "K", next := 249986, root := ~, start := 249975, stop := 249985 ), rec( content := " or ", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "K", next := 250002, root := ~, start := 249990, stop := 250001 ), rec( content := ".", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 250013, root := ~, start := 249966, stop := 250012 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ultidigraph", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "E", next := 250055, root := ~, start := 250036, stop := 250054 ), rec( content := " is one \ that has at least two\n edges with equal source and range.", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "P", next := 250127, root := ~, start := 250123, stop := 250126 ), rec( content := "\n\n \ If the argument ", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "A", next := 250163, root := ~, start := 250149, stop := 250162 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \"b\", \"b\"], [\"b\", \"c\"\ , \"a\"]);\n\ngap> IsMultiDigraph(\ D);\nfalse\ngap> D := DigraphFromDigraph6String(\"&Bug\");\n\ngap> IsDuplicateFree(DigraphEdges(D));\ntrue\ngap>\ IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1, 2, 3, 2], [2, 1], [3]]);\n<\ immutable multidigraph with 3 vertices, 7 edges>\ngap> IsDuplicateFree(Digraph\ Edges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\ngap> D := DigraphMutableCopy\ (D); \n\ngap> IsMultiDigraph(D)\ ;\ntrue", count := [ 6, 1, 10, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 5 ], mathmode := "Text", name := "Example", next := 250887, root := ~, start := 250264, stop := 250886 ), rec( content := "\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 250904, root := ~, start := 250016, stop := 250903 ) ], count := [ 6, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 250918, root := ~, start := 249905, stop := 250917 ), rec( content := "\n\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsReflexiveDigr\ aph" ), content := 0, count := [ 6, 1, 11, 2 ], mathmode := "Text", name := "Prop", next := 250986, root := ~, start := 250939, stop := 250985 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 251009, root := ~, start := 250998, stop := 251008 ), rec( content := " or ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 251025, root := ~, start := 251013, stop := 251024 ), rec( content := ".", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Returns", next := 251036, root := ~, start := 250989, stop := 251035 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 251085, root := ~, start := 251074, stop := 251084 ), rec( content := " if the \ digraph ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "A", next := 251115, root := ~, start := 251101, stop := 251114 ), rec( content := " is\n \ reflexive, and ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 251150, root := ~, start := 251138, stop := 251149 ), rec( content := " if it i\ s not.\n A digraph is ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eflexive", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "E", next := 251198, root := ~, start := 251182, stop := 251197 ), rec( content := " if it h\ as a loop at every vertex. ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 11, 4 ], mathmode := "Text", name := "P", next := 251237, root := ~, start := 251233, stop := 251236 ), rec( content := "\n ", count := [ 6, 1, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "P", next := 251246, root := ~, start := 251242, stop := 251245 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "A", next := 251288, root := ~, start := 251274, stop := 251287 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [2]]);\n\ngap> IsReflexiveDigraph(D);\ntrue\ngap> D := Digraph([[3, 1], [4, 2], [3],\ [2, 1]]);\n\ngap> IsReflexiveDigr\ aph(D);\nfalse\n", count := [ 6, 1, 11, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 6 ], mathmode := "Text", name := "Example", next := 251661, root := ~, start := 251388, stop := 251660 ), rec( content := "\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Description", next := 251678, root := ~, start := 251039, stop := 251677 ) ], count := [ 6, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 251692, root := ~, start := 250924, stop := 251691 ), rec( content := "\n\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsSymmetricDigr\ aph" ), content := 0, count := [ 6, 1, 12, 2 ], mathmode := "Text", name := "Prop", next := 251760, root := ~, start := 251713, stop := 251759 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 251783, root := ~, start := 251772, stop := 251782 ), rec( content := " or ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 251799, root := ~, start := 251787, stop := 251798 ), rec( content := ".", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 251810, root := ~, start := 251763, stop := 251809 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 251859, root := ~, start := 251848, stop := 251858 ), rec( content := " if the \ digraph ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "A", next := 251889, root := ~, start := 251875, stop := 251888 ), rec( content := "\n is\ symmetric, and ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 251924, root := ~, start := 251912, stop := 251923 ), rec( content := " if it i\ s not.", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "P", next := 251942, root := ~, start := 251938, stop := 251941 ), rec( content := "\n\n \ A ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ymmetric digraph", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "E", next := 251974, root := ~, start := 251950, stop := 251973 ), rec( content := " is one \ where for each non-loop edge, having\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252038, root := ~, start := 252030, stop := 252037 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252057, root := ~, start := 252049, stop := 252056 ), rec( content := ", there \ is a corresponding edge with\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252113, root := ~, start := 252105, stop := 252112 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252132, root := ~, start := 252124, stop := 252131 ), rec( content := ". If th\ ere are ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252156, root := ~, start := 252148, stop := 252155 ), rec( content := " edges w\ ith\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252187, root := ~, start := 252179, stop := 252186 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252206, root := ~, start := 252198, stop := 252205 ), rec( content := ", then t\ here must be precisely ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252245, root := ~, start := 252237, stop := 252244 ), rec( content := "\n ed\ ges with source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252276, root := ~, start := 252268, stop := 252275 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 252295, root := ~, start := 252287, stop := 252294 ), rec( content := ". In ot\ her words, a symmetric\n digraph has a symmetric adjacency matrix ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "Ref", next := 252400, root := ~, start := 252371, stop := 252399 ), rec( content := ".\n " , count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "P", next := 252410, root := ~, start := 252406, stop := 252409 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "A", next := 252452, root := ~, start := 252438, stop := 252451 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr1);\ntrue\ngap> adj1 := AdjacencyMatri\ x(gr1);;\ngap> Display(adj1);\n[ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, \ 1, 1 ] ]\ngap> adj1 = TransposedMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(g\ r1);\ntrue\ngap> gr2 := Digraph([[2, 3], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 :=\ AdjacencyMatrix(gr2);;\ngap> Display(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, \ 1 ],\n [ 0, 1, 1 ] ]\ngap> adj2 = TransposedMat(adj2);\nfalse\n", count := [ 6, 1, 12, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 12, 6 ], mathmode := "Text", name := "Example", next := 253172, root := ~, start := 252552, stop := 253171 ), rec( content := "\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 253189, root := ~, start := 251813, stop := 253188 ) ], count := [ 6, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 253203, root := ~, start := 251698, stop := 253202 ), rec( content := "\n\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsTournament" ) , content := 0, count := [ 6, 1, 13, 2 ], mathmode := "Text", name := "Prop", next := 253265, root := ~, start := 253224, stop := 253264 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 253288, root := ~, start := 253277, stop := 253287 ), rec( content := " or ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 253304, root := ~, start := 253292, stop := 253303 ), rec( content := ".", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 253315, root := ~, start := 253268, stop := 253314 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 253364, root := ~, start := 253353, stop := 253363 ), rec( content := " if the \ digraph ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "A", next := 253394, root := ~, start := 253380, stop := 253393 ), rec( content := " is a to\ urnament,\n and ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 253432, root := ~, start := 253420, stop := 253431 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 13, 4 ], mathmode := "Text", name := "P", next := 253451, root := ~, start := 253447, stop := 253450 ), rec( content := "\n\n \ A tournament is an orientation of a complete (undirected) graph.\n Specific\ ally, a tournament is a digraph which has a unique directed edge\n (of some\ orientation) between any pair of distinct vertices, and no loops.\n ", count := [ 6, 1, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "P", next := 253686, root := ~, start := 253682, stop := 253685 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "A", next := 253728, root := ~, start := 253714, stop := 253727 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3, 4], [3, 4], [4], []]);\n\ngap> IsTournament(D);\ntrue\ngap> D := Digraph([[2], [1], [\ 3]]);\n\ngap> IsTournament(D);\nfa\ lse\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> IsTournament(D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\ \n\ngap> IsTournament(D);\nfalse\n", count := [ 6, 1, 13, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 13, 6 ], mathmode := "Text", name := "Example", next := 254309, root := ~, start := 253828, stop := 254308 ), rec( content := "\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 254326, root := ~, start := 253318, stop := 254325 ) ], count := [ 6, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 254340, root := ~, start := 253209, stop := 254339 ), rec( content := "\n\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsTransitiveDig\ raph" ), content := 0, count := [ 6, 1, 14, 2 ], mathmode := "Text", name := "Prop", next := 254409, root := ~, start := 254361, stop := 254408 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 254432, root := ~, start := 254421, stop := 254431 ), rec( content := " or ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 254448, root := ~, start := 254436, stop := 254447 ), rec( content := ".", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 254459, root := ~, start := 254412, stop := 254458 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 254508, root := ~, start := 254497, stop := 254507 ), rec( content := " if the \ digraph ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "A", next := 254538, root := ~, start := 254524, stop := 254537 ), rec( content := "\n is\ transitive, and ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 254574, root := ~, start := 254562, stop := 254573 ), rec( content := " if it i\ s not.\n\n A digraph is ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransitive", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "E", next := 254624, root := ~, start := 254607, stop := 254623 ), rec( content := " if when\ ever ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 254652, root := ~, start := 254637, stop := 254651 ), rec( content := " and\n \ ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, k ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 254676, root := ~, start := 254661, stop := 254675 ), rec( content := " are edg\ es of the digraph, then ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, k ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 254723, root := ~, start := 254708, stop := 254722 ), rec( content := " is also\ an\n edge of the digraph. ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "P", next := 254764, root := ~, start := 254760, stop := 254763 ), rec( content := "\n\n \ Let ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 254782, root := ~, start := 254774, stop := 254781 ), rec( content := " be the \ number of vertices of an arbitrary digraph, and let\n ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 254858, root := ~, start := 254850, stop := 254857 ), rec( content := " be the \ number of edges.\n For general digraphs, the methods used for this property\ use a version\n of the Floyd-Warshall algorithm, and have complexity ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 255028, root := ~, start := 255015, stop := 255027 ), rec( content := ".\n\n \ However for digraphs which are topologically sortable\n [", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "Ref", next := 255130, root := ~, start := 255094, stop := 255129 ), rec( content := "], then \ methods with\n complexity ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 255193, root := ~, start := 255166, stop := 255192 ), rec( content := " will be\ used when appropriate.\n ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "P", next := 255233, root := ~, start := 255229, stop := 255232 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "A", next := 255275, root := ~, start := 255261, stop := 255274 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 6 ], mathmode := "Text", name := "P", next := 255379, root := ~, start := 255375, stop := 255378 ), rec( content := "\n ", count := [ 6, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [3]]);\n\ngap> IsTransitiveDigraph(D);\nfalse\ngap> gr2 := Digraph([[1, 2, 3], \ [3], [3]]);\n\ngap> IsTransitiveDi\ graph(gr2);\ntrue\ngap> gr2 = DigraphTransitiveClosure(D);\ntrue\ngap> gr3 := \ Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ngap> IsTransitiveDigraph(gr3);\ntrue\n", count := [ 6, 1, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 14, 7 ], mathmode := "Text", name := "Example", next := 255841, root := ~, start := 255384, stop := 255840 ), rec( content := "\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 255858, root := ~, start := 254462, stop := 255857 ) ], count := [ 6, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 255872, root := ~, start := 254346, stop := 255871 ), rec( content := "\n\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPreorderDigra\ ph" ), content := 0, count := [ 6, 1, 15, 2 ], mathmode := "Text", name := "Prop", next := 255939, root := ~, start := 255893, stop := 255938 ), rec( attributes := rec( Arg := "digraph", Name := "IsQuasiorderDig\ raph" ), content := 0, count := [ 6, 1, 15, 4 ], mathmode := "Text", name := "Prop", next := 255990, root := ~, start := 255942, stop := 255989 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "K", next := 256013, root := ~, start := 256002, stop := 256012 ), rec( content := " or ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "K", next := 256029, root := ~, start := 256017, stop := 256028 ), rec( content := ".", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 256040, root := ~, start := 255993, stop := 256039 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is a preorder digraph if and only if the digraph satisifies both\n " , count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Ref", next := 256172, root := ~, start := 256140, stop := 256171 ), rec( content := " and ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Ref", next := 256210, root := ~, start := 256177, stop := 256209 ), rec( content := ".\n A\ preorder digraph (or quasiorder digraph) ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "A", next := 256273, root := ~, start := 256259, stop := 256272 ), rec( content := " corresp\ onds to\n the preorder relation ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "M", next := 256326, root := ~, start := 256315, stop := 256325 ), rec( content := " defined\ by ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "M", next := 256353, root := ~, start := 256338, stop := 256352 ), rec( content := " if and \ only\n if ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "C", next := 256386, root := ~, start := 256373, stop := 256385 ), rec( content := " is an e\ dge of ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "A", next := 256415, root := ~, start := 256401, stop := 256414 ), rec( content := ".\n " , count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "P", next := 256425, root := ~, start := 256421, stop := 256424 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "A", next := 256467, root := ~, start := 256453, stop := 256466 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [2, 3], [2, 3]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[1 .. 4], [1 \ .. 4], [1 .. 4], [1 .. 4]]);\n\ng\ ap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], [3], [4], [5], [1]]);\ \n\ngap> IsPreorderDigraph(D);\nfa\ lse\ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n\ngap> IsQuasiorderDigraph(D);\nfalse\n", count := [ 6, 1, 15, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 15, 7 ], mathmode := "Text", name := "Example", next := 257100, root := ~, start := 256564, stop := 257099 ), rec( content := "\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 257117, root := ~, start := 256043, stop := 257116 ) ], count := [ 6, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 257131, root := ~, start := 255878, stop := 257130 ), rec( content := "\n\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPartialOrderD\ igraph" ), content := 0, count := [ 6, 1, 16, 2 ], mathmode := "Text", name := "Prop", next := 257202, root := ~, start := 257152, stop := 257201 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "K", next := 257225, root := ~, start := 257214, stop := 257224 ), rec( content := " or ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "K", next := 257241, root := ~, start := 257229, stop := 257240 ), rec( content := ".", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Returns", next := 257252, root := ~, start := 257205, stop := 257251 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is a partial order digraph if and only if the digraph satisifies\n a\ ll of ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 257391, root := ~, start := 257359, stop := 257390 ), rec( content := ",\n " , count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAntis\ ymmetricDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 257433, root := ~, start := 257397, stop := 257432 ), rec( content := " and ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 257471, root := ~, start := 257438, stop := 257470 ), rec( content := ".\n A\ partial order ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "A", next := 257507, root := ~, start := 257493, stop := 257506 ), rec( content := " corresp\ onds\n to the partial order relation ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "M", next := 257565, root := ~, start := 257554, stop := 257564 ), rec( content := " defined\ by ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "M", next := 257592, root := ~, start := 257577, stop := 257591 ), rec( content := " if and\\ n only if ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "C", next := 257625, root := ~, start := 257612, stop := 257624 ), rec( content := " is an e\ dge of ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "A", next := 257654, root := ~, start := 257640, stop := 257653 ), rec( content := ".\n " , count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "P", next := 257664, root := ~, start := 257660, stop := 257663 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "A", next := 257706, root := ~, start := 257692, stop := 257705 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\ngap> D := CycleDigraph(5);\n\ \ngap> IsPartialOrderDigraph(D);\nfal\ se\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\n", count := [ 6, 1, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 16, 5 ], mathmode := "Text", name := "Example", next := 258210, root := ~, start := 257803, stop := 258209 ), rec( content := "\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 258227, root := ~, start := 257255, stop := 258226 ) ], count := [ 6, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 258241, root := ~, start := 257137, stop := 258240 ), rec( content := "\n\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsMeetSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 2 ], mathmode := "Text", name := "Prop", next := 258315, root := ~, start := 258262, stop := 258314 ), rec( attributes := rec( Arg := "digraph", Name := "IsJoinSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 4 ], mathmode := "Text", name := "Prop", next := 258371, root := ~, start := 258318, stop := 258370 ), rec( attributes := rec( Arg := "digraph", Name := "IsLatticeDigrap\ h" ), content := 0, count := [ 6, 1, 17, 6 ], mathmode := "Text", name := "Prop", next := 258419, root := ~, start := 258374, stop := 258418 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 258442, root := ~, start := 258431, stop := 258441 ), rec( content := " or ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 258458, root := ~, start := 258446, stop := 258457 ), rec( content := ".", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Returns", next := 258469, root := ~, start := 258422, stop := 258468 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMeetSemilatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 258521, root := ~, start := 258490, stop := 258520 ), rec( content := " returns\ ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 258541, root := ~, start := 258530, stop := 258540 ), rec( content := " if the \ digraph\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 258575, root := ~, start := 258561, stop := 258574 ), rec( content := " is a me\ et semilattice; ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sJoinSemilatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 258630, root := ~, start := 258599, stop := 258629 ), rec( content := "\n re\ turns ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 258654, root := ~, start := 258643, stop := 258653 ), rec( content := " if the \ digraph ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 258684, root := ~, start := 258670, stop := 258683 ), rec( content := " is a jo\ in semilattice;\n and ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sLatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 258740, root := ~, start := 258717, stop := 258739 ), rec( content := " returns\ ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 258760, root := ~, start := 258749, stop := 258759 ), rec( content := " if the \ digraph\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 258794, root := ~, start := 258780, stop := 258793 ), rec( content := " is both\ a meet and a join semilattice.\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "P", next := 258842, root := ~, start := 258838, stop := 258841 ), rec( content := "\n\n \ For a partial order digraph ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "Ref", next := 258911, root := ~, start := 258876, stop := 258910 ), rec( content := " the\n \ corresponding partial order is the relation ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "M", next := 258975, root := ~, start := 258964, stop := 258974 ), rec( content := ", define\ d by\n ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "M", next := 259007, root := ~, start := 258992, stop := 259006 ), rec( content := " if and \ only if ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "C", next := 259036, root := ~, start := 259023, stop := 259035 ), rec( content := " is an e\ dge.\n A digraph is a ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ eet semilattice", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "E", next := 259091, root := ~, start := 259068, stop := 259090 ), rec( content := " if it i\ s a partial order and every\n pair of vertices has a greatest lower bound (\ meet) with respect to the\n aforementioned relation. A ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ oin semilattice", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "E", next := 259256, root := ~, start := 259233, stop := 259255 ), rec( content := " is a pa\ rtial order where\n every pair of vertices has a least upper bound (join) w\ ith respect to\n the relation.\n ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "P", next := 259382, root := ~, start := 259378, stop := 259381 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "A", next := 259424, root := ~, start := 259410, stop := 259423 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsMeetSemilatticeDigraph(D);\nfalse\ngap> IsJoinSemilatticeDi\ graph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D := Digraph([[1], [2]\ , [1 .. 3]]);\n\ngap> IsJoinSemila\ tticeDigraph(D);\nfalse\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsLatti\ ceDigraph(D);\nfalse\ngap> D := Digraph([[1 .. 4], [2, 4], [3, 4], [4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\n\ true\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\ \ngap> D := Digraph([[1, 1, 1], [1, 1, 2, 2],\n> [1, 3, 3], \ [1, 2, 3, 3, 4]]);\n\ngap> I\ sMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\nga\ p> IsLatticeDigraph(D);\ntrue\n", count := [ 6, 1, 17, 10 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 10 ], mathmode := "Text", name := "Example", next := 260412, root := ~, start := 259525, stop := 260411 ), rec( content := "\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Description", next := 260429, root := ~, start := 258472, stop := 260428 ) ], count := [ 6, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 260443, root := ~, start := 258247, stop := 260442 ), rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 260457, root := ~, start := 240383, stop := 260456 ) , rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Regularity", count := [ 6, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 260499, root := ~, start := 260470, stop := 260498 ), rec( content := "\n ", count := [ 6, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsInRegularDigr\ aph" ), content := 0, count := [ 6, 2, 1, 2 ], mathmode := "Text", name := "Prop", next := 260566, root := ~, start := 260519, stop := 260565 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 260589, root := ~, start := 260578, stop := 260588 ), rec( content := " or ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 260605, root := ~, start := 260593, stop := 260604 ), rec( content := ".", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 260616, root := ~, start := 260569, stop := 260615 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 260665, root := ~, start := 260654, stop := 260664 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 260697, root := ~, start := 260689, stop := 260696 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 260737, root := ~, start := 260729, stop := 260736 ), rec( content := " of digr\ aph ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "A", next := 260763, root := ~, start := 260749, stop := 260762 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 260794, root := ~, start := 260786, stop := 260793 ), rec( content := " edges t\ erminating in ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 260824, root := ~, start := 260816, stop := 260823 ), rec( content := ".\n\n \ See also ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Ref", next := 260873, root := ~, start := 260840, stop := 260872 ), rec( content := " and\n \ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegul\ arDigraph" ), content := 0, count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Ref", next := 260912, root := ~, start := 260882, stop := 260911 ), rec( content := ".\n " , count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "P", next := 260922, root := ~, start := 260918, stop := 260921 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "A", next := 260964, root := ~, start := 260950, stop := 260963 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsInRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInRegularDigraph(C\ hainDigraph(4));\nfalse\n", count := [ 6, 2, 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 5 ], mathmode := "Text", name := "Example", next := 261194, root := ~, start := 261064, stop := 261193 ), rec( content := "\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 261211, root := ~, start := 260619, stop := 261210 ) ], count := [ 6, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 261225, root := ~, start := 260504, stop := 261224 ), rec( content := "\n\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsOutRegularDig\ raph" ), content := 0, count := [ 6, 2, 2, 2 ], mathmode := "Text", name := "Prop", next := 261294, root := ~, start := 261246, stop := 261293 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 261317, root := ~, start := 261306, stop := 261316 ), rec( content := " or ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 261333, root := ~, start := 261321, stop := 261332 ), rec( content := ".", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 261344, root := ~, start := 261297, stop := 261343 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 261393, root := ~, start := 261382, stop := 261392 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 261425, root := ~, start := 261417, stop := 261424 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 261465, root := ~, start := 261457, stop := 261464 ), rec( content := " of digr\ aph ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "A", next := 261491, root := ~, start := 261477, stop := 261490 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 261522, root := ~, start := 261514, stop := 261521 ), rec( content := " edges s\ tarting at ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 261549, root := ~, start := 261541, stop := 261548 ), rec( content := ".\n " , count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "P", next := 261559, root := ~, start := 261555, stop := 261558 ), rec( content := "\n\n \ See also ", count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInReg\ ularDigraph" ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 261606, root := ~, start := 261574, stop := 261605 ), rec( content := " and\n \ ", count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegul\ arDigraph" ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 261645, root := ~, start := 261615, stop := 261644 ), rec( content := ".\n " , count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "P", next := 261655, root := ~, start := 261651, stop := 261654 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "A", next := 261697, root := ~, start := 261683, stop := 261696 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsOutRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph\ (ChainDigraph(4));\nfalse\n", count := [ 6, 2, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 6 ], mathmode := "Text", name := "Example", next := 261929, root := ~, start := 261797, stop := 261928 ), rec( content := "\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 261946, root := ~, start := 261347, stop := 261945 ) ], count := [ 6, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 261960, root := ~, start := 261231, stop := 261959 ), rec( content := "\n\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsRegularDigrap\ h" ), content := 0, count := [ 6, 2, 3, 2 ], mathmode := "Text", name := "Prop", next := 262026, root := ~, start := 261981, stop := 262025 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 262049, root := ~, start := 262038, stop := 262048 ), rec( content := " or ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 262065, root := ~, start := 262053, stop := 262064 ), rec( content := ".", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 262076, root := ~, start := 262029, stop := 262075 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 262125, root := ~, start := 262114, stop := 262124 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 262157, root := ~, start := 262149, stop := 262156 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 262197, root := ~, start := 262189, stop := 262196 ), rec( content := " of digr\ aph ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "A", next := 262223, root := ~, start := 262209, stop := 262222 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 262254, root := ~, start := 262246, stop := 262253 ), rec( content := " edges s\ tarting and terminating at ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 262297, root := ~, start := 262289, stop := 262296 ), rec( content := ". In oth\ er words,\n the property is ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 262346, root := ~, start := 262335, stop := 262345 ), rec( content := " if ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "A", next := 262364, root := ~, start := 262350, stop := 262363 ), rec( content := " is both\ in-regular and\n and out-regular.\n\n See also ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInReg\ ularDigraph" ), content := 0, count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Ref", next := 262455, root := ~, start := 262423, stop := 262454 ), rec( content := " and\n \ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Ref", next := 262497, root := ~, start := 262464, stop := 262496 ), rec( content := ".\n " , count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "P", next := 262507, root := ~, start := 262503, stop := 262506 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "A", next := 262549, root := ~, start := 262535, stop := 262548 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsRegularDigraph(Chain\ Digraph(4));\nfalse\n", count := [ 6, 2, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 5 ], mathmode := "Text", name := "Example", next := 262775, root := ~, start := 262649, stop := 262774 ), rec( content := "\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 262792, root := ~, start := 262079, stop := 262791 ) ], count := [ 6, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 262806, root := ~, start := 261966, stop := 262805 ), rec( content := "\n\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDistanceRegul\ arDigraph" ), content := 0, count := [ 6, 2, 4, 2 ], mathmode := "Text", name := "Prop", next := 262880, root := ~, start := 262827, stop := 262879 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 262903, root := ~, start := 262892, stop := 262902 ), rec( content := " or ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 262919, root := ~, start := 262907, stop := 262918 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 262930, root := ~, start := 262883, stop := 262929 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 262968, root := ~, start := 262954, stop := 262967 ), rec( content := " is a co\ nnected symmetric graph, this property returns\n ", count := [ 6, 2, 4, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 263038, root := ~, start := 263027, stop := 263037 ), rec( content := " if for \ any two vertices ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263071, root := ~, start := 263063, stop := 263070 ), rec( content := " and ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263084, root := ~, start := 263076, stop := 263083 ), rec( content := " of ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 263102, root := ~, start := 263088, stop := 263101 ), rec( content := "\n an\ d any two integers ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263136, root := ~, start := 263128, stop := 263135 ), rec( content := " and ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263149, root := ~, start := 263141, stop := 263148 ), rec( content := " between\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263166, root := ~, start := 263158, stop := 263165 ), rec( content := " and the\ \n diameter of ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 263205, root := ~, start := 263191, stop := 263204 ), rec( content := ", the nu\ mber of vertices at distance ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263250, root := ~, start := 263242, stop := 263249 ), rec( content := "\n fr\ om ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263268, root := ~, start := 263260, stop := 263267 ), rec( content := " and dis\ tance ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263290, root := ~, start := 263282, stop := 263289 ), rec( content := " from ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263304, root := ~, start := 263296, stop := 263303 ), rec( content := " depends\ only on\n ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263333, root := ~, start := 263325, stop := 263332 ), rec( content := ", ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263343, root := ~, start := 263335, stop := 263342 ), rec( content := ", and th\ e distance between vertices ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263387, root := ~, start := 263379, stop := 263386 ), rec( content := " and\n \ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 263404, root := ~, start := 263396, stop := 263403 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "P", next := 263409, root := ~, start := 263405, stop := 263408 ), rec( content := "\n\n \ Alternatively, a distance regular graph is a graph for which there exist\n \ integers ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263511, root := ~, start := 263501, stop := 263510 ), rec( content := ", ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263523, root := ~, start := 263513, stop := 263522 ), rec( content := ", and ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263537, root := ~, start := 263529, stop := 263536 ), rec( content := " such th\ at for any two\n vertices ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263581, root := ~, start := 263573, stop := 263580 ), rec( content := ", ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263591, root := ~, start := 263583, stop := 263590 ), rec( content := " in ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "A", next := 263609, root := ~, start := 263595, stop := 263608 ), rec( content := " which a\ re distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263637, root := ~, start := 263629, stop := 263636 ), rec( content := "\n ap\ art, there are exactly ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263677, root := ~, start := 263667, stop := 263676 ), rec( content := " neighbo\ rs of ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263699, root := ~, start := 263691, stop := 263698 ), rec( content := " which a\ re at\n distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263738, root := ~, start := 263726, stop := 263737 ), rec( content := " away fr\ om ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263757, root := ~, start := 263749, stop := 263756 ), rec( content := ", and ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263773, root := ~, start := 263763, stop := 263772 ), rec( content := " neighbo\ rs of\n ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263799, root := ~, start := 263791, stop := 263798 ), rec( content := " which a\ re at distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263834, root := ~, start := 263822, stop := 263833 ), rec( content := " away fr\ om ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 263853, root := ~, start := 263845, stop := 263852 ), rec( content := ". This\n\ definition is used to check whether ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "A", next := 263914, root := ~, start := 263900, stop := 263913 ), rec( content := " is dist\ ance regular.", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "P", next := 263939, root := ~, start := 263935, stop := 263938 ), rec( content := "\n\n \ In the case where ", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "A", next := 263977, root := ~, start := 263963, stop := 263976 ), rec( content := " is not \ symmetric or not connected, the\n property is ", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "K", next := 264045, root := ~, start := 264033, stop := 264044 ), rec( content := ".\n " , count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "P", next := 264055, root := ~, start := 264051, stop := 264054 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "A", next := 264097, root := ~, start := 264083, stop := 264096 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> IsDistanceRegularD\ igraph(D);\nfalse\ngap> D := Digraph([[2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, \ 3]]);\n\ngap> IsDistanceRegularDi\ graph(D);\ntrue\n", count := [ 6, 2, 4, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 7 ], mathmode := "Text", name := "Example", next := 264473, root := ~, start := 264198, stop := 264472 ), rec( content := "\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 264490, root := ~, start := 262933, stop := 264489 ) ], count := [ 6, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 264504, root := ~, start := 262812, stop := 264503 ), rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 264518, root := ~, start := 260461, stop := 264517 ) , rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Connectivity and \ cycles", count := [ 6, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 264573, root := ~, start := 264531, stop := 264572 ), rec( content := "\n ", count := [ 6, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAcyclicDigrap\ h" ), content := 0, count := [ 6, 3, 1, 2 ], mathmode := "Text", name := "Prop", next := 264638, root := ~, start := 264593, stop := 264637 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 264661, root := ~, start := 264650, stop := 264660 ), rec( content := " or ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 264677, root := ~, start := 264665, stop := 264676 ), rec( content := ".", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 264688, root := ~, start := 264641, stop := 264687 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 264737, root := ~, start := 264726, stop := 264736 ), rec( content := " if the \ digraph ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "A", next := 264767, root := ~, start := 264753, stop := 264766 ), rec( content := " is acyc\ lic, and\n ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 264800, root := ~, start := 264788, stop := 264799 ), rec( content := " if it i\ s not. A digraph is ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ cyclic", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "E", next := 264842, root := ~, start := 264828, stop := 264841 ), rec( content := " if ever\ y directed\n cycle on the digraph is trivial. See Section ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Ref", next := 264960, root := ~, start := 264910, stop := 264959 ), rec( content := " for the\ definition of a directed cycle, and of a trivial\n directed cycle.", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "P", next := 265041, root := ~, start := 265037, stop := 265040 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 265109, root := ~, start := 265096, stop := 265108 ), rec( content := " where\n\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 265128, root := ~, start := 265120, stop := 265127 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 265201, root := ~, start := 265193, stop := 265200 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "P", next := 265248, root := ~, start := 265244, stop := 265247 ), rec( content := "\n\n \ If the argument ", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "A", next := 265284, root := ~, start := 265270, stop := 265283 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y)\ \n> return IsEmpty(Intersection(x, y));\n> end);;\ngap> D := Digraph(Peterse\ n);\n\ngap> IsAcyclicDigraph(D);\ \nfalse\ngap> D := DigraphFromDiSparse6String(\n> \".b_OGCIDBaPGkULEbQHCeRIdrH\ cuZMfRyDAbPhTi|zF\");\n\ngap> Is\ AcyclicDigraph(D);\ntrue\ngap> IsAcyclicDigraph(ChainDigraph(10));\ntrue\ngap>\ D := CompleteDigraph(IsMutableDigraph, 4);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> IsAcyclicDigraph(CycleDigra\ ph(10));\nfalse", count := [ 6, 3, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 6 ], mathmode := "Text", name := "Example", next := 266033, root := ~, start := 265385, stop := 266032 ), rec( content := "\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 266050, root := ~, start := 264691, stop := 266049 ) ], count := [ 6, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 266064, root := ~, start := 264578, stop := 266063 ), rec( content := "\n\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsChainDigraph"\ ), content := 0, count := [ 6, 3, 2, 2 ], mathmode := "Text", name := "Prop", next := 266128, root := ~, start := 266085, stop := 266127 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 266151, root := ~, start := 266140, stop := 266150 ), rec( content := " or ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 266167, root := ~, start := 266155, stop := 266166 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 266178, root := ~, start := 266131, stop := 266177 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sChainDigraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "C", next := 266220, root := ~, start := 266199, stop := 266219 ), rec( content := " returns\ ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 266240, root := ~, start := 266229, stop := 266239 ), rec( content := " if the \ digraph ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "A", next := 266270, root := ~, start := 266256, stop := 266269 ), rec( content := " is\n \ isomorphic to the chain digraph with the same number of vertices as\n ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "A", next := 266364, root := ~, start := 266350, stop := 266363 ), rec( content := ", and ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 266382, root := ~, start := 266370, stop := 266381 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ChainDi\ graph" ), content := 0, count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Ref", next := 266433, root := ~, start := 266401, stop := 266432 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "P", next := 266438, root := ~, start := 266434, stop := 266437 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ hain", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "E", next := 266471, root := ~, start := 266459, stop := 266470 ), rec( content := " if and \ only if it is a directed tree, in which\n every vertex has out degree at mo\ st one; see ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDirec\ tedTree" ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "Ref", next := 266596, root := ~, start := 266568, stop := 266595 ), rec( content := "\n an\ d ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ ees" ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "Ref", next := 266629, root := ~, start := 266605, stop := 266628 ), rec( content := ".\n " , count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "P", next := 266639, root := ~, start := 266635, stop := 266638 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "A", next := 266681, root := ~, start := 266667, stop := 266680 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsChainDigraph(D);\nfalse\ngap> D := ChainDigraph(5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D := \ DigraphReverse(D);\n\ngap> IsChain\ Digraph(D);\ntrue\ngap> D := ChainDigraph(IsMutableDigraph, 5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphReve\ rse(D);\n\ngap> IsChainDigraph(D);\n\ true", count := [ 6, 3, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 6 ], mathmode := "Text", name := "Example", next := 267340, root := ~, start := 266778, stop := 267339 ), rec( content := "\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 267357, root := ~, start := 266181, stop := 267356 ) ], count := [ 6, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 267371, root := ~, start := 266070, stop := 267370 ), rec( content := "\n\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsConnectedDigr\ aph" ), content := 0, count := [ 6, 3, 3, 2 ], mathmode := "Text", name := "Prop", next := 267439, root := ~, start := 267392, stop := 267438 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 267462, root := ~, start := 267451, stop := 267461 ), rec( content := " or ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 267478, root := ~, start := 267466, stop := 267477 ), rec( content := ".", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 267489, root := ~, start := 267442, stop := 267488 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 267538, root := ~, start := 267527, stop := 267537 ), rec( content := " if the \ digraph ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "A", next := 267568, root := ~, start := 267554, stop := 267567 ), rec( content := "\n is\ weakly connected and ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 267609, root := ~, start := 267597, stop := 267608 ), rec( content := " if it i\ s not. A digraph\n ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "A", next := 267653, root := ~, start := 267639, stop := 267652 ), rec( content := " is ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ eakly connected", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "E", next := 267680, root := ~, start := 267657, stop := 267679 ), rec( content := " if it i\ s possible to travel\n from any vertex to any other vertex by traversing ed\ ges in either\n direction (possibly against the orientation of some of them\ ). ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "P", next := 267849, root := ~, start := 267845, stop := 267848 ), rec( content := "\n\n \ The method used in this function has complexity ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m)", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 267914, root := ~, start := 267903, stop := 267913 ), rec( content := " if the\\ n digraph's ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Source" ), content := 0, count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "Ref", next := 267963, root := ~, start := 267936, stop := 267962 ), rec( content := " attribu\ te is set, otherwise it has\n complexity ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 268027, root := ~, start := 268014, stop := 268026 ), rec( content := " (where\\ n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 268047, root := ~, start := 268039, stop := 268046 ), rec( content := " is the \ number of edges and\n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 268087, root := ~, start := 268079, stop := 268086 ), rec( content := " is the \ number of vertices of the digraph).\n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "P", next := 268139, root := ~, start := 268135, stop := 268138 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "A", next := 268181, root := ~, start := 268167, stop := 268180 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> \ D := Digraph([[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\nga\ p> D := Digraph(IsMutableDigraph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D\ );\ntrue\ngap> D := Digraph(IsMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> \ IsConnectedDigraph(D);\nfalse", count := [ 6, 3, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 6 ], mathmode := "Text", name := "Example", next := 268643, root := ~, start := 268282, stop := 268642 ), rec( content := "\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 268660, root := ~, start := 267492, stop := 268659 ) ], count := [ 6, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 268674, root := ~, start := 267377, stop := 268673 ), rec( content := "\n\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsBiconnectedDi\ graph" ), content := 0, count := [ 6, 3, 4, 2 ], mathmode := "Text", name := "Prop", next := 268744, root := ~, start := 268695, stop := 268743 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "K", next := 268767, root := ~, start := 268756, stop := 268766 ), rec( content := " or ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "K", next := 268783, root := ~, start := 268771, stop := 268782 ), rec( content := ".", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 268794, root := ~, start := 268747, stop := 268793 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ iconnected", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "E", next := 268856, root := ~, start := 268838, stop := 268855 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 268935, root := ~, start := 268903, stop := 268934 ), rec( content := ") when a\ ny vertex is removed. \n If ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "A", next := 268983, root := ~, start := 268975, stop := 268982 ), rec( content := " has at \ least 3 vertices, then ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "C", next := 269041, root := ~, start := 269014, stop := 269040 ), rec( content := "\n \ implies ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 269089, root := ~, start := 269056, stop := 269088 ), rec( content := ";\n s\ ee ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 269131, root := ~, start := 269099, stop := 269130 ), rec( content := " or ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 269156, root := ~, start := 269135, stop := 269155 ), rec( content := " for a m\ ore\n detailed explanation.\n ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "P", next := 269202, root := ~, start := 269198, stop := 269201 ), rec( content := "\n\n \ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "C", next := 269235, root := ~, start := 269208, stop := 269234 ), rec( content := " returns\ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 269255, root := ~, start := 269244, stop := 269254 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "A", next := 269289, root := ~, start := 269275, stop := 269288 ), rec( content := " is bico\ nnected, and ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 269322, root := ~, start := 269310, stop := 269321 ), rec( content := " if it i\ s not. In\n particular, ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "C", next := 269383, root := ~, start := 269356, stop := 269382 ), rec( content := " returns\ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 269404, root := ~, start := 269392, stop := 269403 ), rec( content := " if\n \ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "A", next := 269426, root := ~, start := 269412, stop := 269425 ), rec( content := " is not \ connected. ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "P", next := 269449, root := ~, start := 269445, stop := 269448 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "A", next := 269491, root := ~, start := 269477, stop := 269490 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "P", next := 269590, root := ~, start := 269586, stop := 269589 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "P", next := 269643, root := ~, start := 269639, stop := 269642 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 269711, root := ~, start := 269698, stop := 269710 ), rec( content := " where\n\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 269730, root := ~, start := 269722, stop := 269729 ), rec( content := " is the \ number of edges and ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 269766, root := ~, start := 269758, stop := 269765 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "P", next := 269821, root := ~, start := 269817, stop := 269820 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 269857, root := ~, start := 269836, stop := 269856 ), rec( content := ", ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 269891, root := ~, start := 269859, stop := 269890 ), rec( content := ", and \n\ ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 269935, root := ~, start := 269902, stop := 269934 ), rec( content := ".\n " , count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "P", next := 269945, root := ~, start := 269941, stop := 269944 ), rec( content := "\n \ \n", count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsBiconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBic\ onnectedDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2],\ [3], [3, 3, 4, 4]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := Comple\ teBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBiconnectedDigraph(D);\ntrue", count := [ 6, 3, 4, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 10 ], mathmode := "Text", name := "Example", next := 270333, root := ~, start := 269953, stop := 270332 ), rec( content := "\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 270350, root := ~, start := 268797, stop := 270349 ) ], count := [ 6, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 270364, root := ~, start := 268680, stop := 270363 ), rec( content := "\n\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "IsBridgelessDig\ raph" ), content := 0, count := [ 6, 3, 5, 2 ], mathmode := "Text", name := "Prop", next := 270427, root := ~, start := 270385, stop := 270426 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "K", next := 270450, root := ~, start := 270439, stop := 270449 ), rec( content := " or ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "K", next := 270466, root := ~, start := 270454, stop := 270465 ), rec( content := ".", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Returns", next := 270477, root := ~, start := 270430, stop := 270476 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ ridgeless", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "E", next := 270538, root := ~, start := 270521, stop := 270537 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 270617, root := ~, start := 270585, stop := 270616 ), rec( content := ") when a\ ny edge is removed. \n If ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "A", next := 270661, root := ~, start := 270653, stop := 270660 ), rec( content := " has at \ least 3 vertices, then ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 270730, root := ~, start := 270692, stop := 270729 ), rec( content := " implies\ ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "C", next := 270765, root := ~, start := 270739, stop := 270764 ), rec( content := ";\n s\ ee ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 270807, root := ~, start := 270775, stop := 270806 ), rec( content := " or ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 270832, root := ~, start := 270811, stop := 270831 ), rec( content := " for a m\ ore\n detailed explanation.\n ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "P", next := 270878, root := ~, start := 270874, stop := 270877 ), rec( content := "\n\n \ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "C", next := 270910, root := ~, start := 270884, stop := 270909 ), rec( content := " returns\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 270930, root := ~, start := 270919, stop := 270929 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 270958, root := ~, start := 270950, stop := 270957 ), rec( content := " is brid\ geless, and ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 270990, root := ~, start := 270978, stop := 270989 ), rec( content := " if it i\ s not. In\n particular, ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "C", next := 271050, root := ~, start := 271024, stop := 271049 ), rec( content := " returns\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 271071, root := ~, start := 271059, stop := 271070 ), rec( content := " if\n \ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 271087, root := ~, start := 271079, stop := 271086 ), rec( content := " is not \ connected. ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "P", next := 271110, root := ~, start := 271106, stop := 271109 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "A", next := 271146, root := ~, start := 271138, stop := 271145 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "P", next := 271245, root := ~, start := 271241, stop := 271244 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "P", next := 271298, root := ~, start := 271294, stop := 271297 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 271366, root := ~, start := 271353, stop := 271365 ), rec( content := " where\n\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 271385, root := ~, start := 271377, stop := 271384 ), rec( content := " is the \ number of edges and ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 271421, root := ~, start := 271413, stop := 271420 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "P", next := 271476, root := ~, start := 271472, stop := 271475 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 271512, root := ~, start := 271491, stop := 271511 ), rec( content := ", ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 271546, root := ~, start := 271514, stop := 271545 ), rec( content := ", and \n\ ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 271591, root := ~, start := 271557, stop := 271590 ), rec( content := ".\n \ \n", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsBridgelessDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBrid\ gelessDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [\ 3], [3, 3, 4, 4]]);;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> D := CompleteB\ ipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> D := Digraph([[2, 5], [1, 3,\ 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> IsBiconnectedDigraph(D);\nfalse\n\ gap> D := Digraph([[2], [3], [4], [2]]);\n\ngap> IsBridgelessDigraph(D);\nfalse\ngap> IsBiconnectedDigraph(D);\n\ false\ngap> IsBridgelessDigraph(ChainDigraph(2));\nfalse\ngap> IsBiconnectedDi\ graph(ChainDigraph(2));\ntrue\n", count := [ 6, 3, 5, 9 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 9 ], mathmode := "Text", name := "Example", next := 272415, root := ~, start := 271600, stop := 272414 ), rec( content := "\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Description", next := 272432, root := ~, start := 270480, stop := 272431 ) ], count := [ 6, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 272446, root := ~, start := 270370, stop := 272445 ), rec( content := "\n\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsStronglyConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 6, 2 ], mathmode := "Text", name := "Prop", next := 272522, root := ~, start := 272467, stop := 272521 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 272545, root := ~, start := 272534, stop := 272544 ), rec( content := " or ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 272561, root := ~, start := 272549, stop := 272560 ), rec( content := ".", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 272572, root := ~, start := 272525, stop := 272571 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 272621, root := ~, start := 272610, stop := 272620 ), rec( content := " if the \ digraph ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "A", next := 272651, root := ~, start := 272637, stop := 272650 ), rec( content := " is stro\ ngly\n connected and ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 272694, root := ~, start := 272682, stop := 272693 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "P", next := 272713, root := ~, start := 272709, stop := 272712 ), rec( content := "\n\n \ A digraph ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "A", next := 272743, root := ~, start := 272729, stop := 272742 ), rec( content := " is ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trongly connected", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "E", next := 272772, root := ~, start := 272747, stop := 272771 ), rec( content := " if ther\ e is a directed\n path from every vertex to every other vertex. See Sectio\ n ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "Ref", next := 272909, root := ~, start := 272859, stop := 272908 ), rec( content := " for the\ definition of a directed\n path. ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "P", next := 272960, root := ~, start := 272956, stop := 272959 ), rec( content := "\n\n \ The method used in this operation is based on Gabow's Algorithm ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "Cite", next := 273055, root := ~, start := 273030, stop := 273054 ), rec( content := " and has\ complexity ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 273088, root := ~, start := 273075, stop := 273087 ), rec( content := ", where \ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 273104, root := ~, start := 273096, stop := 273103 ), rec( content := " is\n \ the number of edges (counting multiple edges as one) and ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 273177, root := ~, start := 273169, stop := 273176 ), rec( content := " is the\\ n number of vertices in the digraph.\n ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "P", next := 273232, root := ~, start := 273228, stop := 273231 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "A", next := 273274, root := ~, start := 273260, stop := 273273 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> D := DigraphRemoveEdges(D,\ [[250000, 1]]);\n\ngap>\ IsStronglyConnectedDigraph(D);\nfalse\ngap> D := CycleDigraph(IsMutableDigrap\ h, 250000);\n\ngap> IsStro\ nglyConnectedDigraph(D);\ntrue\ngap> DigraphRemoveEdge(D, [250000, 1]);\n\ngap> IsStronglyConnectedDigra\ ph(D);\nfalse\n", count := [ 6, 3, 6, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 7 ], mathmode := "Text", name := "Example", next := 273951, root := ~, start := 273375, stop := 273950 ), rec( content := "\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 273968, root := ~, start := 272575, stop := 273967 ) ], count := [ 6, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 273982, root := ~, start := 272452, stop := 273981 ), rec( content := "\n\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAperiodicDigr\ aph" ), content := 0, count := [ 6, 3, 7, 2 ], mathmode := "Text", name := "Prop", next := 274050, root := ~, start := 274003, stop := 274049 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 274073, root := ~, start := 274062, stop := 274072 ), rec( content := " or ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 274089, root := ~, start := 274077, stop := 274088 ), rec( content := ".", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 274100, root := ~, start := 274053, stop := 274099 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 274149, root := ~, start := 274138, stop := 274148 ), rec( content := " if the \ digraph ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "A", next := 274179, root := ~, start := 274165, stop := 274178 ), rec( content := "\n is\ aperiodic, i.e. if its ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Period" ), content := 0, count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Ref", next := 274239, root := ~, start := 274210, stop := 274238 ), rec( content := " is equa\ l to 1.\n Otherwise, the property is ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 274298, root := ~, start := 274286, stop := 274297 ), rec( content := ".\n " , count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "P", next := 274308, root := ~, start := 274304, stop := 274307 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "A", next := 274350, root := ~, start := 274336, stop := 274349 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Dig\ raph([[2], [3, 5], [4], [5], [1, 2]]);\n\ngap> IsAperiodicDigraph(D);\ntrue\ngap> D := Digraph(IsMutableDigraph,\ [[2], [3, 5], [4], [5], [1, 2]]);\n\ \ngap> IsAperiodicDigraph(D);\ntrue", count := [ 6, 3, 7, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "Example", next := 274895, root := ~, start := 274450, stop := 274894 ), rec( content := "\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 274912, root := ~, start := 274103, stop := 274911 ) ], count := [ 6, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 274926, root := ~, start := 273988, stop := 274925 ), rec( content := "\n\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDirectedTree"\ ), content := 0, count := [ 6, 3, 8, 2 ], mathmode := "Text", name := "Prop", next := 274990, root := ~, start := 274947, stop := 274989 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 275013, root := ~, start := 275002, stop := 275012 ), rec( content := " or ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 275029, root := ~, start := 275017, stop := 275028 ), rec( content := ".", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 275040, root := ~, start := 274993, stop := 275039 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 275080, root := ~, start := 275069, stop := 275079 ), rec( content := " if the \ digraph ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "A", next := 275110, root := ~, start := 275096, stop := 275109 ), rec( content := " is a di\ rected tree, and\n ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 275151, root := ~, start := 275139, stop := 275150 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "P", next := 275170, root := ~, start := 275166, stop := 275169 ), rec( content := "\n\n \ A ", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ irected tree", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "E", next := 275198, root := ~, start := 275178, stop := 275197 ), rec( content := " is an a\ cyclic digraph with precisely 1 source,\n such that no two vertices share a\ n out-neighbour.\n Note the empty digraph is not considered a directed\n \ tree as it has no source. ", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "P", next := 275390, root := ~, start := 275386, stop := 275389 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sources" ), content := 0, count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "Ref", next := 275433, root := ~, start := 275405, stop := 275432 ), rec( content := ".\n " , count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "P", next := 275443, root := ~, start := 275439, stop := 275442 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "A", next := 275485, root := ~, start := 275471, stop := 275484 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [2]]);\n\ng\ ap> IsDirectedTree(D);\nfalse\ngap> D := Digraph([[3], [3], []]);\n\ngap> IsDirectedTree(D);\nfalse\ngap> D := D\ igraph([[2], [3], []]);\n\ngap> Is\ DirectedTree(D);\ntrue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [], []]);\ \n\ngap> IsDirectedTree(D);\ntrue\ \n", count := [ 6, 3, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "Example", next := 276069, root := ~, start := 275586, stop := 276068 ), rec( content := "\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 276086, root := ~, start := 275043, stop := 276085 ) ], count := [ 6, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 276100, root := ~, start := 274932, stop := 276099 ), rec( content := "\n\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedTre\ e" ), content := 0, count := [ 6, 3, 9, 2 ], mathmode := "Text", name := "Prop", next := 276166, root := ~, start := 276121, stop := 276165 ), rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedFor\ est" ), content := 0, count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "Prop", next := 276216, root := ~, start := 276169, stop := 276215 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 276239, root := ~, start := 276228, stop := 276238 ), rec( content := " or ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 276255, root := ~, start := 276243, stop := 276254 ), rec( content := ".", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 276266, root := ~, start := 276219, stop := 276265 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ property ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedTree", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "C", next := 276323, root := ~, start := 276300, stop := 276322 ), rec( content := " returns\ ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 276343, root := ~, start := 276332, stop := 276342 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "A", next := 276377, root := ~, start := 276363, stop := 276376 ), rec( content := " is an u\ ndirected tree, and the property\n ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedForest", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "C", next := 276447, root := ~, start := 276422, stop := 276446 ), rec( content := " returns\ ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 276467, root := ~, start := 276456, stop := 276466 ), rec( content := " if ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "A", next := 276485, root := ~, start := 276471, stop := 276484 ), rec( content := " is an\n\ undirected forest; otherwise, these properties return ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 276562, root := ~, start := 276550, stop := 276561 ), rec( content := ". ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "P", next := 276568, root := ~, start := 276564, stop := 276567 ), rec( content := "\n\n \ An ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected tree", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "E", next := 276599, root := ~, start := 276577, stop := 276598 ), rec( content := " is a sy\ mmetric digraph without loops, in which for\n any pair of distinct vertices\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 276693, root := ~, start := 276685, stop := 276692 ), rec( content := " and ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 276706, root := ~, start := 276698, stop := 276705 ), rec( content := ", there \ is exactly one\n directed path from ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 276760, root := ~, start := 276752, stop := 276759 ), rec( content := " to ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 276772, root := ~, start := 276764, stop := 276771 ), rec( content := ". See ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 276816, root := ~, start := 276778, stop := 276815 ), rec( content := " and ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 276850, root := ~, start := 276821, stop := 276849 ), rec( content := ", and se\ e\n Section ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 276916, root := ~, start := 276872, stop := 276915 ), rec( content := " for the\ definition of\n directed path. This definition implies that an undirected \ tree has\n no multiple edges. ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "P", next := 277037, root := ~, start := 277033, stop := 277036 ), rec( content := "\n\n \ An ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected forest", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "E", next := 277070, root := ~, start := 277046, stop := 277069 ), rec( content := " is a di\ graph, each of whose connected components\n is an undirected tree. In other\ words, an undirected forest is isomorphic to\n a disjoint union of undirec\ ted trees. See ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "Ref", next := 277294, root := ~, start := 277247, stop := 277293 ), rec( content := " and ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "Ref", next := 277371, root := ~, start := 277299, stop := 277370 ), rec( content := ". In par\ ticular, every\n undirected tree is an undirected forest. ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 8 ], mathmode := "Text", name := "P", next := 277443, root := ~, start := 277439, stop := 277442 ), rec( content := "\n\n \ Please note that the digraph with zero vertices is considered to be neither\n \ an undirected tree nor an undirected forest.\n ", count := [ 6, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "P", next := 277582, root := ~, start := 277578, stop := 277581 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "A", next := 277624, root := ~, start := 277610, stop := 277623 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [3], [1, 2]]);\n\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmetricDigraph(D) and not D\ igraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], [1, 4], [3], [2]]);\n\ngap> IsConnectedDigraph(D);\nfalse\ \ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D);\ntrue\ngap> D \ := Digraph([[1, 2], [1], [2]]);\n\ \ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfalse\ngap> IsSymmetricD\ igraph(D) or not DigraphHasLoops(D);\nfalse", count := [ 6, 3, 9, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 9, 10 ], mathmode := "Text", name := "Example", next := 278325, root := ~, start := 277725, stop := 278324 ), rec( content := "\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 278342, root := ~, start := 276269, stop := 278341 ) ], count := [ 6, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 278356, root := ~, start := 276106, stop := 278355 ), rec( content := "\n\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEulerianDigra\ ph" ), content := 0, count := [ 6, 3, 10, 2 ], mathmode := "Text", name := "Prop", next := 278423, root := ~, start := 278377, stop := 278422 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "K", next := 278446, root := ~, start := 278435, stop := 278445 ), rec( content := " or ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "K", next := 278462, root := ~, start := 278450, stop := 278461 ), rec( content := ".", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 278473, root := ~, start := 278426, stop := 278472 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns true if the digraph ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "A", next := 278550, root := ~, start := 278536, stop := 278549 ), rec( content := " is Eule\ rian.\n ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "P", next := 278572, root := ~, start := 278568, stop := 278571 ), rec( content := "\n\n \ A connected digraph is called ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ulerian", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "E", next := 278623, root := ~, start := 278608, stop := 278622 ), rec( content := " if ther\ e exists a directed\n circuit on the digraph which includes every edge exac\ tly once. See\n Section ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "Ref", next := 278779, root := ~, start := 278735, stop := 278778 ), rec( content := " for the\ definition of\n a directed circuit. Note that the empty digraph with at mo\ st one vertex is\n considered to be Eulerian.\n ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "P", next := 278920, root := ~, start := 278916, stop := 278919 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "A", next := 278962, root := ~, start := 278948, stop := 278961 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[]]);\n\ngap> IsEul\ erianDigraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph([[3\ ], [], [2]]);\n\ngap> IsEulerianDi\ graph(D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> IsEulerianDigraph(D);\ntrue\n", count := [ 6, 3, 10, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "Example", next := 279525, root := ~, start := 279063, stop := 279524 ), rec( content := "\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 279542, root := ~, start := 278476, stop := 279541 ) ], count := [ 6, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 279556, root := ~, start := 278362, stop := 279555 ), rec( content := "\n\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsHamiltonianDi\ graph" ), content := 0, count := [ 6, 3, 11, 2 ], mathmode := "Text", name := "Prop", next := 279626, root := ~, start := 279577, stop := 279625 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 279649, root := ~, start := 279638, stop := 279648 ), rec( content := " or ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 279665, root := ~, start := 279653, stop := 279664 ), rec( content := ".", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Returns", next := 279676, root := ~, start := 279629, stop := 279675 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "A", next := 279714, root := ~, start := 279700, stop := 279713 ), rec( content := " is Hami\ ltonian, then this property returns\n ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 279773, root := ~, start := 279762, stop := 279772 ), rec( content := ", and ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 279791, root := ~, start := 279779, stop := 279790 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "P", next := 279810, root := ~, start := 279806, stop := 279809 ), rec( content := "\n\n \ A digraph with ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "C", next := 279839, root := ~, start := 279831, stop := 279838 ), rec( content := " vertice\ s is ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "E", next := 279870, root := ~, start := 279852, stop := 279869 ), rec( content := " if it h\ as a\n directed cycle of length ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "C", next := 279920, root := ~, start := 279912, stop := 279919 ), rec( content := ". See Se\ ction ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "Ref", next := 279983, root := ~, start := 279934, stop := 279982 ), rec( content := " for the\ definition of a directed cycle.\n Note the empty digraphs on 0 and 1 verti\ ces are considered to be\n Hamiltonian.", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "P", next := 280113, root := ~, start := 280109, stop := 280112 ), rec( content := "\n\n \ The method used in this operation has the worst case complexity as\n ", count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "Ref", next := 280223, root := ~, start := 280190, stop := 280222 ), rec( content := ".\n " , count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "P", next := 280233, root := ~, start := 280229, stop := 280232 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "A", next := 280275, root := ~, start := 280261, stop := 280274 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> g := Digraph([[]]);\n\ngap> IsHam\ iltonianDigraph(g);\ntrue\ngap> g := Digraph([[2], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g := Digr\ aph([[3], [], [2]]);\n\ngap> IsHam\ iltonianDigraph(g);\nfalse\ngap> g := Digraph([[2], [3], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\n", count := [ 6, 3, 11, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 11, 7 ], mathmode := "Text", name := "Example", next := 280851, root := ~, start := 280376, stop := 280850 ), rec( content := "\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Description", next := 280868, root := ~, start := 279679, stop := 280867 ) ], count := [ 6, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 280882, root := ~, start := 279562, stop := 280881 ), rec( content := "\n\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCycleDigraph"\ ), content := 0, count := [ 6, 3, 12, 2 ], mathmode := "Text", name := "Prop", next := 280946, root := ~, start := 280903, stop := 280945 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 280969, root := ~, start := 280958, stop := 280968 ), rec( content := " or ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 280985, root := ~, start := 280973, stop := 280984 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 280996, root := ~, start := 280949, stop := 280995 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCycleDigraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "C", next := 281038, root := ~, start := 281017, stop := 281037 ), rec( content := " returns\ ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 281058, root := ~, start := 281047, stop := 281057 ), rec( content := " if the \ digraph ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "A", next := 281088, root := ~, start := 281074, stop := 281087 ), rec( content := " is\n \ isomorphic to the cycle digraph with the same number of vertices as\n ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "A", next := 281182, root := ~, start := 281168, stop := 281181 ), rec( content := ", and ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 281200, root := ~, start := 281188, stop := 281199 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CycleDi\ graph" ), content := 0, count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Ref", next := 281251, root := ~, start := 281219, stop := 281250 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "P", next := 281256, root := ~, start := 281252, stop := 281255 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "E", next := 281289, root := ~, start := 281277, stop := 281288 ), rec( content := " if and \ only if it is strongly connected and has\n the same number of edges as vert\ ices. \n ", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "P", next := 281389, root := ~, start := 281385, stop := 281388 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "A", next := 281431, root := ~, start := 281417, stop := 281430 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsCycleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\n\ngap> IsCycleDigraph(D);\ntrue\ngap> D := \ OnDigraphs(D, (1, 2, 3));\n\ngap> \ D = CycleDigraph(5);\nfalse\ngap> IsCycleDigraph(D);\ntrue", count := [ 6, 3, 12, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 12, 6 ], mathmode := "Text", name := "Example", next := 281916, root := ~, start := 281528, stop := 281915 ), rec( content := "\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 281933, root := ~, start := 280999, stop := 281932 ) ], count := [ 6, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 281947, root := ~, start := 280888, stop := 281946 ), rec( content := "\n\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 281961, root := ~, start := 264522, stop := 281960 ) , rec( content := "\n \n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Planarity" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Planarity", count := [ 6, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 282022, root := ~, start := 281994, stop := 282021 ), rec( content := "\n ", count := [ 6, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPlanarDigraph\ " ), content := 0, count := [ 6, 4, 1, 2 ], mathmode := "Text", name := "Prop", next := 282086, root := ~, start := 282042, stop := 282085 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "K", next := 282109, root := ~, start := 282098, stop := 282108 ), rec( content := " or ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "K", next := 282125, root := ~, start := 282113, stop := 282124 ), rec( content := ".", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 282136, root := ~, start := 282089, stop := 282135 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ lanar", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "E", next := 282172, root := ~, start := 282159, stop := 282171 ), rec( content := " digraph\ is a digraph that can be embedded in the plane in\n such a way that its ed\ ges do not intersect. A digraph is planar if and only\n if it does not have\ a subdigraph that is homeomorphic to either the\n complete graph on ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "5\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 282413, root := ~, start := 282405, stop := 282412 ), rec( content := " vertice\ s or the complete bipartite graph with\n vertex sets of sizes ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 282493, root := ~, start := 282485, stop := 282492 ), rec( content := " and ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 282506, root := ~, start := 282498, stop := 282505 ), rec( content := ".\n " , count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "P", next := 282516, root := ~, start := 282512, stop := 282515 ), rec( content := "\n \n\ ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "C", next := 282548, root := ~, start := 282526, stop := 282547 ), rec( content := " returns\ ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "K", next := 282568, root := ~, start := 282557, stop := 282567 ), rec( content := " if the \ digraph ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "A", next := 282598, root := ~, start := 282584, stop := 282597 ), rec( content := " is\n \ planar and ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "K", next := 282629, root := ~, start := 282617, stop := 282628 ), rec( content := " if it i\ s not. The directions and multiplicities of\n any edges in ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "A", next := 282712, root := ~, start := 282698, stop := 282711 ), rec( content := " are ign\ ored by ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "C", next := 282750, root := ~, start := 282728, stop := 282749 ), rec( content := ".\n " , count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "P", next := 282760, root := ~, start := 282756, stop := 282759 ), rec( content := " \n\n \ See also ", count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "Ref", next := 282810, root := ~, start := 282776, stop := 282809 ), rec( content := ".\n " , count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "P", next := 282820, root := ~, start := 282816, stop := 282819 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 6, 4, 1, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 6, 4, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "Cite", next := 282978, root := ~, start := 282960, stop := 282977 ), rec( content := ".\n\n \ ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigraph(Complet\ eDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));\nt\ rue\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsPlan\ arDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanarDigraph(C\ ompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(CompleteBip\ artiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteBi\ partiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n", count := [ 6, 4, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 4, 1, 7 ], mathmode := "Text", name := "Example", next := 283517, root := ~, start := 282985, stop := 283516 ), rec( content := "\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 283534, root := ~, start := 282139, stop := 283533 ) ], count := [ 6, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 283548, root := ~, start := 282027, stop := 283547 ), rec( content := "\n\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsOuterPlanarDi\ graph" ), content := 0, count := [ 6, 4, 2, 2 ], mathmode := "Text", name := "Prop", next := 283618, root := ~, start := 283569, stop := 283617 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "K", next := 283641, root := ~, start := 283630, stop := 283640 ), rec( content := " or ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "K", next := 283657, root := ~, start := 283645, stop := 283656 ), rec( content := ".", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 283668, root := ~, start := 283621, stop := 283667 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ uter planar", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "E", next := 283711, root := ~, start := 283692, stop := 283710 ), rec( content := " digraph\ is a digraph that can be embedded in the\n plane in such a way that its ed\ ges do not intersect, and all vertices\n belong to the unbounded face of th\ e embedding. A digraph is outer planar\n if and only if it does not have a\ subdigraph that is homeomorphic to either\n the complete graph on ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 284027, root := ~, start := 284019, stop := 284026 ), rec( content := " vertice\ s or the complete bipartite graph\n with vertex sets of sizes ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 284107, root := ~, start := 284099, stop := 284106 ), rec( content := " and ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 284120, root := ~, start := 284112, stop := 284119 ), rec( content := ".\n " , count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "P", next := 284130, root := ~, start := 284126, stop := 284129 ), rec( content := "\n \n\ ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sOuterPlanarDigraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "C", next := 284167, root := ~, start := 284140, stop := 284166 ), rec( content := " returns\ ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "K", next := 284187, root := ~, start := 284176, stop := 284186 ), rec( content := " if the \ digraph\n ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "A", next := 284221, root := ~, start := 284207, stop := 284220 ), rec( content := " is oute\ r planar and ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "K", next := 284254, root := ~, start := 284242, stop := 284253 ), rec( content := " if it i\ s not. The\n directions and multiplicities of any edges in ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "A", next := 284337, root := ~, start := 284323, stop := 284336 ), rec( content := " are ign\ ored by\n ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "C", next := 284379, root := ~, start := 284357, stop := 284378 ), rec( content := ". ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "P", next := 284386, root := ~, start := 284382, stop := 284385 ), rec( content := "\n\n \ See also ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Ref", next := 284430, root := ~, start := 284401, stop := 284429 ), rec( content := ".\n\n \ This method uses the reference implementation in\n ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 5 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 6, 4, 2, 5 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 6, 4, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 5 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Cite", next := 284589, root := ~, start := 284571, stop := 284588 ), rec( content := ".\n\n", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsOuterPlanarDigraph(CompleteDigraph(4));\nfalse\ngap> IsOuterPlanarDigr\ aph(CompleteDigraph(5));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDi\ graph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(3, 3)\ );\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(10));\ntrue\ngap> IsOuterPla\ narDigraph(CompleteDigraph(IsMutableDigraph, 4));\nfalse\ngap> IsOuterPlanarDi\ graph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsOuterPlanarDigraph\ (CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteD\ igraph(IsMutableDigraph,\n> 3\ , 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(IsMutableDigraph, 10));\ \ntrue\n\n", count := [ 6, 4, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 6 ], mathmode := "Text", name := "Example", next := 285389, root := ~, start := 284592, stop := 285388 ), rec( content := "\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 285406, root := ~, start := 283671, stop := 285405 ) ], count := [ 6, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 285420, root := ~, start := 283554, stop := 285419 ), rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 285434, root := ~, start := 281967, stop := 285433 ) , rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms and\ transformations", count := [ 6, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 285499, root := ~, start := 285447, stop := 285498 ), rec( content := "\n ", count := [ 6, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDigraphCore" ), content := 0, count := [ 6, 5, 1, 2 ], mathmode := "Text", name := "Prop", next := 285561, root := ~, start := 285519, stop := 285560 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 285584, root := ~, start := 285573, stop := 285583 ), rec( content := " or ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 285600, root := ~, start := 285588, stop := 285599 ), rec( content := ".", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 285611, root := ~, start := 285564, stop := 285610 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 285665, root := ~, start := 285654, stop := 285664 ), rec( content := " if ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "A", next := 285683, root := ~, start := 285669, stop := 285682 ), rec( content := " is a co\ re,\n and ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 285715, root := ~, start := 285703, stop := 285714 ), rec( content := " if it i\ s not.", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "P", next := 285733, root := ~, start := 285729, stop := 285732 ), rec( content := "\n\n \ A digraph ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 285757, root := ~, start := 285749, stop := 285756 ), rec( content := " is a ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "E", next := 285774, root := ~, start := 285763, stop := 285773 ), rec( content := " if and \ only if it has no proper\n subdigraphs ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 285831, root := ~, start := 285823, stop := 285830 ), rec( content := " such th\ at there exists a homomorphism from ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 285883, root := ~, start := 285875, stop := 285882 ), rec( content := "\n to\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "A\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 285899, root := ~, start := 285891, stop := 285898 ), rec( content := ". In oth\ er words, a digraph ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 285935, root := ~, start := 285927, stop := 285934 ), rec( content := " is a co\ re if and only if\n every endomorphism on ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 285995, root := ~, start := 285987, stop := 285994 ), rec( content := " is an a\ utomorphism on ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 286026, root := ~, start := 286018, stop := 286025 ), rec( content := ".\n " , count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "P", next := 286036, root := ~, start := 286032, stop := 286035 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "A", next := 286078, root := ~, start := 286064, stop := 286077 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(6);\n\n\ gap> IsDigraphCore(D);\ntrue\ngap> D := DigraphSymmetricClosure(CycleDigraph(6\ ));\n\ngap> DigraphHomo\ morphism(D, CompleteDigraph(2));\nTransformation( [ 1, 2, 1, 2, 1, 2 ] )\ngap>\ IsDigraphCore(D);\nfalse\n", count := [ 6, 5, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 6 ], mathmode := "Text", name := "Example", next := 286539, root := ~, start := 286179, stop := 286538 ), rec( content := "\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 286556, root := ~, start := 285614, stop := 286555 ) ], count := [ 6, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 286570, root := ~, start := 285504, stop := 286569 ), rec( content := "\n\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEdgeTransitiv\ e" ), content := 0, count := [ 6, 5, 2, 2 ], mathmode := "Text", name := "Prop", next := 286636, root := ~, start := 286591, stop := 286635 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 286659, root := ~, start := 286648, stop := 286658 ), rec( content := " or ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 286675, root := ~, start := 286663, stop := 286674 ), rec( content := ".", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 286686, root := ~, start := 286639, stop := 286685 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "A", next := 286724, root := ~, start := 286710, stop := 286723 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEdgeTransitive", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "C", next := 286794, root := ~, start := 286771, stop := 286793 ), rec( content := " returns\ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 286814, root := ~, start := 286803, stop := 286813 ), rec( content := " if ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "A", next := 286832, root := ~, start := 286818, stop := 286831 ), rec( content := "\n is\ edge transitive, and ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 286873, root := ~, start := 286861, stop := 286872 ), rec( content := " otherwi\ se. A digraph is\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge transitive", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "E", next := 286924, root := ~, start := 286902, stop := 286923 ), rec( content := " if its \ automorphism group acts\n transitively on its edges (via the action\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "OnPairs" ), content := 0, count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 287042, root := ~, start := 287006, stop := 287041 ), rec( content := ").\n \ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "P", next := 287053, root := ~, start := 287049, stop := 287052 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "A", next := 287095, root := ~, start := 287081, stop := 287094 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsEdgeTransitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(Chain\ Digraph(3));\nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\nEr\ ror, the argument must be a digraph with no multiple edges,\n", count := [ 6, 5, 2, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 5, 2, 5 ], mathmode := "Text", name := "Example", next := 287442, root := ~, start := 287196, stop := 287441 ), rec( content := "\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 287459, root := ~, start := 286689, stop := 287458 ) ], count := [ 6, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 287473, root := ~, start := 286576, stop := 287472 ), rec( content := "\n\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsVertexTransit\ ive" ), content := 0, count := [ 6, 5, 3, 2 ], mathmode := "Text", name := "Prop", next := 287541, root := ~, start := 287494, stop := 287540 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 287564, root := ~, start := 287553, stop := 287563 ), rec( content := " or ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 287580, root := ~, start := 287568, stop := 287579 ), rec( content := ".", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 287591, root := ~, start := 287544, stop := 287590 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "A", next := 287629, root := ~, start := 287615, stop := 287628 ), rec( content := " is a di\ graph, then ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sVertexTransitive", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "C", next := 287674, root := ~, start := 287649, stop := 287673 ), rec( content := " returns\ \n ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 287698, root := ~, start := 287687, stop := 287697 ), rec( content := " if ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "A", next := 287716, root := ~, start := 287702, stop := 287715 ), rec( content := " is vert\ ex transitive, and ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 287755, root := ~, start := 287743, stop := 287754 ), rec( content := "\n ot\ herwise. A digraph is ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex transitive", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "E", next := 287808, root := ~, start := 287784, stop := 287807 ), rec( content := " if its \ automorphism group\n acts transitively on its vertices.\n ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "P", next := 287882, root := ~, start := 287878, stop := 287881 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "A", next := 287924, root := ~, start := 287910, stop := 287923 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsVertexTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitive(C\ hainDigraph(3));\nfalse\n", count := [ 6, 5, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 5 ], mathmode := "Text", name := "Example", next := 288155, root := ~, start := 288025, stop := 288154 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 288172, root := ~, start := 287594, stop := 288171 ) ], count := [ 6, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 288186, root := ~, start := 287479, stop := 288185 ), rec( content := "\n\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 288200, root := ~, start := 285438, stop := 288199 ) , rec( content := "\n\n \n", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 288215, root := ~, start := 240298, stop := 288214 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " properties ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Homomorphisms" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms", count := [ 7, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 288302, root := ~, start := 288270, stop := 288301 ) , rec( content := "\n ", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "Let $E_a=(E_a^0,E_a^1,\\ra_a,\\s\ o_a)$ and $E_b=(E_b^0,E_b^1,\\ra_b,\\so_b)$ be two\ngraphs, and let $\\phi_0 :\ E_a^0 \\to E_b^0$ and $\\phi_1 : E_a^1 \\to E_b^1$ be\nfunctions. Then the pa\ ir $\\phi = (\\phi_0, \\phi_1)$ is a \\emph{graph homomorphism\nfrom} $E_a$ \\\ emph{to} $E_b$ if $\\phi_0(\\so_a(e)) = \\so_b(\\phi_1(e))$ and\n$\\phi_0(\\ra\ _a(e)) = \\ra_b(\\phi_1(e))$ for every $e \\in E_a^1$. If $\\phi_0$ and\n$\\ph\ i_1$ are in addition bijective, then $\\phi$ is a \\emph{graph isomorphism\nfr\ om} $E_a$ \\emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are\n\\em\ ph{isomorphic} and write $E_a \\cong E_b$.", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Acting on digraph\ s", count := [ 7, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 288946, root := ~, start := 288909, stop := 288945 ), rec( content := "\n ", count := [ 7, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, perm", Label := "for a digraph \ and a perm", Name := "OnDigraphs" ), content := 0, count := [ 7, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 289048, root := ~, start := 288966, stop := 289047 ), rec( attributes := rec( Arg := "digraph, trans", Label := "for a digraph \ and a transformation", Name := "OnDigraphs" ), content := 0, count := [ 7, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 289144, root := ~, start := 289051, stop := 289143 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 289176, root := ~, start := 289147, stop := 289175 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 289214, root := ~, start := 289200, stop := 289213 ), rec( content := " is a di\ graph, and the second argument ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 289264, root := ~, start := 289253, stop := 289263 ), rec( content := " is a\n \ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ermutation", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "E", next := 289292, root := ~, start := 289274, stop := 289291 ), rec( content := " of the \ vertices of ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 289326, root := ~, start := 289312, stop := 289325 ), rec( content := ", then t\ his operation\n returns a digraph constructed by relabelling the vertices o\ f\n ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 289431, root := ~, start := 289417, stop := 289430 ), rec( content := " accordi\ ng to ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 289456, root := ~, start := 289445, stop := 289455 ), rec( content := ". Note \ that for an automorphism\n ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "C", next := 289501, root := ~, start := 289493, stop := 289500 ), rec( content := " of a di\ graph, we have ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 1, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := \ "Text", name := "A", next := 289552, root := ~, start := 289538, stop := 28955\ 1 ), rec( content := ", f) =\n ", count := [ 7, 1, 1, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "C", next := 289567, root := ~, start := 289524, stop := 289566 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 289581, root := ~, start := 289567, stop := 289580 ), rec( content := ".\n " , count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "P", next := 289591, root := ~, start := 289587, stop := 289590 ), rec( content := "\n\n \ If the second argument is a ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransformation", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "E", next := 289646, root := ~, start := 289625, stop := 289645 ), rec( content := " ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 289659, root := ~, start := 289647, stop := 289658 ), rec( content := " of the\\ n vertices of ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 289697, root := ~, start := 289683, stop := 289696 ), rec( content := ", then t\ his operation returns a digraph\n constructed by transforming the source an\ d range of each edge according to\n ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 289832, root := ~, start := 289820, stop := 289831 ), rec( content := ". Thus a\ vertex which does not appear in the image of\n ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 289902, root := ~, start := 289890, stop := 289901 ), rec( content := " will be\ isolated in the returned digraph, and the returned\n digraph may contain m\ ultiple edges, even if ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 290024, root := ~, start := 290010, stop := 290023 ), rec( content := " does no\ t.\n If ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 290054, root := ~, start := 290042, stop := 290053 ), rec( content := " is math\ ematically a permutation, then the result coincides\n with ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 1, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "A", next := 290151, root := ~, start := 290137, stop := 29015\ 0 ), rec( content := ", AsPermutation(", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "trans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 2901\ 79, root := ~, start := 290167, stop := 290178 ), rec( content := "))", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "C", next := 290185, root := ~, start := 290123, stop := 290184 ), rec( content := ".\n " , count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "P", next := 290195, root := ~, start := 290191, stop := 290194 ), rec( content := "\n\n \ The ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "Ref", next := 290238, root := ~, start := 290205, stop := 290237 ), rec( content := " of ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "A", next := 290256, root := ~, start := 290242, stop := 290255 ), rec( content := " will no\ t be retained\n in the returned digraph. ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "P", next := 290311, root := ~, start := 290307, stop := 290310 ), rec( content := "\n\n \ If ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 290334, root := ~, start := 290320, stop := 290333 ), rec( content := " belongs\ to ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "Ref", next := 290376, root := ~, start := 290346, stop := 290375 ), rec( content := ", then\n\ relabelling of the vertices is performed directly on ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 290454, root := ~, start := 290440, stop := 290453 ), rec( content := ".\n I\ f ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 290477, root := ~, start := 290463, stop := 290476 ), rec( content := " belongs\ to ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "Ref", next := 290521, root := ~, start := 290489, stop := 290520 ), rec( content := ", an imm\ utable\n copy of ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 290562, root := ~, start := 290548, stop := 290561 ), rec( content := " with th\ e vertices relabelled is returned.", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 9 ], mathmode := "Text", name := "P", next := 290608, root := ~, start := 290604, stop := 290607 ), rec( content := "\n\n \ ", count := [ 7, 1, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> new := OnDigraphs(D, (1, 2));\n\ngap> OutNeighbours(new);\n[ [ 2, 3, 5 \ ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ]\ngap> D := Digraph([[2], [], [2]])\ ;\n\ngap> t := Transformation([1, \ 2, 1]);;\ngap> new := OnDigraphs(D, t);\n\ngap> OutNeighbours(new);\n[ [ 2, 2 ], [ ], [ ] ]\ngap> ForAll(\ DigraphEdges(D),\n> e -> IsDigraphEdge(new, [e[1] ^ t, e[2] ^ t]));\ntrue\n", count := [ 7, 1, 1, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 1, 1, 10 ], mathmode := "Text", name := "Example", next := 291251, root := ~, start := 290614, stop := 291250 ), rec( content := "\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 291268, root := ~, start := 289179, stop := 291267 ) ], count := [ 7, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 291282, root := ~, start := 288951, stop := 291281 ), rec( content := "\n\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, pair", Name := "OnMultiDigraphs\ " ), content := 0, count := [ 7, 1, 2, 2 ], mathmode := "Text", name := "Oper", next := 291353, root := ~, start := 291303, stop := 291352 ), rec( attributes := rec( Arg := "digraph, perm1, \ perm2", Label := "for a digraph, perm, and perm", Name := "OnMultiDigraphs" ), content := 0, count := [ 7, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 291456, root := ~, start := 291356, stop := 291455 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 291488, root := ~, start := 291459, stop := 291487 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 291526, root := ~, start := 291512, stop := 291525 ), rec( content := " is a di\ graph, and ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ air", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 291556, root := ~, start := 291545, stop := 291555 ), rec( content := " is a pa\ ir consisting of a\n permutation of the vertices and a permutation of the e\ dges of\n ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 291667, root := ~, start := 291653, stop := 291666 ), rec( content := ", then t\ his operation returns a digraph\n constructed by relabelling the vertices a\ nd edges of ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 291778, root := ~, start := 291764, stop := 291777 ), rec( content := "\n ac\ cording to ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm[1]", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 291810, root := ~, start := 291796, stop := 291809 ), rec( content := " and ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm[2]", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 291829, root := ~, start := 291815, stop := 291828 ), rec( content := ", respec\ tively. ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "P", next := 291849, root := ~, start := 291845, stop := 291848 ), rec( content := "\n\n \ In its second form, ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nMultiDigraphs", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "C", next := 291897, root := ~, start := 291875, stop := 291896 ), rec( content := " returns\ a digraph with vertices\n and edges permuted by ", count := [ 7, 1, 2, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm1", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "A", next := 291968, root := ~, start := 291956, stop := 291967 ), rec( content := " and ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm2", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "A", next := 291985, root := ~, start := 291973, stop := 291984 ), rec( content := ", respec\ tively. ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "P", next := 292005, root := ~, start := 292001, stop := 292004 ), rec( content := "\n\n \ Note that ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 2, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := \ "Text", name := "A", next := 292049, root := ~, start := 292035, stop := 29204\ 8 ), rec( content := ",\n perm)=OnMultiDigraphs(", count := [ 7, 1, 2, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 2920\ 93, root := ~, start := 292079, stop := 292092 ), rec( content := ", [perm, ()\ ])", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 292110, root := ~, start := 292021, stop := 292109 ), rec( content := " where " , count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 292128, root := ~, start := 292117, stop := 292127 ), rec( content := " is\n \ a permutation of the vertices of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 292183, root := ~, start := 292169, stop := 292182 ), rec( content := ". If you\ are only interested\n in the action of a permutation on the vertices of a \ digraph, then you can\n use ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 292315, root := ~, start := 292298, stop := 292314 ), rec( content := " instead\ of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nMultiDigraphs", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 292349, root := ~, start := 292327, stop := 292348 ), rec( content := ".\n\n \ If ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 292373, root := ~, start := 292359, stop := 292372 ), rec( content := " belongs\ to ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 292415, root := ~, start := 292385, stop := 292414 ), rec( content := ", then\n\ relabelling of the vertices is performed directly on ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 292493, root := ~, start := 292479, stop := 292492 ), rec( content := ".\n I\ f ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 292516, root := ~, start := 292502, stop := 292515 ), rec( content := " belongs\ to ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 292560, root := ~, start := 292528, stop := 292559 ), rec( content := ", an imm\ utable\n copy of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 292601, root := ~, start := 292587, stop := 292600 ), rec( content := " with th\ e vertices relabelled is returned.", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 8 ], mathmode := "Text", name := "P", next := 292647, root := ~, start := 292643, stop := 292646 ), rec( content := "\n\n \ ", count := [ 7, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([\n> [3, 6, 3], [], [3], [9, 10], [9], [], [], [10, 4, 10\ ], [], []]);\n\ngap> p := B\ lissCanonicalLabelling(D1);\n[ (1,7)(3,6)(4,10)(5,9), () ]\ngap> D2 := OnMulti\ Digraphs(D1, p);\n\ngap> Ou\ tNeighbours(D2);\n[ [ ], [ ], [ ], [ ], [ ], [ 6 ], [ 6, 3, 6 ], [ 4, 10,\ 4 ], \n [ 5 ], [ 5, 4 ] ]", count := [ 7, 1, 2, 9 ], name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 9 ], mathmode := "Text", name := "Example", next := 293092, root := ~, start := 292653, stop := 293091 ), rec( content := "\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 293109, root := ~, start := 291491, stop := 293108 ) ], count := [ 7, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 293123, root := ~, start := 291288, stop := 293122 ), rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 293137, root := ~, start := 288900, stop := 293136 ) , rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Isomorphisms and canonical lab\ ellings" ), content := [ rec( content := "\n ", count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Isomorphisms and \ canonical labellings", count := [ 7, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 293257, root := ~, start := 293201, stop := 293256 ), rec( content := "\n From version 0.11.\ 0 of ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " it is possible to use e\ ither ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " or\n ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "https://p\ allini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 125, root := ~, start := 101, stop := 124 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " (via ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/gap-packages/NautyT\ racesInterface", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := ") to calculate canonical\ labellings and\n automorphism groups of digraphs; see ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Cite", next := 293474, root := ~, start := 293456, stop := 293473 ), rec( content := " and ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Cite", next := 293503, root := ~, start := 293479, stop := 293502 ), rec( content := " for more details about \ ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "https://p\ allini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 125, root := ~, start := 101, stop := 124 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := ",\n respectively. \n\\ n ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsUseNaut\ y" ), content := 0, count := [ 7, 2, 1, 2 ], mathmode := "Text", name := "Func", next := 293625, root := ~, start := 293587, stop := 293624 ), rec( attributes := rec( Arg := "", Name := "DigraphsUseBlis\ s" ), content := 0, count := [ 7, 2, 1, 4 ], mathmode := "Text", name := "Func", next := 293666, root := ~, start := 293628, stop := 293665 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 293696, root := ~, start := 293669, stop := 293695 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se functions can be used to specify whether ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " or ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " should \ be\n used by default by ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ". If ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 1, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is not \ available,\n then these functions do nothing. Otherwise, by calling\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsUseNauty", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "C", next := 293958, root := ~, start := 293935, stop := 293957 ), rec( content := " subsequ\ ent computations will default to using\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " rather \ than ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ", where \ possible.\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "P", next := 294062, root := ~, start := 294058, stop := 294061 ), rec( content := "\n\n \ You can call these functions at any point in a ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " session\ , as many\n times as you like, it is guaranteed that existing digraphs rema\ in\n valid, and that comparison of existing digraphs and newly created digr\ aphs\n via\n ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 294353, root := ~, start := 294299, stop := 294352 ), rec( content := ",\n " , count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 294441, root := ~, start := 294359, stop := 294440 ), rec( content := ",\n " , count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 294501, root := ~, start := 294447, stop := 294500 ), rec( content := ", and\n \ ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 294593, root := ~, start := 294511, stop := 294592 ), rec( content := " are als\ o valid.", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "P", next := 294613, root := ~, start := 294609, stop := 294612 ), rec( content := "\n\n \ It is also possible to compute the automorphism group of a specific digraph\n \ using both ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 7 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " and ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 7 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " using " , count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 294772, root := ~, start := 294736, stop := 294771 ), rec( content := "\n an\ d ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 294839, root := ~, start := 294781, stop := 294838 ), rec( content := ", respec\ tively.\n ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 294871, root := ~, start := 293699, stop := 294870 ) ], count := [ 7, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 294885, root := ~, start := 293572, stop := 294884 ), rec( content := "\n\n ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 2, 2 ], mathmode := "Text", name := "Attr", next := 294974, root := ~, start := 294906, stop := 294973 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 295016, root := ~, start := 294977, stop := 295015 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 295054, root := ~, start := 295040, stop := 295053 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 295146, root := ~, start := 295132, stop := 295145 ), rec( content := ". An ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "E", next := 295171, root := ~, start := 295152, stop := 295170 ), rec( content := " of ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 295189, root := ~, start := 295175, stop := 295188 ), rec( content := "\n is\ an isomorphism from ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 295231, root := ~, start := 295217, stop := 295230 ), rec( content := " to itse\ lf. See ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 295308, root := ~, start := 295247, stop := 295307 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "P", next := 295370, root := ~, start := 295366, stop := 295369 ), rec( content := "\n\n \ If ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "A", next := 295393, root := ~, start := 295379, stop := 295392 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "A", next := 295531, root := ~, start := 295517, stop := 295530 ), rec( content := ". ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "P", next := 295537, root := ~, start := 295533, stop := 295536 ), rec( content := "\n\n \ If ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 295560, root := ~, start := 295546, stop := 295559 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 295716, root := ~, start := 295702, stop := 295715 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 295787, root := ~, start := 295773, stop := 295786 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 295913, root := ~, start := 295824, stop := 295912 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "P", next := 295944, root := ~, start := 295940, stop := 295943 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 2, 6 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 2, 6 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 2, 6 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 296229, root := ~, start := 296171, stop := 296228 ), rec( content := ", \n \ ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 296272, root := ~, start := 296236, stop := 296271 ), rec( content := ", ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 296304, root := ~, start := 296274, stop := 296303 ), rec( content := ",\n a\ nd ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 296344, root := ~, start := 296314, stop := 296343 ), rec( content := ".\n " , count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "P", next := 296354, root := ~, start := 296350, stop := 296353 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "A", next := 296390, root := ~, start := 296376, stop := 296389 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> johnson := DigraphFromGraph6String(\"E}lw\");\n\ngap> G := AutomorphismGroup(johnson);\nGroup([\ (3,4), (2,3)(4,5), (1,2)(5,6) ])\ngap> cycle := CycleDigraph(9);\n\ngap> G := AutomorphismGroup(cycle);\nGroup([ (\ 1,2,3,4,5,6,7,8,9) ])\ngap> IsCyclic(G) and Size(G) = 9;\ntrue", count := [ 7, 2, 2, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 2, 8 ], mathmode := "Text", name := "Example", next := 296886, root := ~, start := 296492, stop := 296885 ), rec( content := "\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 296905, root := ~, start := 295019, stop := 296904 ) ], count := [ 7, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 296921, root := ~, start := 294891, stop := 296920 ), rec( content := "\n\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 2 ], mathmode := "Text", name := "Attr", next := 297015, root := ~, start := 296942, stop := 297014 ), rec( attributes := rec( Arg := "digraph, vertex_\ colours", Label := "for a digraph and homogenous list", Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 297132, root := ~, start := 297018, stop := 297131 ), rec( attributes := rec( Arg := "digraph, vertex_\ colours, edge_colours", Label := "for a digraph, homogenous list, and list", Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 297270, root := ~, start := 297135, stop := 297269 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Returns", next := 297312, root := ~, start := 297273, stop := 297311 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "A", next := 297350, root := ~, start := 297336, stop := 297349 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "A", next := 297442, root := ~, start := 297428, stop := 297441 ), rec( content := " as calc\ ulated using ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 3, 7 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 3, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "P", next := 297516, root := ~, start := 297512, stop := 297515 ), rec( content := "\n\n \ The attribute ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 3, 8 ] , mathmode := "Text", name := "Ref", next := 297589, root := ~, start := 297536, stop := 297588 ), rec( content := " and\n \ operation ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "Ref", next := 297691, root := ~, start := 297608, stop := 297690 ), rec( content := " returns\ \n the value of either ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissAutomorphismGroup", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "C", next := 297753, root := ~, start := 297724, stop := 297752 ), rec( content := " or ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "Ref", next := 297799, root := ~, start := 297757, stop := 297798 ), rec( content := ". These \ groups are, of course, equal but\n their generating sets may differ. ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "P", next := 297882, root := ~, start := 297878, stop := 297881 ), rec( content := "\n Th\ e attribute ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph, homogeneous list, and list" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 297989, root := ~, start := 297901, stop := 297988 ), rec( content := " returns\ the value of\n ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissAutomorphismGroup", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "C", next := 298044, root := ~, start := 298015, stop := 298043 ), rec( content := " as it i\ s not implemented for ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 3, 9 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 3, 9 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " The \n \ requirements for the optional arguments ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex_colours", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "A", next := 298152, root := ~, start := 298131, stop := 298151 ), rec( content := " and \n \ ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "A", next := 298181, root := ~, start := 298162, stop := 298180 ), rec( content := " are doc\ umented in ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph, homogeneous list, and list" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 298288, root := ~, start := 298200, stop := 298287 ), rec( content := ".\n\n \ See also ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 298334, root := ~, start := 298304, stop := 298333 ), rec( content := ", and ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 298370, root := ~, start := 298340, stop := 298369 ), rec( content := ".\n " , count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "P", next := 298380, root := ~, start := 298376, stop := 298379 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "A", next := 298416, root := ~, start := 298402, stop := 298415 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := BlissAutomorphismGroup(JohnsonDigraph(5, 2));;\ngap> IsSymmetricGro\ up(G);\ntrue\ngap> Size(G);\n120", count := [ 7, 2, 3, 11 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 11 ], mathmode := "Text", name := "Example", next := 298655, root := ~, start := 298518, stop := 298654 ), rec( content := "\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Description", next := 298674, root := ~, start := 297315, stop := 298673 ) ], count := [ 7, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 298690, root := ~, start := 296927, stop := 298689 ), rec( content := "\n\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, vert_c\ olours]", Name := "NautyAutomorphismGroup" ), content := 0, count := [ 7, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 298778, root := ~, start := 298711, stop := 298777 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 298820, root := ~, start := 298781, stop := 298819 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "A", next := 298858, root := ~, start := 298844, stop := 298857 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "A", next := 298950, root := ~, start := 298936, stop := 298949 ), rec( content := " as calc\ ulated using ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 4, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno via ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 4, 3 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 4, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := ".\n\n \ ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( content := " FIXME b\ e more explicit about what the second arg is ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ The attribute ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 4, 3 ] , mathmode := "Text", name := "Ref", next := 299185, root := ~, start := 299132, stop := 299184 ), rec( content := " and\n \ operation ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 299287, root := ~, start := 299204, stop := 299286 ), rec( content := " returns\ \n the value of either ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyAutomorphismGroup", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "C", next := 299349, root := ~, start := 299320, stop := 299348 ), rec( content := " or ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 299418, root := ~, start := 299354, stop := 299417 ), rec( content := ".\n T\ hese groups are, of course, equal but their generating sets may differ.", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "P", next := 299500, root := ~, start := 299496, stop := 299499 ), rec( content := "\n\n \ See also ", count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "Ref", next := 299545, root := ~, start := 299515, stop := 299544 ), rec( content := ", and ", count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "Ref", next := 299581, root := ~, start := 299551, stop := 299580 ), rec( content := ".\n " , count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "P", next := 299591, root := ~, start := 299587, stop := 299590 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "A", next := 299627, root := ~, start := 299613, stop := 299626 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> NautyAutomorphismGroup(JohnsonDigraph(5, 2));\nGroup([ (3,4)(6,7)(8,9), \ (2,3)(5,6)(9,10), (2,5)(3,6)(4,7),\n (1,2)(6,8)(7,9) ])", count := [ 7, 2, 4, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 4, 6 ], mathmode := "Text", name := "Log", next := 299883, root := ~, start := 299729, stop := 299882 ), rec( content := "\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 299902, root := ~, start := 298823, stop := 299901 ) ], count := [ 7, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 299918, root := ~, start := 298696, stop := 299917 ), rec( content := "\n\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vert_co\ lours", Label := "for a digraph and a homogeneous list", Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 5, 2 ], mathmode := "Text", name := "Oper", next := 300048, root := ~, start := 299939, stop := 300047 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 300090, root := ~, start := 300051, stop := 300089 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation computes the automorphism group of a vertex-coloured digraph.\n \ A vertex-coloured digraph can be specified by its underlying digraph\n " , count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 300280, root := ~, start := 300266, stop := 300279 ), rec( content := " and its\ colouring ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 300318, root := ~, start := 300299, stop := 300317 ), rec( content := ". Let ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "C", next := 300332, root := ~, start := 300324, stop := 300331 ), rec( content := " be the\\ n number of vertices of ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 300380, root := ~, start := 300366, stop := 300379 ), rec( content := ". The co\ louring ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 300415, root := ~, start := 300396, stop := 300414 ), rec( content := " may\n \ have one of the following two forms:\n\n ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 7, 2, 5, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode\ := "Text", name := "C", next := 300512, root := ~, start := 300504, stop := 3\ 00511 ), rec( content := " integers, where ", count := [ 7, 2, 5, 5 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "vert_colours", count := [ 7, 2, 5, 5 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "T\ ext", name := "A", next := 300548, root := ~, start := 300529, stop := 300547 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 5, 5 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ] , math\ mode := "Text", name := "C", next := 300558, root := ~, start := 300548, stop \ := 300557 ), rec( content := " is the\n colour of vertex ", count := [ \ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "i", count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "C", next := 300599, root := ~, start := 300591, stop \ := 300598 ), rec( content := ", using the colours ", count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 5, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "Tex\ t", name := "C", next := 300634, root := ~, start := 300619, stop := 300633 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "C", next\ := 300668, root := ~, start := 300652, stop := 300667 ), rec( content := "; o\ r\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 5 ], name := "Item", next := 300686, root := ~, start := 300\ 479, stop := 300685 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 5, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 \ ], mathmode := "Text", name := "A", next := 300799, root := ~, start := 300785\ , stop := 300798 ), rec( content := ")", count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "C", next := 300804, root := ~, start := 300766, stop := 30080\ 3 ), rec( content := ", such that\n ", count := [ 7, 2, 5, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "vert_colours", count := [ 7, 2, 5, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := "T\ ext", name := "A", next := 300843, root := ~, start := 300824, stop := 300842 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 5, 7 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ] , math\ mode := "Text", name := "C", next := 300853, root := ~, start := 300843, stop \ := 300852 ), rec( content := " is the list of all vertices with colour\n \ ", count := [ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 5, 7 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], math\ mode := "Text", name := "C", next := 300910, root := ~, start := 300902, stop \ := 300909 ), rec( content := ".\n ", count := [ 7, 2, 5, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], name := "Ite\ m", next := 300925, root := ~, start := 300693, stop := 300924 ) ], count := [ 7, 2, 5, 4 ], mathmode := "Text", name := "List", next := 300937, root := ~, start := 300466, stop := 300936 ), rec( content := "\n\n \ The ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism group", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "E", next := 300972, root := ~, start := 300947, stop := 300971 ), rec( content := " of a co\ loured digraph ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 301009, root := ~, start := 300995, stop := 301008 ), rec( content := " with\n \ colouring ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 301048, root := ~, start := 301029, stop := 301047 ), rec( content := " is the \ group consisting of its automorphisms; an\n ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "E", next := 301121, root := ~, start := 301102, stop := 301120 ), rec( content := " of ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 301139, root := ~, start := 301125, stop := 301138 ), rec( content := " is an i\ somorphism of coloured\n digraphs from ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 301202, root := ~, start := 301188, stop := 301201 ), rec( content := " to itse\ lf. This group is equal to the\n subgroup of ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utomorphismGroup(", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 3012\ 93, root := ~, start := 301279, stop := 301292 ), rec( content := ")", count := [ 7, 2, 5, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "C", next := 301298, root := ~, start := 301258, stop := 301297 ), rec( content := " consist\ ing of those\n automorphisms that preserve the colouring specified by ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 301397, root := ~, start := 301378, stop := 301396 ), rec( content := ". See\n \ ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 5, 9 ] , mathmode := "Text", name := "Ref", next := 301461, root := ~, start := 301407, stop := 301460 ), rec( content := ", and se\ e ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "Ref", next := 301554, root := ~, start := 301471, stop := 301553 ), rec( content := "\n fo\ r more information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "P", next := 301625, root := ~, start := 301621, stop := 301624 ), rec( content := "\n\n \ If ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "A", next := 301648, root := ~, start := 301634, stop := 301647 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "A", next := 301786, root := ~, start := 301772, stop := 301785 ), rec( content := ". ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "P", next := 301792, root := ~, start := 301788, stop := 301791 ), rec( content := "\n\n \ If ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 301815, root := ~, start := 301801, stop := 301814 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 301971, root := ~, start := 301957, stop := 301970 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 302042, root := ~, start := 302028, stop := 302041 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "Ref", next := 302168, root := ~, start := 302079, stop := 302167 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "P", next := 302199, root := ~, start := 302195, stop := 302198 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 5, 12 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 5, 12 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 5, 12 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 5, 12 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph and homogenous list" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 302510, root := ~, start := 302426, stop := 302509 ), rec( content := ",\n " , count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 302552, root := ~, start := 302516, stop := 302551 ), rec( content := ", ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 302584, root := ~, start := 302554, stop := 302583 ), rec( content := ",\n a\ nd ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 302624, root := ~, start := 302594, stop := 302623 ), rec( content := ".\n\n " , count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> cycle := CycleDigraph(9);\n\nga\ p> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and Size(G) = 9;\ntrue\ng\ ap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := AutomorphismGrou\ p(cycle, colours);;\ngap> Size(H);\n3\ngap> H = AutomorphismGroup(cycle, [1, 2\ , 3, 1, 2, 3, 1, 2, 3]);\ntrue\ngap> H = SubgroupByProperty(G, p -> OnTuplesSe\ ts(colours, p) = colours);\ntrue\ngap> IsTrivial(AutomorphismGroup(cycle, [1, \ 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue", count := [ 7, 2, 5, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 13 ], mathmode := "Text", name := "Example", next := 303146, root := ~, start := 302629, stop := 303145 ), rec( content := "\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 303163, root := ~, start := 300093, stop := 303162 ) ], count := [ 7, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 303177, root := ~, start := 299924, stop := 303176 ), rec( content := "\n\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vert_co\ lours, edge_colours", Label := "for a digraph, homogeneous list, and list", Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 6, 2 ], mathmode := "Text", name := "Oper", next := 303330, root := ~, start := 303198, stop := 303329 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Returns", next := 303372, root := ~, start := 303333, stop := 303371 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation computes the automorphism group of a vertex- and/or\n edge-co\ loured digraph.\n A coloured digraph can be specified by its underlying dig\ raph\n ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 303572, root := ~, start := 303558, stop := 303571 ), rec( content := " and col\ ourings ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 303607, root := ~, start := 303588, stop := 303606 ), rec( content := ", ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 303628, root := ~, start := 303609, stop := 303627 ), rec( content := ".\n L\ et ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "C", next := 303646, root := ~, start := 303638, stop := 303645 ), rec( content := " be the \ number of vertices of ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 303690, root := ~, start := 303676, stop := 303689 ), rec( content := ". The co\ lourings\n must have the following forms:\n\n ", count := [ 7, 2, 6, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 6, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "vert_colours", count := [ 7, 2, 6, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := \ "Text", name := "A", next := 303794, root := ~, start := 303775, stop := 30379\ 3 ), rec( content := " must be ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "K", next :\ = 303814, root := ~, start := 303803, stop := 303813 ), rec( content := " or a\ list of ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "n", count :\ = [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 303836, root := ~, star\ t := 303828, stop := 303835 ), rec( content := " integers,\n where ", coun\ t := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "vert_colours", count := [ 7, \ 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 2, 6, 5 ], mathmode := "Text", name := "A", next := 303880, root := ~, start :\ = 303861, stop := 303879 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "[i]", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next :\ = 303890, root := ~, start := 303880, stop := 303889 ), rec( content := " is t\ he colour of vertex ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "i", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 303923, root\ := ~, start := 303915, stop := 303922 ), rec( content := ",\n using th\ e colours ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[1 .. m]", \ count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 303966, root := \ ~, start := 303951, stop := 303965 ), rec( content := " for some ", count := [ \ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "m <= n", count := [ 7, 2, 6, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], math\ mode := "Text", name := "C", next := 303992, root := ~, start := 303976, stop \ := 303991 ), rec( content := ";\n ", count := [ 7, 2, 6, 5 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], name := "Ite\ m", next := 304007, root := ~, start := 303760, stop := 304006 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 6, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "edge_colours", count := [ 7, 2, 6, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := \ "Text", name := "A", next := 304048, root := ~, start := 304029, stop := 30404\ 7 ), rec( content := " must be ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "K", next :\ = 304068, root := ~, start := 304057, stop := 304067 ), rec( content := " or a\ list of ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "n", count :\ = [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 6, 7 ], mathmode := "Text", name := "C", next := 304090, root := ~, star\ t := 304082, stop := 304089 ), rec( content := " lists of\n integers of\ the same shape as ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ OutNeighbours(digraph)", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := \ "C", next := 304167, root := ~, start := 304138, stop := 304166 ), rec( conten\ t := ", where\n ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "edge_colours", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "A", next\ := 304202, root := ~, start := 304183, stop := 304201 ), rec( attributes := re\ c( ), content := [ rec( content := "[i][j]", count := [ 7, 2, 6, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmod\ e := "Text", name := "C", next := 304215, root := ~, start := 304202, stop := \ 304214 ), rec( content := " is the colour of the edge\n ", count := [ 7\ , 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "OutNeighbours(digraph)[i][j]", cou\ nt := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 6, 7 ], mathmode := "Text", name := "C", next := 304285, root := \ ~, start := 304250, stop := 304284 ), rec( content := ", using the colours ", coun\ t := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[1 .. k]", count := [ 7, 2, 6\ , 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6\ , 7 ], mathmode := "Text", name := "C", next := 304320, root := ~, start := 30\ 4305, stop := 304319 ), rec( content := "\n for some ", count := [ 7, 2\ , 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "k <= n", count := [ 7, 2, 6, 7 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], math\ mode := "Text", name := "C", next := 304354, root := ~, start := 304338, stop \ := 304353 ), rec( content := ";\n ", count := [ 7, 2, 6, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], name := "Ite\ m", next := 304369, root := ~, start := 304014, stop := 304368 ) ], count := [ 7, 2, 6, 4 ], mathmode := "Text", name := "List", next := 304381, root := ~, start := 303747, stop := 304380 ), rec( content := "\n\n \ Giving ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "A", next := 304413, root := ~, start := 304394, stop := 304412 ), rec( content := " [", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "A", next := 304434, root := ~, start := 304415, stop := 304433 ), rec( content := "] as ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "C", next := 304450, root := ~, start := 304439, stop := 304449 ), rec( content := " is\n \ equivalent to setting all vertices [edges] to be the same colour. ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "P", next := 304528, root := ~, start := 304524, stop := 304527 ), rec( content := "\n\n \ Unlike ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 304594, root := ~, start := 304541, stop := 304593 ), rec( content := ", it is\\ n possible to obtain the automorphism group of an edge-coloured multidigrap\ h\n (see ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 304719, root := ~, start := 304690, stop := 304718 ), rec( content := ") when n\ o two edges share the same\n source, range, and colour.\n\n The ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism group", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "E", next := 304819, root := ~, start := 304794, stop := 304818 ), rec( content := " of a ve\ rtex/edge-coloured digraph\n ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 304872, root := ~, start := 304858, stop := 304871 ), rec( content := " with co\ louring ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 304896, root := ~, start := 304888, stop := 304895 ), rec( content := " is the \ group consisting of its\n vertex/edge-colour preserving automorphisms; an\n\ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "E", next := 305003, root := ~, start := 304984, stop := 305002 ), rec( content := " of ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 305021, root := ~, start := 305007, stop := 305020 ), rec( content := " is an i\ somorphism of vertex/edge-coloured\n digraphs from ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 305096, root := ~, start := 305082, stop := 305095 ), rec( content := " to itse\ lf. This group is equal to the\n subgroup of ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utomorphismGroup(", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 3051\ 87, root := ~, start := 305173, stop := 305186 ), rec( content := ")", count := [ 7, 2, 6, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "C", next := 305192, root := ~, start := 305152, stop := 305191 ), rec( content := " consist\ ing of those\n automorphisms that preserve the colouring specified by ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 305286, root := ~, start := 305272, stop := 305285 ), rec( content := ". See\n \ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 305350, root := ~, start := 305296, stop := 305349 ), rec( content := ", and se\ e ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 305443, root := ~, start := 305360, stop := 305442 ), rec( content := "\n fo\ r more information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "P", next := 305514, root := ~, start := 305510, stop := 305513 ), rec( content := "\n\n \ If ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "A", next := 305537, root := ~, start := 305523, stop := 305536 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "A", next := 305675, root := ~, start := 305661, stop := 305674 ), rec( content := ". ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "P", next := 305681, root := ~, start := 305677, stop := 305680 ), rec( content := "\n\n \ If ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 305704, root := ~, start := 305690, stop := 305703 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 305860, root := ~, start := 305846, stop := 305859 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 305931, root := ~, start := 305917, stop := 305930 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "Ref", next := 306057, root := ~, start := 305968, stop := 306056 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "P", next := 306088, root := ~, start := 306084, stop := 306087 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 6, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 6, 13 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 6, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 6, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph, homogenous list, and list" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 306408, root := ~, start := 306316, stop := 306407 ), rec( content := ",\n \ ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 306451, root := ~, start := 306415, stop := 306450 ), rec( content := ", ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 306483, root := ~, start := 306453, stop := 306482 ), rec( content := ", and\n \ ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 306524, root := ~, start := 306494, stop := 306523 ), rec( content := ".\n\n " , count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> cycle := CycleDigraph(12);\n\n\ gap> vert_colours := List([1 .. 12], x -> x mod 3 + 1);;\ngap> edge_colours :=\ List([1 .. 12], x -> [x mod 2 + 1]);;\ngap> Size(AutomorphismGroup(cycle));\n\ 12\ngap> Size(AutomorphismGroup(cycle, vert_colours));\n4\ngap> Size(Automorph\ ismGroup(cycle, fail, edge_colours));\n6\ngap> Size(AutomorphismGroup(cycle, v\ ert_colours, edge_colours));\n2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> ve\ rt_colours, List([1 .. 12], x -> [x mod 4 + 1])));\ntrue\n", count := [ 7, 2, 6, 14 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 6, 14 ], mathmode := "Text", name := "Example", next := 307071, root := ~, start := 306529, stop := 307070 ), rec( content := "\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 307088, root := ~, start := 303375, stop := 307087 ) ], count := [ 7, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 307102, root := ~, start := 303183, stop := 307101 ), rec( content := "\n\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "BlissCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 2 ], mathmode := "Text", name := "Attr", next := 307197, root := ~, start := 307123, stop := 307196 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "NautyCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 307274, root := ~, start := 307200, stop := 307273 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion, or a list of two permutations.", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Returns", next := 307341, root := ~, start := 307277, stop := 307340 ), rec( attributes := rec( ), content := [ rec( content := "\n A f\ unction ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 307384, root := ~, start := 307373, stop := 307383 ), rec( content := " that ma\ ps a digraph to a digraph is a ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical\n representative map", count := [ 7, 2, 7, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "E", next := 307464, root := ~, start := 307423, stop := 307463 ), rec( content := " if the \ following two conditions hold for all\n digraphs ", count := [ 7, 2, 7, 5 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 307531, root := ~, start := 307523, stop := 307530 ), rec( content := " and ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 307544, root := ~, start := 307536, stop := 307543 ), rec( content := ": ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 6 ], mathmode := "Text", name := "P", next := 307550, root := ~, start := 307546, stop := 307549 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 7, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\\rho(G)", count := [ 7, 2, 7, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 8 ], mathmode := "Tex\ t", name := "M", next := 307598, root := ~, start := 307584, stop := 307597 ), rec( cont\ ent := " and ", count := [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 7, 8 ], mathmode := "Text", name := "M", next := 307611, root := ~, star\ t := 307603, stop := 307610 ), rec( content := " are isomorphic as digraphs; a\ nd\n ", count := [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 8 ], name := "Item", next := 307657, root := ~, start := 307\ 569, stop := 307656 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\rho(G)=\\\ rho(H)", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "M", next := 3\ 07701, root := ~, start := 307679, stop := 307700 ), rec( content := " if and \ only if ", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 7, 10 ], mathmode := "Text", name := "M", next := 307725, root := ~ , star\ t := 307717, stop := 307724 ), rec( content := " and ", count := [ 7, 2, 7, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "H", count := [ 7, 2, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 10 ], mathmode := \ "Text", name := "M", next := 307738, root := ~, start := 307730, stop := 30773\ 7 ), rec( content := " are\n isomorphic as digraphs.\n ", count :=\ [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 10 ], name := "Ite\ m", next := 307788, root := ~, start := 307664, stop := 307787 ) ], count := [ 7, 2, 7, 7 ], mathmode := "Text", name := "List", next := 307800, root := ~, start := 307556, stop := 307799 ), rec( content := "\n\n \ A ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical labelling", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "E", next := 307834, root := ~, start := 307808, stop := 307833 ), rec( content := " of a di\ graph ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 307856, root := ~, start := 307848, stop := 307855 ), rec( content := " (under \ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 307875, root := ~, start := 307864, stop := 307874 ), rec( content := ") is an\\ n isomorphism of ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 307910, root := ~, start := 307902, stop := 307909 ), rec( content := " onto it\ s ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical representative", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "E", next := 307951, root := ~, start := 307920, stop := 307950 ), rec( content := ",\n " , count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho(G)", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 307971, root := ~, start := 307957, stop := 307970 ), rec( content := ". See " , count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "Ref", next := 308033, root := ~, start := 307978, stop := 308032 ), rec( content := "\n fo\ r more information about isomorphisms of digraphs. ", count := [ 7, 2, 7, 12 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "P", next := 308095, root := ~, start := 308091, stop := 308094 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "C", next := 308131, root := ~, start := 308101, stop := 308130 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "A", next := 308195, root := ~, start := 308181, stop := 308194 ), rec( content := " found u\ sing ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila and Petteri Kaski.\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "C", next := 308287, root := ~, start := 308257, stop := 308286 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "A", next := 308351, root := ~, start := 308337, stop := 308350 ), rec( content := " found u\ sing ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by Bren\ dan McKay and Adolfo Piperno.\n Note that the canonical labellings returned\ by ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " are not\ \n usually the same (and may depend of the version used).", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "P", next := 308550, root := ~, start := 308546, stop := 308549 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "C", next := 308586, root := ~, start := 308556, stop := 308585 ), rec( content := " can onl\ y be computed if ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "A", next := 308625, root := ~, start := 308611, stop := 308624 ), rec( content := " has\n \ no multiple edges; see ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "Ref", next := 308686, root := ~, start := 308657, stop := 308685 ), rec( content := ". ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 15 ], mathmode := "Text", name := "P", next := 308692, root := ~, start := 308688, stop := 308691 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> BlissCanonicalLabelling(digraph1);\n(1\ ,9,5,7)(3,6,4,10)\ngap> p := (1, 2, 7, 5)(3, 9)(6, 10, 8);;\ngap> digraph2 := \ OnDigraphs(digraph1, p);\n\ngap> \ digraph1 = digraph2;\nfalse\ngap> OnDigraphs(digraph1, BlissCanonicalLabelling\ (digraph1)) =\n> OnDigraphs(digraph2, BlissCanonicalLabelling(digraph2));\n\ true", count := [ 7, 2, 7, 16 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 16 ], mathmode := "Text", name := "Example", next := 309188, root := ~, start := 308698, stop := 309187 ), rec( content := "\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Description", next := 309205, root := ~, start := 307344, stop := 309204 ) ], count := [ 7, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 309219, root := ~, start := 307108, stop := 309218 ), rec( content := "\n\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, colours\ ", Label := "for a digraph and a list", Name := "BlissCanonicalLabelling" ), content := 0, count := [ 7, 2, 8, 2 ], mathmode := "Text", name := "Oper", next := 309338, root := ~, start := 309240, stop := 309337 ), rec( attributes := rec( Arg := "digraph, colours\ ", Label := "for a digraph and a list", Name := "NautyCanonicalLabelling" ), content := 0, count := [ 7, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 309439, root := ~, start := 309341, stop := 309438 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion.", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Returns", next := 309475, root := ~, start := 309442, stop := 309474 ), rec( attributes := rec( ), content := [ rec( content := "\n A f\ unction ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 309518, root := ~, start := 309507, stop := 309517 ), rec( content := " that ma\ ps a coloured digraph to a coloured digraph is\n a ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical representative map", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "E", next := 309614, root := ~, start := 309579, stop := 309613 ), rec( content := " if the \ following two conditions hold\n for all coloured digraphs ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 309690, root := ~, start := 309682, stop := 309689 ), rec( content := " and ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 309703, root := ~, start := 309695, stop := 309702 ), rec( content := ":\n\n \ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 8, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\\rho(G)", count := [ 7, 2, 8, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 7 ], mathmode := "Tex\ t", name := "M", next := 309752, root := ~, start := 309738, stop := 309751 ), rec( cont\ ent := " and ", count := [ 7, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 8, 7 ], mathmode := "Text", name := "M", next := 309765, root := ~, star\ t := 309757, stop := 309764 ), rec( content := " are isomorphic as coloured di\ graphs; and\n ", count := [ 7, 2, 8, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 7, 2, 8, 7 ], name := "Item", next := 309820, root := ~, start := 309\ 723, stop := 309819 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\rho(G)=\\\ rho(H)", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "M", next := 309\ 864, root := ~, start := 309842, stop := 309863 ), rec( content := " if and on\ ly if ", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count := \ [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 2, 8, 9 ], mathmode := "Text", name := "M", next := 309888, root := ~, star\ t := 309880, stop := 309887 ), rec( content := " and ", count := [ 7, 2, 8, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "H", count := [ 7, 2, 8, 9 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 9 ], mathmode := "Text", name\ := "M", next := 309901, root := ~, start := 309893, stop := 309900 ), rec( cont\ ent := " are\n isomorphic as coloured digraphs.\n ", count := [ 7,\ 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 9 ], name := "Ite\ m", next := 309960, root := ~, start := 309827, stop := 309959 ) ], count := [ 7, 2, 8, 6 ], mathmode := "Text", name := "List", next := 309972, root := ~, start := 309710, stop := 309971 ), rec( content := "\n\n \ A ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical labelling", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "E", next := 310006, root := ~, start := 309980, stop := 310005 ), rec( content := " of a co\ loured digraph ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 310037, root := ~, start := 310029, stop := 310036 ), rec( content := " (under\\ n ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 310060, root := ~, start := 310049, stop := 310059 ), rec( content := ") is an \ isomorphism of ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 310091, root := ~, start := 310083, stop := 310090 ), rec( content := " onto it\ s ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical\n representative", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "E", next := 310138, root := ~, start := 310101, stop := 310137 ), rec( content := ", ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho(G)", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 310154, root := ~, start := 310140, stop := 310153 ), rec( content := ". See " , count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "Ref", next := 310244, root := ~, start := 310161, stop := 310243 ), rec( content := " for mor\ e information about\n isomorphisms of coloured digraphs. ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "P", next := 310315, root := ~, start := 310311, stop := 310314 ), rec( content := "\n\n \ A coloured digraph can be specified by its underlying digraph ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 310397, root := ~, start := 310383, stop := 310396 ), rec( content := "\n an\ d its colouring ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 310434, root := ~, start := 310420, stop := 310433 ), rec( content := ". Let " , count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "C", next := 310449, root := ~, start := 310441, stop := 310448 ), rec( content := " be the \ number of vertices of\n ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 310497, root := ~, start := 310483, stop := 310496 ), rec( content := ". The co\ louring ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 310527, root := ~, start := 310513, stop := 310526 ), rec( content := " may hav\ e one of the following\n two forms: ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 13 ], mathmode := "Text", name := "P", next := 310577, root := ~, start := 310573, stop := 310576 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 8, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 8, 15 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathm\ ode := "Text", name := "C", next := 310629, root := ~, start := 310621, stop :\ = 310628 ), rec( content := " integers, where ", count := [ 7, 2, 8, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 8, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "Te\ xt", name := "A", next := 310660, root := ~, start := 310646, stop := 310659 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 8, 15 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15\ ], mathmode := "Text", name := "C", next := 310670, root := ~, start := 31066\ 0, stop := 310669 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 8, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ] , math\ mode := "Text", name := "C", next := 310711, root := ~, start := 310703, stop \ := 310710 ), rec( content := ", using the colours ", count := [ 7, 2, 8, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 8, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "T\ ext", name := "C", next := 310746, root := ~, start := 310731, stop := 310745 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "C", next\ := 310780, root := ~, start := 310764, stop := 310779 ), rec( content := "; o\ r\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 15 ], name := "Item", next := 310798, root := ~, start := 310\ 596, stop := 310797 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 8, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, \ 17 ], mathmode := "Text", name := "A", next := 310911, root := ~, start := 310\ 897, stop := 310910 ), rec( content := ")", count := [ 7, 2, 8, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmod\ e := "Text", name := "C", next := 310916, root := ~, start := 310878, stop := \ 310915 ), rec( content := ", such that\n ", count := [ 7, 2, 8, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 8, 17 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmode := "Te\ xt", name := "A", next := 310950, root := ~, start := 310936, stop := 310949 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 8, 17 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17\ ], mathmode := "Text", name := "C", next := 310960, root := ~, start := 31095\ 0, stop := 310959 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 8, 17 ], mathmode := "Text", name := "C", next := 311017, root := ~ , star\ t := 311009, stop := 311016 ), rec( content := ".\n ", count := [ 7, 2, 8\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], name := "Ite\ m", next := 311032, root := ~, start := 310805, stop := 311031 ) ], count := [ 7, 2, 8, 14 ], mathmode := "Text", name := "List", next := 311044, root := ~, start := 310583, stop := 311043 ), rec( content := "\n\n \ If ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 311067, root := ~, start := 311053, stop := 311066 ), rec( content := " and ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 311086, root := ~, start := 311072, stop := 311085 ), rec( content := " togethe\ r form a coloured digraph,\n ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "C", next := 311155, root := ~, start := 311125, stop := 311154 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 311219, root := ~, start := 311205, stop := 311218 ), rec( content := " found u\ sing ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila and Petteri Kaski.\n Similarly, ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "C", next := 311322, root := ~, start := 311292, stop := 311321 ), rec( content := " returns\ a canonical labelling of\n the digraph ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 311386, root := ~, start := 311372, stop := 311385 ), rec( content := " found u\ sing ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by Bren\ dan McKay and Adolfo\n Piperno. Note that the canonical labellings returne\ d by ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and\n \ ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " are not\ usually the same (and may depend of the version used).", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "P", next := 311586, root := ~, start := 311582, stop := 311585 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "C", next := 311622, root := ~, start := 311592, stop := 311621 ), rec( content := " can onl\ y be computed if ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 311661, root := ~, start := 311647, stop := 311660 ), rec( content := " has\n \ no multiple edges; see ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "Ref", next := 311722, root := ~, start := 311693, stop := 311721 ), rec( content := ". The ca\ nonical\n labelling of ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 311769, root := ~, start := 311755, stop := 311768 ), rec( content := " is give\ n as a permutation of its vertices.\n The canonical representative of ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 311863, root := ~, start := 311849, stop := 311862 ), rec( content := " can be \ created from\n ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 311902, root := ~, start := 311888, stop := 311901 ), rec( content := " and its\ canonical labelling ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "C", next := 311939, root := ~, start := 311931, stop := 311938 ), rec( content := " by usin\ g the\n operation ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a perm", Oper := "OnDigraphs" ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "Ref", next := 312025, root := ~, start := 311967, stop := 312024 ), rec( content := ":\n " , count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> OnDigraphs(digraph, p);", count := [ 7, 2, 8, 21 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 21 ], mathmode := "Text", name := "Log", next := 312070, root := ~, start := 312031, stop := 312069 ), rec( content := "\n\n \ The colouring of the canonical representative can easily be\n constructed. \ A vertex ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 312170, root := ~, start := 312162, stop := 312169 ), rec( content := " (in ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 312189, root := ~, start := 312175, stop := 312188 ), rec( content := ") has co\ lour ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 312210, root := ~, start := 312202, stop := 312209 ), rec( content := " if\n \ and only if the vertex ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ^ p", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 312253, root := ~, start := 312241, stop := 312252 ), rec( content := " (in the\ canonical representative) has\n colour ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 312311, root := ~, start := 312303, stop := 312310 ), rec( content := ", where \ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 312327, root := ~, start := 312319, stop := 312326 ), rec( content := " is the \ permutation of the canonical\n labelling that acts on the vertices of ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 312421, root := ~, start := 312407, stop := 312420 ), rec( content := ". In par\ ticular, if\n ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 312459, root := ~, start := 312445, stop := 312458 ), rec( content := " has the\ first form that is described above, then the\n colouring of the canonical \ representative is given by:\n\n ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> List(DigraphVertices(digraph), i -> colours[i / p]);", count := [ 7, 2, 8, 23 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 23 ], mathmode := "Text", name := "Log", next := 312645, root := ~, start := 312577, stop := 312644 ), rec( content := "\n\n \ On the other hand, if ", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "A", next := 312687, root := ~, start := 312673, stop := 312686 ), rec( content := " has the\ second form above, then the\n canonical representative has colouring:\n\n \ ", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> OnTuplesSets(colours, p);", count := [ 7, 2, 8, 25 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 25 ], mathmode := "Text", name := "Log", next := 312814, root := ~, start := 312773, stop := 312813 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 27 ], mathmode := "Text", name := "P", next := 312824, root := ~, start := 312820, stop := 312823 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> colours := [[1, 2, 8, 9, 10], [3, 4, 5,\ 6, 7]];;\ngap> p := BlissCanonicalLabelling(digraph, colours);\n(1,5,8,4,10,3\ ,9)(6,7)\ngap> OnDigraphs(digraph, p);\n\ngap> OnTuplesSets(colours, p);\n[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10\ ] ]\ngap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ngap> p := BlissCanonic\ alLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnDigraphs(digraph\ , p);\n\ngap> List(DigraphVertice\ s(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ]", count := [ 7, 2, 8, 28 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 28 ], mathmode := "Text", name := "Example", next := 313535, root := ~, start := 312830, stop := 313534 ), rec( content := "\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 313552, root := ~, start := 309478, stop := 313551 ) ], count := [ 7, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 313566, root := ~, start := 309225, stop := 313565 ), rec( content := "\n\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BlissCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 2 ], mathmode := "Text", name := "Attr", next := 313637, root := ~, start := 313587, stop := 313636 ), rec( attributes := rec( Arg := "digraph", Name := "NautyCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 313690, root := ~, start := 313640, stop := 313689 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Returns", next := 313722, root := ~, start := 313693, stop := 313721 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ attribute ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "C", next := 313787, root := ~, start := 313757, stop := 313786 ), rec( content := " returns\ the canonical\n representative found by applying ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissCa\ nonicalLabelling", Label := "for a digraph" ), content := 0, count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Ref", next := 313912, root := ~, start := 313847, stop := 313911 ), rec( content := ". The di\ graph returned is canonical in the sense\n that\n\n ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 9, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "BlissCanonicalDigraph(", count := [ 7, 2, 9, 7 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), c\ ontent := [ rec( content := "digraph", count := [ 7, 2, 9, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "T\ ext", name := "A", next := 314042, root := ~, start := 314028, stop := 314041 \ ), rec( content := ")", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ C", next := 314047, root := ~, start := 314003, stop := 314046 ), rec( content\ := " and ", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 9, 7 ], mathmode := "Text", name := "A", next := 314066, root := \ ~, start := 314052, stop := 314065 ), rec( content := " are\n isomorphi\ c as digraphs; and\n ", count := [ 7, 2, 9, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], name := "Item", next := 3141\ 20, root := ~, start := 313988, stop := 314119 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n If ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "gr", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next := 31415\ 4, root := ~, start := 314145, stop := 314153 ), rec( content := " is any digr\ aph then ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "BlissCanoni\ calDigraph(gr)", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next\ := 314207, root := ~, start := 314175, stop := 314206 ), rec( content := "\n \ and ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "BlissCanoni\ calDigraph(", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph"\ , count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ]\ , count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "A", next := 314259, ro\ ot := ~, start := 314245, stop := 314258 ), rec( content := ")", count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "C", next := 314264, root := ~, start \ := 314220, stop := 314263 ), rec( content := " are equal if and only\n \ if ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "gr", count := [ \ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next := 314307, root := ~, star\ t := 314298, stop := 314306 ), rec( content := " and ", count := [ 7, 2, 9, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph", count := [ 7, 2, 9, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text\ ", name := "A", next := 314326, root := ~, start := 314312, stop := 314325 ), rec( cont\ ent := " are isomorphic as digraphs.\n ", count := [ 7, 2, 9, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 9 ], name := "Ite\ m", next := 314368, root := ~, start := 314127, stop := 314367 ) ], count := [ 7, 2, 9, 6 ], mathmode := "Text", name := "List", next := 314380, root := ~, start := 313975, stop := 314379 ), rec( content := "\n\n \ Analogously, the attribute ", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "C", next := 314443, root := ~, start := 314413, stop := 314442 ), rec( content := " returns\ the\n canonical representative found by applying ", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyCa\ nonicalLabelling", Label := "for a digraph" ), content := 0, count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "Ref", next := 314568, root := ~, start := 314503, stop := 314567 ), rec( content := ".\n " , count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "P", next := 314578, root := ~, start := 314574, stop := 314577 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "A", next := 314614, root := ~, start := 314600, stop := 314613 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);\n\ngap> canon := BlissCanonicalDigraph(digraph);\n\ngap> OutNeighbours(canon);\n[ [ 1 ], \ [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]\n", count := [ 7, 2, 9, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 13 ], mathmode := "Text", name := "Example", next := 315007, root := ~, start := 314716, stop := 315006 ), rec( content := "\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Description", next := 315024, root := ~, start := 313725, stop := 315023 ) ], count := [ 7, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 315038, root := ~, start := 313572, stop := 315037 ), rec( content := "\n\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphGroup" ) , content := 0, count := [ 7, 2, 10, 2 ], mathmode := "Text", name := "Attr", next := 315100, root := ~, start := 315059, stop := 315099 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 315142, root := ~, start := 315103, stop := 315141 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 315180, root := ~, start := 315166, stop := 315179 ), rec( content := " is immu\ table and was created knowing a subgroup of\n its automorphism group, then \ this group is stored in the attribute\n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "C", next := 315326, root := ~, start := 315307, stop := 315325 ), rec( content := ". If ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 315345, root := ~, start := 315331, stop := 315344 ), rec( content := " is muta\ ble, or was not created\n knowing a subgroup of its automorphism group, the\ n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "C", next := 315451, root := ~, start := 315432, stop := 315450 ), rec( content := "\n re\ turns the entire automorphism group of ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 315511, root := ~, start := 315497, stop := 315510 ), rec( content := ". \n \ Note that if ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 315548, root := ~, start := 315534, stop := 315547 ), rec( content := " is muta\ ble, then the automorphism group is\n recomputed every time this function i\ s called.\n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "P", next := 315651, root := ~, start := 315647, stop := 315650 ), rec( content := "\n\n \ Note that certain other constructor operations such as ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 315746, root := ~, start := 315713, stop := 315745 ), rec( content := ", ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Biparti\ teDoubleDigraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 315784, root := ~, start := 315748, stop := 315783 ), rec( content := ", and ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DoubleD\ igraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 315823, root := ~, start := 315790, stop := 315822 ), rec( content := ", may no\ t require a group as one of the arguments,\n but use the standard construct\ or method using a group, and hence set the\n ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "C", next := 315974, root := ~, start := 315955, stop := 315973 ), rec( content := " attribu\ te for the resulting digraph.\n\n ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1\ ) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermG\ roup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1\ .. n], \\^, adj);\n\ngap> HasDigrap\ hGroup(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> A\ utomorphismGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> D := Digraph(group, [\ 1 .. n], \\^, adj);\n\ngap> HasDig\ raphGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,4) ])\ngap> D := Do\ ubleDigraph(D);\n\ngap> HasDigrap\ hGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,4)(5,6,7,8), (1,5)(2,6\ )(3,7)(4,8) ])\ngap> AutomorphismGroup(D) =\n> Group([(6, 8), (5, 7), (4, 6), \ (3, 5), (2, 4),\n> (1, 2)(3, 4)(5, 6)(7, 8)]);\ntrue\ngap> D := Digraph\ ([[2, 3], [], []]);\n\ngap> HasDig\ raphGroup(D);\nfalse\ngap> HasAutomorphismGroup(D);\nfalse\ngap> DigraphGroup(\ D);\nGroup([ (2,3) ])\ngap> HasAutomorphismGroup(D);\ntrue\ngap> group := Dihe\ dralGroup(8);\n\ngap> D := CayleyDigraph\ (group);\n\ngap> HasDigraphGroup(\ D);\ntrue\ngap> GeneratorsOfGroup(DigraphGroup(D));\n[ (1,2)(3,5)(4,6)(7,8), (\ 1,7,4,3)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]", count := [ 7, 2, 10, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 10, 5 ], mathmode := "Text", name := "Example", next := 317444, root := ~, start := 316017, stop := 317443 ), rec( content := "\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 317461, root := ~, start := 315145, stop := 317460 ) ], count := [ 7, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 317475, root := ~, start := 315044, stop := 317474 ), rec( content := "\n\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOrbits" ), content := 0, count := [ 7, 2, 11, 2 ], mathmode := "Text", name := "Attr", next := 317538, root := ~, start := 317496, stop := 317537 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of lists of integers.\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Returns", next := 317607, root := ~, start := 317541, stop := 317606 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOrbits", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "C", next := 317648, root := ~, start := 317628, stop := 317647 ), rec( content := " returns\ the orbits of the action of the\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Ref", next := 317719, root := ~, start := 317693, stop := 317718 ), rec( content := " on the \ set of vertices of ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "A", next := 317760, root := ~, start := 317746, stop := 317759 ), rec( content := ".\n\n \ ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := Group([(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := Edg\ eOrbitsDigraph(G, [1, 2]);\n\ngap>\ DigraphOrbits(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := Digra\ phMutableCopy(D);\n\ngap> DigraphOrb\ its(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6, 7, 8, 9 ] ]\n", count := [ 7, 2, 11, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 11, 4 ], mathmode := "Text", name := "Example", next := 318149, root := ~, start := 317767, stop := 318148 ), rec( content := "\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 318166, root := ~, start := 317610, stop := 318165 ) ], count := [ 7, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 318180, root := ~, start := 317481, stop := 318179 ), rec( content := "\n\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOrbitRep\ s" ), content := 0, count := [ 7, 2, 12, 2 ], mathmode := "Text", name := "Attr", next := 318246, root := ~, start := 318201, stop := 318245 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of integers.\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Returns", next := 318306, root := ~, start := 318249, stop := 318305 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOrbitReps", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "C", next := 318350, root := ~, start := 318327, stop := 318349 ), rec( content := " returns\ a list of orbit representatives\n of the action of the ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Ref", next := 318442, root := ~, start := 318416, stop := 318441 ), rec( content := " on the \ set of vertices of\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "A", next := 318487, root := ~, start := 318473, stop := 318486 ), rec( content := ".\n\n \ ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CayleyDigraph(AlternatingGroup(4));\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphMutableCo\ py(D);\n\ngap> DigraphOrbitReps(D)\ ;\n[ 1 ]\ngap> D := DigraphFromDigraph6String(\"&IGO??S?`?_@?a?CK?O\");\n\ngap> DigraphOrbitReps(D);\n[ 1, 2, \ 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> DigraphOrbitReps(DigraphMutableCopy(D));\n[ 1,\ 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n", count := [ 7, 2, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 4 ], mathmode := "Text", name := "Example", next := 319008, root := ~, start := 318494, stop := 319007 ), rec( content := "\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 319025, root := ~, start := 318309, stop := 319024 ) ], count := [ 7, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 319039, root := ~, start := 318186, stop := 319038 ), rec( content := "\n\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSchreier\ Vector" ), content := 0, count := [ 7, 2, 13, 2 ], mathmode := "Text", name := "Attr", next := 319110, root := ~, start := 319060, stop := 319109 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of integers.\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Returns", next := 319170, root := ~, start := 319113, stop := 319169 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphSchreierVector", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 319219, root := ~, start := 319191, stop := 319218 ), rec( content := " returns\ the so-called ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ chreier vector", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "E", next := 319264, root := ~, start := 319242, stop := 319263 ), rec( content := "\n of\ the action of the ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Ref", next := 319316, root := ~, start := 319290, stop := 319315 ), rec( content := " on the \ set of vertices of\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "A", next := 319361, root := ~, start := 319347, stop := 319360 ), rec( content := ". The Sc\ hreier vector is a list ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ch", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 319403, root := ~, start := 319393, stop := 319402 ), rec( content := " of inte\ gers with\n length ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "A", next := 3194\ 67, root := ~, start := 319453, stop := 319466 ), rec( content := ")", count := [ 7, 2, 13, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 319472, root := ~, start := 319432, stop := 319471 ), rec( content := " where:\\ n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "sch[i] < 0:", count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "C", next\ := 319524, root := ~, start := 319503, stop := 319523 ) ], count := [ 7, 2, 13, 4 ], name := "Mar\ k", next := 319531, root := ~, start := 319497, stop := 319530 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n implies that ", count := [ \ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 13, 5 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "C", next := 319574, root := ~, start := 319566, stop \ := 319573 ), rec( content := " is an orbit representative and\n ", coun\ t := [ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphOrbitReps(", count := \ [ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attri\ butes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 13,\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13\ , 5 ], mathmode := "Text", name := "A", next := 319648, root := ~, start := 31\ 9634, stop := 319647 ), rec( content := ")[-sch[i]] = i", count := [ 7, 2, 13,\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13\ , 5 ], mathmode := "Text", name := "C", next := 319666, root := ~, start := 31\ 9614, stop := 319665 ), rec( content := ".\n ", count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], name := "Ite\ m", next := 319681, root := ~, start := 319538, stop := 319680 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "sch[i] > 0:", count := [ 7, 2, 13, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 6 ], mathmode := "Text", name := "C", next\ := 319716, root := ~, start := 319695, stop := 319715 ) ], count := [ 7, 2, 13, 6 ], name := "Mar\ k", next := 319723, root := ~, start := 319689, stop := 319722 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n implies that ", count := [ \ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "i / gens[sch[i]]", count := [ 7,\ 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7\ , 2, 13, 7 ], mathmode := "Text", name := "C", next := 319781, root := ~, star\ t := 319758, stop := 319780 ), rec( content := "\n is one step closer t\ o the root (or representative) of the tree, where\n ", count := [ 7, 2,\ 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "gens", count := [ 7, 2, 13, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 7 ], math\ mode := "Text", name := "C", next := 319879, root := ~, start := 319868, stop \ := 319878 ), rec( content := " is the generators of ", count := [ 7, 2, 13, 7 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphGroup(", count := [ 7, 2, 13, 7 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "digraph", count := [ 7, 2, 13, 7 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", \ name := "A", next := 319931, root := ~, start := 319917, stop := 319930 ), rec\ ( content := ")", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "C", next\ := 319936, root := ~, start := 319901, stop := 319935 ), rec( content := ".\n\ ", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 7 ], name := "Item", next := 319951, root := ~, start := 319\ 730, stop := 319950 ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "List", next := 319963, root := ~, start := 319484, stop := 319962 ), rec( content := "\n\n \ ", count := [ 7, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1\ ) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermG\ roup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1\ .. n], \\^, adj);\n\ngap> sch := Di\ graphSchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(AlternatingG\ roup(4));\n\ngap> sch := Digraph\ SchreierVector(D);\n[ -1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1 ]\ngap> DigraphOrbit\ Reps(D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,7,5)(2\ ,10,9)(3,4,11)(6,8,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12) ]\ngap> 10 / gens[sch[\ 10]];\n2\ngap> 7 / gens[sch[7]];\n1\ngap> 5 / gens[sch[5]];\n7", count := [ 7, 2, 13, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 10 ], mathmode := "Text", name := "Example", next := 320740, root := ~, start := 319969, stop := 320739 ), rec( content := "\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Description", next := 320757, root := ~, start := 319173, stop := 320756 ) ], count := [ 7, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 320771, root := ~, start := 319045, stop := 320770 ), rec( content := "\n\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphStabiliz\ er" ), content := 0, count := [ 7, 2, 14, 2 ], mathmode := "Text", name := "Oper", next := 320841, root := ~, start := 320792, stop := 320840 ), rec( attributes := rec( ), content := [ rec( content := "\n A p\ ermutation group.\n ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Returns", next := 320891, root := ~, start := 320844, stop := 320890 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphStabilizer", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "C", next := 320936, root := ~, start := 320912, stop := 320935 ), rec( content := " returns\ the stabilizer of the vertex ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "A", next := 320982, root := ~, start := 320974, stop := 320981 ), rec( content := "\n un\ der of the action of the ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 321040, root := ~, start := 321014, stop := 321039 ), rec( content := " on the \ set of\n vertices of ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "A", next := 321085, root := ~, start := 321071, stop := 321084 ), rec( content := ".\n\n \ ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> Digr\ aphStabilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\nga\ p> DigraphStabilizer(D, 2);\nGroup(())\n", count := [ 7, 2, 14, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 14, 4 ], mathmode := "Text", name := "Example", next := 321461, root := ~, start := 321092, stop := 321460 ), rec( content := "\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 321478, root := ~, start := 320894, stop := 321477 ) ], count := [ 7, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 321492, root := ~, start := 320777, stop := 321491 ), rec( content := "\n\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Label := "for digraphs", Name := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 15, 2 ], mathmode := "Text", name := "Oper", next := 321593, root := ~, start := 321513, stop := 321592 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 321616, root := ~, start := 321605, stop := 321615 ), rec( content := " or ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 321632, root := ~, start := 321620, stop := 321631 ), rec( content := ".", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Returns", next := 321643, root := ~, start := 321596, stop := 321642 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 321698, root := ~, start := 321687, stop := 321697 ), rec( content := " if ther\ e exists an isomorphism from the\n digraph ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "A", next := 321766, root := ~, start := 321751, stop := 321765 ), rec( content := " to the \ digraph ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "A", next := 321797, root := ~, start := 321782, stop := 321796 ), rec( content := ". See " , count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Ref", next := 321865, root := ~, start := 321804, stop := 321864 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. ", count := [ 7, 2, 15, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "P", next := 321927, root := ~, start := 321923, stop := 321926 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 15, 4 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 15, 4 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 15, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 15, 4 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Ref", next := 322235, root := ~, start := 322205, stop := 322234 ), rec( content := ", and ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Ref", next := 322271, root := ~, start := 322241, stop := 322270 ), rec( content := ".\n\n \ ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := CycleDigraph(4);\n\ \ngap> digraph2 := CycleDigraph(5);\n\ \ngap> IsIsomorphicDigraph(digraph1, digraph2);\nfalse\ngap> digraph2 := Digra\ phReverse(digraph1);\n\ngap> IsIso\ morphicDigraph(digraph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], \ []]);\n\ngap> digraph2 := Digraph([\ [], [], [2]]);\n\ngap> IsIsomorphic\ Digraph(digraph1, digraph2);\ntrue", count := [ 7, 2, 15, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 15, 5 ], mathmode := "Text", name := "Example", next := 322873, root := ~, start := 322278, stop := 322872 ), rec( content := "\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Description", next := 322890, root := ~, start := 321646, stop := 322889 ) ], count := [ 7, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 322904, root := ~, start := 321498, stop := 322903 ), rec( content := "\n\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2, colours1, colours2", Label := "for digraphs and homogeneous lists", Name := "IsIsomorphicDig\ raph" ), content := 0, count := [ 7, 2, 16, 2 ], mathmode := "Text", name := "Oper", next := 323051, root := ~, start := 322925, stop := 323050 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "K", next := 323074, root := ~, start := 323063, stop := 323073 ), rec( content := " or ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "K", next := 323090, root := ~, start := 323078, stop := 323089 ), rec( content := ".", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 323101, root := ~, start := 323054, stop := 323100 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation tests for isomorphism of coloured digraphs. A coloured\n digra\ ph can be specified by its underlying digraph ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 323263, root := ~, start := 323248, stop := 323262 ), rec( content := " and its\ \n colouring ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 323301, root := ~, start := 323286, stop := 323300 ), rec( content := ". Let " , count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "C", next := 323316, root := ~, start := 323308, stop := 323315 ), rec( content := " be the \ number of vertices of\n ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 323365, root := ~, start := 323350, stop := 323364 ), rec( content := ". The co\ louring ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 323396, root := ~, start := 323381, stop := 323395 ), rec( content := " may hav\ e one of the following\n two forms:\n\n ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 16, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathm\ ode := "Text", name := "C", next := 323493, root := ~, start := 323485, stop :\ = 323492 ), rec( content := " integers, where ", count := [ 7, 2, 16, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 16, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "Te\ xt", name := "A", next := 323524, root := ~, start := 323510, stop := 323523 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 16, 5 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5\ ], mathmode := "Text", name := "C", next := 323534, root := ~, start := 32352\ 4, stop := 323533 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 16, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ] , math\ mode := "Text", name := "C", next := 323575, root := ~, start := 323567, stop \ := 323574 ), rec( content := ", using the colours ", count := [ 7, 2, 16, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 16, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "T\ ext", name := "C", next := 323610, root := ~, start := 323595, stop := 323609 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "C", next\ := 323644, root := ~, start := 323628, stop := 323643 ), rec( content := "; o\ r\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 5 ], name := "Item", next := 323662, root := ~, start := 323\ 460, stop := 323661 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 16, \ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16,\ 7 ], mathmode := "Text", name := "A", next := 323775, root := ~, start := 323\ 761, stop := 323774 ), rec( content := ")", count := [ 7, 2, 16, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmod\ e := "Text", name := "C", next := 323780, root := ~, start := 323742, stop := \ 323779 ), rec( content := ", such that\n ", count := [ 7, 2, 16, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 16, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmode := "Te\ xt", name := "A", next := 323814, root := ~, start := 323800, stop := 323813 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 16, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7\ ], mathmode := "Text", name := "C", next := 323824, root := ~, start := 32381\ 4, stop := 323823 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 16, 7 ], mathmode := "Text", name := "C", next := 323881, root := ~ , star\ t := 323873, stop := 323880 ), rec( content := ".\n ", count := [ 7, 2, 1\ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], name := "Ite\ m", next := 323896, root := ~, start := 323669, stop := 323895 ) ], count := [ 7, 2, 16, 4 ], mathmode := "Text", name := "List", next := 323908, root := ~, start := 323447, stop := 323907 ), rec( content := "\n\n \ If ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 323932, root := ~, start := 323917, stop := 323931 ), rec( content := " and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 323952, root := ~, start := 323937, stop := 323951 ), rec( content := " are dig\ raphs without multiple edges,\n and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 324013, root := ~, start := 323998, stop := 324012 ), rec( content := " and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 324033, root := ~, start := 324018, stop := 324032 ), rec( content := " are col\ ourings of ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 324067, root := ~, start := 324052, stop := 324066 ), rec( content := "\n an\ d ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 324091, root := ~, start := 324076, stop := 324090 ), rec( content := ", respec\ tively, then this operation returns ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "K", next := 324146, root := ~, start := 324135, stop := 324145 ), rec( content := "\n if\ there exists an isomorphism between these two coloured digraphs. See\n ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "Ref", next := 324311, root := ~, start := 324228, stop := 324310 ), rec( content := " for mor\ e information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "P", next := 324378, root := ~, start := 324374, stop := 324377 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL\ ", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 16, 10 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 16, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 16, 10 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 16, 10 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Ref", next := 324686, root := ~, start := 324656, stop := 324685 ), rec( content := ", and ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Ref", next := 324722, root := ~, start := 324692, stop := 324721 ), rec( content := ".\n\n \ ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := ChainDigraph(4);\n\ \ngap> digraph2 := ChainDigraph(3);\n\ \ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], \ [3]]);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [1\ , 1, 1, 1], [1, 1, 1, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, digraph2,\ \n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, di\ graph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse", count := [ 7, 2, 16, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 16, 11 ], mathmode := "Text", name := "Example", next := 325335, root := ~, start := 324729, stop := 325334 ), rec( content := "\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 325352, root := ~, start := 323104, stop := 325351 ) ], count := [ 7, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 325366, root := ~, start := 322910, stop := 325365 ), rec( content := "\n\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Label := "for digraphs", Name := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 17, 2 ], mathmode := "Text", name := "Oper", next := 325467, root := ~, start := 325387, stop := 325466 ), rec( attributes := rec( ), content := [ rec( content := " A permut\ ation, or a pair of permutations, or ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "K", next := 325536, root := ~, start := 325525, stop := 325535 ), rec( content := ".", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 325547, root := ~, start := 325470, stop := 325546 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns an isomorphism between the digraphs ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "A", next := 325642, root := ~, start := 325627, stop := 325641 ), rec( content := "\n an\ d ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "A", next := 325666, root := ~, start := 325651, stop := 325665 ), rec( content := " if one \ exists, else this operation returns ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "K", next := 325721, root := ~, start := 325710, stop := 325720 ), rec( content := ".\n " , count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "P", next := 325731, root := ~, start := 325727, stop := 325730 ), rec( content := "\n\n \ An ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "E", next := 325758, root := ~, start := 325740, stop := 325757 ), rec( content := " from a \ digraph ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 325789, root := ~, start := 325774, stop := 325788 ), rec( content := " to a di\ graph\n ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 325822, root := ~, start := 325807, stop := 325821 ), rec( content := " is a bi\ jection ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 325846, root := ~, start := 325838, stop := 325845 ), rec( content := " from th\ e vertices of\n ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 325887, root := ~, start := 325872, stop := 325886 ), rec( content := " to the \ vertices of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 325922, root := ~, start := 325907, stop := 325921 ), rec( content := " with th\ e following\n property: for all vertices ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 325981, root := ~, start := 325973, stop := 325980 ), rec( content := " and ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 325994, root := ~, start := 325986, stop := 325993 ), rec( content := " of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 326013, root := ~, start := 325998, stop := 326012 ), rec( content := ",\n " , count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 326032, root := ~, start := 326019, stop := 326031 ), rec( content := " is an e\ dge of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 326062, root := ~, start := 326047, stop := 326061 ), rec( content := " if and \ only if ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i ^ p, j\n ^ p]", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 326105, root := ~, start := 326078, stop := 326104 ), rec( content := " is an e\ dge of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 326135, root := ~, start := 326120, stop := 326134 ), rec( content := ". ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "P", next := 326141, root := ~, start := 326137, stop := 326140 ), rec( content := "\n\n \ If there exists such an isomorphism, then this operation returns one.\n The\ form of this isomorphism is a permutation ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "C", next := 326275, root := ~, start := 326267, stop := 326274 ), rec( content := " of the \ vertices of\n ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "A", next := 326314, root := ~, start := 326299, stop := 326313 ), rec( content := " such th\ at ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "P", next := 326329, root := ~, start := 326325, stop := 326328 ), rec( content := "\n\n \ ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph1", count := [ 7, 2, 17, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "A", next := 326364, root := ~, start := 326349, stop := 32636\ 3 ), rec( content := ", p) = digraph2", count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "C", next := 326383, root := ~, start := 326335, stop := 326382 ), rec( content := ".\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 17, 6 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 17, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 17, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 17, 6 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Ref", next := 326692, root := ~, start := 326662, stop := 326691 ), rec( content := ", and ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Ref", next := 326728, root := ~, start := 326698, stop := 326727 ), rec( content := ".\n\n \ ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := CycleDigraph(4);\n\ \ngap> digraph2 := CycleDigraph(5);\n\ \ngap> IsomorphismDigraphs(digraph1, digraph2);\nfail\ngap> digraph1 := Comple\ teBipartiteDigraph(10, 5);\n\ngap> digraph2 := CompleteBipartiteDigraph(5, 10);\n\ngap> p := I\ somorphismDigraphs(digraph1, digraph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10\ ,15)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\n", count := [ 7, 2, 17, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 17, 7 ], mathmode := "Text", name := "Example", next := 327351, root := ~, start := 326735, stop := 327350 ), rec( content := "\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 327368, root := ~, start := 325550, stop := 327367 ) ], count := [ 7, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 327382, root := ~, start := 325372, stop := 327381 ), rec( content := "\n\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2, colours1, colours2", Label := "for digraphs and homogeneous lists", Name := "IsomorphismDigr\ aphs" ), content := 0, count := [ 7, 2, 18, 2 ], mathmode := "Text", name := "Oper", next := 327529, root := ~, start := 327403, stop := 327528 ), rec( attributes := rec( ), content := [ rec( content := " A permut\ ation, or ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "K", next := 327571, root := ~, start := 327560, stop := 327570 ), rec( content := ".", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Returns", next := 327582, root := ~, start := 327532, stop := 327581 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation searches for an isomorphism between coloured digraphs. A\n col\ oured digraph can be specified by its underlying digraph ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 327755, root := ~, start := 327740, stop := 327754 ), rec( content := "\n an\ d its colouring ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 327793, root := ~, start := 327778, stop := 327792 ), rec( content := ". Let " , count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "C", next := 327808, root := ~, start := 327800, stop := 327807 ), rec( content := " be the \ number of vertices\n of ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 327857, root := ~, start := 327842, stop := 327856 ), rec( content := ". The co\ louring ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 327888, root := ~, start := 327873, stop := 327887 ), rec( content := " may hav\ e one of the\n following two forms:\n\n ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 18, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathm\ ode := "Text", name := "C", next := 327985, root := ~, start := 327977, stop :\ = 327984 ), rec( content := " integers, where ", count := [ 7, 2, 18, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 18, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "Te\ xt", name := "A", next := 328016, root := ~, start := 328002, stop := 328015 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 18, 5 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5\ ], mathmode := "Text", name := "C", next := 328026, root := ~, start := 32801\ 6, stop := 328025 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 18, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ] , math\ mode := "Text", name := "C", next := 328067, root := ~, start := 328059, stop \ := 328066 ), rec( content := ", using the colours ", count := [ 7, 2, 18, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 18, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "T\ ext", name := "C", next := 328102, root := ~, start := 328087, stop := 328101 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "C", next\ := 328136, root := ~, start := 328120, stop := 328135 ), rec( content := "; o\ r\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 5 ], name := "Item", next := 328154, root := ~, start := 327\ 952, stop := 328153 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 18, \ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18,\ 7 ], mathmode := "Text", name := "A", next := 328267, root := ~, start := 328\ 253, stop := 328266 ), rec( content := ")", count := [ 7, 2, 18, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmod\ e := "Text", name := "C", next := 328272, root := ~, start := 328234, stop := \ 328271 ), rec( content := ", such that\n ", count := [ 7, 2, 18, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 18, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmode := "Te\ xt", name := "A", next := 328306, root := ~, start := 328292, stop := 328305 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 18, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7\ ], mathmode := "Text", name := "C", next := 328316, root := ~, start := 32830\ 6, stop := 328315 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 18, 7 ], mathmode := "Text", name := "C", next := 328373, root := ~ , star\ t := 328365, stop := 328372 ), rec( content := ".\n ", count := [ 7, 2, 1\ 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], name := "Ite\ m", next := 328388, root := ~, start := 328161, stop := 328387 ) ], count := [ 7, 2, 18, 4 ], mathmode := "Text", name := "List", next := 328400, root := ~, start := 327939, stop := 328399 ), rec( content := "\n\n \ An ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "E", next := 328427, root := ~, start := 328409, stop := 328426 ), rec( content := " between\ coloured digraphs is an isomorphism between\n the underlying digraphs that\ preserves the colourings. See ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "Ref", next := 328604, root := ~, start := 328544, stop := 328603 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. More precisely, let ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 328690, root := ~, start := 328682, stop := 328689 ), rec( content := " be an\n\ isomorphism of digraphs from the digraph ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 328757, root := ~, start := 328742, stop := 328756 ), rec( content := " (with c\ olouring\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 328793, root := ~, start := 328778, stop := 328792 ), rec( content := ") to the\ digraph ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 328825, root := ~, start := 328810, stop := 328824 ), rec( content := " (with c\ olouring\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 328861, root := ~, start := 328846, stop := 328860 ), rec( content := "), and l\ et ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 328880, root := ~, start := 328872, stop := 328879 ), rec( content := " be the \ permutation of the vertices of\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 328938, root := ~, start := 328923, stop := 328937 ), rec( content := " that co\ rresponds to ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 328967, root := ~, start := 328959, stop := 328966 ), rec( content := ". Then \ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 328983, root := ~, start := 328975, stop := 328982 ), rec( content := " preserv\ es the\n colourings of ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 329031, root := ~, start := 329016, stop := 329030 ), rec( content := " and ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 329051, root := ~, start := 329036, stop := 329050 ), rec( content := " – and\ hence is an\n isomorphism of coloured digraphs – if ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "colours1", count := [ 7, 2, 18, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "A", next := 329134, root := ~, start := 329119, stop := 32913\ 3 ), rec( content := "[i] =\n ", count := [ 7, 2, 18, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "colours2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 3291\ 61, root := ~, start := 329146, stop := 329160 ), rec( content := "[i ^ p]", count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 329172, root := ~, start := 329116, stop := 329171 ), rec( content := " for all\ vertices ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 329198, root := ~, start := 329190, stop := 329197 ), rec( content := " in ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 329217, root := ~, start := 329202, stop := 329216 ), rec( content := ".\n " , count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "P", next := 329227, root := ~, start := 329223, stop := 329226 ), rec( content := "\n\n \ This operation returns such an isomorphism if one exists, else it returns\n \ ", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "K", next := 329322, root := ~, start := 329311, stop := 329321 ), rec( content := ". ", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "P", next := 329328, root := ~, start := 329324, stop := 329327 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL\ ", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 18, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 18, 11 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 18, 11 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "https://pallini.di.unir\ oma1.it/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL", next := 75, root := ~, start := 17, stop := 74 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 81, root := ~, start := 1, stop := 80 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 18, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Package", next := 125, root := ~, start := 101 , stop := 124 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 131, root := ~, start := 81, stop := 130 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Ref", next := 329636, root := ~, start := 329606, stop := 329635 ), rec( content := ", and ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Ref", next := 329672, root := ~, start := 329642, stop := 329671 ), rec( content := ".\n\n \ ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := ChainDigraph(4);\n\ \ngap> digraph2 := ChainDigraph(3);\n\ \ngap> IsomorphismDigraphs(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], \ [3]]);\nfail\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> colours1 := [1, 1, 1, 1];;\ngap> colours2 := [1\ , 1, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colours1, colo\ urs2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> List(\ DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> colou\ rs1 := [1, 1, 2, 2];;\ngap> colours2 := [2, 2, 1, 1];;\ngap> p := IsomorphismD\ igraphs(digraph1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(\ digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> colo\ urs1[i ^ p]) = colours2;\ntrue\ngap> IsomorphismDigraphs(digraph1, digraph2,\n\ > [1, 1, 2, 2], [1, 1, 2, 2]);\nfail", count := [ 7, 2, 18, 12 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 18, 12 ], mathmode := "Text", name := "Example", next := 330655, root := ~, start := 329679, stop := 330654 ), rec( content := "\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Description", next := 330672, root := ~, start := 327585, stop := 330671 ) ], count := [ 7, 2, 18, 1 ], mathmode := "Text", name := "ManSection", next := 330686, root := ~, start := 327388, stop := 330685 ), rec( content := "\n\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "RepresentativeO\ utNeighbours" ), content := 0, count := [ 7, 2, 19, 2 ], mathmode := "Text", name := "Attr", next := 330763, root := ~, start := 330707, stop := 330762 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists.", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Returns", next := 330812, root := ~, start := 330766, stop := 330811 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "C", next := 330874, root := ~, start := 330864, stop := 330873 ), rec( content := " of ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut-neighbours", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "E", next := 330899, root := ~, start := 330878, stop := 330898 ), rec( content := " of each\ \n representative of the orbits of the action of ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Ref", next := 330984, root := ~, start := 330958, stop := 330983 ), rec( content := " on\n \ the vertex set of the digraph ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "A", next := 331036, root := ~, start := 331022, stop := 331035 ), rec( content := ". ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "P", next := 331042, root := ~, start := 331038, stop := 331041 ), rec( content := "\n\n \ More specifically, if ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eps", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 331081, root := ~, start := 331070, stop := 331080 ), rec( content := " is the \ list of orbit representatives,\n then a vertex ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 331146, root := ~, start := 331138, stop := 331145 ), rec( content := " appears\ in ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 331171, root := ~, start := 331158, stop := 331170 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eps[i]", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 331233, root := ~, start := 331219, stop := 331232 ), rec( content := " and ran\ ge ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 331252, root := ~, start := 331244, stop := 331251 ), rec( content := " in\n \ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "A", next := 331274, root := ~, start := 331260, stop := 331273 ), rec( content := ".\n " , count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "P", next := 331284, root := ~, start := 331280, stop := 331283 ), rec( content := "\n\n \ If ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "Ref", next := 331319, root := ~, start := 331293, stop := 331318 ), rec( content := " is triv\ ial, then ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "Ref", next := 331364, root := ~, start := 331337, stop := 331363 ), rec( content := "\n is\ returned.\n\n ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3,\ 4]]);\n\ngap> DigraphGroup(D);\n\ Group(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ]\ , [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := Digraph(IsMutableDigraph,\ [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);\n\ngap> DigraphGroup(D);\nGroup(())\ngap> Repr\ esentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, \ 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\ \n\ngap> G := DigraphGroup(D);;\n\ gap> GeneratorsOfGroup(G);\n[ (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1,5)(\ 2,6)(3,8)(4,7) ]\ngap> Set(RepresentativeOutNeighbours(D), Set);\n[ [ 2, 3, 5 \ ] ]", count := [ 7, 2, 19, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 6 ], mathmode := "Text", name := "Example", next := 332257, root := ~, start := 331387, stop := 332256 ), rec( content := "\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Description", next := 332274, root := ~, start := 330815, stop := 332273 ) ], count := [ 7, 2, 19, 1 ], mathmode := "Text", name := "ManSection", next := 332288, root := ~, start := 330692, stop := 332287 ), rec( content := "\n\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd transformation or permutation", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 2 ], mathmode := "Text", name := "Oper", next := 332426, root := ~, start := 332309, stop := 332425 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 4 ], mathmode := "Text", name := "Oper", next := 332494, root := ~, start := 332429, stop := 332493 ), rec( attributes := rec( Arg := "digraph, x", Label := "for a digraph \ and a transformation or permutation", Name := "IsDigraphAutomorphism" ), content := 0, count := [ 7, 2, 20, 6 ], mathmode := "Text", name := "Oper", next := 332617, root := ~, start := 332497, stop := 332616 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphAutomo\ rphism" ), content := 0, count := [ 7, 2, 20, 8 ], mathmode := "Text", name := "Oper", next := 332678, root := ~, start := 332620, stop := 332677 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 332701, root := ~, start := 332690, stop := 332700 ), rec( content := " or ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 332717, root := ~, start := 332705, stop := 332716 ), rec( content := ".", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Returns", next := 332728, root := ~, start := 332681, stop := 332727 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphIsomorphism", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "C", next := 332776, root := ~, start := 332749, stop := 332775 ), rec( content := " returns\ ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 332796, root := ~, start := 332785, stop := 332795 ), rec( content := " if the \ permutation or\n transformation ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 332846, root := ~, start := 332838, stop := 332845 ), rec( content := " is an i\ somorphism from the digraph ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 332892, root := ~, start := 332882, stop := 332891 ), rec( content := " to\n \ the digraph ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 332922, root := ~, start := 332912, stop := 332921 ), rec( content := ".\n " , count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "P", next := 332932, root := ~, start := 332928, stop := 332931 ), rec( content := "\n\n \ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphAutomorphism", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "C", next := 332966, root := ~, start := 332938, stop := 332965 ), rec( content := " returns\ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "K", next := 332986, root := ~, start := 332975, stop := 332985 ), rec( content := " if the \ permutation or\n transformation ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "A", next := 333036, root := ~, start := 333028, stop := 333035 ), rec( content := " is an a\ utomorphism of the digraph ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "A", next := 333085, root := ~, start := 333071, stop := 333084 ), rec( content := ".\n " , count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "P", next := 333095, root := ~, start := 333091, stop := 333094 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 333141, root := ~, start := 333133, stop := 333140 ), rec( content := " is an " , count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "E", next := 333166, root := ~, start := 333148, stop := 333165 ), rec( content := " from a\\ n digraph ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 333196, root := ~, start := 333186, stop := 333195 ), rec( content := " to a di\ graph ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 333220, root := ~, start := 333210, stop := 333219 ), rec( content := " if the \ following hold:\n ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 13 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "x", count := [ 7, 2, 20, 13 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text", name\ := "A", next := 333284, root := ~, start := 333276, stop := 333283 ), rec( cont\ ent := " is a bijection from the vertices of ", count := [ 7, 2, 20, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "src", count := [ 7, 2, 20, 13 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text\ ", name := "A", next := 333331, root := ~, start := 333321, stop := 333330 ), rec( cont\ ent := " to those of\n ", count := [ 7, 2, 20, 13 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "ran", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "A", next\ := 333362, root := ~, start := 333352, stop := 333361 ), rec( content := ";\n\ ", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 13 ], name := "Item", next := 333377, root := ~, start := 333\ 261, stop := 333376 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[u ^\ ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "x", count := [ 7\ , 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [\ 7, 2, 20, 15 ], mathmode := "Text", name := "A", next := 333415, root := ~, s\ tart := 333407, stop := 333414 ), rec( content := ", v ^ ", count := [ 7, 2, 2\ 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "x", count := [ 7, 2, 20, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathm\ ode := "Text", name := "A", next := 333429, root := ~, start := 333421, stop :\ = 333428 ), rec( content := "]", count := [ 7, 2, 20, 15 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "C", next := 333434, root := ~, start := 333399, stop := 333433 ), rec( cont\ ent := " is an edge of\n ", count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "ran", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := \ "A", next := 333467, root := ~, start := 333457, stop := 333466 ), rec( conten\ t := " if and only if ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content\ := "[u, v]", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "C", next\ := 333496, root := ~, start := 333483, stop := 333495 ), rec( content := " is\ an\n edge of ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "src", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "A", next :\ = 333529, root := ~, start := 333519, stop := 333528 ), rec( content := "; and\ \n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 15 ], name := "Item", next := 333548, root := ~, start := 333\ 384, stop := 333547 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", \ count := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 7, 2, 20, 17 ], mathmode := "Text", name := "A", next := 333578, root :\ = ~, start := 333570, stop := 333577 ), rec( content := " fixes every ", count\ := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 20, 17 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17\ ], mathmode := "Text", name := "C", next := 333599, root := ~, start := 33359\ 1, stop := 333598 ), rec( content := " which is not a vertex of ", count := [ \ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "src", count := [ 7, 2, 20, 17 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17\ ], mathmode := "Text", name := "A", next := 333635, root := ~, start := 33362\ 5, stop := 333634 ), rec( content := ".\n ", count := [ 7, 2, 20, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17 ], name := "Ite\ m", next := 333650, root := ~, start := 333555, stop := 333649 ) ], count := [ 7, 2, 20, 12 ], mathmode := "Text", name := "List", next := 333662, root := ~, start := 333248, stop := 333661 ), rec( content := "\n Se\ e also ", count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "Ref", next := 333736, root := ~, start := 333676, stop := 333735 ), rec( content := ".", count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "P", next := 333741, root := ~, start := 333737, stop := 333740 ), rec( content := "\n \n\ If ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 333765, root := ~, start := 333754, stop := 333764 ), rec( content := " and ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 333781, root := ~, start := 333770, stop := 333780 ), rec( content := ", or ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 333796, root := ~, start := 333786, stop := 333795 ), rec( content := ", are gi\ ven, then they must\n represent vertex colourings; see \n ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "Ref", next := 333942, root := ~, start := 333866, stop := 333941 ), rec( content := " \n f\ or details of the permissible values for\n these arguments. The homomorphis\ m must then also have the property:\n \n ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 22 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col1[i] = col2[i ^ x]", count := [ 7, 2, 20, 22 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ], math\ mode := "Text", name := "C", next := 334125, root := ~, start := 334097, stop \ := 334124 ), rec( content := " for all vertices ", count := [ 7, 2, 20, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 2, 20, 22 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ], mathmode := "Text", name\ := "C", next := 334151, root := ~, start := 334143, stop := 334150 ), rec( cont\ ent := " of ", count := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "src", count\ := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "A", next := 334165, root :\ = ~, start := 334155, stop := 334164 ), rec( content := ",\n for ", coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "IsDigraphIsomorphism", count \ := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "C", next := 334206, root :\ = ~, start := 334179, stop := 334205 ), rec( content := ". ", count := [ 7, 2,\ 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ] , name := "Ite\ m", next := 334215, root := ~, start := 334082, stop := 334214 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 24 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col[i] = col[i ^ x]", count := [ 7, 2, 20, 24 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ], math\ mode := "Text", name := "C", next := 334263, root := ~, start := 334237, stop \ := 334262 ), rec( content := " for all vertices ", count := [ 7, 2, 20, 24 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 2, 20, 24 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ], mathmode := "Text", name\ := "C", next := 334289, root := ~, start := 334281, stop := 334288 ), rec( cont\ ent := " of ", count := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "A", next := 334307, root :\ = ~, start := 334293, stop := 334306 ), rec( content := ",\n for ", coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "IsDigraphAutomorphism", count\ := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "C", next := 334349, root :\ = ~, start := 334321, stop := 334348 ), rec( content := ". ", count := [ 7, 2,\ 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ] , name := "Ite\ m", next := 334358, root := ~, start := 334222, stop := 334357 ) ], count := [ 7, 2, 20, 21 ], mathmode := "Text", name := "List", next := 334370, root := ~, start := 334069, stop := 334369 ), rec( content := "\n\n \ For some digraphs, it can be faster to use ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphAutomorphism", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "C", next := 334447, root := ~, start := 334419, stop := 334446 ), rec( content := "\n th\ an to test membership in the automorphism group of ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "A", next := 334519, root := ~, start := 334505, stop := 334518 ), rec( content := ".\n\n \ ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigra\ phAutomorphism(src, (2, 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2\ , 1, 1]);\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ng\ ap> IsDigraphAutomorphism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphi\ sm(src, (1, 4));\nfalse\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran \ := Digraph([[2, 1], [2], [2, 3]]);\n\ngap> IsDigraphIsomorphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorph\ ism(ran, src, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntr\ ue\ngap> IsDigraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nf\ alse\ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ \ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n" , count := [ 7, 2, 20, 27 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 27 ], mathmode := "Text", name := "Example", next := 335462, root := ~, start := 334526, stop := 335461 ), rec( content := "\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Description", next := 335479, root := ~, start := 332731, stop := 335478 ) ], count := [ 7, 2, 20, 1 ], mathmode := "Text", name := "ManSection", next := 335493, root := ~, start := 332294, stop := 335492 ), rec( content := "\n\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsDigraphColour\ ing" ), content := 0, count := [ 7, 2, 21, 2 ], mathmode := "Text", name := "Oper", next := 335571, root := ~, start := 335518, stop := 335570 ), rec( attributes := rec( Arg := "digraph, t", Label := "for a transfor\ mation", Name := "IsDigraphColouring" ), content := 0, count := [ 7, 2, 21, 4 ], mathmode := "Text", name := "Oper", next := 335662, root := ~, start := 335576, stop := 335661 ), rec( attributes := rec( ), content := [ rec( content := " ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "K", next := 335688, root := ~, start := 335677, stop := 335687 ), rec( content := " or ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "K", next := 335704, root := ~, start := 335692, stop := 335703 ), rec( content := ". ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Returns", next := 335716, root := ~, start := 335667, stop := 335715 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphColouring", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "C", next := 335780, root := ~, start := 335755, stop := 335779 ), rec( content := " verifie\ s whether or not \n the list ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "A", next := 335832, root := ~, start := 335821, stop := 335831 ), rec( content := " describ\ es a proper colouring of the digraph \n ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "A", next := 335898, root := ~, start := 335884, stop := 335897 ), rec( content := ". \n \ ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "P", next := 335911, root := ~, start := 335907, stop := 335910 ), rec( content := "\n\n \ A list ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 335937, root := ~, start := 335926, stop := 335936 ), rec( content := " describ\ es a ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "E", next := 335973, root := ~, start := 335950, stop := 335972 ), rec( content := " of a di\ graph\n ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 336007, root := ~, start := 335993, stop := 336006 ), rec( content := " if ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 336022, root := ~, start := 336011, stop := 336021 ), rec( content := " consist\ s of positive integers, the length of\n ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 336085, root := ~, start := 336074, stop := 336084 ), rec( content := " equals \ the number of vertices in ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 336133, root := ~, start := 336119, stop := 336132 ), rec( content := ", and fo\ r any\n vertices ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ , v", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 336173, root := ~, start := 336162, stop := 336172 ), rec( content := " of ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 336191, root := ~, start := 336177, stop := 336190 ), rec( content := " if ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 336203, root := ~, start := 336195, stop := 336202 ), rec( content := " and ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 336216, root := ~, start := 336208, stop := 336215 ), rec( content := " are adj\ acent,\n then ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 7, 2, 21, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "A", next := 336256, root := ~, start := 336245, stop := 33625\ 5 ), rec( content := "[u] >< ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 3362\ 80, root := ~, start := 336269, stop := 336279 ), rec( content := "[v]", count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 336287, root := ~, start := 336242, stop := 336286 ), rec( content := ".\n \ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "P", next := 336299, root := ~, start := 336295, stop := 336298 ), rec( content := "\n\n \ A transformation ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 336332, root := ~, start := 336324, stop := 336331 ), rec( content := " describ\ es a proper colouring of a digraph\n ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 336395, root := ~, start := 336381, stop := 336394 ), rec( content := ", if ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ mageListOfTransformation(", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "A", next := 336437, root := ~, start := 336429, stop := 33643\ 6 ), rec( content := ",\n DigraphNrVertices(", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 3364\ 77, root := ~, start := 336463, stop := 336476 ), rec( content := "))", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "C", next := 336483, root := ~, start := 336400, stop := 336482 ), rec( content := " is a pr\ oper colouring of \n ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 336530, root := ~, start := 336516, stop := 336529 ), rec( content := ". ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "P", next := 336536, root := ~, start := 336532, stop := 336535 ), rec( content := "\n\n \ See also ", count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phHomomorphism" ), content := 0, count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "Ref", next := 336588, root := ~, start := 336553, stop := 336587 ), rec( content := ".\n \ \n ", count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := JohnsonDigraph(5, 3);\n\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);\ntru\ e\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, 7]);\nfalse\ngap> Is\ DigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);\nfalse\ngap> IsDigraphCo\ louring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphColouring(D, IdentityTransformati\ on);\ntrue\n", count := [ 7, 2, 21, 9 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 9 ], mathmode := "Text", name := "Example", next := 337024, root := ~, start := 336603, stop := 337023 ), rec( content := "\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Description", next := 337043, root := ~, start := 335721, stop := 337042 ) ], count := [ 7, 2, 21, 1 ], mathmode := "Text", name := "ManSection", next := 337059, root := ~, start := 335501, stop := 337058 ), rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 337073, root := ~, start := 293141, stop := 337072 ) , rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms of \ digraphs", count := [ 7, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 337130, root := ~, start := 337086, stop := 337129 ), rec( content := "\n\n The following me\ thods exist to find homomorphisms between digraphs.\n If an argument to one\ of these methods is a digraph with multiple edges,\n then the multiplicity\ of edges will be ignored in order to perform the\n calculation; the digrap\ h will be treated as if it has no multiple edges.\n\n ", count := [ 7, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, hook, us\ er_param, max_results, hint, injective, image, partial_map, colors1, colors2[,\ order, aut_grp]", Name := "HomomorphismDigraphsFinder" ), content := 0, count := [ 7, 3, 1, 2 ], mathmode := "Text", name := "Func", next := 337611, root := ~, start := 337453, stop := 337610 ), rec( attributes := rec( ), content := [ rec( content := "The argum\ ent ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ser_param", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 337653, root := ~, start := 337636, stop := 337652 ), rec( content := ".", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 337664, root := ~, start := 337614, stop := 337663 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function finds homomorphisms from the digraph ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 337745, root := ~, start := 337736, stop := 337744 ), rec( content := " to the \ digraph\n ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 337774, root := ~, start := 337765, stop := 337773 ), rec( content := " subject\ to the conditions imposed by the other arguments as\n described below.", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "P", next := 337859, root := ~, start := 337855, stop := 337858 ), rec( content := "\n\n \ If ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 337876, root := ~, start := 337868, stop := 337875 ), rec( content := " and ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 337889, root := ~, start := 337881, stop := 337888 ), rec( content := " are hom\ omorphisms found by\n ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismDigraphsFinder", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 337954, root := ~, start := 337921, stop := 337953 ), rec( content := ", then " , count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 337969, root := ~, start := 337961, stop := 337968 ), rec( content := " cannot \ be obtained from\n ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 338006, root := ~, start := 337998, stop := 338005 ), rec( content := " by righ\ t multiplying by an automorphism of ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "A", next := 338059, root := ~, start := 338050, stop := 338058 ), rec( content := " in\n \ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "a\ ut_grp", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "A", next := 338081, root := ~, start := 338067, stop := 338080 ), rec( content := ".\n\n \ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "hook", count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "A", next := 33\ 8118, root := ~, start := 338107, stop := 338117 ) ], count := [ 7, 3, 1, 5 ], name := "Mar\ k", next := 338125, root := ~, start := 338101, stop := 338124 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a f\ unction or ", count := [ 7, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 7, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 6 ], mathmode := "Text", name := "K", next := 338196, root := \ ~, start := 338185, stop := 338195 ), rec( content := ".", count := [ 7, 3, 1,\ 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "P", next\ := 338201, root := ~, start := 338197, stop := 338200 ), rec( content := "\n\ \n If ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "hook", coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "A", next := 338225, root := \ ~, start := 338214, stop := 338224 ), rec( content := " is a function, then it\ must have two arguments\n ", count := [ 7, 3, 1, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "user_param", count := [ 7, 3, 1, 7 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name \ := "A", next := 338298, root := ~, start := 338281, stop := 338297 ), rec( cont\ ent := " (see below) and a transformation ", count := [ 7, 3, 1, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "t", count := [ 7, 3, 1, 7 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name :\ = "C", next := 338340, root := ~, start := 338332, stop := 338339 ), rec( cont\ ent := ". The\n function ", count := [ 7, 3, 1, 7 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attri\ butes := rec( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 7 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ]\ , mathmode := "Text", name := "A", next := 338377, root := ~, start := 338366,\ stop := 338376 ), rec( content := "(", count := [ 7, 3, 1, 7 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [\ rec( content := "user_param", count := [ 7, 3, 1, 7 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", na\ me := "A", next := 338395, root := ~, start := 338378, stop := 338394 ), rec( \ content := ", t)", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 338403, root := ~, start := 338363, stop := 338402 ), rec( content := " is\ called every time\n a new homomorphism ", count := [ 7, 3, 1, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "t", count := [ 7, 3, 1, 7 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name\ := "C", next := 338460, root := ~, start := 338452, stop := 338459 ), rec( cont\ ent := " is found by\n ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "HomomorphismDigraphsFinder", count := [ 7, 3, 1, 7 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text\ ", name := "C", next := 338514, root := ~, start := 338481, stop := 338513 ), rec( cont\ ent := ". If the function returns ", count := [ 7, 3, 1, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "true", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "K" , next\ := 338551, root := ~, start := 338540, stop := 338550 ), rec( content := ",\n\ then ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Homomorphis\ mDigraphsFinder", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 338599, root := ~, start := 338566, stop := 338598 ), rec( content := " st\ ops and does not find any\n further homomorphisms. This feature might b\ e useful if you are\n searching for a homomorphism that satisfies some \ condition that you\n cannot specify via the other arguments to\n \ ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "HomomorphismDigraphsFinder", \ count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next := 338866, root := \ ~, start := 338833, stop := 338865 ), rec( content := ".\n ", count := [ \ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := 0, count := [ 7, 3, 1, 8 ], mathmode := "Text", name\ := "P", next := 338880, root := ~, start := 338876, stop := 338879 ), rec( cont\ ent := "\n\n If ", count := [ 7, 3, 1, 8 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 1, 8 ], mathmode := "Text", name := "A", next := 3\ 38904, root := ~, start := 338893, stop := 338903 ), rec( content := " is ", coun\ t := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 7, 3, 1, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 8 ] , math\ mode := "Text", name := "K", next := 338919, root := ~, start := 338908, stop \ := 338918 ), rec( content := ", then a default function is used which\n \ simply adds every new homomorphism found by\n ", count := [ 7, 3, 1, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "HomomorphismDigraphsFinder", count := [ 7, \ 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 3, 1, 8 ], mathmode := "Text", name := "C", next := 339052, root := ~, start :\ = 339019, stop := 339051 ), rec( content := " to ", count := [ 7, 3, 1, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 7, 3, 1, 8 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 8 ], mathmode := "T\ ext", name := "A", next := 339073, root := ~, start := 339056, stop := 339072 ), rec( cont\ ent := ", which must be a\n mutable list in this case.\n ", count \ := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 6 ], name := "Ite\ m", next := 339139, root := ~, start := 338132, stop := 339138 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "user_param", count := [ 7, 3, 1, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 3, 1, 9 ], mathmode := "Text", name := "A", next\ := 339170, root := ~, start := 339153, stop := 339169 ) ], count := [ 7, 3, 1, 9 ], name := "Mar\ k", next := 339177, root := ~, start := 339147, stop := 339176 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If ", count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode\ := "Text", name := "A", next := 339213, root := ~, start := 339202, stop := 3\ 39212 ), rec( content := " is a function, then ", count := [ 7, 3, 1, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := \ "Text", name := "A", next := 339251, root := ~, start := 339234, stop := 33925\ 0 ), rec( content := " can be any ", count := [ 7, 3, 1, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "GAP", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "\ Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := "\n\ object. The object ", count := [ 7, 3, 1, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "A" , next\ := 339313, root := ~, start := 339296, stop := 339312 ), rec( content := " is\ used as the first argument of\n the function ", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "hook", count := [ 7, 3, 1, 10 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "A", next := 339379, root := ~, start := 339368, stop := 33\ 9378 ), rec( content := ". For example, ", count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Tex\ t", name := "A", next := 339411, root := ~, start := 339394, stop := 339410 ), rec( cont\ ent := " might be a\n transformation semigroup, and ", count := [ 7, 3,\ 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "A", next :=\ 339475, root := ~, start := 339464, stop := 339474 ), rec( content := "(", co\ unt := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec\ ( attributes := rec( ), content := [ rec( content := "user_param", count := [\ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 7, 3, 1, 10 ], mathmode := "Text", name := "A", next := 339493, root := ~, s\ tart := 339476, stop := 339492 ), rec( content := ", t)", count := [ 7, 3, 1, \ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1,\ 10 ], mathmode := "Text", name := "C", next := 339501, root := ~, start := 33\ 9461, stop := 339500 ), rec( content := "\n might set ", count := [ 7, \ 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "user_param", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "A", next := 339537, root := ~, start := 3395\ 20, stop := 339536 ), rec( content := " to be the closure of ", count := [ 7, \ 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "user_param", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "A", next := 339576, root := ~, start := 3395\ 59, stop := 339575 ), rec( content := " and\n ", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "t", count := [ 7, 3, 1, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := \ "Text", name := "C", next := 339597, root := ~, start := 339589, stop := 33959\ 6 ), rec( content := ". ", count := [ 7, 3, 1, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 7, 3, 1, 11 ], mathmode := "Text", name := "P", next := 339603, root := ~, star\ t := 339599, stop := 339602 ), rec( content := "\n\n If the value of ", coun\ t := [ 7, 3, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 1 ], mathmode := "Text", name := "A", next := 339640, root := ~, start := 3396\ 29, stop := 339639 ), rec( content := " is ", count := [ 7, 3, 1, 11 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "fail", count := [ 7, 3, 1, 11 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 11 ], mathmode := "Text", name\ := "K", next := 339655, root := ~, start := 339644, stop := 339654 ), rec( cont\ ent := ", then the value of\n ", count := [ 7, 3, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 7, 3, 1, 11 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 11 ], mathmode := "Text", name\ := "A", next := 339700, root := ~, start := 339683, stop := 339699 ), rec( cont\ ent := " must be a mutable list.\n ", count := [ 7, 3, 1, 11 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], name := "Ite\ m", next := 339738, root := ~, start := 339184, stop := 339737 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "max_results", count := [ 7, 3, 1, 12 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 12 ], mathmode := "Text", name := "A", next\ := 339770, root := ~, start := 339752, stop := 339769 ) ], count := [ 7, 3, 1, 12 ], name := "Mar\ k", next := 339777, root := ~, start := 339746, stop := 339776 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a p\ ositive integer or ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "infinity", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "K", next :\ = 339860, root := ~, start := 339845, stop := 339859 ), rec( content := ".\n \ ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Homomorphism\ DigraphsFinder", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "C", next\ := 339903, root := ~, start := 339870, stop := 339902 ), rec( content := " wi\ ll return after it has found\n ", count := [ 7, 3, 1, 13 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "max_results", count := [ 7, 3, 1, 13 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text\ ", name := "A", next := 339961, root := ~, start := 339943, stop := 339960 ), rec( cont\ ent := " homomorphisms or the search is complete, whichever\n happens f\ irst.\n ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 1, 13 ], name := "Item", next := 340049, root := ~, start := 339\ 784, stop := 340048 ), rec( attributes := rec( ), content := [ rec( attribute\ s := rec( ), content := [ rec( content := "hint", count := [ 7, 3, 1, 14 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 14 ], math\ mode := "Text", name := "A", next := 340074, root := ~, start := 340063, stop \ := 340073 ) ], count := [ 7, 3, 1, 14 ], name := "Mark", next := 340081, root := ~, start := 340\ 057, stop := 340080 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n This argument should be a positive integer or ", count := [ 7, 3,\ 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "fail", count := [ 7, 3, 1, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 15 ], math\ mode := "Text", name := "K", next := 340160, root := ~, start := 340149, stop \ := 340159 ), rec( content := ". ", count := [ 7, 3, 1, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, coun\ t := [ 7, 3, 1, 16 ], mathmode := "Text", name := "P", next := 340166, root :=\ ~, start := 340162, stop := 340165 ), rec( content := "\n\n If ", coun\ t := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hint", count := [ 7, 3, 1, 16\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 6 ], mathmode := "Text", name := "A", next := 340190, root := ~, start := 3401\ 79, stop := 340189 ), rec( content := " is a positive integer, then only homor\ phisms of rank\n ", count := [ 7, 3, 1, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "hint", count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "A", next \ := 340263, root := ~, start := 340252, stop := 340262 ), rec( content := " are\ found.", count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 1, 17 ], math\ mode := "Text", name := "P", next := 340278, root := ~, start := 340274, stop \ := 340277 ), rec( content := "\n\n If ", count := [ 7, 3, 1, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "hint", count := [ 7, 3, 1, 17 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 17 ], mathmode := "Text" , name\ := "A", next := 340302, root := ~, start := 340291, stop := 340301 ), rec( cont\ ent := " is ", count := [ 7, 3, 1, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 7, 3, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 17 ], mathmode := "Text", name := "K", next := 340317, root :=\ ~, start := 340306, stop := 340316 ), rec( content := ", then no restriction \ is put on the rank\n of homomorphisms found.\n ", count := [ 7, 3,\ 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 15 ], name := "Ite\ m", next := 340403, root := ~, start := 340088, stop := 340402 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "injective", count := [ 7, 3, 1, 18 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 3, 1, 18 ], mathmode := "Text", name := "A", next\ := 340433, root := ~, start := 340417, stop := 340432 ) ], count := [ 7, 3, 1, 18 ], name := "Mar\ k", next := 340440, root := ~, start := 340411, stop := 340439 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "0", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 340494, root := ~, start := 340486, stop \ := 340493 ), rec( content := ", ", count := [ 7, 3, 1, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "1", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next\ := 340504, root := ~, start := 340496, stop := 340503 ), rec( content := ", o\ r ", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "2", count := [ 7\ , 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next := 340517, root := ~, star\ t := 340509, stop := 340516 ), rec( content := ". If it is\n ", count \ := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "2", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 340545, root := ~, start := 340537, stop \ := 340544 ), rec( content := ", then only embeddings are found, if it is ", coun\ t := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 340596, root := ~, start := 340588, stop \ := 340595 ), rec( content := ", then only\n injective homomorphisms are\ found, and if it is ", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "0", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next := 340\ 672, root := ~, start := 340664, stop := 340671 ), rec( content := " there are\ no\n restrictions imposed by this argument.\n ", count := [ 7, \ 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := 0, count := [ 7, 3, 1, 20 ], mathmode := "Text", name \ := "P", next := 340745, root := ~, start := 340741, stop := 340744 ), rec( cont\ ent := "\n\n For backwards compatibility, ", count := [ 7, 3, 1, 20 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "injective", count := [ 7, 3, 1, 20 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "\ Text", name := "A", next := 340800, root := ~, start := 340784, stop := 340799\ ), rec( content := " can also be ", count := [ 7, 3, 1, 20 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "false", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name :=\ "K", next := 340825, root := ~, start := 340813, stop := 340824 ), rec( conte\ nt := "\n or ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "true", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "K", next := 3\ 40848, root := ~, start := 340837, stop := 340847 ), rec( content := " which c\ orrespond to the values ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "0", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "C", next := \ 340888, root := ~, start := 340880, stop := 340887 ), rec( content := " and ", coun\ t := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 7, 3, 1, 20 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ] , math\ mode := "Text", name := "C", next := 340901, root := ~, start := 340893, stop \ := 340900 ), rec( content := "\n described in the previous paragraph, r\ espectively. \n ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ], name := "Item", next := 3409\ 75, root := ~, start := 340447, stop := 340974 ), rec( attributes := rec( ), content := [ rec( attr\ ibutes := rec( ), content := [ rec( content := "image", count := [ 7, 3, 1, 2\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, \ 21 ], mathmode := "Text", name := "A", next := 341001, root := ~, start := 340\ 989, stop := 341000 ) ], count := [ 7, 3, 1, 21 ], name := "Mark", next := 3410\ 08, root := ~, start := 340983, stop := 341007 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n This argument should be a subset of the vertices of the grap\ h ", count := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "D2", count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "A", next := 341101, root := ~, star\ t := 341092, stop := 341100 ), rec( content := ".\n ", count := [ 7, 3,\ 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "HomomorphismDigraphsFinder", count :=\ [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 3, 1, 22 ], mathmode := "Text", name := "C", next := 341144, root := ~, star\ t := 341111, stop := 341143 ), rec( content := " only finds homomorphisms from\ \n ", count := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1", count \ := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 1, 22 ], mathmode := "Text", name := "A", next := 341192, root := \ ~, start := 341183, stop := 341191 ), rec( content := " to the subgraph of ", coun\ t := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 22 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ] , math\ mode := "Text", name := "A", next := 341221, root := ~, start := 341212, stop \ := 341220 ), rec( content := " induced by the vertices\n ", count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "image", count := [ 7, 3, 1, 22 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ] , math\ mode := "Text", name := "A", next := 341266, root := ~, start := 341254, stop \ := 341265 ), rec( content := ".\n ", count := [ 7, 3, 1, 22 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ], name := "Ite\ m", next := 341281, root := ~, start := 341015, stop := 341280 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "partial_map", count := [ 7, 3, 1, 23 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 23 ], mathmode := "Text", name := "A", next\ := 341313, root := ~, start := 341295, stop := 341312 ) ], count := [ 7, 3, 1, 23 ], name := "Mar\ k", next := 341320, root := ~, start := 341289, stop := 341319 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a p\ artial map from ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1\ ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "A", next := 341394, root\ := ~, start := 341385, stop := 341393 ), rec( content := " to ", count := [ 7\ , 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 24 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], math\ mode := "Text", name := "A", next := 341407, root := ~, start := 341398, stop \ := 341406 ), rec( content := ",\n that is, a (not necessarily dense) li\ st of vertices of the digraph\n ", count := [ 7, 3, 1, 24 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "D2", count := [ 7, 3, 1, 24 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], mathmode := "Text", name \ := "A", next := 341501, root := ~, start := 341492, stop := 341500 ), rec( cont\ ent := " of length no greater than the number vertices in the digraph\n \ ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "D1", count := [ 7, 3, 1, 24 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ] , math\ mode := "Text", name := "A", next := 341580, root := ~, start := 341571, stop \ := 341579 ), rec( content := ". ", count := [ 7, 3, 1, 24 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "HomomorphismDigraphsFinder", count := [ 7, 3, 1, 24 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], mathmode\ := "Text", name := "C", next := 341615, root := ~, start := 341582, stop := 3\ 41614 ), rec( content := " only finds homomorphisms\n extending ", coun\ t := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "partial_map", count := [ 7, 3\ , 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 3, 1, 24 ], mathmode := "Text", name := "A", next := 341677, root := ~, start \ := 341659, stop := 341676 ), rec( content := " (if any).\n ", count := [ \ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], name := "Ite\ m", next := 341701, root := ~, start := 341327, stop := 341700 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "colors1", count := [ 7, 3, 1, 25 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 3, 1, 25 ], mathmode := "Text", name := "A", next\ := 341735, root := ~, start := 341721, stop := 341734 ) ], count := [ 7, 3, 1, 25 ], name := "Mar\ k", next := 341742, root := ~, start := 341715, stop := 341741 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This should be a list repre\ senting possible colours of vertices in the\n digraph ", count := [ 7, \ 3, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "D1", count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "A", next := 341860, root := ~, start := 341851, stop \ := 341859 ), rec( content := "; see \n ", count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label :\ = "for a digraph and a homogeneous list", Oper := "AutomorphismGroup" ), conte\ nt := 0, count := [ 7, 3, 1, 26 ], mathmode := "Text", name := "Ref", next := \ 341962, root := ~, start := 341875, stop := 341961 ), rec( content := " \n \ for details of the permissible values for this argument.\n ", count :\ = [ 7, 3, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 26 ], name := "Ite\ m", next := 342042, root := ~, start := 341749, stop := 342041 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "colors2", count := [ 7, 3, 1, 27 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 3, 1, 27 ], mathmode := "Text", name := "A", next\ := 342076, root := ~, start := 342062, stop := 342075 ) ], count := [ 7, 3, 1, 27 ], name := "Mar\ k", next := 342083, root := ~, start := 342056, stop := 342082 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This should be a list repre\ senting possible colours of vertices in the\n digraph ", count := [ 7, \ 3, 1, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "A", next := 342201, root := ~, start := 342192, stop \ := 342200 ), rec( content := "; see \n ", count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label :\ = "for a digraph and a homogeneous list", Oper := "AutomorphismGroup" ), conte\ nt := 0, count := [ 7, 3, 1, 28 ], mathmode := "Text", name := "Ref", next := \ 342303, root := ~, start := 342216, stop := 342302 ), rec( content := " \n \ for details of the permissible values for this argument.\n ", count :\ = [ 7, 3, 1, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 28 ], name := "Ite\ m", next := 342383, root := ~, start := 342090, stop := 342382 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "order", count := [ 7, 3, 1, 29 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 7, 3, 1, 29 ], mathmode := "Text", name := "A", next :\ = 342408, root := ~, start := 342396, stop := 342407 ) ], count := [ 7, 3, 1, 29 ], name := "Mar\ k", next := 342415, root := ~, start := 342390, stop := 342414 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The optional argument ", coun\ t := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "order", count := [ 7, 3, 1, 3\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, \ 30 ], mathmode := "Text", name := "A", next := 342471, root := ~, start := 342\ 459, stop := 342470 ), rec( content := " specifies the order the\n vert\ ices in ", count := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1", count \ := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 1, 30 ], mathmode := "Text", name := "A", next := 342525, root := \ ~, start := 342516, stop := 342524 ), rec( content := " appear in the search f\ or homomorphisms. \n The value of this parameter can have a large impac\ t\n on the runtime of the function. It seems in many cases to be a good\ \n idea for this to be the ", count := [ 7, 3, 1, 30 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphWe\ lshPowellOrder" ), content := 0, count := [ 7, 3, 1, 30 ], mathmode := "Text", name\ := "Ref", next := 342772, root := ~, start := 342735, stop := 342771 ), rec( cont\ ent := ", i.e.\n vertices ordered from highest to lowest degree.\n \ ", count := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 30 ], name := "Ite\ m", next := 342848, root := ~, start := 342422, stop := 342847 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The optional argument ", coun\ t := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "aut_grp", count := [ 7, 3, 1,\ 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1\ , 32 ], mathmode := "Text", name := "A", next := 342906, root := ~, start := 3\ 42892, stop := 342905 ), rec( content := " should be a subgroup of the\n \ automorphism group of ", count := [ 7, 3, 1, 32 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "D2", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "A", next :=\ 342974, root := ~, start := 342965, stop := 342973 ), rec( content := ". This\ function returns unique\n representatives of the homomorphisms found u\ p to right multiplication\n by ", count := [ 7, 3, 1, 32 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "aut_grp", count := [ 7, 3, 1, 32 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name\ := "A", next := 343108, root := ~, start := 343094, stop := 343107 ), rec( cont\ ent := ". If this argument is not specific, it defaults to the\n full a\ utomorphism group of ", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "D2", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "A", next := 34\ 3207, root := ~, start := 343198, stop := 343206 ), rec( content := ", which m\ ay be costly to calculate.\n ", count := [ 7, 3, 1, 32 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 32 ], name := "Item" , next := 3432\ 56, root := ~, start := 342855, stop := 343255 ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "List", next := 343268, root := ~, start := 338088, stop := 343267 ), rec( content := "\n\n \ ", count := [ 7, 3, 1, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(10);\n\ngap>\ D := DigraphSymmetricClosure(D);\n\ngap> HomomorphismDigraphsFinder(D, D, fail, [], infinity, 2, 0,\ \n> [3, 4], [], fail, fail);\n[ Transformation( [ 3, 4, 3, 4, 3, 4, 3, 4, 3, 4\ ] ), \n Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ] ) ]\ngap> D2 := Com\ pleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, fail, [], 1, fail, 0\ ,\n> [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformation( [ 1, 2, 1, 3, 4, 5, \ 6, 1, 2, 1 ] ) ]\ngap> func := function(user_param, t)\n> Add(user_param, t * \ user_param[1]);\n> end;;\ngap> HomomorphismDigraphsFinder(D, D2, func, [Transf\ ormation([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fail, fail);\n[ Transfo\ rmation( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 2 ] ), \ \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n Transformation( [ \ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFinder(NullDigrap\ h(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], fail, fail, f\ ail, fail,\n> Group(()));\n[ IdentityTransformation, Transformation( [ 1, 3, 3\ ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), \n Trans\ formation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> HomomorphismDi\ graphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [\ 1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityTransformation\ , Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ) ]", count := [ 7, 3, 1, 35 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 35 ], mathmode := "Text", name := "Example", next := 344823, root := ~, start := 343274, stop := 344822 ), rec( content := "\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 344840, root := ~, start := 337667, stop := 344839 ) ], count := [ 7, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 344854, root := ~, start := 337438, stop := 344853 ), rec( content := "\n\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphHomomorphism" ), content := 0, count := [ 7, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 344934, root := ~, start := 344875, stop := 344933 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "K", next := 344979, root := ~, start := 344968, stop := 344978 ), rec( content := ".", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 344990, root := ~, start := 344937, stop := 344989 ), rec( attributes := rec( ), content := [ rec( content := "\n A h\ omomorphism from ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 345046, root := ~, start := 345031, stop := 345045 ), rec( content := " to ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 345065, root := ~, start := 345050, stop := 345064 ), rec( content := " is a ma\ pping from\n the vertex set of ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 345121, root := ~, start := 345106, stop := 345120 ), rec( content := " to a su\ bset of the vertices of\n ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 345172, root := ~, start := 345157, stop := 345171 ), rec( content := ", such t\ hat every pair of vertices ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 345219, root := ~, start := 345207, stop := 345218 ), rec( content := " which h\ as\n an edge ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ->j", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 345253, root := ~, start := 345242, stop := 345252 ), rec( content := " is mapp\ ed to a pair of vertices ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ a,b]", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 345298, root := ~, start := 345286, stop := 345297 ), rec( content := " which\n\ has an edge ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ->b", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 345332, root := ~, start := 345321, stop := 345331 ), rec( content := ". Note \ that non-adjacent vertices can still be\n mapped to adjacent vertices. ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "P", next := 345417, root := ~, start := 345413, stop := 345416 ), rec( content := "\n\n \ ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphHomomorphism", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "C", next := 345449, root := ~, start := 345423, stop := 345448 ), rec( content := " returns\ a single homomorphism between\n ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "A", next := 345507, root := ~, start := 345492, stop := 345506 ), rec( content := " and ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "A", next := 345527, root := ~, start := 345512, stop := 345526 ), rec( content := " if it e\ xists, otherwise it returns\n ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "K", next := 345578, root := ~, start := 345567, stop := 345577 ), rec( content := ".\n", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> DigraphHomomorphism\ (gr1, gr1);\nIdentityTransformation\ngap> map := DigraphHomomorphism(gr1, gr2)\ ;\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap> IsDigraphHomomorphism(gr1, gr2, m\ ap);\ntrue\n", count := [ 7, 3, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 5 ], mathmode := "Text", name := "Example", next := 345926, root := ~, start := 345580, stop := 345925 ), rec( content := "\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 345943, root := ~, start := 344993, stop := 345942 ) ], count := [ 7, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 345957, root := ~, start := 344860, stop := 345956 ), rec( content := "\n\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphs" ), content := 0, count := [ 7, 3, 3, 2 ], mathmode := "Text", name := "Oper", next := 346039, root := ~, start := 345978, stop := 346038 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 3, 4 ], mathmode := "Text", name := "Oper", next := 346118, root := ~, start := 346042, stop := 346117 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f transformations.", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Returns", next := 346167, root := ~, start := 346121, stop := 346166 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphsRepresentatives", count := [ 7, 3, 3, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 346231, root := ~, start := 346188, stop := 346230 ), rec( content := " finds e\ very\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 346281, root := ~, start := 346248, stop := 346280 ), rec( content := " between\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 346305, root := ~, start := 346290, stop := 346304 ), rec( content := " and\n \ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 346329, root := ~, start := 346314, stop := 346328 ), rec( content := ", up to \ right multiplication by an element of the\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 3, 3, 5 ] , mathmode := "Text", name := "Ref", next := 346436, root := ~, start := 346383, stop := 346435 ), rec( content := " of ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 346455, root := ~, start := 346440, stop := 346454 ), rec( content := ".\n I\ n other words, every homomorphism ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 346504, root := ~, start := 346496, stop := 346503 ), rec( content := " between\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 346528, root := ~, start := 346513, stop := 346527 ), rec( content := " and\n \ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 346552, root := ~, start := 346537, stop := 346551 ), rec( content := " can be \ written as the composition ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ = g * x", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 346603, root := ~, start := 346587, stop := 346602 ), rec( content := ",\n w\ here ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 346623, root := ~, start := 346615, stop := 346622 ), rec( content := " is one \ of the ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphsRepresentatives", count := [ 7, 3, 3, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 346681, root := ~, start := 346638, stop := 346680 ), rec( content := "\n an\ d ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 346698, root := ~, start := 346690, stop := 346697 ), rec( content := " is an a\ utomorphism of ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 346736, root := ~, start := 346721, stop := 346735 ), rec( content := ".\n " , count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "P", next := 346746, root := ~, start := 346742, stop := 346745 ), rec( content := "\n\n \ ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphs", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "C", next := 346780, root := ~, start := 346752, stop := 346779 ), rec( content := " returns\ all homomorphisms between\n ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "A", next := 346834, root := ~, start := 346819, stop := 346833 ), rec( content := " and ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "A", next := 346854, root := ~, start := 346839, stop := 346853 ), rec( content := ".\n " , count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> HomomorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ] ), Transformation( [ 1, 3, 3 ] ),\ \n Transformation( [ 1, 5, 4, 4, 5 ] ), Transformation( [ 2, 2, 2 ] ), \n T\ ransformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1, 5, 4, 5 ] ), \n Transfo\ rmation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] ) ]\ngap> HomomorphismsDig\ raphsRepresentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ), \ Transformation( [ 2, 1, 2 ] ) ]\n", count := [ 7, 3, 3, 7 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 7 ], mathmode := "Text", name := "Example", next := 347460, root := ~, start := 346860, stop := 347459 ), rec( content := "\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Description", next := 347477, root := ~, start := 346170, stop := 347476 ) ], count := [ 7, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 347491, root := ~, start := 345963, stop := 347490 ), rec( content := "\n\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphMonomorphism" ), content := 0, count := [ 7, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 347571, root := ~, start := 347512, stop := 347570 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "K", next := 347615, root := ~, start := 347604, stop := 347614 ), rec( content := ".", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 347626, root := ~, start := 347574, stop := 347625 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphMonomorphism", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "C", next := 347673, root := ~, start := 347647, stop := 347672 ), rec( content := " returns\ a single ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ njective", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "E", next := 347707, root := ~, start := 347691, stop := 347706 ), rec( content := " ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 347747, root := ~, start := 347708, stop := 347746 ), rec( content := " between\ ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "A", next := 347771, root := ~, start := 347756, stop := 347770 ), rec( content := " and ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "A", next := 347791, root := ~, start := 347776, stop := 347790 ), rec( content := "\n if\ one exists, otherwise it returns ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "K", next := 347843, root := ~, start := 347832, stop := 347842 ), rec( content := ".\n " , count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> DigraphMonomorphism\ (gr1, gr1);\nIdentityTransformation\ngap> DigraphMonomorphism(gr1, gr2);\nTran\ sformation( [ 3, 1, 5, 4, 5 ] )\n", count := [ 7, 3, 4, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 4 ], mathmode := "Text", name := "Example", next := 348140, root := ~, start := 347849, stop := 348139 ), rec( content := "\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 348157, root := ~, start := 347629, stop := 348156 ) ], count := [ 7, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 348171, root := ~, start := 347497, stop := 348170 ), rec( content := "\n\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphs" ), content := 0, count := [ 7, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 348253, root := ~, start := 348192, stop := 348252 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 348332, root := ~, start := 348256, stop := 348331 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Returns", next := 348380, root := ~, start := 348335, stop := 348379 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Ref", next := 348472, root := ~, start := 348437, stop := 348471 ), rec( content := " and\n \ ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Ref", next := 348531, root := ~, start := 348481, stop := 348530 ), rec( content := ", except\ they only return\n ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ njective", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "E", next := 348577, root := ~, start := 348561, stop := 348576 ), rec( content := " homomor\ phisms.\n ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> MonomorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 5, 4, 4, 5 ] ), \n Transformation( [ 3,\ 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigraphsRepresentatives(gr1, CompleteDigr\ aph(3));\n[ Transformation( [ 2, 1 ] ) ]\n", count := [ 7, 3, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 5, 6 ], mathmode := "Text", name := "Example", next := 348974, root := ~, start := 348597, stop := 348973 ), rec( content := "\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 348991, root := ~, start := 348383, stop := 348990 ) ], count := [ 7, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 349005, root := ~, start := 348177, stop := 349004 ), rec( content := "\n\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphEpimorphism" ), content := 0, count := [ 7, 3, 6, 2 ], mathmode := "Text", name := "Oper", next := 349084, root := ~, start := 349026, stop := 349083 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "K", next := 349128, root := ~, start := 349117, stop := 349127 ), rec( content := ".", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 349139, root := ~, start := 349087, stop := 349138 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEpimorphism", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "C", next := 349185, root := ~, start := 349160, stop := 349184 ), rec( content := " returns\ a single ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ urjective", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "E", next := 349220, root := ~, start := 349203, stop := 349219 ), rec( content := " ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 349260, root := ~, start := 349221, stop := 349259 ), rec( content := " between\ ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "A", next := 349284, root := ~, start := 349269, stop := 349283 ), rec( content := " and ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "A", next := 349304, root := ~, start := 349289, stop := 349303 ), rec( content := "\n if\ one exists, otherwise it returns ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "K", next := 349356, root := ~, start := 349345, stop := 349355 ), rec( content := ".\n " , count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphRemoveEdge(CompleteDigraph(3), [1, 2]);\n\ngap> DigraphEpimorphism(gr2, gr1);\ \nfail\ngap> DigraphEpimorphism(gr1, gr2);\nTransformation( [ 3, 1, 2, 3 ] )\n\ ", count := [ 7, 3, 6, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 4 ], mathmode := "Text", name := "Example", next := 349696, root := ~, start := 349362, stop := 349695 ), rec( content := "\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 349713, root := ~, start := 349142, stop := 349712 ) ], count := [ 7, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 349727, root := ~, start := 349011, stop := 349726 ), rec( content := "\n\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphs" ), content := 0, count := [ 7, 3, 7, 2 ], mathmode := "Text", name := "Oper", next := 349808, root := ~, start := 349748, stop := 349807 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 4 ], mathmode := "Text", name := "Oper", next := 349886, root := ~, start := 349811, stop := 349885 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Returns", next := 349934, root := ~, start := 349889, stop := 349933 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 350026, root := ~, start := 349991, stop := 350025 ), rec( content := " and\n \ ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 350085, root := ~, start := 350035, stop := 350084 ), rec( content := ", except\ they only return\n ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ urjective", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "E", next := 350132, root := ~, start := 350115, stop := 350131 ), rec( content := " homomor\ phisms.\n ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(3));\n\ngap> EpimorphismsDigraphsRepre\ sentatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3\ , 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> EpimorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 2, 1, 3 ] ), Transformation( [ 1, 2, 3, \ 1 ] ), \n Transformation( [ 1, 2, 3, 2 ] ), Transformation( [ 1, 3, 1, 2 ] ),\ \n Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2, 3 ] ), \n T\ ransformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] ), \n Transfo\ rmation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), \n Transformatio\ n( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), \n Transformation( [ 3\ , 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), \n Transformation( [ 3, 1, 3\ , 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), \n Transformation( [ 3, 2, 1, 3 ] \ ), Transformation( [ 3, 2, 3, 1 ] ) ]\n", count := [ 7, 3, 7, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 7, 6 ], mathmode := "Text", name := "Example", next := 351219, root := ~, start := 350152, stop := 351218 ), rec( content := "\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Description", next := 351236, root := ~, start := 349937, stop := 351235 ) ], count := [ 7, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 351250, root := ~, start := 349733, stop := 351249 ), rec( content := "\n\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphEmbedding" ), content := 0, count := [ 7, 3, 8, 2 ], mathmode := "Text", name := "Oper", next := 351327, root := ~, start := 351271, stop := 351326 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "K", next := 351372, root := ~, start := 351361, stop := 351371 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 351383, root := ~, start := 351330, stop := 351382 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ embedding of a digraph ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 351445, root := ~, start := 351430, stop := 351444 ), rec( content := " into an\ other digraph\n ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 351486, root := ~, start := 351471, stop := 351485 ), rec( content := " is a ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Ref", next := 351525, root := ~, start := 351492, stop := 351524 ), rec( content := " from ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 351546, root := ~, start := 351531, stop := 351545 ), rec( content := "\n to\ ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 351569, root := ~, start := 351554, stop := 351568 ), rec( content := " which h\ as the additional property that a pair of\n vertices ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 351645, root := ~, start := 351632, stop := 351644 ), rec( content := " which h\ ave no edge ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 351678, root := ~, start := 351665, stop := 351677 ), rec( content := " in ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 351697, root := ~, start := 351682, stop := 351696 ), rec( content := " are\n\n\ mapped to a pair of vertices ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ a, b]", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 351749, root := ~, start := 351736, stop := 351748 ), rec( content := " which h\ ave no edge ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ->b", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 351780, root := ~, start := 351769, stop := 351779 ), rec( content := " in\n \ ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 351803, root := ~, start := 351788, stop := 351802 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "P", next := 351808, root := ~, start := 351804, stop := 351807 ), rec( content := "\n\n \ In other words, an embedding ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "C", next := 351851, root := ~, start := 351843, stop := 351850 ), rec( content := " is an i\ somorphism from\n ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "A", next := 351894, root := ~, start := 351879, stop := 351893 ), rec( content := " to the \ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "Ref", next := 351933, root := ~, start := 351902, stop := 351932 ), rec( content := " of ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "A", next := 351952, root := ~, start := 351937, stop := 351951 ), rec( content := "\n on\ the image of ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "C", next := 351981, root := ~, start := 351973, stop := 351980 ), rec( content := ". ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "P", next := 351987, root := ~, start := 351983, stop := 351986 ), rec( content := "\n\n \ ", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEmbedding", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "C", next := 352016, root := ~, start := 351993, stop := 352015 ), rec( content := " returns\ a single embedding if one exists, otherwise\n it returns ", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "K", next := 352095, root := ~, start := 352084, stop := 352094 ), rec( content := ".\n " , count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(3);\n\ngap> \ DigraphEmbedding(gr, CompleteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Dig\ raph([[3], [1, 4], [1], [3]]));\nTransformation( [ 2, 4, 3, 4 ] )\n", count := [ 7, 3, 8, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 8, 6 ], mathmode := "Text", name := "Example", next := 352349, root := ~, start := 352101, stop := 352348 ), rec( content := "\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 352366, root := ~, start := 351386, stop := 352365 ) ], count := [ 7, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 352380, root := ~, start := 351256, stop := 352379 ), rec( content := "\n\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phs" ), content := 0, count := [ 7, 3, 9, 2 ], mathmode := "Text", name := "Oper", next := 352447, root := ~, start := 352401, stop := 352446 ), rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phsRepresentatives" ), content := 0, count := [ 7, 3, 9, 4 ], mathmode := "Text", name := "Oper", next := 352511, root := ~, start := 352450, stop := 352510 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 352559, root := ~, start := 352514, stop := 352558 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 352651, root := ~, start := 352616, stop := 352650 ), rec( content := " and\n \ ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 352710, root := ~, start := 352660, stop := 352709 ), rec( content := ", except\ they only return\n embeddings of ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "A", next := 352763, root := ~, start := 352754, stop := 352762 ), rec( content := " into ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "A", next := 352778, root := ~, start := 352769, stop := 352777 ), rec( content := ".", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "P", next := 352783, root := ~, start := 352779, stop := 352782 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phEmbedding" ), content := 0, count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 352830, root := ~, start := 352798, stop := 352829 ), rec( content := ".\n " , count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := NullDigraph(2);\n\ngap> D\ 2 := CycleDigraph(5);\n\ngap> Embeddi\ ngsDigraphsRepresentatives(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transfor\ mation( [ 1, 4, 3, 4 ] ) ]\ngap> EmbeddingsDigraphs(D1, D2);\n[ Transformation\ ( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ), \n Transformation( [ 2, 4,\ 4, 5, 1 ] ), \n Transformation( [ 2, 5, 4, 5, 1 ] ), \n Transformation( [ 3\ , 1, 5, 1, 2 ] ), \n Transformation( [ 3, 5, 5, 1, 2 ] ), \n Transformation(\ [ 4, 1, 1, 2, 3 ] ), \n Transformation( [ 4, 2, 1, 2, 3 ] ), \n Transformat\ ion( [ 5, 2, 2, 3, 4 ] ), \n Transformation( [ 5, 3, 2, 3, 4 ] ) ]\n", count := [ 7, 3, 9, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 9, 7 ], mathmode := "Text", name := "Example", next := 353544, root := ~, start := 352836, stop := 353543 ), rec( content := "\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 353561, root := ~, start := 352562, stop := 353560 ) ], count := [ 7, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 353575, root := ~, start := 352386, stop := 353574 ), rec( content := "\n\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 353716, root := ~, start := 353596, stop := 353715 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 4 ], mathmode := "Text", name := "Oper", next := 353785, root := ~, start := 353719, stop := 353784 ), rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 6 ], mathmode := "Text", name := "Oper", next := 353906, root := ~, start := 353788, stop := 353905 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 8 ], mathmode := "Text", name := "Oper", next := 353974, root := ~, start := 353909, stop := 353973 ), rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 10 ], mathmode := "Text", name := "Oper", next := 354096, root := ~, start := 353977, stop := 354095 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 12 ], mathmode := "Text", name := "Oper", next := 354165, root := ~, start := 354099, stop := 354164 ), rec( attributes := rec( Arg := "digraph, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEndomorphism" ), content := 0, count := [ 7, 3, 10, 14 ], mathmode := "Text", name := "Oper", next := 354286, root := ~, start := 354168, stop := 354285 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphEndomo\ rphism" ), content := 0, count := [ 7, 3, 10, 16 ], mathmode := "Text", name := "Oper", next := 354347, root := ~, start := 354289, stop := 354346 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 354370, root := ~, start := 354359, stop := 354369 ), rec( content := " or ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 354386, root := ~, start := 354374, stop := 354385 ), rec( content := ".", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Returns", next := 354397, root := ~, start := 354350, stop := 354396 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphHomomorphism", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "C", next := 354446, root := ~, start := 354418, stop := 354445 ), rec( content := " returns\ ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 354466, root := ~, start := 354455, stop := 354465 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 354516, root := ~, start := 354508, stop := 354515 ), rec( content := " is a ho\ momorphism from the digraph\n ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 354566, root := ~, start := 354556, stop := 354565 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 354592, root := ~, start := 354582, stop := 354591 ), rec( content := ".\n " , count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "P", next := 354602, root := ~, start := 354598, stop := 354601 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEpimorphism", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "C", next := 354635, root := ~, start := 354608, stop := 354634 ), rec( content := " returns\ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "K", next := 354655, root := ~, start := 354644, stop := 354654 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 354705, root := ~, start := 354697, stop := 354704 ), rec( content := " is a su\ rjective homomorphism from the digraph\n ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 354766, root := ~, start := 354756, stop := 354765 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 354792, root := ~, start := 354782, stop := 354791 ), rec( content := ".\n " , count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "P", next := 354802, root := ~, start := 354798, stop := 354801 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphMonomorphism", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "C", next := 354836, root := ~, start := 354808, stop := 354835 ), rec( content := " returns\ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "K", next := 354856, root := ~, start := 354845, stop := 354855 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 354906, root := ~, start := 354898, stop := 354905 ), rec( content := " is an i\ njective homomorphism from the digraph\n ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 354967, root := ~, start := 354957, stop := 354966 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 354993, root := ~, start := 354983, stop := 354992 ), rec( content := ".\n " , count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "P", next := 355003, root := ~, start := 354999, stop := 355002 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEndomorphism", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "C", next := 355037, root := ~, start := 355009, stop := 355036 ), rec( content := " returns\ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "K", next := 355057, root := ~, start := 355046, stop := 355056 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "A", next := 355107, root := ~, start := 355099, stop := 355106 ), rec( content := " is an e\ ndomorphism of the digraph\n ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "A", next := 355160, root := ~, start := 355146, stop := 355159 ), rec( content := ".\n " , count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "P", next := 355170, root := ~, start := 355166, stop := 355169 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 355216, root := ~, start := 355208, stop := 355215 ), rec( content := " is a ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ omomorphism", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "E", next := 355241, root := ~, start := 355222, stop := 355240 ), rec( content := " from a\\ n digraph ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 355271, root := ~, start := 355261, stop := 355270 ), rec( content := " to a di\ graph ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 355295, root := ~, start := 355285, stop := 355294 ), rec( content := " if the \ following hold:\n ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[u ^ ", count := [ 7, 3, 10, 23 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "x", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "A"\ , next := 355367, root := ~, start := 355359, stop := 355366 ), rec( content :\ = ", v ^ ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", coun\ t := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], co\ unt := [ 7, 3, 10, 23 ], mathmode := "Text", name := "A", next := 355381, root\ := ~, start := 355373, stop := 355380 ), rec( content := "]", count := [ 7, 3\ , 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 3, 10, 23 ], mathmode := "Text", name := "C", next := 355386, root := ~, star\ t := 355351, stop := 355385 ), rec( content := " is an edge of\n ", coun\ t := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "ran", count := [ 7, 3, 10, 23\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, \ 23 ], mathmode := "Text", name := "A", next := 355419, root := ~, start := 355\ 409, stop := 355418 ), rec( content := " whenever ", count := [ 7, 3, 10, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[u, v]", count := [ 7, 3, 10, 23 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "T\ ext", name := "C", next := 355442, root := ~, start := 355429, stop := 355441 ), rec( cont\ ent := " is an\n edge of ", count := [ 7, 3, 10, 23 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "src", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "Text", name := \ "A", next := 355475, root := ~, start := 355465, stop := 355474 ), rec( conten\ t := "; and ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 10, 23 ], name := "Item", next := 355488, root := ~, start := 355\ 336, stop := 355487 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", \ count := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 7, 3, 10, 25 ], mathmode := "Text", name := "A", next := 355518, root :\ = ~, start := 355510, stop := 355517 ), rec( content := " fixes every ", count\ := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 3, 10, 25 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25\ ], mathmode := "Text", name := "C", next := 355539, root := ~, start := 35553\ 1, stop := 355538 ), rec( content := " which is not a vertex of ", count := [ \ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "src", count := [ 7, 3, 10, 25 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25\ ], mathmode := "Text", name := "A", next := 355575, root := ~, start := 35556\ 5, stop := 355574 ), rec( content := ".\n ", count := [ 7, 3, 10, 25 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25 ], name := "Ite\ m", next := 355590, root := ~, start := 355495, stop := 355589 ) ], count := [ 7, 3, 10, 22 ], mathmode := "Text", name := "List", next := 355602, root := ~, start := 355323, stop := 355601 ), rec( content := "\n Se\ e also ", count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Generat\ orsOfEndomorphismMonoid" ), content := 0, count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "Ref", next := 355660, root := ~, start := 355616, stop := 355659 ), rec( content := ".", count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "P", next := 355665, root := ~, start := 355661, stop := 355664 ), rec( content := "\n\n\n \ If ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 355686, root := ~, start := 355675, stop := 355685 ), rec( content := " and ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 355702, root := ~, start := 355691, stop := 355701 ), rec( content := ", or ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 355717, root := ~, start := 355707, stop := 355716 ), rec( content := ", are gi\ ven, then they must\n represent vertex colourings; see \n ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "Ref", next := 355863, root := ~, start := 355787, stop := 355862 ), rec( content := " \n f\ or details of the permissible values for these argument. The\n homomorphism\ must then also have the property:\n \n ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 30 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col[i] = col[i ^ x]", count := [ 7, 3, 10, 30 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ], math\ mode := "Text", name := "C", next := 356043, root := ~, start := 356017, stop \ := 356042 ), rec( content := " for all vertices ", count := [ 7, 3, 10, 30 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 3, 10, 30 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ], mathmode := "Text", name\ := "C", next := 356069, root := ~, start := 356061, stop := 356068 ), rec( cont\ ent := " of ", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "A", next := 356087, root :\ = ~, start := 356073, stop := 356086 ), rec( content := ",\n in the cas\ e of ", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "IsDigraphEnd\ omorphism", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "C", next\ := 356140, root := ~, start := 356112, stop := 356139 ), rec( content := ".", coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ] , name := "Ite\ m", next := 356148, root := ~, start := 356002, stop := 356147 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 32 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col1[i] = col2[i ^ x]", count := [ 7, 3, 10, 32 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], math\ mode := "Text", name := "C", next := 356198, root := ~, start := 356170, stop \ := 356197 ), rec( content := " for all vertices ", count := [ 7, 3, 10, 32 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 3, 10, 32 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], mathmode := "Text", name\ := "C", next := 356224, root := ~, start := 356216, stop := 356223 ), rec( cont\ ent := " of ", count := [ 7, 3, 10, 32 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "src", count\ := [ 7, 3, 10, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 10, 32 ], mathmode := "Text", name := "A", next := 356238, root :\ = ~, start := 356228, stop := 356237 ), rec( content := ",\n in the cas\ es of the other operations.", count := [ 7, 3, 10, 32 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], name := "Item", next := 3562\ 92, root := ~, start := 356155, stop := 356291 ) ], count := [ 7, 3, 10, 29 ], mathmode := "Text", name := "List", next := 356304, root := ~, start := 355989, stop := 356303 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ sRespectsColouring" ), content := 0, count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "Ref", next := 356358, root := ~, start := 356319, stop := 356357 ), rec( content := ".\n \\ n ", count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> ran := Digraph([[1], [1, 2]]);\n\ngap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, \ 2]));\ntrue\ngap> IsDigraphHomomorphism(src, ran, Transformation([2, 1, 2]));\ \nfalse\ngap> IsDigraphHomomorphism(src, ran, Transformation([3, 3, 3]));\nfal\ se\ngap> IsDigraphHomomorphism(src, src, Transformation([3, 3, 3]));\ntrue\nga\ p> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),\n> \ [1, 2, 2], [1, 2]);\ntrue\ngap> IsDigraphHomomorphism(src, ran, Tr\ ansformation([1, 2, 2]),\n> [2, 1, 1], [1, 2]);\nfals\ e\ngap> IsDigraphEndomorphism(src, Transformation([3, 3, 3]));\ntrue\ngap> IsD\ igraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsD\ igraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> Is\ DigraphEpimorphism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigrap\ hMonomorphism(src, ran, Transformation([1, 2, 2]));\nfalse\ngap> IsDigraphEpim\ orphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphis\ m(ran, src, ());\ntrue", count := [ 7, 3, 10, 35 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 35 ], mathmode := "Text", name := "Example", next := 357560, root := ~, start := 356369, stop := 357559 ), rec( content := "\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Description", next := 357577, root := ~, start := 354400, stop := 357576 ) ], count := [ 7, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 357591, root := ~, start := 353581, stop := 357590 ), rec( content := "\n\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 2 ], mathmode := "Text", name := "Oper", next := 357728, root := ~, start := 357612, stop := 357727 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 4 ], mathmode := "Text", name := "Oper", next := 357794, root := ~, start := 357731, stop := 357793 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 357817, root := ~, start := 357806, stop := 357816 ), rec( content := " or ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 357833, root := ~, start := 357821, stop := 357832 ), rec( content := ".", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 357844, root := ~, start := 357797, stop := 357843 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEmbedding", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "C", next := 357890, root := ~, start := 357865, stop := 357889 ), rec( content := " returns\ ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 357910, root := ~, start := 357899, stop := 357909 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 357960, root := ~, start := 357952, stop := 357959 ), rec( content := " is a em\ bedding of the digraph\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 358005, root := ~, start := 357995, stop := 358004 ), rec( content := " into th\ e digraph ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 358033, root := ~, start := 358023, stop := 358032 ), rec( content := ", while \ respecting the colourings\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 358082, root := ~, start := 358071, stop := 358081 ), rec( content := " and ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 358098, root := ~, start := 358087, stop := 358097 ), rec( content := " if give\ n.\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "P", next := 358117, root := ~, start := 358113, stop := 358116 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358163, root := ~, start := 358155, stop := 358162 ), rec( content := " is a ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mbedding", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "E", next := 358185, root := ~, start := 358169, stop := 358184 ), rec( content := " of a di\ graph\n ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358213, root := ~, start := 358203, stop := 358212 ), rec( content := " into a \ digraph ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358239, root := ~, start := 358229, stop := 358238 ), rec( content := " if ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358251, root := ~, start := 358243, stop := 358250 ), rec( content := " is a mo\ nomorphism from\n ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358289, root := ~, start := 358279, stop := 358288 ), rec( content := " to ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358303, root := ~, start := 358293, stop := 358302 ), rec( content := " and the\ inverse of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358331, root := ~, start := 358323, stop := 358330 ), rec( content := " is a mo\ nomorphism from\n the subdigraph of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358387, root := ~, start := 358377, stop := 358386 ), rec( content := " induced\ by the image of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358420, root := ~, start := 358412, stop := 358419 ), rec( content := " to\n \ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 358438, root := ~, start := 358428, stop := 358437 ), rec( content := ".\n\n \ See also ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phHomomorphism" ), content := 0, count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 358489, root := ~, start := 358454, stop := 358488 ), rec( content := ".", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 11, 7 ], mathmode := "Text", name := "P", next := 358494, root := ~, start := 358490, stop := 358493 ), rec( content := "\n\n \ ", count := [ 7, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDig\ raphEmbedding(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, \ 1], [2, 1, 1]);\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]\ );\nfalse\ngap> ran := Digraph([[1, 2], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, IdentityTransf\ ormation);\ntrue\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\n\ false", count := [ 7, 3, 11, 8 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 8 ], mathmode := "Text", name := "Example", next := 359143, root := ~, start := 358500, stop := 359142 ), rec( content := "\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 359160, root := ~, start := 357847, stop := 359159 ) ], count := [ 7, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 359174, root := ~, start := 357597, stop := 359173 ), rec( content := "\n\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "DigraphsRespectsColouring" ), content := 0, count := [ 7, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 359269, root := ~, start := 359199, stop := 359268 ), rec( attributes := rec( ), content := [ rec( content := " ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 359295, root := ~, start := 359284, stop := 359294 ), rec( content := " or ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 359311, root := ~, start := 359299, stop := 359310 ), rec( content := ". ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 359323, root := ~, start := 359274, stop := 359322 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsRespectsColouring", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 359394, root := ~, start := 359362, stop := 359393 ), rec( content := " verifie\ s whether or not \n the permutation or transformation ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 359468, root := ~, start := 359460, stop := 359467 ), rec( content := " respect\ s the vertex colourings\n ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 359517, root := ~, start := 359506, stop := 359516 ), rec( content := " and ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 359533, root := ~, start := 359522, stop := 359532 ), rec( content := " of the \ digraphs ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 359560, root := ~, start := 359550, stop := 359559 ), rec( content := " and ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ ange", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 359577, root := ~, start := 359565, stop := 359576 ), rec( content := ".\n \ That is, ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsRespectsColouring", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 359626, root := ~, start := 359594, stop := 359625 ), rec( content := " returns\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 359646, root := ~, start := 359635, stop := 359645 ), rec( content := " if and \ only if for\n all vertices ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 359693, root := ~, start := 359685, stop := 359692 ), rec( content := " of ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 359707, root := ~, start := 359697, stop := 359706 ), rec( content := ", ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1[i] = col2[i ^ x]", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 359737, root := ~, start := 359709, stop := 359736 ), rec( content := ".\n \ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 12, 4 ], mathmode := "Text", name := "P", next := 359749, root := ~, start := 359745, stop := 359748 ), rec( content := "\n\n \ ", count := [ 7, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [1\ , 2, 1]);\ntrue\ngap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [2, \ 1, 1]);\nfalse\n", count := [ 7, 3, 12, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 5 ], mathmode := "Text", name := "Example", next := 360108, root := ~, start := 359757, stop := 360107 ), rec( content := "\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 360127, root := ~, start := 359328, stop := 360126 ) ], count := [ 7, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 360143, root := ~, start := 359182, stop := 360142 ), rec( content := "\n\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, colors\ ][, limit]", Name := "GeneratorsOfEndomorphismMonoid" ), content := 0, count := [ 7, 3, 13, 2 ], mathmode := "Text", name := "Func", next := 360242, root := ~, start := 360164, stop := 360241 ), rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfEnd\ omorphismMonoidAttr" ), content := 0, count := [ 7, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 360308, root := ~, start := 360245, stop := 360307 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f transformations.", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Returns", next := 360357, root := ~, start := 360311, stop := 360356 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ endomorphism of ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 360411, root := ~, start := 360397, stop := 360410 ), rec( content := " is a ho\ momorphism\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 360467, root := ~, start := 360434, stop := 360466 ), rec( content := " from ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 360487, root := ~, start := 360473, stop := 360486 ), rec( content := " back to\ itself.\n\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "C", next := 360546, root := ~, start := 360509, stop := 360545 ), rec( content := ", called\ with a single argument,\n returns a generating set for the monoid of all e\ ndomorphisms of\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 360665, root := ~, start := 360651, stop := 360664 ), rec( content := ". If ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 360684, root := ~, start := 360670, stop := 360683 ), rec( content := " belong\ s to ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 360735, root := ~, start := 360697, stop := 360734 ), rec( content := ", then t\ he value of\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "C", next := 360796, root := ~, start := 360759, stop := 360795 ), rec( content := " will no\ t be recomputed on future\n calls.", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "P", next := 360844, root := ~, start := 360840, stop := 360843 ), rec( content := "\n\n \ If the ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olors", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "A", next := 360870, root := ~, start := 360857, stop := 360869 ), rec( content := " argumen\ t is specified, then\n ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "C", next := 360940, root := ~, start := 360903, stop := 360939 ), rec( content := " will re\ turn a generating\n set for the monoid of endomorphisms which respect the g\ iven colouring.\n The colouring ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olors", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "A", next := 361072, root := ~, start := 361059, stop := 361071 ), rec( content := " can be \ in one of two forms: ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 7 ], mathmode := "Text", name := "P", next := 361105, root := ~, start := 361101, stop := 361104 ), rec( content := "\n\n \ ", count := [ 7, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n A list of positive integers \ of size the number of vertices of\n ", count := [ 7, 3, 13, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "digraph", count := [ 7, 3, 13, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Tex\ t", name := "A", next := 361223, root := ~, start := 361209, stop := 361222 ), rec( cont\ ent := ", where ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "co\ lors", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "A", next := 361\ 244, root := ~, start := 361231, stop := 361243 ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[i]", count := [ 7, 3, 13, 9 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "C", next := 361254, root := ~, start := 361244, stop := 361253 ), rec( cont\ ent := " is the colour of vertex\n ", count := [ 7, 3, 13, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "i", count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "C", next := 361295, root := ~, start := 361287, stop := 361294 ), rec( cont\ ent := ".\n ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], name := "Item", next := 361310, root := ~, start := 361\ 124, stop := 361309 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n A list of lists, such that ", count := [ 7, 3, 13, 11 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "colors", count := [ 7, 3, 13, 11 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ], mathmode := "Text\ ", name := "A", next := 361372, root := ~, start := 361359, stop := 361371 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 3, 13, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, \ 11 ], mathmode := "Text", name := "C", next := 361382, root := ~, start := 361\ 372, stop := 361381 ), rec( content := " is a list of all\n vertices wi\ th colour ", count := [ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 13, 11 ], mathmode := "Text", name := "C", next := 361437, root :=\ ~, start := 361429, stop := 361436 ), rec( content := ".\n ", count := [ \ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ] , name := "Ite\ m", next := 361452, root := ~, start := 361317, stop := 361451 ) ], count := [ 7, 3, 13, 8 ], mathmode := "Text", name := "List", next := 361464, root := ~, start := 361111, stop := 361463 ), rec( content := "\n\n \ If the ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 361489, root := ~, start := 361477, stop := 361488 ), rec( content := " argumen\ t is specified, then it will return only the\n first ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 361564, root := ~, start := 361552, stop := 361563 ), rec( content := " homomor\ phisms, where ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 361598, root := ~, start := 361586, stop := 361597 ), rec( content := " must be\ a positive\n integer or ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "C", next := 361648, root := ~, start := 361633, stop := 361647 ), rec( content := ". ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 14 ], mathmode := "Text", name := "P", next := 361654, root := ~, start := 361650, stop := 361653 ), rec( content := "\n\n \ ", count := [ 7, 3, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph(List([1 .. 3], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomor\ phismMonoid(gr);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ),\ \n IdentityTransformation, Transformation( [ 1, 2, 1 ] ), \n Transformation\ ( [ 1, 2, 2 ] ), Transformation( [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 \ ] ) ]\ngap> GeneratorsOfEndomorphismMonoid(gr, 3);\n[ Transformation( [ 1, 3, \ 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap> gr := Co\ mpleteDigraph(3);;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ Transformation\ ( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap\ > GeneratorsOfEndomorphismMonoid(gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 \ ] ), IdentityTransformation ]\ngap> GeneratorsOfEndomorphismMonoid(gr, [[1], [\ 2, 3]]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]\n", count := [ 7, 3, 13, 15 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 15 ], mathmode := "Text", name := "Example", next := 362523, root := ~, start := 361660, stop := 362522 ), rec( content := "\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Description", next := 362540, root := ~, start := 360360, stop := 362539 ) ], count := [ 7, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 362554, root := ~, start := 360149, stop := 362553 ), rec( content := "\n\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, n", Label := "for a digraph \ and a number of colours", Name := "DigraphColouring" ), content := 0, count := [ 7, 3, 14, 2 ], mathmode := "Text", name := "Oper", next := 362673, root := ~, start := 362575, stop := 362672 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "K", next := 362718, root := ~, start := 362707, stop := 362717 ), rec( content := ".", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 362729, root := ~, start := 362676, stop := 362728 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 362775, root := ~, start := 362752, stop := 362774 ), rec( content := " of a di\ graph is a labelling of its vertices in\n such a way that adjacent vertices\ have different labels. A ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper\n ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 7, 3, 14, 3 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := \ "Text", name := "C", next := 362910, root := ~, start := 362902, stop := 36290\ 9 ), rec( content := "-colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 362924, root := ~, start := 362886, stop := 362923 ), rec( content := " is a pr\ oper colouring that uses exactly ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 362973, root := ~, start := 362965, stop := 362972 ), rec( content := "\n co\ lours. Equivalently, a proper (", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 363020, root := ~, start := 363012, stop := 363019 ), rec( content := "-)colour\ ing of a digraph can be\n defined to be a ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Epimorphism" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 363104, root := ~, start := 363072, stop := 363103 ), rec( content := " from a \ digraph onto the\n complete digraph (with ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 363164, root := ~, start := 363156, stop := 363163 ), rec( content := " vertice\ s); see ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eDigraph" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 363215, root := ~, start := 363180, stop := 363214 ), rec( content := ". Note \ that a digraph with loops (", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 363285, root := ~, start := 363250, stop := 363284 ), rec( content := ") does n\ ot have a proper ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 363318, root := ~, start := 363310, stop := 363317 ), rec( content := "-colouri\ ng for\n any value ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 363355, root := ~, start := 363347, stop := 363354 ), rec( content := ".\n " , count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "P", next := 363365, root := ~, start := 363361, stop := 363364 ), rec( content := "\n\n \ If ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 363388, root := ~, start := 363374, stop := 363387 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 363414, root := ~, start := 363406, stop := 363413 ), rec( content := " is a no\ n-negative integer, then\n ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphColouring(", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 3634\ 85, root := ~, start := 363471, stop := 363484 ), rec( content := ", ", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "n", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 3634\ 95, root := ~, start := 363487, stop := 363494 ), rec( content := ")", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "C", next := 363500, root := ~, start := 363451, stop := 363499 ), rec( content := " returns\ an epimorphism\n from ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 363547, root := ~, start := 363533, stop := 363546 ), rec( content := " onto th\ e complete digraph with ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 363587, root := ~, start := 363579, stop := 363586 ), rec( content := " vertice\ s if one\n exists, else it returns ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "K", next := 363643, root := ~, start := 363632, stop := 363642 ), rec( content := ". ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "P", next := 363649, root := ~, start := 363645, stop := 363648 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 363722, root := ~, start := 363664, stop := 363721 ), rec( content := " and \n \ ", count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "P", next := 363736, root := ~, start := 363732, stop := 363735 ), rec( content := "\n\n \ Note that a digraph with at least two vertices has a 2-colouring if and only\n\ if it is bipartite, see ", count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "Ref", next := 363879, root := ~, start := 363847, stop := 363878 ), rec( content := ".\n\n \ ", count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphColouring(Ch\ ainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := DigraphColo\ uring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigra\ phColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransformation( [ \ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n", count := [ 7, 3, 14, 7 ], name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 364252, root := ~, start := 363886, stop := 364251 ), rec( content := "\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 364269, root := ~, start := 362732, stop := 364268 ) ], count := [ 7, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 364283, root := ~, start := 362560, stop := 364282 ), rec( content := "\n\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, order", Label := "for a digraph \ and vertex order", Name := "DigraphGreedyColouring" ), content := 0, count := [ 7, 3, 15, 2 ], mathmode := "Text", name := "Oper", next := 364419, root := ~, start := 364304, stop := 364418 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and vertex order function", Name := "DigraphGreedyColouring" ), content := 0, count := [ 7, 3, 15, 4 ], mathmode := "Text", name := "Oper", next := 364545, root := ~, start := 364422, stop := 364544 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "DigraphGreedyCo\ louring" ), content := 0, count := [ 7, 3, 15, 6 ], mathmode := "Text", name := "Attr", next := 364639, root := ~, start := 364548, stop := 364638 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "K", next := 364684, root := ~, start := 364673, stop := 364683 ), rec( content := ".", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Returns", next := 364695, root := ~, start := 364642, stop := 364694 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "E", next := 364741, root := ~, start := 364718, stop := 364740 ), rec( content := " of a di\ graph is a labelling of its vertices in\n such a way that adjacent vertices\ have different labels. Note that a digraph \n with loops (", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 364916, root := ~, start := 364887, stop := 364915 ), rec( content := ") does n\ ot have any proper \n colouring.\n ", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "P", next := 364967, root := ~, start := 364963, stop := 364966 ), rec( content := "\n \n\ If ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 364994, root := ~, start := 364980, stop := 364993 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rder", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 365024, root := ~, start := 365012, stop := 365023 ), rec( content := " is a de\ nse list consisting \n of all of the vertices of ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 365097, root := ~, start := 365083, stop := 365096 ), rec( content := " (in any\ order), then \n ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "C", next := 365154, root := ~, start := 365125, stop := 365153 ), rec( content := "\n us\ es a greedy algorithm with the specified order to obtain some proper\n colo\ uring of ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 365261, root := ~, start := 365247, stop := 365260 ), rec( content := ", which \ may not use the minimal number of\n colours. ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "P", next := 365320, root := ~, start := 365316, stop := 365319 ), rec( content := "\n \n\ If ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 365347, root := ~, start := 365333, stop := 365346 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 365376, root := ~, start := 365365, stop := 365375 ), rec( content := " is a fu\ nction whose argument \n is a digraph, and that returns a dense list ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rder", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 365467, root := ~, start := 365455, stop := 365466 ), rec( content := ", then \\ n ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring(", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3655\ 19, root := ~, start := 365505, stop := 365518 ), rec( content := ", ", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "func", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3655\ 32, root := ~, start := 365521, stop := 365531 ), rec( content := ")", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 365537, root := ~, start := 365479, stop := 365536 ), rec( content := " returns\ \n ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring(", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3655\ 90, root := ~, start := 365576, stop := 365589 ), rec( content := ", ", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "func", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3656\ 03, root := ~, start := 365592, stop := 365602 ), rec( content := "(", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3656\ 18, root := ~, start := 365604, stop := 365617 ), rec( content := "))", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 365624, root := ~, start := 365550, stop := 365623 ), rec( content := ".\n " , count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "P", next := 365634, root := ~, start := 365630, stop := 365633 ), rec( content := "\n\n \ If the optional second argument (either a list or a function), is not \n sp\ ecified, then ", count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ WelshPowellOrder" ), content := 0, count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "Ref", next := 365768, root := ~, start := 365731, stop := 365767 ), rec( content := " is used\ by default. \n ", count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "P", next := 365798, root := ~, start := 365794, stop := 365797 ), rec( content := "\n \n\ See also\n ", count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a number of colours", Oper := "DigraphColouring" ), content := 0, count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "Ref", next := 365907, root := ~, start := 365821, stop := 365906 ), rec( content := ".\n " , count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 12 ], mathmode := "Text", name := "P", next := 365917, root := ~, start := 365913, stop := 365916 ), rec( content := "\n\n \ ", count := [ 7, 3, 15, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphGreedyColouring(ChainDigraph(10));\nTransformation( [ 2, 1, 2, 1,\ 2, 1, 2, 1, 2, 1 ] )\ngap> DigraphGreedyColouring(ChainDigraph(10), [1 .. 10]\ );\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\n", count := [ 7, 3, 15, 13 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 15, 13 ], mathmode := "Text", name := "Example", next := 366162, root := ~, start := 365923, stop := 366161 ), rec( content := "\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Description", next := 366179, root := ~, start := 364698, stop := 366178 ) ], count := [ 7, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 366193, root := ~, start := 364289, stop := 366192 ), rec( content := "\n\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphWelshPow\ ellOrder" ), content := 0, count := [ 7, 3, 16, 2 ], mathmode := "Text", name := "Attr", next := 366266, root := ~, start := 366214, stop := 366265 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f the vertices.", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Returns", next := 366312, root := ~, start := 366269, stop := 366311 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphWelshPowellOrder", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "C", next := 366363, root := ~, start := 366333, stop := 366362 ), rec( content := " returns\ a list of all of the vertices of \n the digraph ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "A", next := 366436, root := ~, start := 366422, stop := 366435 ), rec( content := " ordered\ according to the sum of the number of \n out- and in-neighbours, from high\ est to lowest. \n ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 16, 4 ], mathmode := "Text", name := "P", next := 366545, root := ~, start := 366541, stop := 366544 ), rec( content := "\n\n \ ", count := [ 7, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphWelshPowellOrder(Digraph([[4], [9], [9], [], \n> \ [4, 6, 9], [1], [], [], \n> \ [4, 5], [4, 5]]));\n[ 5, 9, 4, 1, 6, 10, 2, 3, 7, 8 ]\n", count := [ 7, 3, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 16, 5 ], mathmode := "Text", name := "Example", next := 366795, root := ~, start := 366551, stop := 366794 ), rec( content := "\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Description", next := 366812, root := ~, start := 366315, stop := 366811 ) ], count := [ 7, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 366826, root := ~, start := 366199, stop := 366825 ), rec( content := "\n\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ChromaticNumber\ " ), content := 0, count := [ 7, 3, 17, 2 ], mathmode := "Text", name := "Attr", next := 366891, root := ~, start := 366847, stop := 366890 ), rec( attributes := rec( ), content := [ rec( content := " A non-ne\ gative integer.", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 366937, root := ~, start := 366894, stop := 366936 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "E", next := 366983, root := ~, start := 366960, stop := 366982 ), rec( content := " of a di\ graph is a labelling of its\n vertices in such a way that adjacent vertices\ have different labels.\n Equivalently, a proper digraph colouring can be d\ efined to be a ", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Epimorphism" ), content := 0, count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 367198, root := ~, start := 367160, stop := 367197 ), rec( content := " from a \ digraph onto a complete digraph. ", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "P", next := 367243, root := ~, start := 367239, stop := 367242 ), rec( content := "\n\n \ If ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 367266, root := ~, start := 367252, stop := 367265 ), rec( content := " is a di\ graph without loops (see ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 367334, root := ~, start := 367299, stop := 367333 ), rec( content := ", then " , count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ hromaticNumber", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 367363, root := ~, start := 367341, stop := 367362 ), rec( content := " returns\ the least\n non-negative integer ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 367415, root := ~, start := 367407, stop := 367414 ), rec( content := " such th\ at there is a proper colouring of\n ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 367475, root := ~, start := 367461, stop := 367474 ), rec( content := " with ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 367489, root := ~, start := 367481, stop := 367488 ), rec( content := " colours\ . In other words, for a digraph with at\n least one vertex, ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ hromaticNumber", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 367582, root := ~, start := 367560, stop := 367581 ), rec( content := " returns\ the least number ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 367616, root := ~, start := 367608, stop := 367615 ), rec( content := "\n su\ ch that ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphColouring(", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 3676\ 65, root := ~, start := 367651, stop := 367664 ), rec( content := ", n)", count := [ 7, 3, 17, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 367673, root := ~, start := 367631, stop := 367672 ), rec( content := " does no\ t return\n ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "K", next := 367705, root := ~, start := 367694, stop := 367704 ), rec( content := ". See ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a number of colours", Oper := "DigraphColouring" ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 367793, root := ~, start := 367711, stop := 367792 ), rec( content := ".\n\n \ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(CompleteDigra\ ph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));\n2\ngap> C\ hromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> ChromaticNumber\ (NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> ChromaticNumber(D);\n3\n", count := [ 7, 3, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 368192, root := ~, start := 367800, stop := 368191 ), rec( content := "\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 368209, root := ~, start := 366940, stop := 368208 ) ], count := [ 7, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 368223, root := ~, start := 366832, stop := 368222 ), rec( content := "\n\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "DigraphCore" ), content := 0, count := [ 7, 3, 18, 2 ], mathmode := "Text", name := "Attr", next := 368286, root := ~, start := 368248, stop := 368285 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers.", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 368338, root := ~, start := 368291, stop := 368337 ), rec( attributes := rec( ), content := [ rec( content := "\n I\ f ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 368374, root := ~, start := 368366, stop := 368373 ), rec( content := " is a di\ graph, then ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCore", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "C", next := 368412, root := ~, start := 368394, stop := 368411 ), rec( content := " returns\ a list of vertices\n corresponding to the ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "C", next := 368478, root := ~, start := 368467, stop := 368477 ), rec( content := " of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 368490, root := ~, start := 368482, stop := 368489 ), rec( content := ". In par\ ticular, the subdigraph\n of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 368539, root := ~, start := 368531, stop := 368538 ), rec( content := " induced\ by this list is isomorphic to the core of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 368598, root := ~, start := 368590, stop := 368597 ), rec( content := ".", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "P", next := 368603, root := ~, start := 368599, stop := 368602 ), rec( content := "\n\n \ The ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "E", next := 368624, root := ~, start := 368613, stop := 368623 ), rec( content := " of a di\ graph ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 368646, root := ~, start := 368638, stop := 368645 ), rec( content := " is the \ minimal subdigraph ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "A", next := 368681, root := ~, start := 368673, stop := 368680 ), rec( content := " of\n \ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 368697, root := ~, start := 368689, stop := 368696 ), rec( content := " which i\ s a homomorphic image of ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 368738, root := ~, start := 368730, stop := 368737 ), rec( content := ". The co\ re of a digraph \n is unique up to isomorphism.\n ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(CycleDigraph(8));\n\ngap> DigraphCore(D);\n[ 1, 2 ]\ngap> D := Pet\ ersenGraph();\n\ngap> DigraphCor\ e(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMutableDigraph, [[3], [3], [4], [5], [\ 2]]);\n\ngap> DigraphCore(D);\n[ 2, \ 3, 4, 5 ]\n", count := [ 7, 3, 18, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 5 ], mathmode := "Text", name := "Example", next := 369222, root := ~, start := 368802, stop := 369221 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 369241, root := ~, start := 368343, stop := 369240 ) ], count := [ 7, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 369257, root := ~, start := 368231, stop := 369256 ), rec( content := "\n\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 369271, root := ~, start := 337077, stop := 369270 ) , rec( content := "\n\n", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 369283, root := ~, start := 288239, stop := 369282 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " homomorphisms ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Finding cliques and independent sets" ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Finding cliques and indep\ endent sets", count := [ 8, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 369422, root := ~, start := 369367, stop := 369421 ) , rec( content := "\n\n In ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "clique", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 369458, root := ~, start := 369445, stop := 369457 ), rec( content := " of a digraph is a set of mutual\ ly adjacent\n vertices of the digraph, and an ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "independent set", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 369560, root := ~, start := 369538, stop := 369559 ), rec( content := " is a set of mutually\n non-a\ djacent vertices of the digraph. A ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal clique", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 369647, root := ~, start := 369626, stop := 369646 ), rec( content := " is a clique\n which is not p\ roperly contained in another clique, and a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal\n independen\ t set", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 369757, root := ~, start := 369721, stop := 369756 ), rec( content := " is an independent set which is \ not properly contained\n in another independent set. Using this definition\ in ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", cliques\n and independent s\ ets are both permitted, but not required, to contain vertices\n at which th\ ere is a loop. Another name for a clique is a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "complete\n subgraph" , count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 370065, root := ~, start := 370035, stop := 370064 ), rec( content := ".\n ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "P", next := 370075, root := ~, start := 370071, stop := 370074 ), rec( content := "\n\n ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " provides extensive functionalit\ y for computing cliques and\n independent sets of a digraph, whether maxima\ l or not. The fundamental\n algorithm used in most of the methods in ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to calculate cliques\n and i\ ndependent sets is a version of the Bron-Kerbosch algorithm.\n Calculating \ the cliques and independent sets of a digraph is a well-known\n and hard pr\ oblem, and searching for cliques or independent sets in a\n digraph can be \ very lengthy, even for a digraph with a small number of\n vertices. ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " uses several strategies to incr\ ease the performance\n of these calculations.\n ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "P", next := 370713, root := ~, start := 370709, stop := 370712 ), rec( content := "\n\n From the definition of c\ liques and independent sets, it follows that the\n presence of loops and mu\ ltiple edges in a digraph is irrelevant to the\n existence of cliques and i\ ndependent sets in the digraph. See ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "Ref", next := 370969, root := ~, start := 370934, stop := 370968 ), rec( content := " and ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMultiDigraph" ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "Ref", next := 371002, root := ~, start := 370974, stop := 371001 ), rec( content := " for more\n information about\ these properties. Therefore given a digraph\n ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "A", next := 371097, root := ~, start := 371083, stop := 371096 ), rec( content := ", the cliques and independent se\ ts of ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "A", next := 371149, root := ~, start := 371135, stop := 371148 ), rec( content := " are\n equal to the cliques a\ nd independent sets of the digraph:\n\n ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphRe\ moveLoops(DigraphRemoveAllMultipleEdges(", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "C", next := 371305, root := ~, start := 371249, stop := 371304 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "A", next := 371319, root := ~, start := 371305, stop := 371318 ), rec( attributes := rec( ), content := [ rec( content := "))", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "C", next := 371328, root := ~, start := 371319, stop := 371327 ), rec( content := ".\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], name := "Item", next := 371343, root := ~, start := 371234, stop := 371342 ) ], count := [ 8, 0, 0, 6 ], mathmode := "Text", name := "List", next := 371355, root := ~, start := 371221, stop := 371354 ), rec( content := "\n\n See ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphRemoveLoops" ), content := 0, count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "Ref", next := 371397, root := ~, start := 371365, stop := 371396 ), rec( content := " and ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphRemoveAllMultipleEdges" ), content := 0, count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "Ref", next := 371451, root := ~, start := 371402, stop := 371450 ), rec( content := " for more information about thes\ e\n attributes. Furthermore, the cliques of this digraph are equal to the\ \n cliques of the digraph formed by removing any edge ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[u,v]", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "C", next := 371627, root := ~, start := 371615, stop := 371626 ), rec( content := " for which\n the correspondin\ g reverse edge ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[v,u]", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "C", next := 371685, root := ~, start := 371673, stop := 371684 ), rec( content := " is not present. Therefore,\n \ overall, the cliques of ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "A", next := 371756, root := ~, start := 371742, stop := 371755 ), rec( content := " are equal to the cliques of the\ \n symmetric digraph:\n\n ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "MaximalSy\ mmetricSubdigraphWithoutLoops(", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "C", next := 371891, root := ~, start := 371845, stop := 371890 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "A", next := 371905, root := ~, start := 371891, stop := 371904 ), rec( attributes := rec( ), content := [ rec( content := ")", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "C", next := 371913, root := ~, start := 371905, stop := 371912 ), rec( content := ".\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], name := "Item", next := 371928, root := ~, start := 371830, stop := 371927 ) ], count := [ 8, 0, 0, 10 ], mathmode := "Text", name := "List", next := 371940, root := ~, start := 371817, stop := 371939 ), rec( content := "\n\n See ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "MaximalSymmetricSubdigraphWitho\ utLoops" ), content := 0, count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "Ref", next := 372002, root := ~, start := 371950, stop := 372001 ), rec( content := " for more\n information about\ this. The ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "AutomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "Ref", next := 372104, root := ~, start := 372044, stop := 372103 ), rec( content := " of this symmetric digraph conta\ ins the\n automorphism group of ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "A", next := 372184, root := ~, start := 372170, stop := 372183 ), rec( content := " as a subgroup. By performing t\ he\n search for maximal cliques with the help of this larger automorphism g\ roup\n to reduce the search space, the computation time may be reduced. Th\ e\n functions and attributes which return representatives of cliques of\n \ ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "A", next := 372462, root := ~, start := 372448, stop := 372461 ), rec( content := " will return orbit representativ\ es of cliques under the\n action of the automorphism group of the ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal symmetric subdigr\ aph\n without loops", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "E", next := 372617, root := ~, start := 372562, stop := 372616 ), rec( content := " on sets of vertices.", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "P", next := 372642, root := ~, start := 372638, stop := 372641 ), rec( content := "\n\n The independent sets of \ a digraph are equal to the independent sets of the\n ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphSymmetricClosure" ), content := 0, count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "Ref", next := 372764, root := ~, start := 372727, stop := 372763 ), rec( content := ". Therefore, overall, the indep\ endent\n sets of ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "A", next := 372829, root := ~, start := 372815, stop := 372828 ), rec( content := " are equal to the independent se\ ts of the symmetric\n digraph:\n\n ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphSy\ mmetricClosure(DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "C", next := 373016, root := ~, start := 372927, stop := 373015 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "A", next := 373030, root := ~, start := 373016, stop := 373029 ), rec( attributes := rec( ), content := [ rec( content := ")))", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "C", next := 373040, root := ~, start := 373030, stop := 373039 ), rec( content := ".\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], name := "Item", next := 373055, root := ~, start := 372912, stop := 373054 ) ], count := [ 8, 0, 0, 15 ], mathmode := "Text", name := "List", next := 373067, root := ~, start := 372899, stop := 373066 ), rec( content := "\n\n Again, the automorphism \ group of this symmetric digraph contains the\n automorphism group of ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "A", next := 373182, root := ~, start := 373168, stop := 373181 ), rec( content := ". Since a search for independent\ sets\n is equal to a search for cliques in the ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphDual" ), content := 0, count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "Ref", next := 373289, root := ~, start := 373264, stop := 373288 ), rec( content := ", the\n methods used in ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " usually transform a search for \ independent sets\n into a search for cliques, as described above. The func\ tions and\n attributes which return representatives of independent sets of\ \n ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "A", next := 373529, root := ~, start := 373515, stop := 373528 ), rec( content := " will return orbit representativ\ es of independent sets under\n the action of the automorphism group of the \ ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "symmetric closure", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "E", next := 373662, root := ~, start := 373638, stop := 373661 ), rec( content := " of the\n digraph formed by r\ emoving loops and multiple edges.", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "P", next := 373730, root := ~, start := 373726, stop := 373729 ), rec( content := "\n\n Please note that in ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", cliques and independent sets a\ re not\n required to be maximal. Some authors use the word clique to mean\n\ ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "E", next := 373892, root := ~, start := 373878, stop := 373891 ), rec( content := " clique, and some authors use th\ e phrase independent set to\n mean ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "E", next := 373975, root := ~, start := 373961, stop := 373974 ), rec( content := " independent set, but please not\ e that ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " does\n not use this definiti\ on.\n ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 20 ], mathmode := "Text", name := "P", next := 374067, root := ~, start := 374063, stop := 374066 ), rec( content := "\n\n ", count := [ 8, 0, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Finding cliques", count := [ 8, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 374114, root := ~, start := 374080, stop := 374113 ), rec( content := "\n ", count := [ 8, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, l", Name := "IsClique" ), content := 0, count := [ 8, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 374172, root := ~, start := 374132, stop := 374171 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalClique\ " ), content := 0, count := [ 8, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 374220, root := ~, start := 374173, stop := 374219 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 374241, root := ~, start := 374230, stop := 374240 ), rec( content := " or ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 374257, root := ~, start := 374245, stop := 374256 ), rec( content := ".", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 374268, root := ~, start := 374221, stop := 374267 ), rec( attributes := rec( ), content := [ rec( content := "\n If ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374302, root := ~, start := 374288, stop := 374301 ), rec( content := " is a di\ graph and ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374328, root := ~, start := 374320, stop := 374327 ), rec( content := " is a du\ plicate-free list of\n vertices of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374385, root := ~, start := 374371, stop := 374384 ), rec( content := ", then\n\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sClique(", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 374410, root := ~, start := 374394, stop := 374409 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374424, root := ~, start := 374410, stop := 374423 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 374432, root := ~, start := 374424, stop := 374431 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374440, root := ~, start := 374432, stop := 374439 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 374448, root := ~, start := 374440, stop := 374447 ), rec( content := " returns\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 374468, root := ~, start := 374457, stop := 374467 ), rec( content := "\n if\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374484, root := ~, start := 374476, stop := 374483 ), rec( content := " is a ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "E", next := 374503, root := ~, start := 374490, stop := 374502 ), rec( content := " of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374521, root := ~, start := 374507, stop := 374520 ), rec( content := " and ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 374538, root := ~, start := 374526, stop := 374537 ), rec( content := " if it i\ s\n not. Similarly,\n ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalClique(", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 374596, root := ~, start := 374573, stop := 374595 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374610, root := ~, start := 374596, stop := 374609 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 374618, root := ~, start := 374610, stop := 374617 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374626, root := ~, start := 374618, stop := 374625 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 374634, root := ~, start := 374626, stop := 374633 ), rec( content := " returns\ \n ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 374658, root := ~, start := 374647, stop := 374657 ), rec( content := " if ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374670, root := ~, start := 374662, stop := 374669 ), rec( content := " is a ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal clique", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "E", next := 374697, root := ~, start := 374676, stop := 374696 ), rec( content := " of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 374715, root := ~, start := 374701, stop := 374714 ), rec( content := " and\n \ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 374736, root := ~, start := 374724, stop := 374735 ), rec( content := " if it i\ s not. ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "P", next := 374756, root := ~, start := 374752, stop := 374755 ), rec( content := "\n\n \ A ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "E", next := 374777, root := ~, start := 374764, stop := 374776 ), rec( content := " of a di\ graph is a set of mutually adjacent vertices of the\n digraph. A ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal clique", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "E", next := 374873, root := ~, start := 374852, stop := 374872 ), rec( content := " is a cl\ ique that is not properly\n contained in another clique. A clique is permit\ ted, but not required, to\n contain vertices at which there is a loop.\n \ ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> IsClique(D, [1, 3, 2]);\ntrue\ngap> IsMa\ ximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMaximalClique(D, DigraphVertices(D))\ ;\ntrue\ngap> D := Digraph([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> IsClique(D, [2, 3, 4]);\nfa\ lse\ngap> IsMaximalClique(D, [1, 2, 4]);\ntrue\ngap> D := CompleteDigraph(IsMu\ tableDigraph, 4);;\ngap> IsClique(D, [1, 3, 2]);\ntrue", count := [ 8, 1, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 1, 7 ], mathmode := "Text", name := "Example", next := 375495, root := ~, start := 375035, stop := 375494 ), rec( content := "\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 375512, root := ~, start := 374269, stop := 375511 ) ], count := [ 8, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 375526, root := ~, start := 374119, stop := 375525 ), rec( content := "\n\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, hook, u\ ser_param, limit, include,\n exclude, max, size, reps", Name := "CliquesFinder" ), content := 0, count := [ 8, 1, 2, 2 ], mathmode := "Text", name := "Func", next := 375653, root := ~, start := 375547, stop := 375652 ), rec( attributes := rec( ), content := [ rec( content := "The argum\ ent ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ser_param", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "A", next := 375695, root := ~, start := 375678, stop := 375694 ), rec( content := ".", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 375706, root := ~, start := 375656, stop := 375705 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function finds cliques of the digraph ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "A", next := 375784, root := ~, start := 375770, stop := 375783 ), rec( content := " subject\ to the\n conditions imposed by the other arguments as described below. Not\ e\n that a clique is represented by the immutable list of the vertices that\ \n it contains.\n ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "P", next := 375972, root := ~, start := 375968, stop := 375971 ), rec( content := "\n\n \ Let ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "C", next := 375990, root := ~, start := 375982, stop := 375989 ), rec( content := " denote \ the automorphism group of the maximal symmetric\n subdigraph of ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "A", next := 376078, root := ~, start := 376064, stop := 376077 ), rec( content := " without\ loops (see ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 1, 2, 4 ] , mathmode := "Text", name := "Ref", next := 376157, root := ~, start := 376098, stop := 376156 ), rec( content := " and ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraphWithoutLoops" ), content := 0, count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "Ref", next := 376220, root := ~, start := 376162, stop := 376219 ), rec( content := ").\n\n \ ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "hook", count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "A", next := 37\ 6258, root := ~, start := 376247, stop := 376257 ) ], count := [ 8, 1, 2, 5 ], name := "Mar\ k", next := 376265, root := ~, start := 376241, stop := 376264 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a f\ unction or ", count := [ 8, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 6 ], mathmode := "Text", name := "K", next := 376336, root := \ ~, start := 376325, stop := 376335 ), rec( content := ".", count := [ 8, 1, 2,\ 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "P", next\ := 376341, root := ~, start := 376337, stop := 376340 ), rec( content := "\n\ \n If ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "hook", coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "A", next := 376365, root := \ ~, start := 376354, stop := 376364 ), rec( content := " is a function, then it\ should have two arguments\n ", count := [ 8, 1, 2, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 8, 1, 2, 7 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name\ := "A", next := 376440, root := ~, start := 376423, stop := 376439 ), rec( cont\ ent := " (see below) and a clique ", count := [ 8, 1, 2, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "c", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next\ := 376474, root := ~, start := 376466, stop := 376473 ), rec( content := ". T\ he function\n ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes\ := rec( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 7 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mat\ hmode := "Text", name := "A", next := 376511, root := ~, start := 376500, stop\ := 376510 ), rec( content := "(", count := [ 8, 1, 2, 7 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec(\ content := "user_param", count := [ 8, 1, 2, 7 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name :=\ "A", next := 376529, root := ~, start := 376512, stop := 376528 ), rec( conte\ nt := ", c)", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next :\ = 376537, root := ~, start := 376497, stop := 376536 ), rec( content := " is c\ alled every time a new\n clique ", count := [ 8, 1, 2, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "c", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ C", next := 376588, root := ~, start := 376580, stop := 376587 ), rec( content\ := " is found by ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ liquesFinder", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next \ := 376621, root := ~, start := 376601, stop := 376620 ), rec( content := ".", coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 1, 2, 8 ], mathmode := "Text", name\ := "P", next := 376626, root := ~, start := 376622, stop := 376625 ), rec( cont\ ent := "\n\n If ", count := [ 8, 1, 2, 8 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "A", next := 3\ 76650, root := ~, start := 376639, stop := 376649 ), rec( content := " is ", coun\ t := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 8 ] , math\ mode := "Text", name := "K", next := 376665, root := ~, start := 376654, stop \ := 376664 ), rec( content := ", then a default function is used that\n \ simply adds every new clique found by ", count := [ 8, 1, 2, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "CliquesFinder", count := [ 8, 1, 2, 8 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name\ := "C", next := 376770, root := ~, start := 376750, stop := 376769 ), rec( cont\ ent := " to\n ", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "user_param", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "A", next \ := 376799, root := ~, start := 376782, stop := 376798 ), rec( content := ", wh\ ich must be a list in this case.\n ", count := [ 8, 1, 2, 8 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 6 ], name := "Ite\ m", next := 376849, root := ~, start := 376272, stop := 376848 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "user_param", count := [ 8, 1, 2, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 8, 1, 2, 9 ], mathmode := "Text", name := "A", next\ := 376880, root := ~, start := 376863, stop := 376879 ) ], count := [ 8, 1, 2, 9 ], name := "Mar\ k", next := 376887, root := ~, start := 376857, stop := 376886 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If ", count := [ 8, 1, 2, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "A", next := 376923, root := ~, start := 376912, stop := 3\ 76922 ), rec( content := " is a function, then ", count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := \ "Text", name := "A", next := 376961, root := ~, start := 376944, stop := 37696\ 0 ), rec( content := " can be any ", count := [ 8, 1, 2, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "GAP", count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "\ Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := "\n\ object. The object ", count := [ 8, 1, 2, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "A" , next\ := 377023, root := ~, start := 377006, stop := 377022 ), rec( content := " is\ used as the first argument for\n the function ", count := [ 8, 1, 2, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "A", next := 377090, root := ~, start := 377079, stop := 3\ 77089 ), rec( content := ". For example, ", count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Te\ xt", name := "A", next := 377122, root := ~, start := 377105, stop := 377121 ) , rec( cont\ ent := " might be a\n list, and ", count := [ 8, 1, 2, 10 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( attributes := rec( ), content := [ rec( content := "hook", count := \ [ 8, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 8, 1, 2, 10 ], mathmode := "Text", name := "A", next := 377166, root := ~, \ start := 377155, stop := 377165 ), rec( content := "(", count := [ 8, 1, 2, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mat\ hmode := "Text", name := "A", next := 377184, root := ~, start := 377167, stop\ := 377183 ), rec( content := ", c)", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "C", next := 377192, root := ~, start := 377152, stop := 377191\ ), rec( content := " might add the size\n of the clique ", count := [ \ 8, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "c", count := [ 8, 1, 2, 10 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "C", next := 377242, root := ~, start := 377234, stop \ := 377241 ), rec( content := " to the list ", count := [ 8, 1, 2, 10 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "A", next := 377272, root := ~, start := 377255, stop := 377271\ ), rec( content := ". ", count := [ 8, 1, 2, 10 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 8, 1, 2, 11 ], mathmode := "Text", name := "P", next := 377278, root := ~, star\ t := 377274, stop := 377277 ), rec( content := "\n\n If the value of ", coun\ t := [ 8, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 1\ 1 ], mathmode := "Text", name := "A", next := 377315, root := ~, start := 3773\ 04, stop := 377314 ), rec( content := " is ", count := [ 8, 1, 2, 11 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "fail", count := [ 8, 1, 2, 11 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 11 ], mathmode := "Text", name\ := "K", next := 377330, root := ~, start := 377319, stop := 377329 ), rec( cont\ ent := ", then the value of\n ", count := [ 8, 1, 2, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 8, 1, 2, 11 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 11 ], mathmode := "Text", name\ := "A", next := 377375, root := ~, start := 377358, stop := 377374 ), rec( cont\ ent := " must be a list.\n ", count := [ 8, 1, 2, 11 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], name := "Item", next := 3774\ 05, root := ~, start := 376894, stop := 377404 ), rec( attributes := rec( ), content := [ rec( attr\ ibutes := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 2, 1\ 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, \ 12 ], mathmode := "Text", name := "A", next := 377431, root := ~, start := 377\ 419, stop := 377430 ) ], count := [ 8, 1, 2, 12 ], name := "Mark", next := 3774\ 38, root := ~, start := 377413, stop := 377437 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n This argument should be a positive integer or ", count := [ \ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "infinity", count := [ 8, 1, 2, 1\ 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, \ 13 ], mathmode := "Text", name := "K", next := 377521, root := ~, start := 377\ 506, stop := 377520 ), rec( content := ".\n ", count := [ 8, 1, 2, 13 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "CliquesFinder", count := [ 8, 1, 2, 13 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ], mathmode \ := "Text", name := "C", next := 377551, root := ~, start := 377531, stop := 37\ 7550 ), rec( content := " will return after it has found\n ", count := [ \ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 2, 13 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ] , math\ mode := "Text", name := "A", next := 377603, root := ~, start := 377591, stop \ := 377602 ), rec( content := " cliques or the search is complete.\n ", coun\ t := [ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ], name := "Ite\ m", next := 377652, root := ~, start := 377445, stop := 377651 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "include", count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "A", next\ := 377680, root := ~, start := 377666, stop := 377679 ), rec( content := " an\ d ", count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "exclude", count \ := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 8, 1, 2, 14 ], mathmode := "Text", name := "A", next := 377699, root := \ ~, start := 377685, stop := 377698 ) ], count := [ 8, 1, 2, 14 ], name := "Mar\ k", next := 377706, root := ~, start := 377660, stop := 377705 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n These arguments should each\ be a (possibly empty) duplicate-free list\n of vertices of ", count :=\ [ 8, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2\ , 15 ], mathmode := "Text", name := "A", next := 377835, root := ~, start := 3\ 77821, stop := 377834 ), rec( content := " (i.e. positive integers less than t\ he\n number of vertices of ", count := [ 8, 1, 2, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "digraph", count := [ 8, 1, 2, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 15 ], mathmode := "Text", name \ := "A", next := 377918, root := ~, start := 377904, stop := 377917 ), rec( cont\ ent := "). ", count := [ 8, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 16 ], math\ mode := "Text", name := "P", next := 377925, root := ~, start := 377921, stop \ := 377924 ), rec( content := "\n\n ", count := [ 8, 1, 2, 16 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "CliquesFinder", count := [ 8, 1, 2, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 16 ], mathmode :=\ "Text", name := "C", next := 377955, root := ~, start := 377935, stop := 3779\ 54 ), rec( content := " will only look for cliques containing all of the\n \ vertices in ", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "in\ clude", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "A", next := 37\ 8039, root := ~, start := 378025, stop := 378038 ), rec( content := " and cont\ aining none of the vertices in\n ", count := [ 8, 1, 2, 16 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "exclude", count := [ 8, 1, 2, 16 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 16 ], mathmode := "Text", name\ := "A", next := 378101, root := ~, start := 378087, stop := 378100 ), rec( cont\ ent := ". ", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 17 ], math\ mode := "Text", name := "P", next := 378107, root := ~, start := 378103, stop \ := 378106 ), rec( content := "\n\n Note that the search may be much mor\ e efficient if each of these lists\n is invariant under the action of " , coun\ t := [ 8, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 17 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 17 ] , math\ mode := "Text", name := "C", next := 378237, root := ~, start := 378229, stop \ := 378236 ), rec( content := " on sets of vertices.\n ", count := [ 8, 1,\ 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 15 ], name := "Ite\ m", next := 378272, root := ~, start := 377713, stop := 378271 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "max", count := [ 8, 1, 2, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 18 ], mathmode := "Text", name := "A", next := \ 378296, root := ~, start := 378286, stop := 378295 ) ], count := [ 8, 1, 2, 18 ], name := "Mar\ k", next := 378303, root := ~, start := 378280, stop := 378302 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "true", count := [ 8, 1, 2, 19\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 1\ 9 ], mathmode := "Text", name := "K", next := 378360, root := ~, start := 3783\ 49, stop := 378359 ), rec( content := " or ", count := [ 8, 1, 2, 19 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "false", count := [ 8, 1, 2, 19 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], mathmode := "Text" , name\ := "K", next := 378376, root := ~, start := 378364, stop := 378375 ), rec( cont\ ent := ". If ", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "max"\ , count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 8, 1, 2, 19 ], mathmode := "Text", name := "A", next := 378392, root :=\ ~, start := 378382, stop := 378391 ), rec( content := " is\n true then\ ", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 19 ], mathmode := "Text", name := "C", next := 378434, root := ~, star\ t := 378414, stop := 378433 ), rec( content := " will only search for ", count\ := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "maximal", count := [ 8, 1, 2,\ 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2\ , 19 ], mathmode := "Text", name := "E", next := 378470, root := ~, start := 3\ 78456, stop := 378469 ), rec( content := "\n cliques. If ", count := [ \ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "max", count := [ 8, 1, 2, 19 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], math\ mode := "Text", name := "K", next := 378501, root := ~, start := 378491, stop \ := 378500 ), rec( content := " is ", count := [ 8, 1, 2, 19 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "false", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], mathmode := "Text", name :=\ "K", next := 378517, root := ~, start := 378505, stop := 378516 ), rec( conte\ nt := " then non-maximal cliques may be\n found.\n ", count := [ 8\ , 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], name := "Ite\ m", next := 378578, root := ~, start := 378310, stop := 378577 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "size", count := [ 8, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 8, 1, 2, 20 ], mathmode := "Text", name := "A", next :=\ 378603, root := ~, start := 378592, stop := 378602 ) ], count := [ 8, 1, 2, 20 ], name := "Mar\ k", next := 378610, root := ~, start := 378586, stop := 378609 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "K", next := 378667, root := ~, start := 3786\ 56, stop := 378666 ), rec( content := " or a positive integer.\n If ", coun\ t := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "size", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "A", next := 378713, root := ~, start := 3787\ 02, stop := 378712 ), rec( content := " is a positive integer then ", count :=\ [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 21 ], mathmode := "Text", name := "C", next := 378761, root := ~, star\ t := 378741, stop := 378760 ), rec( content := " will\n only search for\ cliques that contain precisely ", count := [ 8, 1, 2, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "size", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "A\ ", next := 378833, root := ~, start := 378822, stop := 378832 ), rec( content \ := " vertices.\n If ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "size", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "A", next\ := 378866, root := ~, start := 378855, stop := 378865 ), rec( content := " is\ ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "K", next := 378881, root := ~, start := 3788\ 70, stop := 378880 ), rec( content := " then cliques of any size may be found.\ \n ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 21 ], name := "Item", next := 378934, root := ~, start := 378\ 617, stop := 378933 ), rec( attributes := rec( ), content := [ rec( attribute\ s := rec( ), content := [ rec( content := "reps", count := [ 8, 1, 2, 22 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 22 ], math\ mode := "Text", name := "A", next := 378959, root := ~, start := 378948, stop \ := 378958 ) ], count := [ 8, 1, 2, 22 ], name := "Mark", next := 378966, root := ~, start := 378\ 942, stop := 378965 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n This argument should be ", count := [ 8, 1, 2, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "true", count := [ 8, 1, 2, 23 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 23 ], mathmode := "Text", name :=\ "K", next := 379023, root := ~, start := 379012, stop := 379022 ), rec( conte\ nt := " or ", count := [ 8, 1, 2, 23 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "false", cou\ nt := [ 8, 1, 2, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 23 ], mathmode := "Text", name := "K", next := 379039, root :=\ ~, start := 379027, stop := 379038 ), rec( content := ".", count := [ 8, 1, 2\ , 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "P" , next\ := 379044, root := ~, start := 379040, stop := 379043 ), rec( content := "\n\ \n If ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "reps", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next := 379068, root :=\ ~, start := 379057, stop := 379067 ), rec( content := " is ", count := [ 8, 1\ , 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "true", count := [ 8, 1, 2, 24 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "K", next := 379083, root := ~, start := 379072, stop \ := 379082 ), rec( content := " then the arguments ", count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 1, 2, 24 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Te\ xt", name := "A", next := 379117, root := ~, start := 379103, stop := 379116 ) , rec( cont\ ent := " and\n ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next\ := 379144, root := ~, start := 379130, stop := 379143 ), rec( content := " ar\ e each required to be invariant under the action of\n ", count := [ 8, \ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 24 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "C", next := 379215, root := ~, start := 379207, stop \ := 379214 ), rec( content := " on sets of vertices. In this case, ", count :=\ [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 24 ], mathmode := "Text", name := "C", next := 379272, root := ~, star\ t := 379252, stop := 379271 ), rec( content := " will\n find representa\ tives of the orbits of the desired cliques under the\n action of ", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 24 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ] , math\ mode := "Text", name := "C", next := 379380, root := ~, start := 379372, stop \ := 379379 ), rec( content := ", ", count := [ 8, 1, 2, 24 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "although representatives may be returned that\n are in \ the same orbit", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "E", next\ := 379466, root := ~, start := 379382, stop := 379465 ), rec( content := ".\n\ If ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "reps", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next := 379490, root :=\ ~, start := 379479, stop := 379489 ), rec( content := " is false then ", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 24 ], mathmode := "Text", name := "C", next := 379525, root := ~, star\ t := 379505, stop := 379524 ), rec( content := " will not take this into\n \ consideration.", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1,\ 2, 25 ], mathmode := "Text", name := "P", next := 379576, root := ~, start :=\ 379572, stop := 379575 ), rec( content := "\n\n For a digraph such tha\ t ", count := [ 8, 1, 2, 25 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "G", count := [ 8\ , 1, 2, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 8, 1, 2, 25 ], mathmode := "Text", name := "C", next := 379618, root := ~, star\ t := 379610, stop := 379617 ), rec( content := " is non-trivial, the search fo\ r\n clique representatives can be much more efficient than the search f\ or\n all cliques.\n ", count := [ 8, 1, 2, 25 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 23 ], name := "Item", next := 3797\ 62, root := ~, start := 378973, stop := 379761 ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "List", next := 379774, root := ~, start := 376228, stop := 379773 ), rec( content := "\n\n \ This function uses a version of the Bron-Kerbosch algorithm.\n ", count := [ 8, 1, 2, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> user_param := [];;\ngap> f := function(a, b) # Calculate size of clique\ \n> AddSet(user_param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_par\ am, infinity, [], [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5\ ]\ngap> CliquesFinder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ \ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail\ , [], 2, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> Cl\ iquesFinder(D, fail, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5\ ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], [], false, 3, false);\ \n[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], i\ nfinity, [1, 3], [], true, 3, false);\n[ ]\ngap> D := CompleteDigraph(IsMutab\ leDigraph, 5);\n\ngap> user_param :\ = [];;\ngap> f := function(a, b) # Calculate size of clique\n> AddSet(user_\ param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], \ [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFi\ nder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 \ ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [\ 3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fai\ l, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> Cliques\ Finder(D, fail, [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ \ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], []\ , true, 3, false);\n[ ]", count := [ 8, 1, 2, 28 ], name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 28 ], mathmode := "Text", name := "Example", next := 381499, root := ~, start := 379845, stop := 381498 ), rec( content := "\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 381516, root := ~, start := 375709, stop := 381515 ) ], count := [ 8, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 381530, root := ~, start := 375532, stop := 381529 ), rec( content := "\n\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, size]]]", Name := "DigraphClique" ), content := 0, count := [ 8, 1, 3, 2 ], mathmode := "Text", name := "Func", next := 381623, root := ~, start := 381551, stop := 381622 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalClique" ), content := 0, count := [ 8, 1, 3, 4 ], mathmode := "Text", name := "Func", next := 381709, root := ~, start := 381626, stop := 381708 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of positive integers, or ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "K", next := 381775, root := ~, start := 381764, stop := 381774 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Returns", next := 381786, root := ~, start := 381712, stop := 381785 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "A", next := 381824, root := ~, start := 381810, stop := 381823 ), rec( content := " is a di\ graph, then these functions returns a clique\n of ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "A", next := 381898, root := ~, start := 381884, stop := 381897 ), rec( content := " if one \ exists that satisfies the arguments, else it\n returns ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "K", next := 381974, root := ~, start := 381963, stop := 381973 ), rec( content := ". A cli\ que is defined by the set of vertices that it\n contains; see ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsCliqu\ e" ), content := 0, count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 382068, root := ~, start := 382046, stop := 382067 ), rec( content := " and ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alClique" ), content := 0, count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 382102, root := ~, start := 382073, stop := 382101 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "P", next := 382107, root := ~, start := 382103, stop := 382106 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382150, root := ~, start := 382136, stop := 382149 ), rec( content := " and ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382169, root := ~, start := 382155, stop := 382168 ), rec( content := " must ea\ ch be a\n (possibly empty) duplicate-free list of vertices of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382255, root := ~, start := 382241, stop := 382254 ), rec( content := ", and\n \ the optional argument ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382298, root := ~, start := 382287, stop := 382297 ), rec( content := " must be\ a positive integer. By default,\n ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382358, root := ~, start := 382344, stop := 382357 ), rec( content := " and ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382377, root := ~, start := 382363, stop := 382376 ), rec( content := " are emp\ ty. These functions will search for\n a clique of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382451, root := ~, start := 382437, stop := 382450 ), rec( content := " that in\ cludes the vertices of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382496, root := ~, start := 382482, stop := 382495 ), rec( content := "\n bu\ t does not include any vertices in ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382552, root := ~, start := 382538, stop := 382551 ), rec( content := "; if the\ argument\n ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382585, root := ~, start := 382574, stop := 382584 ), rec( content := " is supp\ lied, then additionally the clique will be required to\n contain precisely \ ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 382681, root := ~, start := 382670, stop := 382680 ), rec( content := " vertice\ s.", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "P", next := 382695, root := ~, start := 382691, stop := 382694 ), rec( content := "\n\n \ If ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "A", next := 382718, root := ~, start := 382704, stop := 382717 ), rec( content := " is not \ a clique, then these functions return ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "K", next := 382775, root := ~, start := 382764, stop := 382774 ), rec( content := ".\n O\ therwise, the functions behave in the following way, depending on the\n num\ ber of arguments:\n\n ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "One or two arguments", count := [ 8, 1\ , 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 8 ], name := "Mar\ k", next := 382928, root := ~, start := 382895, stop := 382927 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If one or two arguments are\ supplied, then ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Dig\ raphClique", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "C", next :=\ 383013, root := ~, start := 382993, stop := 383012 ), rec( content := " and\n\ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malClique", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "C", next := \ 383053, root := ~, start := 383026, stop := 383052 ), rec( content := " greedi\ ly enlarge the clique ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "include", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "A", next :=\ 383096, root := ~, start := 383082, stop := 383095 ), rec( content := "\n \ until it cannot continue. The result is guaranteed to be a maximal\n \ clique. This may or may not return an answer more quickly than using\n \ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( Func := "DigraphMaximalCliques" ), content := 0, coun\ t := [ 8, 1, 3, 9 ], mathmode := "Text", name := "Ref", next := 383293, root :\ = ~, start := 383258, stop := 383292 ), rec( content := " with a limit of 1.\n\ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 8, 1, 3, 9 ], name := "Item", next := 383326, root := ~, start := 382\ 935, stop := 383325 ), rec( attributes := rec( ), content := [ rec( content :\ = "Three arguments", count := [ 8, 1, 3, 10 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 10 ], name := "Mark", next := 383361 , root := ~, start := 383\ 333, stop := 383360 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n If three arguments are supplied, then ", count := [ 8, 1, 3, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphClique", count := [ 8, 1, 3, 11 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 11 ], mathmode \ := "Text", name := "C", next := 383441, root := ~, start := 383421, stop := 38\ 3440 ), rec( content := " greedily\n enlarges the clique ", count := [ \ 8, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "include", count := [ 8, 1, 3, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 1\ 1 ], mathmode := "Text", name := "A", next := 383493, root := ~, start := 3834\ 79, stop := 383492 ), rec( content := " until it cannot continue, although\n \ this clique may not be maximal.", count := [ 8, 1, 3, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := \ 0, count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "P", next := 383572, root\ := ~, start := 383568, stop := 383571 ), rec( content := "\n\n Given t\ hree arguments, ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Di\ graphMaximalClique", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "C\ ", next := 383632, root := ~, start := 383605, stop := 383631 ), rec( content \ := " returns the maximal\n clique returned by ", count := [ 8, 1, 3, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "DigraphMaximalCliques(", count := [ 8, 1, 3\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, \ 3, 12 ], mathmode := "Text", name := "C", next := 383709, root := ~, start := \ 383680, stop := 383708 ), rec( attributes := rec( ), content := [ rec( conten\ t := "digraph", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "A", next\ := 383723, root := ~, start := 383709, stop := 383722 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 1, 3, 12 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], math\ mode := "Text", name := "C", next := 383740, root := ~, start := 383723, stop \ := 383739 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "A", next := 383754, root\ := ~, start := 383740, stop := 383753 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name\ := "C", next := 383763, root := ~, start := 383754, stop := 383762 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 1, 3,\ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3\ , 12 ], mathmode := "Text", name := "A", next := 383777, root := ~, start := 3\ 83763, stop := 383776 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1)", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "C", next :\ = 383788, root := ~, start := 383777, stop := 383787 ), rec( content := " if o\ ne exists,\n else ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "K", next\ := 383828, root := ~, start := 383817, stop := 383827 ), rec( content := ".\n\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 3, 11 ], name := "Item", next := 383843, root := ~, start := 383\ 368, stop := 383842 ), rec( attributes := rec( ), content := [ rec( content :\ = "Four arguments", count := [ 8, 1, 3, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 13 ], name := "Mark", next := 383877, root := ~, start := 383\ 850, stop := 383876 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n If four arguments are supplied, then ", count := [ 8, 1, 3, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphClique", count := [ 8, 1, 3, 14 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode \ := "Text", name := "C", next := 383956, root := ~, start := 383936, stop := 38\ 3955 ), rec( content := " returns the\n clique returned by ", count := [ \ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "DigraphCliques(", count := [ 8, \ 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8,\ 1, 3, 14 ], mathmode := "Text", name := "C", next := 384018, root := ~, start\ := 383996, stop := 384017 ), rec( attributes := rec( ), content := [ rec( co\ ntent := "digraph", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "A" , next\ := 384032, root := ~, start := 384018, stop := 384031 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 1, 3, 14 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ] , math\ mode := "Text", name := "C", next := 384051, root := ~, start := 384032, stop \ := 384050 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "A", next := 384065, root\ := ~, start := 384051, stop := 384064 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name\ := "C", next := 384074, root := ~, start := 384065, stop := 384073 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 1, 3,\ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3\ , 14 ], mathmode := "Text", name := "A", next := 384088, root := ~, start := 3\ 84074, stop := 384087 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1,\n ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "\ C", next := 384108, root := ~, start := 384088, stop := 384107 ), rec( attribu\ tes := rec( ), content := [ rec( content := "size", count := [ 8, 1, 3, 14 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ] , math\ mode := "Text", name := "A", next := 384119, root := ~, start := 384108, stop \ := 384118 ), rec( attributes := rec( ), content := [ rec( content := ")", cou\ nt := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 14 ], mathmode := "Text", name := "C", next := 384127, root :=\ ~, start := 384119, stop := 384126 ), rec( content := " if one exists, else " , coun\ t := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 3, 14\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 1\ 4 ], mathmode := "Text", name := "K", next := 384159, root := ~, start := 3841\ 48, stop := 384158 ), rec( content := ". This clique may\n not be maxim\ al.", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 15 ], mathm\ ode := "Text", name := "P", next := 384204, root := ~, start := 384200, stop :\ = 384203 ), rec( content := " Given four arguments, ", count := [ 8, 1, 3, 15 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalClique", count := [ 8, 1, 3, 15 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], math\ mode := "Text", name := "C", next := 384254, root := ~, start := 384227, stop \ := 384253 ), rec( content := "\n returns the maximal clique returned by\ \n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malCliques(", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C", next\ := 384339, root := ~, start := 384310, stop := 384338 ), rec( attributes := re\ c( ), content := [ rec( content := "digraph", count := [ 8, 1, 3, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], math\ mode := "Text", name := "A", next := 384353, root := ~, start := 384339, stop \ := 384352 ), rec( attributes := rec( ), content := [ rec( content := ", ", co\ unt := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C", next := 384362, root :=\ ~, start := 384353, stop := 384361 ), rec( attributes := rec( ), content := [ \ rec( content := "include", count := [ 8, 1, 3, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name\ := "A", next := 384376, root := ~, start := 384362, stop := 384375 ), rec( attr\ ibutes := rec( ), content := [ rec( content := ",\n ", count := [ 8, 1\ , 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, \ 1, 3, 15 ], mathmode := "Text", name := "C", next := 384393, root := ~, start \ := 384376, stop := 384392 ), rec( attributes := rec( ), content := [ rec( con\ tent := "exclude", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "A", next\ := 384407, root := ~, start := 384393, stop := 384406 ), rec( attributes := re\ c( ), content := [ rec( content := ", 1, ", count := [ 8, 1, 3, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmo\ de := "Text", name := "C", next := 384419, root := ~, start := 384407, stop :=\ 384418 ), rec( attributes := rec( ), content := [ rec( content := "size", co\ unt := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 15 ], mathmode := "Text", name := "A", next := 384430, root :=\ ~, start := 384419, stop := 384429 ), rec( attributes := rec( ), content := [ \ rec( content := ")", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C\ ", next := 384438, root := ~, start := 384430, stop := 384437 ), rec( content \ := " if one exists, else\n ", count := [ 8, 1, 3, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "fail", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := \ "K", next := 384478, root := ~, start := 384467, stop := 384477 ), rec( conten\ t := ".\n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 3, 14 ], name := "Item", next := 384493, root := ~, start := 383\ 884, stop := 384492 ) ], count := [ 8, 1, 3, 8 ], mathmode := "Text", name := "List", next := 384505, root := ~, start := 382882, stop := 384504 ), rec( content := "\n\n \ ", count := [ 8, 1, 3, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> Digraph\ Clique(D);\n[ 5 ]\ngap> DigraphMaximalClique(D);\n[ 5 ]\ngap> DigraphClique(D,\ [1, 2]);\n[ 1, 2, 3 ]\ngap> DigraphMaximalClique(D, [1, 3]);\n[ 1, 3, 2 ]\nga\ p> DigraphClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalClique(D, [1], [3\ , 4]);\nfail\ngap> DigraphClique(D, [1, 5]);\nfail\ngap> DigraphClique(D, [], \ [], 2);\n[ 1, 2 ]\ngap> D := Digraph(IsMutableDigraph,\n> [[2,\ 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> DigraphClique(D);\n[ 5 ]", count := [ 8, 1, 3, 18 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 18 ], mathmode := "Text", name := "Example", next := 385210, root := ~, start := 384511, stop := 385209 ), rec( content := "\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Description", next := 385227, root := ~, start := 381789, stop := 385226 ) ], count := [ 8, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 385241, root := ~, start := 381536, stop := 385240 ), rec( content := "\n\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, limit[, size]]]]", Name := "DigraphMaximalCliques" ), content := 0, count := [ 8, 1, 4, 2 ], mathmode := "Text", name := "Func", next := 385351, root := ~, start := 385262, stop := 385350 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphMaximalCliquesReps" ), content := 0, count := [ 8, 1, 4, 4 ], mathmode := "Text", name := "Func", next := 385451, root := ~, start := 385354, stop := 385450 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphCliques" ), content := 0, count := [ 8, 1, 4, 6 ], mathmode := "Text", name := "Func", next := 385540, root := ~, start := 385454, stop := 385539 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphCliquesReps" ), content := 0, count := [ 8, 1, 4, 8 ], mathmode := "Text", name := "Func", next := 385633, root := ~, start := 385543, stop := 385632 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesAttr" ), content := 0, count := [ 8, 1, 4, 10 ], mathmode := "Text", name := "Attr", next := 385690, root := ~, start := 385636, stop := 385689 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesRepsAttr" ), content := 0, count := [ 8, 1, 4, 12 ], mathmode := "Text", name := "Attr", next := 385751, root := ~, start := 385693, stop := 385750 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists of positive integers.", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Returns", next := 385821, root := ~, start := 385754, stop := 385820 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "A", next := 385859, root := ~, start := 385845, stop := 385858 ), rec( content := " is digr\ aph, then these functions and attributes use ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Cliques\ Finder" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 385945, root := ~, start := 385912, stop := 385944 ), rec( content := " to retu\ rn cliques of ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "A", next := 385981, root := ~, start := 385967, stop := 385980 ), rec( content := ". A\n \ clique is defined by the set of vertices that it contains; see ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsCliqu\ e" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 386080, root := ~, start := 386052, stop := 386079 ), rec( content := " and ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alClique" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 386114, root := ~, start := 386085, stop := 386113 ), rec( content := ".", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "P", next := 386119, root := ~, start := 386115, stop := 386118 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386162, root := ~, start := 386148, stop := 386161 ), rec( content := " and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386181, root := ~, start := 386167, stop := 386180 ), rec( content := " must ea\ ch be a\n (possibly empty) list of vertices of ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386252, root := ~, start := 386238, stop := 386251 ), rec( content := ", the op\ tional argument\n ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386292, root := ~, start := 386280, stop := 386291 ), rec( content := " must be\ either a positive integer or ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "C", next := 386345, root := ~, start := 386330, stop := 386344 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386388, root := ~, start := 386377, stop := 386387 ), rec( content := " must be\ a positive integer. If not\n specified, then ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386459, root := ~, start := 386445, stop := 386458 ), rec( content := " and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386478, root := ~, start := 386464, stop := 386477 ), rec( content := " are cho\ sen to be empty\n lists, and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 386529, root := ~, start := 386517, stop := 386528 ), rec( content := " is set \ to ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "C", next := 386555, root := ~, start := 386540, stop := 386554 ), rec( content := ". ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "P", next := 386561, root := ~, start := 386557, stop := 386560 ), rec( content := "\n\n \ The functions will return as many suitable cliques as possible, up to the\n \ number ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 386664, root := ~, start := 386652, stop := 386663 ), rec( content := ". These\ functions will find cliques that contain all\n of the vertices of ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 386755, root := ~, start := 386741, stop := 386754 ), rec( content := " but do \ not contain any of the\n vertices of ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 386816, root := ~, start := 386802, stop := 386815 ), rec( content := ". The a\ rgument ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 386843, root := ~, start := 386832, stop := 386842 ), rec( content := " restric\ ts the search\n to those cliques that contain precisely ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 386920, root := ~, start := 386909, stop := 386919 ), rec( content := " vertice\ s.\n If the function or attribute has ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximal", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "C", next := 386982, root := ~, start := 386968, stop := 386981 ), rec( content := " in its \ name, then only\n maximal cliques will be returned; otherwise non-maximal c\ liques may be\n returned. ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "P", next := 387099, root := ~, start := 387095, stop := 387098 ), rec( content := "\n\n \ Let ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "C", next := 387117, root := ~, start := 387109, stop := 387116 ), rec( content := " denote \ the automorphism group of maximal symmetric subdigraph\n of ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "A", next := 387201, root := ~, start := 387187, stop := 387200 ), rec( content := " without\ loops (see ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "Ref", next := 387280, root := ~, start := 387221, stop := 387279 ), rec( content := " and ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraphWithoutLoops" ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "Ref", next := 387343, root := ~, start := 387285, stop := 387342 ), rec( content := ").\n\n \ ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Distinct cliques", count := [ 8, 1, 4,\ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 17 ], name := "Mar\ k", next := 387393, root := ~, start := 387364, stop := 387392 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 8, 1, 4, 18 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalCliques", count := [ 8, 1, 4, 18 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], math\ mode := "Text", name := "C", next := 387443, root := ~, start := 387415, stop \ := 387442 ), rec( content := " and ", count := [ 8, 1, 4, 18 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "DigraphCliques", count := [ 8, 1, 4, 18 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], mathmode := "Text\ ", name := "C", next := 387469, root := ~, start := 387448, stop := 387468 ), rec( cont\ ent := " each return a\n duplicate-free list of at most ", count := [ 8\ , 1, 4, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 4, 18 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ] , math\ mode := "Text", name := "A", next := 387535, root := ~, start := 387523, stop \ := 387534 ), rec( content := " cliques of ", count := [ 8, 1, 4, 18 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "digraph", count := [ 8, 1, 4, 18 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], mathmode := "Text\ ", name := "A", next := 387561, root := ~, start := 387547, stop := 387560 ), rec( cont\ ent := "\n that satisfy the arguments.", count := [ 8, 1, 4, 18 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "P", next := 3\ 87601, root := ~, start := 387597, stop := 387600 ), rec( content := "\n\n \ The computation may be significantly faster if ", count := [ 8, 1, 4, 19 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 1, 4, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Te\ xt", name := "A", next := 387672, root := ~, start := 387658, stop := 387671 ) , rec( cont\ ent := " and\n ", count := [ 8, 1, 4, 19 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "A", next\ := 387699, root := ~, start := 387685, stop := 387698 ), rec( content := " ar\ e invariant under the action of ", count := [ 8, 1, 4, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "G", count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "C", next\ := 387742, root := ~, start := 387734, stop := 387741 ), rec( content := "\n \ on sets of vertices.\n ", count := [ 8, 1, 4, 19 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], name := "Item\ ", next := 387785, root := ~, start := 387400, stop := 387784 ), rec( attributes :\ = rec( ), content := [ rec( content := "Orbit representatives of cliques", coun\ t := [ 8, 1, 4, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 20 ], name := "Mar\ k", next := 387838, root := ~, start := 387793, stop := 387837 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n To use ", count := [ 8, 1, \ 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "DigraphMaximalCliquesReps", count := [\ 8, 1, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 8, 1, 4, 21 ], mathmode := "Text", name := "C", next := 387899, root := ~, star\ t := 387867, stop := 387898 ), rec( content := " or ", count := [ 8, 1, 4, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphCliquesReps", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "C", next := 387928, root := ~, start := 387903, stop \ := 387927 ), rec( content := ",\n the arguments ", count := [ 8, 1, 4, \ 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "include", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "A", next := 387966, root := ~, start := 387952, stop \ := 387965 ), rec( content := " and ", count := [ 8, 1, 4, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "exclude", count := [ 8, 1, 4, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], mathmode := "Text", name\ := "A", next := 387985, root := ~, start := 387971, stop := 387984 ), rec( cont\ ent := " must each be invariant\n under the action of ", count := [ 8, \ 1, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "G", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "C", next := 388045, root := ~, start := 388037, stop \ := 388044 ), rec( content := " on sets of vertices.", count := [ 8, 1, 4, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 8, 1, 4, 22 ], mathmode := "Text", name := "P", next := 3\ 88070, root := ~, start := 388066, stop := 388069 ), rec( content := "\n\n \ If this is the case, then ", count := [ 8, 1, 4, 22 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphMaximalCliquesReps", count := [ 8, 1, 4, 22 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ], mathmode := "Tex\ t", name := "C", next := 388138, root := ~, start := 388106, stop := 388137 ), rec( cont\ ent := " and\n ", count := [ 8, 1, 4, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "DigraphCliquesReps", count := [ 8, 1, 4, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ], mathmode := "Text", name :\ = "C", next := 388176, root := ~, start := 388151, stop := 388175 ), rec( cont\ ent := " each return a duplicate-free list of at most\n ", count := [ 8\ , 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 4, 22 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ] , math\ mode := "Text", name := "A", next := 388242, root := ~, start := 388230, stop \ := 388241 ), rec( content := " orbits representatives (under the action of ", coun\ t := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 4, 22 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ] , math\ mode := "Text", name := "C", next := 388295, root := ~, start := 388287, stop \ := 388294 ), rec( content := " on\n sets vertices) of cliques of ", coun\ t := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 4,\ 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4\ , 22 ], mathmode := "Text", name := "A", next := 388350, root := ~, start := 3\ 88336, stop := 388349 ), rec( content := " that satisfy the\n arguments\ .", count := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 1, 4, 23 ], mathmode := "Text", name\ := "P", next := 388390, root := ~, start := 388386, stop := 388389 ), rec( cont\ ent := "\n\n The representatives are not guaranteed to be in distinct o\ rbits.\n However, if fewer than ", count := [ 8, 1, 4, 23 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "lim", count := [ 8, 1, 4, 23 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 23 ], mathmode := "Text", name\ := "A", next := 388506, root := ~, start := 388496, stop := 388505 ), rec( cont\ ent := " results are returned, then there\n will be at least one repres\ entative from each orbit of maximal cliques.\n ", count := [ 8, 1, 4, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], name := "Ite\ m", next := 388633, root := ~, start := 387845, stop := 388632 ) ], count := [ 8, 1, 4, 17 ], mathmode := "Text", name := "List", next := 388645, root := ~, start := 387351, stop := 388644 ), rec( content := "\n\n \ ", count := [ 8, 1, 4, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\ \ntrue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4)\ ])\ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ng\ ap> DigraphMaximalCliquesReps(D);\n[ [ 1, 2, 3 ], [ 3, 4 ] ]\ngap> Orbit(G, [1\ , 2, 3], OnSets);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]\ngap> Orbit(G, [3, 4], OnSets)\ ;\n[ [ 3, 4 ] ]\ngap> DigraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ]\ ]\ngap> DigraphMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCl\ iques(D, [1], [5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph\ (IsMutableDigraph, [\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\nt\ rue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ \ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]", count := [ 8, 1, 4, 26 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 4, 26 ], mathmode := "Text", name := "Example", next := 389660, root := ~, start := 388651, stop := 389659 ), rec( content := "\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Description", next := 389677, root := ~, start := 385824, stop := 389676 ) ], count := [ 8, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 389691, root := ~, start := 385247, stop := 389690 ), rec( content := "\n\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "CliqueNumber" ) , content := 0, count := [ 8, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 389753, root := ~, start := 389712, stop := 389752 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer.", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 389798, root := ~, start := 389756, stop := 389797 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "A", next := 389836, root := ~, start := 389822, stop := 389835 ), rec( content := " is a di\ graph, then ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ liqueNumber(", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 5, 3 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := \ "Text", name := "A", next := 389886, root := ~, start := 389872, stop := 38988\ 5 ), rec( content := ")", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 389891, root := ~, start := 389856, stop := 389890 ), rec( content := "\n re\ turns the largest integer ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 389932, root := ~, start := 389924, stop := 389931 ), rec( content := " such th\ at ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "A", next := 389957, root := ~, start := 389943, stop := 389956 ), rec( content := " contain\ s a\n clique with ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 389993, root := ~, start := 389985, stop := 389992 ), rec( content := " vertice\ s as an induced subdigraph.\n ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "P", next := 390037, root := ~, start := 390033, stop := 390036 ), rec( content := "\n\n \ A ", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "E", next := 390058, root := ~, start := 390045, stop := 390057 ), rec( content := " of a di\ graph is a set of mutually adjacent vertices of the\n digraph. Loops and mu\ ltiple edges are ignored for the purpose of\n determining the clique number\ of a digraph. \n ", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> CliqueNumber(D);\n4\ngap> D := Digraph([\ [1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> CliqueNumber(D);\n3\ngap> D := CompleteDigraph(IsMutabl\ eDigraph, 4);;\ngap> CliqueNumber(D);\n4", count := [ 8, 1, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 5 ], mathmode := "Text", name := "Example", next := 390536, root := ~, start := 390240, stop := 390535 ), rec( content := "\n", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 390551, root := ~, start := 389801, stop := 390550 ) ], count := [ 8, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 390565, root := ~, start := 389697, stop := 390564 ), rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 390579, root := ~, start := 374071, stop := 390578 ) , rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Finding independe\ nt sets", count := [ 8, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 390635, root := ~, start := 390592, stop := 390634 ), rec( content := "\n ", count := [ 8, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, l", Name := "IsIndependentSe\ t" ), content := 0, count := [ 8, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 390703, root := ~, start := 390655, stop := 390702 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalIndepe\ ndentSet" ), content := 0, count := [ 8, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 390761, root := ~, start := 390706, stop := 390760 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 390784, root := ~, start := 390773, stop := 390783 ), rec( content := " or ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 390800, root := ~, start := 390788, stop := 390799 ), rec( content := ".", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 390811, root := ~, start := 390764, stop := 390810 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 390849, root := ~, start := 390835, stop := 390848 ), rec( content := " is a di\ graph and ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 390875, root := ~, start := 390867, stop := 390874 ), rec( content := " is a du\ plicate-free list of\n vertices of ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 390934, root := ~, start := 390920, stop := 390933 ), rec( content := ", then\n\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sIndependentSet(", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 390969, root := ~, start := 390945, stop := 390968 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 390983, root := ~, start := 390969, stop := 390982 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 390991, root := ~, start := 390983, stop := 390990 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 390999, root := ~, start := 390991, stop := 390998 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 391007, root := ~, start := 390999, stop := 391006 ), rec( content := " returns\ \n ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 391031, root := ~, start := 391020, stop := 391030 ), rec( content := " if ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 391043, root := ~, start := 391035, stop := 391042 ), rec( content := " is an " , count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ndependent set", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "E", next := 391072, root := ~, start := 391050, stop := 391071 ), rec( content := " of ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 391090, root := ~, start := 391076, stop := 391089 ), rec( content := " and\n \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 391111, root := ~, start := 391099, stop := 391110 ), rec( content := " if it i\ s not. Similarly,\n ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalIndependentSet(", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 391173, root := ~, start := 391142, stop := 391172 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 391187, root := ~, start := 391173, stop := 391186 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 391195, root := ~, start := 391187, stop := 391194 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 391203, root := ~, start := 391195, stop := 391202 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 391211, root := ~, start := 391203, stop := 391210 ), rec( content := "\n re\ turns ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 391235, root := ~, start := 391224, stop := 391234 ), rec( content := " if ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 391247, root := ~, start := 391239, stop := 391246 ), rec( content := " is a ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal independent set", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "E", next := 391283, root := ~, start := 391253, stop := 391282 ), rec( content := " of\n \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 391305, root := ~, start := 391291, stop := 391304 ), rec( content := " and ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 391322, root := ~, start := 391310, stop := 391321 ), rec( content := " if it i\ s not. ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "P", next := 391342, root := ~, start := 391338, stop := 391341 ), rec( content := "\n\n \ An ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ndependent set", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "E", next := 391373, root := ~, start := 391351, stop := 391372 ), rec( content := " of a di\ graph is a set of mutually non-adjacent\n vertices of the digraph. A ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal independent set", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "E", next := 391482, root := ~, start := 391452, stop := 391481 ), rec( content := " is an i\ ndependent\n set that is not properly contained in another independent set.\ An\n independent set is permitted, but not required, to contain vertices a\ t\n which there is a loop.\n ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsMax\ imalIndependentSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfals\ e\ngap> IsIndependentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, [2\ , 4]);\ntrue\ngap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndependen\ tSet(D, [1]);\ntrue", count := [ 8, 2, 1, 7 ], name := "PCDATA", root := ~ ) ] , count := [ 8, 2, 1, 7 ], mathmode := "Text", name := "Example", next := 392026, root := ~, start := 391677, stop := 392025 ), rec( content := "\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 392043, root := ~, start := 390814, stop := 392042 ) ], count := [ 8, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 392057, root := ~, start := 390640, stop := 392056 ), rec( content := "\n\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, size]]]", Name := "DigraphIndependentSet" ), content := 0, count := [ 8, 2, 2, 2 ], mathmode := "Text", name := "Func", next := 392162, root := ~, start := 392078, stop := 392161 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalIndependentSet" ), content := 0, count := [ 8, 2, 2, 4 ], mathmode := "Text", name := "Func", next := 392260, root := ~, start := 392165, stop := 392259 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of positive integers, or ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "K", next := 392326, root := ~, start := 392315, stop := 392325 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Returns", next := 392337, root := ~, start := 392263, stop := 392336 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "A", next := 392375, root := ~, start := 392361, stop := 392374 ), rec( content := " is a di\ graph, then these functions returns an independent\n set of ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "A", next := 392459, root := ~, start := 392445, stop := 392458 ), rec( content := " if one \ exists that satisfies the arguments, else it\n returns ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "K", next := 392535, root := ~, start := 392524, stop := 392534 ), rec( content := ". An in\ dependent set is defined by the set of vertices\n that it contains; see ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsIndep\ endentSet" ), content := 0, count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 392647, root := ~, start := 392617, stop := 392646 ), rec( content := " and ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alIndependentSet" ), content := 0, count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 392695, root := ~, start := 392652, stop := 392694 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "P", next := 392700, root := ~, start := 392696, stop := 392699 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 392743, root := ~, start := 392729, stop := 392742 ), rec( content := " and ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 392762, root := ~, start := 392748, stop := 392761 ), rec( content := " must ea\ ch be a\n (possibly empty) duplicate-free list of vertices of ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 392848, root := ~, start := 392834, stop := 392847 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 392891, root := ~, start := 392880, stop := 392890 ), rec( content := " must be\ a positive integer. By default,\n ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 392951, root := ~, start := 392937, stop := 392950 ), rec( content := " and ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 392970, root := ~, start := 392956, stop := 392969 ), rec( content := " are emp\ ty. These functions will search\n for an independent set of ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 393054, root := ~, start := 393040, stop := 393053 ), rec( content := " that in\ cludes the vertices of\n ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 393103, root := ~, start := 393089, stop := 393102 ), rec( content := " but doe\ s not include any vertices in ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 393155, root := ~, start := 393141, stop := 393154 ), rec( content := ";\n i\ f the argument ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 393188, root := ~, start := 393177, stop := 393187 ), rec( content := " is supp\ lied, then additionally the independent\n set will be required to contain p\ recisely ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 393293, root := ~, start := 393282, stop := 393292 ), rec( content := " vertice\ s.", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "P", next := 393307, root := ~, start := 393303, stop := 393306 ), rec( content := "\n\n \ If ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "A", next := 393330, root := ~, start := 393316, stop := 393329 ), rec( content := " is not \ an independent set, then these functions return\n ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "K", next := 393401, root := ~, start := 393390, stop := 393400 ), rec( content := ". Other\ wise, the functions behave in the following way,\n depending on the number \ of arguments:\n\n ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "One or two arguments", count := [ 8, 2\ , 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 8 ], name := "Mar\ k", next := 393551, root := ~, start := 393518, stop := 393550 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If one or two arguments are\ supplied, then ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Dig\ raphIndependentSet", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "C", next\ := 393644, root := ~, start := 393616, stop := 393643 ), rec( content := "\n \ and ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malIndependentSet", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "C", next\ := 393692, root := ~, start := 393657, stop := 393691 ), rec( content := " gr\ eedily enlarge the\n independent set ", count := [ 8, 2, 2, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "include", count := [ 8, 2, 2, 9 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text" , name\ := "A", next := 393752, root := ~, start := 393738, stop := 393751 ), rec( cont\ ent := " until it cannot continue. The result\n is guaranteed to be a \ maximal independent set. This may or may not\n return an answer more qu\ ickly than using ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Func := "DigraphMaximalIndependentSe\ ts" ), content := 0, count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "Ref\ ", next := 393968, root := ~, start := 393915, stop := 393967 ), rec( content \ := " with a limit of 1.\n ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], name := "Item", next := 3940\ 01, root := ~, start := 393558, stop := 394000 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "Three arguments", count := [ 8, 2, 2, 10 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 10 ], name := "Mark", next := 3940\ 36, root := ~, start := 394008, stop := 394035 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n If three arguments are supplied, then ", count := [ 8, 2, 2,\ 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "DigraphIndependentSet", count := [ 8, 2, \ 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2,\ 2, 11 ], mathmode := "Text", name := "C", next := 394124, root := ~, start :=\ 394096, stop := 394123 ), rec( content := "\n greedily enlarges the in\ dependent set ", count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "incl\ ude", count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "A", next := 3941\ 85, root := ~, start := 394171, stop := 394184 ), rec( content := " until it c\ annot\n continue, although this independent set may not be maximal.", coun\ t := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "P", next := 394273, root := ~, start := 394269, stop := 394272 ), rec( cont\ ent := "\n\n Given three arguments, ", count := [ 8, 2, 2, 12 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "DigraphMaximalIndependentSet", count := [ 8, 2, 2, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 1\ 2 ], mathmode := "Text", name := "C", next := 394341, root := ~, start := 3943\ 06, stop := 394340 ), rec( content := " returns the\n maximal independe\ nt set returned by\n ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphMaximalIndependentSets(", count := [ 8, 2, 2, 12 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode :=\ "Text", name := "C", next := 394443, root := ~, start := 394406, stop := 3944\ 42 ), rec( attributes := rec( ), content := [ rec( content := "digraph", coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "A", next := 394457, root :=\ ~, start := 394443, stop := 394456 ), rec( attributes := rec( ), content := [ \ rec( content := ",\n ", count := [ 8, 2, 2, 12 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "C", next := 394474, root := ~, start := 394457, stop := 394473 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "include", count := [ 8, 2, 2,\ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 12 ], mathmode := "Text", name := "A", next := 394488, root := ~, start := 3\ 94474, stop := 394487 ), rec( attributes := rec( ), content := [ rec( content\ := ", ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "C", next := \ 394497, root := ~, start := 394488, stop := 394496 ), rec( attributes := rec( \ ), content := [ rec( content := "exclude", count := [ 8, 2, 2, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmod\ e := "Text", name := "A", next := 394511, root := ~, start := 394497, stop := \ 394510 ), rec( attributes := rec( ), content := [ rec( content := ", 1)", cou\ nt := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "C", next := 394522, root :=\ ~, start := 394511, stop := 394521 ), rec( content := " if one exists,\n \ else ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "K", next := 394562, root :=\ ~, start := 394551, stop := 394561 ), rec( content := ".\n ", count := [ \ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 11 ], name := "Ite\ m", next := 394577, root := ~, start := 394043, stop := 394576 ), rec( attributes :\ = rec( ), content := [ rec( content := "Four arguments", count := [ 8, 2, 2, \ 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 13 ], name := "Mar\ k", next := 394611, root := ~, start := 394584, stop := 394610 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If four arguments are suppl\ ied, then ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphInde\ pendentSet", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next \ := 394698, root := ~, start := 394670, stop := 394697 ), rec( content := "\n \ returns the independent set returned by\n ", count := [ 8, 2, 2, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "DigraphIndependentSets(", count := [ 8, 2\ , 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, \ 2, 2, 14 ], mathmode := "Text", name := "C", next := 394785, root := ~, start \ := 394755, stop := 394784 ), rec( attributes := rec( ), content := [ rec( con\ tent := "digraph", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "A", next\ := 394799, root := ~, start := 394785, stop := 394798 ), rec( attributes := re\ c( ), content := [ rec( content := ", ", count := [ 8, 2, 2, 14 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode \ := "Text", name := "C", next := 394808, root := ~, start := 394799, stop := 39\ 4807 ), rec( attributes := rec( ), content := [ rec( content := "include", co\ unt := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "A", next := 394822, root :=\ ~, start := 394808, stop := 394821 ), rec( attributes := rec( ), content := [ \ rec( content := ",\n ", count := [ 8, 2, 2, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name\ := "C", next := 394839, root := ~, start := 394822, stop := 394838 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 2,\ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 14 ], mathmode := "Text", name := "A", next := 394853, root := ~, start := 3\ 94839, stop := 394852 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1, ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next \ := 394865, root := ~, start := 394853, stop := 394864 ), rec( attributes := re\ c( ), content := [ rec( content := "size", count := [ 8, 2, 2, 14 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmod\ e := "Text", name := "A", next := 394876, root := ~, start := 394865, stop := \ 394875 ), rec( attributes := rec( ), content := [ rec( content := ")", count \ := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next := 394884, root := \ ~, start := 394876, stop := 394883 ), rec( content := " if one exists, else\n \ ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "K", next := 394924, root :=\ ~, start := 394913, stop := 394923 ), rec( content := ". This independent set\ may not be maximal.", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, \ 2, 2, 15 ], mathmode := "Text", name := "P", next := 394970, root := ~, start \ := 394966, stop := 394969 ), rec( content := " Given four\n arguments, \ ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphMaximalIndependentSet"\ , count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "C", next := 395036, root :=\ ~, start := 395001, stop := 395035 ), rec( content := " returns the maximal\n\ independent set returned by\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalIndependentSets(", count := [ 8, 2, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 15 ], mathmode := "Text", name := "C", next := 395138, root := ~, start := 3\ 95101, stop := 395137 ), rec( attributes := rec( ), content := [ rec( content\ := "digraph", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "A", next\ := 395152, root := ~, start := 395138, stop := 395151 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 2, 2, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ] , math\ mode := "Text", name := "C", next := 395171, root := ~, start := 395152, stop \ := 395170 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "A", next := 395185, root\ := ~, start := 395171, stop := 395184 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name\ := "C", next := 395194, root := ~, start := 395185, stop := 395193 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 15 ], mathmode := "Text", name := "A", next := 395208, root := ~, start := 3\ 95194, stop := 395207 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1,\n ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "\ C", next := 395228, root := ~, start := 395208, stop := 395227 ), rec( attribu\ tes := rec( ), content := [ rec( content := "size", count := [ 8, 2, 2, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ] , math\ mode := "Text", name := "A", next := 395239, root := ~, start := 395228, stop \ := 395238 ), rec( attributes := rec( ), content := [ rec( content := ")", cou\ nt := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "C", next := 395247, root :=\ ~, start := 395239, stop := 395246 ), rec( content := " if one exists, else " , coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 2, 2, 15\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 1\ 5 ], mathmode := "Text", name := "K", next := 395279, root := ~, start := 3952\ 68, stop := 395278 ), rec( content := ".\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], name := "Ite\ m", next := 395294, root := ~, start := 394618, stop := 395293 ) ], count := [ 8, 2, 2, 8 ], mathmode := "Text", name := "List", next := 395306, root := ~, start := 393505, stop := 395305 ), rec( content := "\n ", count := [ 8, 2, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(6);\n\ngap> D\ igraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n\ [ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> Digraph\ MaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, \ [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nf\ ail\ngap> DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximal\ IndependentSet(D, [1], [], 2);\nfail\ngap> DigraphMaximalIndependentSet(D, [1]\ , [], 3);\n[ 1, 3, 5 ]\ngap> D := ChainDigraph(IsMutableDigraph, 6);\n\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 \ ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependent\ Set(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[\ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> Digrap\ hMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [\ 1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ \ngap> DigraphMaximalIndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]", count := [ 8, 2, 2, 18 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 2, 18 ], mathmode := "Text", name := "Example", next := 396459, root := ~, start := 395311, stop := 396458 ), rec( content := "\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Description", next := 396476, root := ~, start := 392340, stop := 396475 ) ], count := [ 8, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 396490, root := ~, start := 392063, stop := 396489 ), rec( content := "\n\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphMaximalIndependentSets" ), content := 0, count := [ 8, 2, 3, 2 ], mathmode := "Text", name := "Func", next := 396612, root := ~, start := 396511, stop := 396611 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphMaximalIndependentSetsRe\ ps" ), content := 0, count := [ 8, 2, 3, 4 ], mathmode := "Text", name := "Func", next := 396720, root := ~, start := 396615, stop := 396719 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphIndependentSets" ), content := 0, count := [ 8, 2, 3, 6 ], mathmode := "Text", name := "Func", next := 396817, root := ~, start := 396723, stop := 396816 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphIndependentSetsReps" ), content := 0, count := [ 8, 2, 3, 8 ], mathmode := "Text", name := "Func", next := 396918, root := ~, start := 396820, stop := 396917 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsAttr" ), content := 0, count := [ 8, 2, 3, 10 ], mathmode := "Text", name := "Attr", next := 396983, root := ~, start := 396921, stop := 396982 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsRepsAttr" ), content := 0, count := [ 8, 2, 3, 12 ], mathmode := "Text", name := "Attr", next := 397052, root := ~, start := 396986, stop := 397051 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists of positive integers.", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Returns", next := 397122, root := ~, start := 397055, stop := 397121 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "A", next := 397160, root := ~, start := 397146, stop := 397159 ), rec( content := " is digr\ aph, then these functions and attributes use ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Cliques\ Finder" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 397246, root := ~, start := 397213, stop := 397245 ), rec( content := " to retu\ rn independent sets of ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "A", next := 397291, root := ~, start := 397277, stop := 397290 ), rec( content := ". An\n \ independent set is defined by the set of vertices that it contains; see\n \ ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alIndependentSet" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 397413, root := ~, start := 397376, stop := 397412 ), rec( content := " and ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsIndep\ endentSet" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 397454, root := ~, start := 397418, stop := 397453 ), rec( content := ".", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "P", next := 397459, root := ~, start := 397455, stop := 397458 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397502, root := ~, start := 397488, stop := 397501 ), rec( content := " and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397521, root := ~, start := 397507, stop := 397520 ), rec( content := " must ea\ ch be a\n (possibly empty) list of vertices of ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397592, root := ~, start := 397578, stop := 397591 ), rec( content := ", the op\ tional argument\n ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397632, root := ~, start := 397620, stop := 397631 ), rec( content := " must be\ either a positive integer or ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "C", next := 397685, root := ~, start := 397670, stop := 397684 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397728, root := ~, start := 397717, stop := 397727 ), rec( content := " must be\ a positive integer. If not\n specified, then ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397799, root := ~, start := 397785, stop := 397798 ), rec( content := " and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397818, root := ~, start := 397804, stop := 397817 ), rec( content := " are cho\ sen to be empty\n lists, and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 397869, root := ~, start := 397857, stop := 397868 ), rec( content := " is set \ to ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "C", next := 397895, root := ~, start := 397880, stop := 397894 ), rec( content := ". ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "P", next := 397901, root := ~, start := 397897, stop := 397900 ), rec( content := "\n\n \ The functions will return as many suitable independent sets as possible, up\n \ to the number ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 398013, root := ~, start := 398001, stop := 398012 ), rec( content := ". These\ functions will find independent sets\n that contain all of the vertices of\ ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 398113, root := ~, start := 398099, stop := 398112 ), rec( content := " but do \ not\n contain any of the vertices of ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 398174, root := ~, start := 398160, stop := 398173 ), rec( content := " The arg\ ument ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 398199, root := ~, start := 398188, stop := 398198 ), rec( content := "\n re\ stricts the search to those cliques that contain precisely ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 398276, root := ~, start := 398265, stop := 398275 ), rec( content := "\n ve\ rtices. If the function or attribute has ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximal", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "C", next := 398339, root := ~, start := 398325, stop := 398338 ), rec( content := " in its \ name,\n then only maximal independent sets will be returned; otherwise non-\ maximal\n independent sets may be returned.\n ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "P", next := 398478, root := ~, start := 398474, stop := 398477 ), rec( content := "\n\n \ Let ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "C", next := 398496, root := ~, start := 398488, stop := 398495 ), rec( content := " denote \ the ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "Ref", next := 398561, root := ~, start := 398508, stop := 398560 ), rec( content := "\n of\ the ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "Ref", next := 398610, root := ~, start := 398573, stop := 398609 ), rec( content := " of the \ digraph formed from\n ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "A", next := 398656, root := ~, start := 398642, stop := 398655 ), rec( content := " by remo\ ving loops and ignoring the multiplicity of edges.\n\n ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Distinct independent sets", count := [ \ 8, 2, 3, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 17 ], name := "Mar\ k", next := 398771, root := ~, start := 398733, stop := 398770 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 8, 2, 3, 18 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalIndependentSets", count := [ 8, 2, 3, \ 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3,\ 18 ], mathmode := "Text", name := "C", next := 398829, root := ~, start := 39\ 8793, stop := 398828 ), rec( content := " and ", count := [ 8, 2, 3, 18 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphIndependentSets", count := [ 8, 2, 3, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 18 ], math\ mode := "Text", name := "C", next := 398863, root := ~, start := 398834, stop \ := 398862 ), rec( content := "\n each return a duplicate-free list of a\ t most ", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "limit", cou\ nt := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 18 ], mathmode := "Text", name := "A", next := 398929, root :=\ ~, start := 398917, stop := 398928 ), rec( content := " independent\n \ sets of ", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 18 ], mathmode := "Text", name := "A", next := 398972, root :=\ ~, start := 398958, stop := 398971 ), rec( content := " that satisfy the argu\ ments.", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 19 ], math\ mode := "Text", name := "P", next := 399004, root := ~, start := 399000, stop \ := 399003 ), rec( content := "\n\n The computation may be significantly\ faster if ", count := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "include", c\ ount := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 19 ], mathmode := "Text", name := "A", next := 399075, root :=\ ~, start := 399061, stop := 399074 ), rec( content := " and\n ", count\ := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 3,\ 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3\ , 19 ], mathmode := "Text", name := "A", next := 399102, root := ~, start := 3\ 99088, stop := 399101 ), rec( content := " are invariant under the action of " , coun\ t := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 2, 3, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 19 ] , math\ mode := "Text", name := "C", next := 399145, root := ~, start := 399137, stop \ := 399144 ), rec( content := " on sets of\n vertices.\n ", count :\ = [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 18 ], name := "Ite\ m", next := 399188, root := ~, start := 398778, stop := 399187 ), rec( attributes :\ = rec( ), content := [ rec( content := "Representatives of distinct orbits of\ independent sets", count := [ 8, 2, 3, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 3, 20 ], name := "Mark", next := 399263, root := ~, start := 399\ 196, stop := 399262 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n To use ", count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "DigraphMaximalIndependentSetsReps", count := [ 8, 2, 3, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "\ Text", name := "C", next := 399332, root := ~, start := 399292, stop := 399331\ ), rec( content := " or\n ", count := [ 8, 2, 3, 21 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "DigraphIndependentSetsReps", count := [ 8, 2, 3, 21 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmo\ de := "Text", name := "C", next := 399377, root := ~, start := 399344, stop :=\ 399376 ), rec( content := ", the arguments ", count := [ 8, 2, 3, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 2, 3, 21 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Te\ xt", name := "A", next := 399407, root := ~, start := 399393, stop := 399406 ) , rec( cont\ ent := " and\n ", count := [ 8, 2, 3, 21 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "A", next\ := 399434, root := ~, start := 399420, stop := 399433 ), rec( content := " mu\ st each be invariant under the action of ", count := [ 8, 2, 3, 21 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "G", count := [ 8, 2, 3, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Text", name\ := "C", next := 399486, root := ~, start := 399478, stop := 399485 ), rec( cont\ ent := " on\n sets of vertices.", count := [ 8, 2, 3, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ 0, count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "P", next := 399519, root\ := ~, start := 399515, stop := 399518 ), rec( content := "\n\n If this\ is the case, then ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "DigraphMaximalIndependentSetsReps", count := [ 8, 2, 3, 22 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "T\ ext", name := "C", next := 399595, root := ~, start := 399555, stop := 399594 ), rec( cont\ ent := " and\n ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "DigraphIndependentSetsReps", count := [ 8, 2, 3, 22 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "Text" , name\ := "C", next := 399641, root := ~, start := 399608, stop := 399640 ), rec( cont\ ent := " each return a list of\n at most ", count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "limit", count := [ 8, 2, 3, 22 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "Text\ ", name := "A", next := 399692, root := ~, start := 399680, stop := 399691 ), rec( cont\ ent := " orbits representatives (under the action of\n ", count := [ 8,\ 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "G", count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "C", next := 399753, root := ~, start := 399745, stop \ := 399752 ), rec( content := " on sets of vertices) of independent sets of ", coun\ t := [ 8, 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 2, 3,\ 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3\ , 22 ], mathmode := "Text", name := "A", next := 399812, root := ~, start := 3\ 99798, stop := 399811 ), rec( content := "\n that satisfy the arguments\ . ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 23 ], mathmo\ de := "Text", name := "P", next := 399853, root := ~, start := 399849, stop :=\ 399852 ), rec( content := "\n\n The representatives are not guaranteed\ to be in distinct orbits.\n However, if ", count := [ 8, 2, 3, 23 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "lim", count := [ 8, 2, 3, 23 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 23 ], mathmode := "Text", name\ := "A", next := 399958, root := ~, start := 399948, stop := 399957 ), rec( cont\ ent := " is not specified, or fewer than ", count := [ 8, 2, 3, 23 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "lim", count := [ 8, 2, 3, 23 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 23 ], mathmode := "Text", name\ := "A", next := 400001, root := ~, start := 399991, stop := 400000 ), rec( cont\ ent := "\n results are returned, then there will be at least one repres\ entative\n from each orbit of maximal independent sets.\n ", count\ := [ 8, 2, 3, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], name := "Ite\ m", next := 400145, root := ~, start := 399270, stop := 400144 ) ], count := [ 8, 2, 3, 17 ], mathmode := "Text", name := "List", next := 400157, root := ~, start := 398720, stop := 400156 ), rec( content := "\n\n \ ", count := [ 8, 2, 3, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(5);\n\ngap> D\ igraphMaximalIndependentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSet\ sReps(D);\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[\ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndepe\ ndentSets(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [],\ [4, 5]);\n[ [ 1 ], [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [\ ], [4, 5], 1, 2);\n[ [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\ \n\ngap> DigraphMaximalIndependentSe\ tsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ], [ 1, 3\ ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, \ 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\n[ [ 1,\ 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]);\n[ [ 1 ], [ 2 ],\ [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5], 1, 2);\n[ [ 1, \ 3 ] ]", count := [ 8, 2, 3, 26 ], name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 26 ], mathmode := "Text", name := "Example", next := 401185, root := ~, start := 400163, stop := 401184 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Description", next := 401202, root := ~, start := 397125, stop := 401201 ) ], count := [ 8, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 401216, root := ~, start := 396496, stop := 401215 ), rec( content := "\n\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 401230, root := ~, start := 390583, stop := 401229 ) , rec( content := "\n", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 401241, root := ~, start := 369310, stop := 401240 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " cliques ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Visualising and IO" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Visualising and IO", count := [ 9, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 401335, root := ~, start := 401298, stop := 401334 ) , rec( content := "\n\n ", count := [ 9, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Visualising a dig\ raph", count := [ 9, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 401388, root := ~, start := 401348, stop := 401387 ), rec( content := "\n ", count := [ 9, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str, [opts]", Name := "Splash" ), content := 0, count := [ 9, 1, 1, 2 ], mathmode := "Text", name := "Func", next := 401451, root := ~, start := 401412, stop := 401450 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 401483, root := ~, start := 401456, stop := 401482 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function attempts to convert the string ", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "A", next := 401563, root := ~, start := 401553, stop := 401562 ), rec( content := " into a \ pdf\n document and open this document, i.e. to splash it all over your mo\ nitor.", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "P", next := 401658, root := ~, start := 401654, stop := 401657 ), rec( content := "\n\n \ The string ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "A", next := 401687, root := ~, start := 401677, stop := 401686 ), rec( content := " must co\ rrespond to a valid ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 401725, root := ~, start := 401715, stop := 401724 ), rec( content := " or\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ aTeX", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 401747, root := ~, start := 401735, stop := 401746 ), rec( content := " text fi\ le and you must have have ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 401796, root := ~, start := 401781, stop := 401795 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ dflatex", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 401822, root := ~, start := 401807, stop := 401821 ), rec( content := " install\ ed on your computer. For details about these file\n formats, see ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://www.latex-project.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 401940, root := ~, start := 401900, stop := 401939 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://www.graphviz.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 401986, root := ~, start := 401951, stop := 401985 ), rec( content := ".", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "P", next := 401991, root := ~, start := 401987, stop := 401990 ), rec( content := "\n\n \ This function is provided to allow convenient, immediate viewing of the\n \ pictures produced by the function ", count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "Ref", next := 402135, root := ~, start := 402111, stop := 402134 ), rec( content := ".", count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "P", next := 402140, root := ~, start := 402136, stop := 402139 ), rec( content := "\n\n \ The optional second argument ", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ pts", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "A", next := 402188, root := ~, start := 402177, stop := 402187 ), rec( content := " should \ be a record with\n components corresponding to various options, given bel\ ow.\n\n ", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "path", count := [ 9, 1, 1, 7 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 7 ], name := "Mar\ k", next := 402316, root := ~, start := 402299, stop := 402315 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the path to the directory where\n you want ", count := [ \ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "Splash", count := [ 9, 1, 1, 8 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], math\ mode := "Text", name := "C", next := 402443, root := ~, start := 402430, stop \ := 402442 ), rec( content := " to do its work. The default value of this\n \ option is ", count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"~\ /\"", count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "C", next := 402517 , root\ := ~, start := 402506, stop := 402516 ), rec( content := ".\n ", count\ := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], name := "Ite\ m", next := 402534, root := ~, start := 402325, stop := 402533 ), rec( attributes :\ = rec( ), content := [ rec( content := "directory", count := [ 9, 1, 1, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 9 ], name := "Mar\ k", next := 402566, root := ~, start := 402544, stop := 402565 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the directory in\n ", count := [ 9, 1, 1, 10 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "path", count := [ 9, 1, 1, 10 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 10 ], mathmode := "Text" , name\ := "C", next := 402679, root := ~, start := 402668, stop := 402678 ), rec( cont\ ent := " where you want ", count := [ 9, 1, 1, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "Splash", count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "C", next\ := 402708, root := ~, start := 402695, stop := 402707 ), rec( content := " to\ do its work. This function\n will create this directory if does not \ already exist. ", count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1,\ 11 ], mathmode := "Text", name := "P", next := 402807, root := ~, start := 40\ 2803, stop := 402806 ), rec( content := "\n\n The default value of th\ is option is ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"tmp.viz\"\ ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "C", next := 402871, root\ := ~, start := 402855, stop := 402870 ), rec( content := " if the option\n \ ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "path", coun\ t := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 11 ], mathmode := "Text", name := "C", next := 402907, root :=\ ~, start := 402896, stop := 402906 ), rec( content := " is present, and the r\ esult of\n ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "Dire\ ctoryTemporary" ), content := 0, count := [ 9, 1, 1, 11 ], mathmode := "Text", name\ := "Ref", next := 402995, root := ~, start := 402948, stop := 402994 ), rec( cont\ ent := " is used otherwise.\n ", count := [ 9, 1, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 10 ], name := "Ite\ m", next := 403030, root := ~, start := 402575, stop := 403029 ), rec( attributes :\ = rec( ), content := [ rec( content := "filename", count := [ 9, 1, 1, 12 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 12 ], name := "Mar\ k", next := 403061, root := ~, start := 403040, stop := 403060 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the file where\n ", count := [ 9, 1, 1, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "str", count := [ 9, 1, 1, 13 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], mathmode := "Text", name\ := "A", next := 403171, root := ~, start := 403161, stop := 403170 ), rec( cont\ ent := " will be written. The default value of this option is\n ", coun\ t := [ 9, 1, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"vizpicture\"", count := [ 9\ , 1, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 9, 1, 1, 13 ], mathmode := "Text", name := "C", next := 403255, root := ~, star\ t := 403236, stop := 403254 ), rec( content := ".\n ", count := [ 9, 1,\ 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], name := "Ite\ m", next := 403272, root := ~, start := 403070, stop := 403271 ), rec( attributes :\ = rec( ), content := [ rec( content := "viewer", count := [ 9, 1, 1, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 14 ], name := "Mar\ k", next := 403301, root := ~, start := 403282, stop := 403300 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the program which\n should open the files pro\ duced by ", count := [ 9, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "GraphViz", \ count := [ 9, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 15 ], mathmode := "Text", name := "C", next := 403453, root :=\ ~, start := 403438, stop := 403452 ), rec( content := " or ", count := [ 9, 1\ , 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "pdflatex", count := [ 9, 1, 1, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ] , math\ mode := "Text", name := "C", next := 403472, root := ~, start := 403457, stop \ := 403471 ), rec( content := ".\n ", count := [ 9, 1, 1, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ], name := "Ite\ m", next := 403489, root := ~, start := 403310, stop := 403488 ), rec( attributes :\ = rec( ), content := [ rec( content := "type", count := [ 9, 1, 1, 16 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 16 ], name := "Mar\ k", next := 403516, root := ~, start := 403499, stop := 403515 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this option can be used t\ o specify that the string ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "str", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "A", next \ := 403603, root := ~, start := 403593, stop := 403602 ), rec( content := " con\ tains\n a ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( Only := "LaTeX" ), content := [ r\ ec( content := "{\\LaTeX}", count := [ 9, 1, 1, 17 ], name := "PCDATA", root :\ = ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "Alt", next := \ 33, root := ~, start := 1, stop := 32 ), rec( attributes := rec( Not := "LaTeX\ " ), content := [ rec( content := "LaTeX", count := [ 9, 1, 1, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := \ "Alt", next := 61, root := ~, start := 33, stop := 60 ), rec( content := " or \ ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "dot", count := [ 9, 1, 1, 17 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17\ ], mathmode := "Text", name := "C", next := 403646, root := ~, start := 40363\ 6, stop := 403645 ), rec( content := " document. You can specify this option i\ n\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "str", count\ := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "A", next := 403708, root :=\ ~, start := 403698, stop := 403707 ), rec( content := " directly by making th\ e first line ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"%latex\""\ , count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "C", next := 403758, root :=\ ~, start := 403743, stop := 403757 ), rec( content := " or\n ", coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"//dot\"", count := [ 9, 1, \ 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1,\ 1, 17 ], mathmode := "Text", name := "C", next := 403786, root := ~, start :=\ 403772, stop := 403785 ), rec( content := ". There is no default value for t\ his option, this\n option must be specified in ", count := [ 9, 1, 1,\ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "str", count := [ 9, 1, 1, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode\ := "Text", name := "A", next := 403885, root := ~, start := 403875, stop := 4\ 03884 ), rec( content := " or in ", count := [ 9, 1, 1, 17 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "opt.type", count := [ 9, 1, 1, 17 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name \ := "A", next := 403907, root := ~, start := 403892, stop := 403906 ), rec( cont\ ent := ".\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := "Item", next := 40392\ 4, root := ~, start := 403525, stop := 403923 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "engine", count := [ 9, 1, 1, 18 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 18 ], name := "Mark", next := 403953, root := ~, start := 403\ 934, stop := 403952 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n this option can be used to specify the GraphViz engine to use\n\ to render a ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "dot", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := \ 404073, root := ~, start := 404063, stop := 404072 ), rec( content := " docume\ nt. The valid choices are ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "\"dot\"", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next\ := 404118, root := ~, start := 404106, stop := 404117 ), rec( content := ",\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"neato\"",\ count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 404144, root :=\ ~, start := 404130, stop := 404143 ), rec( content := ", ", count := [ 9, 1, \ 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "\"circo\"", count := [ 9, 1, 1, 19 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], math\ mode := "Text", name := "C", next := 404160, root := ~, start := 404146, stop \ := 404159 ), rec( content := ", ", count := [ 9, 1, 1, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "\"twopi\"", count := [ 9, 1, 1, 19 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name \ := "C", next := 404176, root := ~, start := 404162, stop := 404175 ), rec( cont\ ent := ", ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"fdp\"", c\ ount := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 404190, root :=\ ~, start := 404178, stop := 404189 ), rec( content := ",\n ", count \ := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"sfdp\"", count := [ 9, 1, 1\ , 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, \ 1, 19 ], mathmode := "Text", name := "C", next := 404215, root := ~, start := \ 404202, stop := 404214 ), rec( content := ", and ", count := [ 9, 1, 1, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\"patchwork\"", count := [ 9, 1, 1, 19 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode \ := "Text", name := "C", next := 404239, root := ~, start := 404221, stop := 40\ 4238 ), rec( content := ". Please refer to the\n GraphViz documentati\ on for details on these engines.\n The default value for this option \ is ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "\"dot\"", count\ := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 404383, root :=\ ~, start := 404371, stop := 404382 ), rec( content := ", and it\n mu\ st be specified in ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "opt.engine", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "A", next\ := 404440, root := ~, start := 404423, stop := 404439 ), rec( content := ".\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], name := "Item", next := 404457, root := ~, start := 403\ 962, stop := 404456 ), rec( attributes := rec( ), content := [ rec( content :\ = "filetype", count := [ 9, 1, 1, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 20 ], name := "Mark", next := 404488, root := ~, start := 404\ 467, stop := 404487 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n this should be a string representing the type of file which\n \ ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Splash", co\ unt := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 21 ], mathmode := "Text", name := "C", next := 404597, root :=\ ~, start := 404584, stop := 404596 ), rec( content := " should produce. For " , coun\ t := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( Only := "LaTeX" ), content := [ rec( content := "{\\LaTeX}", co\ unt := [ 9, 1, 1, 21 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 2\ 1 ], mathmode := "Text", name := "Alt", next := 33, root := ~, start := 1, stop\ := 32 ), rec( attributes := rec( Not := "LaTeX" ), content := [ rec( content \ := "LaTeX", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Alt", next := 61, root := ~, star\ t := 33, stop := 60 ), rec( content := " files, this option is\n igno\ red and the default value ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "\"pdf\"", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "C", next\ := 404700, root := ~, start := 404688, stop := 404699 ), rec( content := " is\ used.\n ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Item", next := 404725, root := ~, start := 404\ 497, stop := 404724 ) ], count := [ 9, 1, 1, 7 ], mathmode := "Text", name := "List", next := 404739, root := ~, start := 402284, stop := 404738 ), rec( content := "\n\n \ This function was written by Attila Egri-Nagy and Manuel Delgado with some\n\ minor changes by J. D. Mitchell.", count := [ 9, 1, 1, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 24 ], mathmode := "Text", name := "P", next := 404864, root := ~, start := 404860, stop := 404863 ), rec( content := "\n \ ", count := [ 9, 1, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Splash(DotDigraph(RandomDigraph(4)));\n", count := [ 9, 1, 1, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 25 ], mathmode := "Text", name := "Log", next := 404938, root := ~, start := 404871, stop := 404937 ), rec( content := "\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 404957, root := ~, start := 401488, stop := 404956 ) ], count := [ 9, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 404973, root := ~, start := 401395, stop := 404972 ), rec( content := "\n\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotDigraph" ), content := 0, count := [ 9, 1, 2, 2 ], mathmode := "Text", name := "Attr", next := 405037, root := ~, start := 404998, stop := 405036 ), rec( attributes := rec( Arg := "digraph, vert, e\ dge", Name := "DotColoredDigraph" ), content := 0, count := [ 9, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 405100, root := ~, start := 405042, stop := 405099 ), rec( attributes := rec( Arg := "digraph", Name := "DotVertexLabell\ edDigraph" ), content := 0, count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "Oper", next := 405158, root := ~, start := 405105, stop := 405157 ), rec( attributes := rec( Arg := "digraph, vert", Name := "DotVertexColore\ dDigraph" ), content := 0, count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "Oper", next := 405221, root := ~, start := 405163, stop := 405220 ), rec( attributes := rec( Arg := "digraph, edge", Name := "DotEdgeColoredD\ igraph" ), content := 0, count := [ 9, 1, 2, 10 ], mathmode := "Text", name := "Oper", next := 405282, root := ~, start := 405226, stop := 405281 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "Returns", next := 405315, root := ~, start := 405287, stop := 405314 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "C", next := 405357, root := ~, start := 405340, stop := 405356 ), rec( content := " produce\ s a graphical representation of the digraph\n ", count := [ 9, 1, 2, 11 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "A", next := 405429, root := ~, start := 405415, stop := 405428 ), rec( content := ". Vertic\ es are displayed as circles, numbered consistently\n with ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "A", next := 405513, root := ~, start := 405499, stop := 405512 ), rec( content := ". Edges \ are displayed as arrowed lines between\n vertices, with the arrowhead of \ each line pointing towards the range\n of the edge.", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "P", next := 405657, root := ~, start := 405653, stop := 405656 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otColoredDigraph", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "C", next := 405689, root := ~, start := 405665, stop := 405688 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "C", next := 405720, root := ~, start := 405703, stop := 405719 ), rec( content := " only in\ \n that the values in given in the two lists are used to color the vertic\ es and\n edges of the graph when displayed. The list for vertex colours s\ hould be\n a list of length equal to the number of vertices, containing s\ trings that\n are accepted by the graphviz software, which is the one use\ d for graph\n representation. The list for edge colours should be a list \ of lists\n with the same shape of the outneighbours of the digraph that c\ ontains strings \n that correspond to colours accepted by the graphviz so\ ftware. If the lists \n are not the appropriate size, or have holes then \ the function will return\n an error.", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "P", next := 406388, root := ~, start := 406384, stop := 406387 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexColoredDigraph", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "C", next := 406426, root := ~, start := 406396, stop := 406425 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "C", next := 406457, root := ~, start := 406440, stop := 406456 ), rec( content := " only in\ \n that the values in given in the list are used to color the vertices\n \ of the graph when displayed. The list for vertex colours should be\n \ a list of length equal to the number of vertices, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error.", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "P", next := 406896, root := ~, start := 406892, stop := 406895 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otEdgeColoredDigraph", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "C", next := 406932, root := ~, start := 406904, stop := 406931 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "C", next := 406963, root := ~, start := 406946, stop := 406962 ), rec( content := " only in\ \n that the values in given in the list are used to color the vertices an\ d\n edges of the graph when displayed. The list for edge colours should b\ e a list\n of lists with the same shape of the outneighbours of the digra\ ph that contains\n strings that correspond to colours accepted by the gra\ phviz software. If the \n list is not the appropriate size, or has holes \ then the function will return\n an error.", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "P", next := 407405, root := ~, start := 407401, stop := 407404 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "C", next := 407444, root := ~, start := 407413, stop := 407443 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "C", next := 407475, root := ~, start := 407458, stop := 407474 ), rec( content := " only in\ \n that the values in ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "Ref", next := 407542, root := ~, start := 407509, stop := 407541 ), rec( content := " are use\ d to label\n the vertices in the produced picture rather than the numbers\ ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "C", next := 407636, root := ~, start := 407628, stop := 407635 ), rec( content := " to\n \ the number of vertices of the digraph. ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "P", next := 407689, root := ~, start := 407685, stop := 407688 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "C", next := 407724, root := ~, start := 407714, stop := 407723 ), rec( content := " format \ (also known as ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "C", next := 407762, root := ~, start := 407747, stop := 407761 ), rec( content := ")\n \ format. For details about this file format, and information about how to\n \ display or edit this format see ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://www.graphviz.org", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "URL", next := 407916, root := ~, start := 407881, stop := 407915 ), rec( content := ". ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "P", next := 407922, root := ~, start := 407918, stop := 407921 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "C", next := 407970, root := ~, start := 407953, stop := 407969 ), rec( content := " or\n \ ", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "C", next := 408011, root := ~, start := 407980, stop := 408010 ), rec( content := " can be \ written to a file using\n the command ", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "Ref", next := 408103, root := ~, start := 408061, stop := 408102 ), rec( content := ".", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 18 ], mathmode := "Text", name := "P", next := 408108, root := ~, start := 408104, stop := 408107 ), rec( content := "\n \ ", count := [ 9, 1, 2, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> adj := List([1 .. 4], x -> [1, 1, 1, 1]);\n[ [ 1, 1, 1, 1 ], [ 1, 1, 1, \ 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] ]\ngap> adj[1][3] := 0;\n0\ngap> gr := Dig\ raphByAdjacencyMatrix(adj);\n\nga\ p> D := CompleteDigraph(4);\n\ngap\ > vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\ \";; \ngap> vertcolors[3] := \"green\";; vertcolors[4] := \"yellow\";;\ngap> e\ dgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edge\ colors[3] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2] := \"lightblue\ \";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4] := \"purple\"\ ;;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2][3] := \"pink\ \";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3][1] := \"ligh\ tblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolors[3][4] := \"p\ urple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgecolors[4][2] :=\ \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(DotColoredDigra\ ph(D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[c\ olor=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, style=fil\ led]\n4[color=yellow, style=filled]\n1 -> 2[color=lightblue]\n1 -> 3[color=pin\ k]\n1 -> 4[color=purple]\n2 -> 1[color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[\ color=purple]\n3 -> 1[color=lightblue]\n3 -> 2[color=pink]\n3 -> 4[color=purpl\ e]\n4 -> 1[color=lightblue]\n4 -> 2[color=pink]\n4 -> 3[color=purple]\n}\ngap>\ D := EmptyDigraph(3);\n\ngap> vertco\ lors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap\ > vertcolors[3] := \"green\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] :=\ [];; edgecolors[2] := [];; \ngap> edgecolors[3] := [];;\ngap> Print(DotColore\ dDigraph(D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle\ ]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, sty\ le=filled]\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\"\ ;;\ngap> vertcolors[2] := \"pink\";;\ngap> vertcolors[3] := \"purple\";;\ngap>\ edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> ed\ gecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgecolors[2][1] := \ \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2] := \"red\";;\ \ngap> Print(DotSymmetricColoredDigraph(D, vertcolors, edgecolors));\n//dot\ng\ raph hgn{\nnode [shape=circle]\n\n1[color=blue, style=filled]\n2[color=pink, s\ tyle=filled]\n3[color=purple, style=filled]\n1 -- 2[color=green]\n2 -- 3[color\ =red]\n}\ngap> D := Digraph([[2, 3], [1, 3], [1]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";;\ vertcolors[2] := \"red\";;\ngap> vertcolors[3] := \"green\";;\ngap> edgecolor\ s := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edgecolors[3\ ] := [];;\ngap> edgecolors[1][2] := \"orange\";; edgecolors[1][3] := \"yellow\ \";;\ngap> edgecolors[2][1] := \"orange\";; edgecolors[2][3] := \"pink\";;\nga\ p> edgecolors[3][1] := \"yellow\";;\ngap> Print(DotColoredDigraph(D, vertcolor\ s, edgecolors));;\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[color=blue, sty\ le=filled]\n2[color=red, style=filled]\n3[color=green, style=filled]\n1 -> 2[c\ olor=orange]\n1 -> 3[color=yellow]\n2 -> 1[color=orange]\n2 -> 3[color=pink]\n\ 3 -> 1[color=yellow]\n}\ngap> D := Digraph(IsMutableDigraph, [[2, 3], [1, 3], \ [1]]);\n\ngap> vertcolors := [];;\ng\ ap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap> vertcolors[3]\ := \"green\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolor\ s[2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"orange\";\ ; edgecolors[1][3] := \"yellow\";;\ngap> edgecolors[2][1] := \"orange\";; edge\ colors[2][3] := \"pink\";;\ngap> edgecolors[3][1] := \"yellow\";;\ngap> Print(\ DotColoredDigraph(D, vertcolors, edgecolors));;\n//dot\ndigraph hgn{\nnode [sh\ ape=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[color=\ green, style=filled]\n1 -> 2[color=orange]\n1 -> 3[color=yellow]\n2 -> 1[color\ =orange]\n2 -> 3[color=pink]\n3 -> 1[color=yellow]\n}\ngap> D;\n\ngap> DotSymmetricDigraph(gr2){[12 .. 70]};\n\" h\ gn{\\nnode [shape=circle]\\n\\n1\\n2\\n3\\n4\\n1 -- 2\\n2 -- 3\\n3 -- 3\\n3 -\ \"\ngap> DotSymmetricDigraph(gr1);\nError, the argument must be a symmetri\ c digraph,\ngap> D := CompleteDigraph(4);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolor\ s[2] := \"red\";; \ngap> vertcolors[3] := \"green\";; vertcolors[4] := \"yello\ w\";;\ngap> Print(DotVertexColoredDigraph(D, vertcolors));\n//dot\ndigraph hgn\ {\nnode [shape=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled\ ]\n3[color=green, style=filled]\n4[color=yellow, style=filled]\n1 -> 2\n1 -> 3\ \n1 -> 4\n2 -> 1\n2 -> 3\n2 -> 4\n3 -> 1\n3 -> 2\n3 -> 4\n4 -> 1\n4 -> 2\n4 ->\ 3\n}\ngap> D := CompleteDigraph(4);\n\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];\ ;\ngap> edgecolors[3] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2] :=\ \"lightblue\";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4] :\ = \"purple\";;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2][3\ ] := \"pink\";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3][1\ ] := \"lightblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolors[3\ ][4] := \"purple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgecolo\ rs[4][2] := \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(DotE\ dgeColoredDigraph(D, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n\ 1\n2\n3\n4\n1 -> 2[color=lightblue]\n1 -> 3[color=pink]\n1 -> 4[color=purple]\ \n2 -> 1[color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[color=purple]\n3 -> 1[co\ lor=lightblue]\n3 -> 2[color=pink]\n3 -> 4[color=purple]\n4 -> 1[color=lightbl\ ue]\n4 -> 2[color=pink]\n4 -> 3[color=purple]\n}\ngap> FileString(\"dot/k4.dot\ \", DotDigraph(gr));\n154", count := [ 9, 1, 2, 19 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 19 ], mathmode := "Text", name := "Log", next := 414063, root := ~, start := 408115, stop := 414062 ), rec( content := "\n ", count := [ 9, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "Description", next := 414082, root := ~, start := 405320, stop := 414081 ) ], count := [ 9, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 414098, root := ~, start := 404981, stop := 414097 ), rec( content := "\n\n ", count := [ 9, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotSymmetricDig\ raph" ), content := 0, count := [ 9, 1, 3, 2 ], mathmode := "Text", name := "Attr", next := 414171, root := ~, start := 414123, stop := 414170 ), rec( attributes := rec( Arg := "digraph, vert, e\ dge", Name := "DotSymmetricColoredDigraph" ), content := 0, count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "Oper", next := 414243, root := ~, start := 414176, stop := 414242 ), rec( attributes := rec( Arg := "digraph, vert", Name := "DotSymmetricVer\ texColoredDigraph" ), content := 0, count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "Oper", next := 414315, root := ~, start := 414248, stop := 414314 ), rec( attributes := rec( Arg := "digraph, edge", Name := "DotSymmetricEdg\ eColoredDigraph" ), content := 0, count := [ 9, 1, 3, 8 ], mathmode := "Text", name := "Oper", next := 414385, root := ~, start := 414320, stop := 414384 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "Returns", next := 414418, root := ~, start := 414390, stop := 414417 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of the symmetric\n digra\ ph ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "A", next := 414538, root := ~, start := 414524, stop := 414537 ), rec( content := ". ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "C", next := 414566, root := ~, start := 414540, stop := 414565 ), rec( content := " will re\ turn an\n error if ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "A", next := 414611, root := ~, start := 414597, stop := 414610 ), rec( content := " is not \ a symmetric digraph. See\n ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "Ref", next := 414682, root := ~, start := 414650, stop := 414681 ), rec( content := ".", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "P", next := 414687, root := ~, start := 414683, stop := 414686 ), rec( content := "\n\n \ The function ", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricColoredDigraph", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "C", next := 414741, root := ~, start := 414708, stop := 414740 ), rec( content := " differs\ from ", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "C", next := 414772, root := ~, start := 414755, stop := 414771 ), rec( content := "\n \ only in that the values given in the two lists are used to color the vertices\ \n and edges of the graph when displayed. The list for vertex colours sho\ uld be\n a list of length equal to the number of vertices, containing str\ ings that\n are accepted by the graphviz software, which is the one used \ for graph\n representation. The list for edge colours should be a list o\ f lists\n with the same shape of the outneighbours of the digraph that co\ ntains strings \n that correspond to colours accepted by the graphviz sof\ tware.\n If the list is not the appropriate size, or has holes then the f\ unction\n will return an error. ", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "P", next := 415436, root := ~, start := 415432, stop := 415435 ), rec( content := "\n\n \ The function ", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricVertexColoredDigraph", count := [ 9, 1, 3, 11 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "C", next := 415496, root := ~, start := 415457, stop := 415495 ), rec( content := " differs\ from ", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "C", next := 415527, root := ~, start := 415510, stop := 415526 ), rec( content := "\n \ only in that the values in given in the list is used to color the vertices\n \ of the graph when displayed. The list for vertex colours should be\n \ a list of length equal to the number of vertices, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error. ", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "P", next := 415967, root := ~, start := 415963, stop := 415966 ), rec( content := "\n\n \ The function ", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricEdgeColoredDigraph", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "C", next := 416025, root := ~, start := 415988, stop := 416024 ), rec( content := " differs\ from ", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "C", next := 416056, root := ~, start := 416039, stop := 416055 ), rec( content := "\n \ only in that the values given in the list are used to color the edges\n o\ f the graph when displayed. The list for edge colours should be\n a list \ of lists with the same shape of the outneighbours, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error. ", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "P", next := 416497, root := ~, start := 416493, stop := 416496 ), rec( content := "\n\n \ Vertices are displayed as circles, numbered consistently with\n ", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "A", next := 416587, root := ~, start := 416573, stop := 416586 ), rec( content := ". Since \ ", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "A", next := 416609, root := ~, start := 416595, stop := 416608 ), rec( content := " is symm\ etric, for every non-loop\n edge there is a complementary edge with oppos\ ite source and range.\n ", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "C", next := 416748, root := ~, start := 416722, stop := 416747 ), rec( content := " display\ s each pair of complementary edges\n as a single line between the relevan\ t vertices, with no arrowhead.", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "P", next := 416867, root := ~, start := 416863, stop := 416866 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "C", next := 416902, root := ~, start := 416892, stop := 416901 ), rec( content := " format \ (also known as ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "C", next := 416940, root := ~, start := 416925, stop := 416939 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://www.graphviz.org", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "URL", next := 417100, root := ~, start := 417065, stop := 417099 ), rec( content := ". ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "P", next := 417106, root := ~, start := 417102, stop := 417105 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "C", next := 417163, root := ~, start := 417137, stop := 417162 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "Ref", next := 417255, root := ~, start := 417213, stop := 417254 ), rec( content := ".", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 16 ], mathmode := "Text", name := "P", next := 417260, root := ~, start := 417256, stop := 417259 ), rec( content := "\n\n \ ", count := [ 9, 1, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> star := Digraph([[2, 2, 3, 4], [1, 1], [1], [1, 4]]);\n\ngap> IsSymmetricDigraph(star);\ntrue\ngap> F\ ileString(\"dot/star.dot\", DotSymmetricDigraph(gr));\ngap> D := Digraph([[2],\ [1, 3], [2]]);\n\ngap> vertcolors\ := [];;\ngap> vertcolors[1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\n\ gap> vertcolors[3] := \"purple\";;\ngap> edgecolors := [];;\ngap> edgecolors[1\ ] := [];; edgecolors[2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1\ ][2] := \"green\";; edgecolors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \ \"red\";; edgecolors[3][2] := \"red\";;\ngap> Print(DotSymmetricColoredDigraph\ (D, vertcolors, edgecolors));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[col\ or=blue, style=filled]\n2[color=pink, style=filled]\n3[color=purple, style=fil\ led]\n1 -- 2[color=green]\n2 -- 3[color=red]\n}\ngap> D := Digraph([[2], [1, 3\ ], [2]]);\n\ngap> vertcolors := []\ ;;\ngap> vertcolors[1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> v\ ertcolors[3] := \"purple\";;\ngap> Print(DotSymmetricVertexColoredDigraph(D, v\ ertcolors));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[color=blue, style=fi\ lled]\n2[color=pink, style=filled]\n3[color=purple, style=filled]\n1 -- 2\n2 -\ - 3\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors\ [2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; \ edgecolors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors\ [3][2] := \"red\";;\ngap> Print(DotSymmetricEdgeColoredDigraph(D, edgecolors))\ ;\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1\n2\n3\n1 -- 2[color=green]\n2 -\ - 3[color=red]\n}\n83", count := [ 9, 1, 3, 17 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 17 ], mathmode := "Text", name := "Log", next := 419014, root := ~, start := 417268, stop := 419013 ), rec( content := "\n ", count := [ 9, 1, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "Description", next := 419033, root := ~, start := 414423, stop := 419032 ) ], count := [ 9, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 419049, root := ~, start := 414106, stop := 419048 ), rec( content := "\n\n ", count := [ 9, 1, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotPartialOrder\ Digraph" ), content := 0, count := [ 9, 1, 4, 2 ], mathmode := "Text", name := "Attr", next := 419125, root := ~, start := 419074, stop := 419124 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 419158, root := ~, start := 419130, stop := 419157 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of a partial order\n dig\ raph ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "A", next := 419280, root := ~, start := 419266, stop := 419279 ), rec( content := ". ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "C", next := 419311, root := ~, start := 419282, stop := 419310 ), rec( content := " will re\ turn an error\n if ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "A", next := 419356, root := ~, start := 419342, stop := 419355 ), rec( content := " is not \ a partial order digraph. See ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Ref", next := 419436, root := ~, start := 419393, stop := 419435 ), rec( content := ".", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "P", next := 419441, root := ~, start := 419437, stop := 419440 ), rec( content := "\n\n \ Since ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "A", next := 419469, root := ~, start := 419455, stop := 419468 ), rec( content := " is a pa\ rtial order, it is both reflexive and\n transitive. The output of ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "C", next := 419576, root := ~, start := 419547, stop := 419575 ), rec( content := " is the\\ n ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 419614, root := ~, start := 419590, stop := 419613 ), rec( content := " of the\\ n ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveReduction" ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 419677, root := ~, start := 419628, stop := 419676 ), rec( content := " of ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "A", next := 419695, root := ~, start := 419681, stop := 419694 ), rec( content := ".", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "P", next := 419700, root := ~, start := 419696, stop := 419699 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "C", next := 419735, root := ~, start := 419725, stop := 419734 ), rec( content := " format \ (also known as ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "C", next := 419773, root := ~, start := 419758, stop := 419772 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://www.graphviz.org", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "URL", next := 419933, root := ~, start := 419898, stop := 419932 ), rec( content := ". ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "P", next := 419939, root := ~, start := 419935, stop := 419938 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "C", next := 419999, root := ~, start := 419970, stop := 419998 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "Ref", next := 420091, root := ~, start := 420049, stop := 420090 ), rec( content := ".", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 7 ], mathmode := "Text", name := "P", next := 420096, root := ~, start := 420092, stop := 420095 ), rec( content := "\n\n \ ", count := [ 9, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> poset := Digraph([[1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigr\ aph(gr);\ntrue\ngap> FileString(\"dot/poset.dot\", DotPartialOrderDigraph(gr))\ ;\n83", count := [ 9, 1, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 8 ], mathmode := "Text", name := "Log", next := 420282, root := ~, start := 420104, stop := 420281 ), rec( content := "\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 420301, root := ~, start := 419163, stop := 420300 ) ], count := [ 9, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 420317, root := ~, start := 419057, stop := 420316 ), rec( content := "\n\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotPreorderDigr\ aph" ), content := 0, count := [ 9, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 420389, root := ~, start := 420342, stop := 420388 ), rec( attributes := rec( Arg := "digraph", Name := "DotQuasiorderDi\ graph" ), content := 0, count := [ 9, 1, 5, 4 ], mathmode := "Text", name := "Attr", next := 420443, root := ~, start := 420394, stop := 420442 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Returns", next := 420476, root := ~, start := 420448, stop := 420475 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of a preorder\n digraph \ ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "A", next := 420593, root := ~, start := 420579, stop := 420592 ), rec( content := ". ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "C", next := 420620, root := ~, start := 420595, stop := 420619 ), rec( content := " will re\ turn an error\n if ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "A", next := 420665, root := ~, start := 420651, stop := 420664 ), rec( content := " is not \ a preorder digraph. See ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPreor\ derDigraph" ), content := 0, count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Ref", next := 420736, root := ~, start := 420697, stop := 420735 ), rec( content := ".", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "P", next := 420741, root := ~, start := 420737, stop := 420740 ), rec( content := "\n\n \ A preorder digraph is reflexive and transitive but in general it is\n n\ ot anti-symmetric and may have strongly connected components\n containing\ more than one vertex. The ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Quotien\ tDigraph" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 420957, root := ~, start := 420928, stop := 420956 ), rec( content := "\n \ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 420972, root := ~, start := 420964, stop := 420971 ), rec( content := " obtaine\ d by forming the quotient of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 421023, root := ~, start := 421009, stop := 421022 ), rec( content := " by the\\ n partition of its vertices into the strongly connected components\n \ satisfies ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 421153, root := ~, start := 421118, stop := 421152 ), rec( content := ". Thus e\ very vertex of\n ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 421190, root := ~, start := 421182, stop := 421189 ), rec( content := " corresp\ onds to a strongly connected component of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 421254, root := ~, start := 421240, stop := 421253 ), rec( content := ".\n \ The output of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "C", next := 421301, root := ~, start := 421276, stop := 421300 ), rec( content := " display\ s the \n ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveReduction" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 421371, root := ~, start := 421322, stop := 421370 ), rec( content := " of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 421383, root := ~, start := 421375, stop := 421382 ), rec( content := " with\n \ vertices displayed as rounded rectangles labelled by all of the vertices\ \n of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 421491, root := ~, start := 421477, stop := 421490 ), rec( content := " in the \ corresponding strongly connected component. ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "P", next := 421547, root := ~, start := 421543, stop := 421546 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "C", next := 421582, root := ~, start := 421572, stop := 421581 ), rec( content := " format \ (also known as ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "C", next := 421620, root := ~, start := 421605, stop := 421619 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://www.graphviz.org", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "URL", next := 421780, root := ~, start := 421745, stop := 421779 ), rec( content := ". ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "P", next := 421786, root := ~, start := 421782, stop := 421785 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "C", next := 421842, root := ~, start := 421817, stop := 421841 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "Ref", next := 421934, root := ~, start := 421892, stop := 421933 ), rec( content := ".", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 9 ], mathmode := "Text", name := "P", next := 421939, root := ~, start := 421935, stop := 421938 ), rec( content := "\n\n \ ", count := [ 9, 1, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> preset := Digraph([[1, 2, 4, 5], [1, 2, 4, 5], [3, 4], [4], [1, 2, 4, 5]\ );\ngap> IsPreorderDigraph(gr);\ntrue\ngap> FileString(\"dot/preset.dot\", Dot\ ProrderDigraph(gr));\n83", count := [ 9, 1, 5, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 10 ], mathmode := "Text", name := "Log", next := 422144, root := ~, start := 421947, stop := 422143 ), rec( content := "\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Description", next := 422163, root := ~, start := 420481, stop := 422162 ) ], count := [ 9, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 422179, root := ~, start := 420325, stop := 422178 ), rec( content := "\n\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts [\ , colour1, colour2]", Name := "DotHighlightedDigraph" ), content := 0, count := [ 9, 1, 6, 2 ], mathmode := "Text", name := "Oper", next := 422282, root := ~, start := 422204, stop := 422281 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 422315, root := ~, start := 422287, stop := 422314 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "C", next := 422368, root := ~, start := 422340, stop := 422367 ), rec( content := " produce\ s a graphical represenation of the\n digraph ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422439, root := ~, start := 422425, stop := 422438 ), rec( content := ", where \ the vertices in the list ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422484, root := ~, start := 422472, stop := 422483 ), rec( content := ", and\n \ edges between them, are drawn with colour ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422552, root := ~, start := 422538, stop := 422551 ), rec( content := " and all\ other\n vertices and edges in ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422609, root := ~, start := 422595, stop := 422608 ), rec( content := " are dra\ wn with colour ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422646, root := ~, start := 422632, stop := 422645 ), rec( content := ".\n \ If ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422671, root := ~, start := 422657, stop := 422670 ), rec( content := " and ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 422690, root := ~, start := 422676, stop := 422689 ), rec( content := " are not\ given then\n ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "C", next := 422744, root := ~, start := 422716, stop := 422743 ), rec( content := " uses bl\ ack and grey respectively. ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "P", next := 422783, root := ~, start := 422779, stop := 422782 ), rec( content := "\n\n \ Note that ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "C", next := 422829, root := ~, start := 422801, stop := 422828 ), rec( content := " does no\ t validate the colours\n ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "A", next := 422880, root := ~, start := 422866, stop := 422879 ), rec( content := " and ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "A", next := 422899, root := ~, start := 422885, stop := 422898 ), rec( content := " - consu\ lt the GraphViz documentation to\n see what is available.\n\n See ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "Ref", next := 423004, root := ~, start := 422980, stop := 423003 ), rec( content := " for mor\ e details on the output.", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 6, 5 ], mathmode := "Text", name := "P", next := 423040, root := ~, start := 423036, stop := 423039 ), rec( content := "\n\n \ ", count := [ 9, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[2, 3], [2], [1, 3]]);\n\ngap> FileString(\"dot/my_digraph.dot\",\n> DotHighlightedDigraph(digra\ ph, [1, 2], \"red\", \"black\"));\n264", count := [ 9, 1, 6, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 6 ], mathmode := "Text", name := "Log", next := 423255, root := ~, start := 423048, stop := 423254 ), rec( content := "\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 423274, root := ~, start := 422320, stop := 423273 ) ], count := [ 9, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 423290, root := ~, start := 422187, stop := 423289 ), rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 423304, root := ~, start := 401339, stop := 423303 ) , rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Reading and writi\ ng digraphs to a file", count := [ 9, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 423374, root := ~, start := 423317, stop := 423373 ), rec( content := "\n This section descr\ ibes different ways to store and read graphs\n from a file in the ", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package.\n ", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := " Graph6 " , count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 4 ], name := "Mark", next := 423523, root := ~, start := 423502, stop := 423522 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "E", next := 423553, root := ~, start := 423538, stop := 423552 ), rec( content := " is a gr\ aph data format for storing undirected graphs\n\twith no multiple edges nor lo\ ops of size up to ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ 2^{36} - 1 ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "M", next := 423674, root := ~, start := 423655, stop := 423673 ), rec( content := " in\n\tp\ rintable chracters. The format consists of two parts. The first part\n\tuses o\ ne to eight bytes to store the number of vertices. And the second\n\tpart is t\ he upper half of the adjacency matrix converted into ASCII\n\tcharacters. For \ a more detail description see ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "Graph6"\ ), content := [ rec( content := "\n\thttp://cs.anu.edu.au/~bdm/data/formats.t\ xt", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "URL", next := 424007, root := ~, start := 423938, stop := 424006 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], name := "Item", next := 424022, root := ~, start := 423530, stop := 424021 ), rec( attributes := rec( ), content := [ rec( content := " Sparse6 \ ", count := [ 9, 2, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 6 ], name := "Mark", next := 424052, root := ~, start := 424030, stop := 424051 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 424083, root := ~, start := 424067, stop := 424082 ), rec( content := " is a gr\ aph data format for storing undirected graphs\n\twith possibly multiple edges \ or loops. The maximal number of vertices\n\tallowed is ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ 2^{36} - 1 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "M", next := 424239, root := ~, start := 424220, stop := 424238 ), rec( content := ". The fo\ rmat consists of two parts. The\n\tfirst part uses one to eight bytes to store\ the number of vertices.\n\tAnd the second part only stores information about \ the edges.\n\tTherefore, the ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 424442, root := ~, start := 424426, stop := 424441 ), rec( content := " format \ return a more compact encoding\n\tthen ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 424502, root := ~, start := 424487, stop := 424501 ), rec( content := " for spa\ rse graph, i.e. graphs where the number\n\tof edges is much less than the num\ ber of vertices squared. For a more\n\tdetail description see ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "Sparse6\ " ), content := [ rec( content := "\n http://cs.anu.edu.au/~bdm/data/fo\ rmats.txt", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "URL", next := 424724, root := ~, start := 424647, stop := 424723 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], name := "Item", next := 424739, root := ~, start := 424059, stop := 424738 ), rec( attributes := rec( ), content := [ rec( content := " Digraph6\ ", count := [ 9, 2, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 8 ], name := "Mark", next := 424770, root := ~, start := 424747, stop := 424769 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraph6", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "E", next := 424800, root := ~, start := 424785, stop := 424799 ), rec( content := " is a ne\ w format based on ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "E", next := 424841, root := ~, start := 424826, stop := 424840 ), rec( content := ", but de\ signed\n\tfor digraphs. The entire adjacency matrix is stored, and\n\ttherefor\ e there is support for directed edges and single-vertex loops.\n\tHowever, mul\ tiple edges are not supported.\n ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], name := "Item", next := 425043, root := ~, start := 424777, stop := 425042 ), rec( attributes := rec( ), content := [ rec( content := " DiSparse\ 6 ", count := [ 9, 2, 0, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 10 ], name := "Mark", next := 425075, root := ~, start := 425051, stop := 425074 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ iSparse6", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 425106, root := ~, start := 425090, stop := 425105 ), rec( content := " is a ne\ w format based on ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 425148, root := ~, start := 425132, stop := 425147 ), rec( content := ", but\n\\ tdesigned for digraphs. In this format the list of edges is\n\tpartitioned int\ o inceasing and decreasing edges, depending whether the\n\tedge has its source\ bigger than the range. Then both sets of edges are\n\twritten separetly in ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 425396, root := ~, start := 425380, stop := 425395 ), rec( content := " format \ with a separation symbol\n\tin between.\n ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], name := "Item", next := 425455, root := ~, start := 425082, stop := 425454 ) ], count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "List", next := 425467, root := ~, start := 423489, stop := 425466 ), rec( content := "\n ", count := [ 9, 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "String" ), content := 0, count := [ 9, 2, 1, 2 ], mathmode := "Text", name := "Attr", next := 425522, root := ~, start := 425487, stop := 425521 ), rec( attributes := rec( Arg := "digraph", Name := "PrintString" ), content := 0, count := [ 9, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 425565, root := ~, start := 425525, stop := 425564 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 425596, root := ~, start := 425568, stop := 425595 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a string ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "C", next := 425647, root := ~, start := 425634, stop := 425646 ), rec( content := " such th\ at ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ valString(string)", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "C", next := 425683, root := ~, start := 425658, stop := 425682 ), rec( content := "\n is\ equal to ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "A", next := 425714, root := ~, start := 425700, stop := 425713 ), rec( content := ", and ha\ s the same mutability.\n See ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "EvalString" ), content := 0, count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Ref", next := 425793, root := ~, start := 425753, stop := 425792 ), rec( content := ".\n " , count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "P", next := 425803, root := ~, start := 425799, stop := 425802 ), rec( content := "\n\n \ The methods installed for ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ tring", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "C", next := 425848, root := ~, start := 425835, stop := 425847 ), rec( content := " make so\ me attempts to\n ensure that ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "C", next := 425900, root := ~, start := 425887, stop := 425899 ), rec( content := " has as \ short a length as possible, but\n there may exist shorter strings that also\ evaluate to ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "A", next := 426012, root := ~, start := 425998, stop := 426011 ), rec( content := ".\n " , count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "P", next := 426022, root := ~, start := 426018, stop := 426021 ), rec( content := "\n\n \ It is possible that ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 426061, root := ~, start := 426048, stop := 426060 ), rec( content := " may con\ tain escaped special\n characters. To obtain a representation of ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "A", next := 426150, root := ~, start := 426136, stop := 426149 ), rec( content := " that\n \ can be entered as GAP input, please use\n ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "Print" ), content := 0, count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 426239, root := ~, start := 426204, stop := 426238 ), rec( content := ".\n N\ ote that ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ rint", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 426267, root := ~, start := 426255, stop := 426266 ), rec( content := " for a d\ igraph delegates to\n ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ rintString", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 426318, root := ~, start := 426300, stop := 426317 ), rec( content := ", which \ delegates to ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ tring", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 426352, root := ~, start := 426339, stop := 426351 ), rec( content := ".\n " , count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(3);\n\ngap> P\ rint(D);\nCycleDigraph(3);\ngap> G := PetersenGraph(IsMutableDigraph);\n\ngap> String(G);\n\"DigraphFromGraph6St\ ring(IsMutableDigraph, \\\"IheA@GUAo\\\");\"\ngap> Print(last);\nDigraphFromGr\ aph6String(IsMutableDigraph, \"IheA@GUAo\");\ngap> DigraphFromGraph6String(IsM\ utableDigraph, \"IheA@GUAo\");\n\n\ ", count := [ 9, 2, 1, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 8 ], mathmode := "Text", name := "Log", next := 426827, root := ~, start := 426358, stop := 426826 ), rec( content := "\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 426844, root := ~, start := 425599, stop := 426843 ) ], count := [ 9, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 426858, root := ~, start := 425472, stop := 426857 ), rec( content := "\n\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromGrap\ h6String" ), content := 0, count := [ 9, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 426938, root := ~, start := 426879, stop := 426937 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDigr\ aph6String" ), content := 0, count := [ 9, 2, 2, 4 ], mathmode := "Text", name := "Oper", next := 427000, root := ~, start := 426941, stop := 426999 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromSpar\ se6String" ), content := 0, count := [ 9, 2, 2, 6 ], mathmode := "Text", name := "Oper", next := 427062, root := ~, start := 427003, stop := 427061 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDiSp\ arse6String" ), content := 0, count := [ 9, 2, 2, 8 ], mathmode := "Text", name := "Oper", next := 427124, root := ~, start := 427065, stop := 427123 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Returns", next := 427156, root := ~, start := 427127, stop := 427155 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "A", next := 427190, root := ~, start := 427180, stop := 427189 ), rec( content := " is a st\ ring encoding a graph in Graph6, Digraph6, Sparse6 or\n DiSparse6 format, t\ hen the corresponding function returns a digraph. In the\n case of either G\ raph6 or Sparse6, formats which do not support directed\n edges, this will \ be a digraph such that for every edge, the edge going in\n the opposite dir\ ection is also present.", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "P", next := 427533, root := ~, start := 427529, stop := 427532 ), rec( content := "\n\n \ Each of these functions takes an optional first argument ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "A", next := 427607, root := ~, start := 427596, stop := 427606 ), rec( content := ",\n w\ hich should be either ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "Ref", next := 427666, root := ~, start := 427636, stop := 427665 ), rec( content := "\n or\ ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "Ref", next := 427706, root := ~, start := 427674, stop := 427705 ), rec( content := ",\n a\ nd which specifies whether the output digraph shall\n be mutable or immutab\ le.\n If no first argument is provided, then an immutable\n digraph is r\ eturned by default.\n ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphFromGraph6String(\"?\");\n\ngap> DigraphFromGraph6String(\"C]\");\n\ngap> DigraphFromGraph6String(\"H?AAEM{\");\n\ngap> DigraphFromDigraph6String(\"&\ ?\");\n\ngap> DigraphFromDigraph6Stri\ ng(IsMutableDigraph, \"&DOOOW?\");\n\ \ngap> DigraphFromDigraph6String(\"&CQFG\");\n\ngap> DigraphFromDigraph6String(\"&IM[SrKLc~lhesbU[F_\");\n\ngap> DigraphFromDiSparse6String(\".C\ aWBGA?b\");\n\n", count := [ 9, 2, 2, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 2, 11 ], mathmode := "Text", name := "Example", next := 428655, root := ~, start := 427890, stop := 428654 ), rec( content := "\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Description", next := 428674, root := ~, start := 427159, stop := 428673 ) ], count := [ 9, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 428688, root := ~, start := 426864, stop := 428687 ), rec( content := "\n\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph6String" ) , content := 0, count := [ 9, 2, 3, 2 ], mathmode := "Text", name := "Oper", next := 428750, root := ~, start := 428709, stop := 428749 ), rec( attributes := rec( Arg := "digraph", Name := "Digraph6String"\ ), content := 0, count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 428796, root := ~, start := 428753, stop := 428795 ), rec( attributes := rec( Arg := "digraph", Name := "Sparse6String" ), content := 0, count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 428841, root := ~, start := 428799, stop := 428840 ), rec( attributes := rec( Arg := "digraph", Name := "DiSparse6String\ " ), content := 0, count := [ 9, 2, 3, 8 ], mathmode := "Text", name := "Oper", next := 428888, root := ~, start := 428844, stop := 428887 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Returns", next := 428919, root := ~, start := 428891, stop := 428918 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se four functions return a highly compressed string fully describing the\n \ digraph ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "A", next := 429042, root := ~, start := 429028, stop := 429041 ), rec( content := ". ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "P", next := 429048, root := ~, start := 429044, stop := 429047 ), rec( content := "\n\n \ Graph6 and Digraph6 are formats best used on small, dense graphs, if\n appl\ icable. For larger, sparse graphs use ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ parse6", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "E", next := 429183, root := ~, start := 429169, stop := 429182 ), rec( content := " and\n \ ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ isparse6", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "E", next := 429208, root := ~, start := 429192, stop := 429207 ), rec( content := " (this l\ atter also preserves multiple edges). ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "P", next := 429258, root := ~, start := 429254, stop := 429257 ), rec( content := "\n\n \ See ", count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "Ref", next := 429295, root := ~, start := 429268, stop := 429294 ), rec( content := ".\n " , count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> Sparse6String(gr);\n\":Bc\"\ngap> DiSparse6String(gr);\n\".Bc{f\ \"\n", count := [ 9, 2, 3, 12 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 12 ], mathmode := "Text", name := "Example", next := 429482, root := ~, start := 429301, stop := 429481 ), rec( content := "\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Description", next := 429499, root := ~, start := 428922, stop := 429498 ) ], count := [ 9, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 429513, root := ~, start := 428694, stop := 429512 ), rec( content := "\n\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, code\ r][, mode]", Name := "DigraphFile" ), content := 0, count := [ 9, 2, 4, 2 ], mathmode := "Text", name := "Func", next := 429593, root := ~, start := 429534, stop := 429592 ), rec( attributes := rec( ), content := [ rec( content := "An IO fil\ e object.", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 429633, root := ~, start := 429596, stop := 429632 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "A", next := 429672, root := ~, start := 429657, stop := 429671 ), rec( content := " is a st\ ring representing the name of a file, then\n ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 429745, root := ~, start := 429727, stop := 429744 ), rec( content := " returns\ an ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n https://gap-package\ s.github.io/io/\n", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "URL" , next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 9, 2, 4, 3 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 4, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], name := "Package", next := 123, root := ~, start := 102 , stop := 122 ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " package\ file object for that file.\n ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "P", next := 429805, root := ~, start := 429801, stop := 429804 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 429848, root := ~, start := 429836, stop := 429847 ), rec( content := " is spec\ ified\n and is a function which either encodes a digraph as a string, or de\ codes a\n string into a digraph, then this function will be used when readi\ ng or\n writing to the returned file object. If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 430094, root := ~, start := 430082, stop := 430093 ), rec( content := "\n is\ not specified, then the encoding of the digraphs in the returned file\n ob\ ject must be specified in the the file extension. The file extension must\n \ be one of: ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430277, root := ~, start := 430267, stop := 430276 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ s6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430289, root := ~, start := 430279, stop := 430288 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ d6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430301, root := ~, start := 430291, stop := 430300 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ds6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430314, root := ~, start := 430303, stop := 430313 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ txt", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430327, root := ~, start := 430316, stop := 430326 ), rec( content := ",\n " , count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430342, root := ~, start := 430333, stop := 430341 ), rec( content := ", or ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ pickle", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 430361, root := ~, start := 430347, stop := 430360 ), rec( content := "; more d\ etails of these file formats is given\n below. ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "P", next := 430422, root := ~, start := 430418, stop := 430421 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 430464, root := ~, start := 430453, stop := 430463 ), rec( content := " is spec\ ified, then it must be one of:\n ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "w\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 430517, root := ~, start := 430507, stop := 430516 ), rec( content := " (for wr\ ite), ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "a\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 430541, root := ~, start := 430531, stop := 430540 ), rec( content := " (for ap\ pend), or ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 430569, root := ~, start := 430559, stop := 430568 ), rec( content := " (for re\ ad).\n If ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 430600, root := ~, start := 430589, stop := 430599 ), rec( content := " is not \ specified, then ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 430634, root := ~, start := 430624, stop := 430633 ), rec( content := " is used\ by default. ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "P", next := 430659, root := ~, start := 430655, stop := 430658 ), rec( content := "\n\n \ If ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "A", next := 430683, root := ~, start := 430668, stop := 430682 ), rec( content := " ends in\ one of: ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 430710, root := ~, start := 430700, stop := 430709 ), rec( content := ", ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 430723, root := ~, start := 430712, stop := 430722 ), rec( content := ", or\n \ ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 430742, root := ~, start := 430732, stop := 430741 ), rec( content := ", then t\ he digraphs which are read from, or written to, the\n returned file object \ are decompressed, or compressed, appropriately.\n ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "P", next := 430883, root := ~, start := 430879, stop := 430882 ), rec( content := "\n\n \ The file object returned by ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "C", next := 430935, root := ~, start := 430917, stop := 430934 ), rec( content := " can be \ given as the first\n argument for either of the functions ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 431029, root := ~, start := 431003, stop := 431028 ), rec( content := " or ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 431066, root := ~, start := 431033, stop := 431065 ), rec( content := ". The pu\ rpose of this is to reduce the overhead of\n recreating the file object ins\ ide the functions ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 431195, root := ~, start := 431169, stop := 431194 ), rec( content := "\n or\ ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 431230, root := ~, start := 431203, stop := 431229 ), rec( content := " when, f\ or example, reading or writing many\n digraphs in a loop.\n ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "P", next := 431306, root := ~, start := 431302, stop := 431305 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 4, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 9 ], name := "Mar\ k", next := 431437, root := ~, start := 431412, stop := 431436 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 4, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 10 ], name := "Ite\ m", next := 431773, root := ~, start := 431444, stop := 431772 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 4, 1\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 11 ], name := "Mar\ k", next := 431806, root := ~, start := 431780, stop := 431805 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 4, 12 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 12 ], name := "Item", next := 4320\ 41, root := ~, start := 431813, stop := 432040 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 4, 13 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 13 ], name := "Mark", next := 4320\ 75, root := ~, start := 432048, stop := 432074 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 14 ], name := "Ite\ m", next := 432305, root := ~, start := 432082, stop := 432304 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 4\ , 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 15 ], name := "Mar\ k", next := 432341, root := ~, start := 432312, stop := 432340 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 4, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 4, 16 ], name := "Item", next := 432553, root := ~, start := 432\ 348, stop := 432552 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 4, 17 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 4, 17 ], name := "Mark", next := 4325\ 90, root := ~, start := 432560, stop := 432589 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 4, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 4, 18 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 18 ], mathmode := "Text", name\ := "C", next := 432722, root := ~, start := 432677, stop := 432721 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 4, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 19 ], mathmode := "Text", name\ := "P", next := 432946, root := ~, start := 432942, stop := 432945 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 4, 19 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 4, 19 ], mathmode := "Text", name := "Ref\ ", next := 432994, root := ~, start := 432960, stop := 432993 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 4, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 20 ], mathmode := "Text", name\ := "P", next := 433071, root := ~, start := 433067, stop := 433070 ), rec( cont\ ent := "\n ", count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 4, 18 ], name := "Item", next := 433085, root := ~, start := 432\ 597, stop := 433084 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 4, 21 ], mathmode := "Text", name := "F", next := 433117, root := \ ~, start := 433108, stop := 433116 ), rec( content := " or ", count := [ 9, 2,\ 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 4, 21 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], math\ mode := "Text", name := "F", next := 433135, root := ~, start := 433121, stop \ := 433134 ), rec( content := ")", count := [ 9, 2, 4, 21 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], name := "Mark", next := 4331\ 43, root := ~, start := 433093, stop := 433142 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 4, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n https://gap-packages.github.io/io/\n", count := [ 9, 2, \ 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2,\ 4, 22 ], mathmode := "Text", name := "URL", next := 79, root := ~, start := 1\ 7, stop := 78 ) ], count := [ 9, 2, 4, 22 ], name := "Alt", next := 85, root :\ = ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ), conte\ nt := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :\ = [ 9, 2, 4, 22 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 22 ], \ name := "Package", next := 123, root := ~, start := 102, stop := 122 ) ], coun\ t := [ 9, 2, 4, 22 ], mathmode := "Text", name := "Alt", next := 129, root := \ ~, start := 85, stop := 128 ), rec( content := " package. This is particularly\ good\n when the ", count := [ 9, 2, 4, 22 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), cont\ ent := 0, count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "Ref", next :=\ 433279, root := ~, start := 433253, stop := 433278 ), rec( content := " is no\ n-trivial.\n ", count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 9, 2, 4, 22 ], name := "Item", next := 433309 , root := ~, start := 433\ 150, stop := 433308 ) ], count := [ 9, 2, 4, 9 ], mathmode := "Text", name := "List", next := 433321, root := ~, start := 431399, stop := 433320 ), rec( content := "\n\n \ ", count := [ 9, 2, 4, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap\ > file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> WriteD\ igraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\n\ gap> file := DigraphFile(filename, \"r\");;\ngap> ReadDigraphs(file, 9);\n\ngap> IO_Close(file);;\n", count := [ 9, 2, 4, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 4, 25 ], mathmode := "Text", name := "Example", next := 433719, root := ~, start := 433327, stop := 433718 ), rec( content := "\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 433738, root := ~, start := 429636, stop := 433737 ) ], count := [ 9, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 433752, root := ~, start := 429519, stop := 433751 ), rec( content := "\n\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename[, decod\ er][, n]", Name := "ReadDigraphs" ), content := 0, count := [ 9, 2, 5, 2 ], mathmode := "Text", name := "Func", next := 433831, root := ~, start := 433773, stop := 433830 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or a list of digraphs.", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 433886, root := ~, start := 433834, stop := 433885 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 433925, root := ~, start := 433910, stop := 433924 ), rec( content := " is a st\ ring containing the name of a file containing\n encoded digraphs or an ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n https://gap-package\ s.github.io/io/\n", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "URL" , next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 9, 2, 5, 3 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 5, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], name := "Package", next := 123, root := ~, start := 102 , stop := 122 ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Ref", next := 434068, root := ~, start := 434037, stop := 434067 ), rec( content := ", then " , count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 434094, root := ~, start := 434075, stop := 434093 ), rec( content := " returns\ the digraphs\n encoded in the file as a list. Note that if ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 434180, root := ~, start := 434165, stop := 434179 ), rec( content := " is a\n \ compressed file, which has been compressed appropriately to give a filename\ \n extension of ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 434293, root := ~, start := 434283, stop := 434292 ), rec( content := ", ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 434306, root := ~, start := 434295, stop := 434305 ), rec( content := ", or ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 434321, root := ~, start := 434311, stop := 434320 ), rec( content := ", then\n\ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 434351, root := ~, start := 434332, stop := 434350 ), rec( content := " can rea\ d ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 434376, root := ~, start := 434361, stop := 434375 ), rec( content := " without\ it first needing to be\n decompressed.\n ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "P", next := 434434, root := ~, start := 434430, stop := 434433 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 434479, root := ~, start := 434465, stop := 434478 ), rec( content := " is spec\ ified\n and is a function which decodes a string into a digraph,\n then \ ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "C", next := 434582, root := ~, start := 434563, stop := 434581 ), rec( content := " will us\ e ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 434606, root := ~, start := 434592, stop := 434605 ), rec( content := " to deco\ de the digraphs\n contained in ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 434662, root := ~, start := 434647, stop := 434661 ), rec( content := ".", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "P", next := 434667, root := ~, start := 434663, stop := 434666 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 434706, root := ~, start := 434698, stop := 434705 ), rec( content := " is spec\ ified, then ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "C", next := 434745, root := ~, start := 434726, stop := 434744 ), rec( content := "\n re\ turns the ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 434770, root := ~, start := 434762, stop := 434769 ), rec( content := "th digra\ ph encoded in the file ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 434816, root := ~, start := 434801, stop := 434815 ), rec( content := ".\n " , count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "P", next := 434826, root := ~, start := 434822, stop := 434825 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "A", next := 434871, root := ~, start := 434857, stop := 434870 ), rec( content := " is not \ specified, then\n ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 434918, root := ~, start := 434899, stop := 434917 ), rec( content := " will de\ duce which decoder to use based on the filename\n extension of ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "A", next := 435006, root := ~, start := 434991, stop := 435005 ), rec( content := " (after \ removing the compression-related\n filename extensions ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 435081, root := ~, start := 435071, stop := 435080 ), rec( content := ", ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 435094, root := ~, start := 435083, stop := 435093 ), rec( content := ", and ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 435110, root := ~, start := 435100, stop := 435109 ), rec( content := "). For \ example,\n if the filename extension is ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 435170, root := ~, start := 435160, stop := 435169 ), rec( content := ", then " , count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 435196, root := ~, start := 435177, stop := 435195 ), rec( content := " will us\ e\n the graph6 decoder ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ FromGraph6String" ), content := 0, count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "Ref", next := 435266, root := ~, start := 435229, stop := 435265 ), rec( content := ".\n " , count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 7 ], mathmode := "Text", name := "P", next := 435276, root := ~, start := 435272, stop := 435275 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 5, 8 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 8 ], name := "Mar\ k", next := 435407, root := ~, start := 435382, stop := 435406 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 9 ], name := "Ite\ m", next := 435743, root := ~, start := 435414, stop := 435742 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 5, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 10 ], name := "Mar\ k", next := 435776, root := ~, start := 435750, stop := 435775 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 5, 11 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 11 ], name := "Item", next := 4360\ 11, root := ~, start := 435783, stop := 436010 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 5, 12 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 12 ], name := "Mark", next := 4360\ 45, root := ~, start := 436018, stop := 436044 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 5, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 13 ], name := "Ite\ m", next := 436275, root := ~, start := 436052, stop := 436274 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 5\ , 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 14 ], name := "Mar\ k", next := 436311, root := ~, start := 436282, stop := 436310 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 5, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 5, 15 ], name := "Item", next := 436523, root := ~, start := 436\ 318, stop := 436522 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 5, 16 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 5, 16 ], name := "Mark", next := 4365\ 60, root := ~, start := 436530, stop := 436559 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 5, 17 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 17 ], mathmode := "Text", name\ := "C", next := 436692, root := ~, start := 436647, stop := 436691 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 18 ], mathmode := "Text", name\ := "P", next := 436916, root := ~, start := 436912, stop := 436915 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 5, 18 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "Ref\ ", next := 436964, root := ~, start := 436930, stop := 436963 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 19 ], mathmode := "Text", name\ := "P", next := 437041, root := ~, start := 437037, stop := 437040 ), rec( cont\ ent := "\n", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( content := "\n Just so I remember:\n ReadPlainT\ extDigraph(\"data-local/soc-Epinions1.txt.gz\", \"\\t\", 1,\n '#');\ \n", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "XMLCOMMENT", root \ := ~ ), rec( content := "\n ", count := [ 9, 2, 5, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 17 ], name := "Item", next := 4371\ 86, root := ~, start := 436567, stop := 437185 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "pickled (", count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "F", next := 437218 , root\ := ~, start := 437209, stop := 437217 ), rec( content := " or ", count := [ 9\ , 2, 5, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 5, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20\ ], mathmode := "Text", name := "F", next := 437236, root := ~, start := 43722\ 2, stop := 437235 ), rec( content := ")", count := [ 9, 2, 5, 20 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20 ], name := "Mar\ k", next := 437244, root := ~, start := 437194, stop := 437243 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n Digraphs are pickled using \ the ", count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n https://gap-package\ s.github.io/io/\n", count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "UR\ L", next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 9, 2, 5, 21\ ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attri\ butes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), conte\ nt := [ rec( content := "IO", count := [ 9, 2, 5, 21 ], name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 5, 21 ], name := "Package", next := 123, root := ~\ , start := 102, stop := 122 ) ], count := [ 9, 2, 5, 21 ], mathmode := "Text", name\ := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content :=\ " package. This is particularly good\n when the ", count := [ 9, 2, 5,\ 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( Attr := "DigraphGroup" ), content := 0, count := [ 9, 2, 5, 21 ], mathmode \ := "Text", name := "Ref", next := 437380, root := ~, start := 437354, stop := \ 437379 ), rec( content := " is non-trivial.\n ", count := [ 9, 2, 5, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 21 ], name := "Ite\ m", next := 437410, root := ~, start := 437251, stop := 437409 ) ], count := [ 9, 2, 5, 8 ], mathmode := "Text", name := "List", next := 437422, root := ~, start := 435369, stop := 437421 ), rec( content := "\n\n \ ", count := [ 9, 2, 5, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), \ 10);\n\ngap> ReadDigraph\ s(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenat\ ion(DIGRAPHS_Dir(), \"/data/tree9.4.txt\"));\n[ , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n\ , \n , \n , \n ]", count := [ 9, 2, 5, 24 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 5, 24 ], mathmode := "Text", name := "Example", next := 438488, root := ~, start := 437428, stop := 438487 ), rec( content := "\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 438507, root := ~, start := 433889, stop := 438506 ) ], count := [ 9, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 438521, root := ~, start := 433758, stop := 438520 ), rec( content := "\n\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ hs[, encoder] [, mode]", Name := "WriteDigraphs" ), content := 0, count := [ 9, 2, 6, 2 ], mathmode := "Text", name := "Func", next := 438615, root := ~, start := 438542, stop := 438614 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 438654, root := ~, start := 438639, stop := 438653 ), rec( content := " is a li\ st of digraphs or a digraph and ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 438709, root := ~, start := 438694, stop := 438708 ), rec( content := "\n is\ a string or an ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n https://gap-package\ s.github.io/io/\n", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "URL" , next := 79, root := ~, start := 17, stop := 78 ) ], count := [ 9, 2, 6, 3 ], name := "Alt", next := 85, root := ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 6, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], name := "Package", next := 123, root := ~, start := 102 , stop := 122 ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Alt", next := 129, root := ~, start := 85, stop := 128 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Ref", next := 438788, root := ~, start := 438763, stop := 438787 ), rec( content := ",\n t\ hen ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDigraphs", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 438819, root := ~, start := 438799, stop := 438818 ), rec( content := " writes \ the digraphs to the file represented by\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 438886, root := ~, start := 438871, stop := 438885 ), rec( content := ". If th\ e supplied filename ends in one of the extensions\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 438958, root := ~, start := 438948, stop := 438957 ), rec( content := ", ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 438971, root := ~, start := 438960, stop := 438970 ), rec( content := ", or ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 438986, root := ~, start := 438976, stop := 438985 ), rec( content := ", then t\ he file will be compressed\n appropriately. Excluding these extensions, if\ the file ends with an\n extension in the list below, the corresponding gra\ ph format will be used to\n encode it. If such an extension is not include\ d, an appropriate format\n will be chosen intelligently, and an extension a\ ppended, to minimise file\n size.\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "P", next := 439346, root := ~, start := 439342, stop := 439345 ), rec( content := "\n\n \ ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO: e\ ncode and mode. ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ For more verbose information on the progress of the function, set the info\n \ level of ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nfoDigraphs", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "A", next := 439496, root := ~, start := 439477, stop := 439495 ), rec( content := " to 1 or\ higher, using ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etInfoLevel", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "C", next := 439538, root := ~, start := 439519, stop := 439537 ), rec( content := ".", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 5 ], mathmode := "Text", name := "P", next := 439543, root := ~, start := 439539, stop := 439542 ), rec( content := "\n\n \ The currently supported file formats are:\n ", count := [ 9, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 6, 6 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 6 ], name := "Mar\ k", next := 439633, root := ~, start := 439608, stop := 439632 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 7 ], name := "Ite\ m", next := 439969, root := ~, start := 439640, stop := 439968 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 6, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 8 ], name := "Mar\ k", next := 440002, root := ~, start := 439976, stop := 440001 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 6, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 9 ], name := "Item", next := 4402\ 37, root := ~, start := 440009, stop := 440236 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 6, 10 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 10 ], name := "Mark", next := 4402\ 71, root := ~, start := 440244, stop := 440270 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 11 ], name := "Ite\ m", next := 440501, root := ~, start := 440278, stop := 440500 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 6\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 12 ], name := "Mar\ k", next := 440537, root := ~, start := 440508, stop := 440536 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 6, 13 ], name := "Item", next := 440749, root := ~, start := 440\ 544, stop := 440748 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 6, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 6, 14 ], name := "Mark", next := 4407\ 87, root := ~, start := 440757, stop := 440786 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 6, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 15 ], mathmode := "Text", name\ := "C", next := 440919, root := ~, start := 440874, stop := 440918 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 6, 16 ], mathmode := "Text", name\ := "P", next := 441143, root := ~, start := 441139, stop := 441142 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 6, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 6, 16 ], mathmode := "Text", name := "Ref\ ", next := 441191, root := ~, start := 441157, stop := 441190 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 6, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 6, 17 ], mathmode := "Text", name\ := "P", next := 441268, root := ~, start := 441264, stop := 441267 ), rec( cont\ ent := "\n", count := [ 9, 2, 6, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( content := "\n Just so I remember:\n ReadPlainT\ extDigraph(\"data-local/soc-Epinions1.txt.gz\", \"\\t\", 1, '#');\n", coun\ t := [ 9, 2, 6, 17 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( cont\ ent := "\n ", count := [ 9, 2, 6, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 6, 15 ], name := "Item", next := 441405, root := ~, start := 440\ 794, stop := 441404 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 6, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 6, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 6, 18 ], mathmode := "Text", name := "F", next := 441436, root := \ ~, start := 441427, stop := 441435 ), rec( content := " or ", count := [ 9, 2,\ 6, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 6, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], math\ mode := "Text", name := "F", next := 441454, root := ~, start := 441440, stop \ := 441453 ), rec( content := ")", count := [ 9, 2, 6, 18 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], name := "Mark", next := 4414\ 62, root := ~, start := 441412, stop := 441461 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 6, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n https://gap-packages.github.io/io/\n", count := [ 9, 2, \ 6, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2,\ 6, 19 ], mathmode := "Text", name := "URL", next := 79, root := ~, start := 1\ 7, stop := 78 ) ], count := [ 9, 2, 6, 19 ], name := "Alt", next := 85, root :\ = ~, start := 1, stop := 84 ), rec( attributes := rec( Only := "Text" ), conte\ nt := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :\ = [ 9, 2, 6, 19 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 19 ], \ name := "Package", next := 123, root := ~, start := 102, stop := 122 ) ], coun\ t := [ 9, 2, 6, 19 ], mathmode := "Text", name := "Alt", next := 129, root := \ ~, start := 85, stop := 128 ), rec( content := " package. This is particularly\ good\n when the ", count := [ 9, 2, 6, 19 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), cont\ ent := 0, count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "Ref", next :=\ 441598, root := ~, start := 441572, stop := 441597 ), rec( content := " is no\ n-trivial.\n ", count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 9, 2, 6, 19 ], name := "Item", next := 441628 , root := ~, start := 441\ 469, stop := 441627 ) ], count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "List", next := 441640, root := ~, start := 439595, stop := 441639 ), rec( content := "\n\n \ ", count := [ 9, 2, 6, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> grs := [];;\ngap> grs[1] := Digraph([]);\n\ngap> grs[2] := Digraph([[1, 3], [2], [1, 2]]);\n\ngap> grs[3] := Digraph([\n> [6, 7], [6, 9], [1, 3\ , 4, 5, 8, 9],\n> [1, 2, 3, 4, 5, 6, 7, 10], [1, 5, 6, 7, 10], [2, 4, 5, 9, 10\ ],\n> [3, 4, 5, 6, 7, 8, 9, 10], [1, 3, 5, 7, 8, 9], [1, 2, 5],\n> [1, 2, 4, 6\ , 7, 8]]);\n\ngap> filename := C\ oncatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap> WriteDigraphs(fil\ ename, grs, \"w\");\nIO_OK\ngap> ReadDigraphs(filename);\n[ , \n , \n \ ]", count := [ 9, 2, 6, 22 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 6, 22 ], mathmode := "Text", name := "Example", next := 442407, root := ~, start := 441646, stop := 442406 ), rec( content := "\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 442426, root := ~, start := 438618, stop := 442425 ) ], count := [ 9, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 442440, root := ~, start := 438527, stop := 442439 ), rec( content := "\n\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, deco\ der]", Name := "IteratorFromDigraphFile" ), content := 0, count := [ 9, 2, 7, 2 ], mathmode := "Text", name := "Func", next := 442526, root := ~, start := 442461, stop := 442525 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Returns", next := 442560, root := ~, start := 442529, stop := 442559 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "A", next := 442599, root := ~, start := 442584, stop := 442598 ), rec( content := " is a st\ ring representing the name of a file containing\n encoded digraphs, then ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "C", next := 442712, root := ~, start := 442682, stop := 442711 ), rec( content := " returns\ an iterator\n for which the value of ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Oper := "NextIterator" ), content := 0, count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Ref", next := 442801, root := ~, start := 442760, stop := 442800 ), rec( content := " is the\\ n next digraph encoded in the file.\n ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "P", next := 442855, root := ~, start := 442851, stop := 442854 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 442900, root := ~, start := 442886, stop := 442899 ), rec( content := " is spec\ ified and is a function\n which decodes a string into a digraph, then ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "C", next := 443010, root := ~, start := 442980, stop := 443009 ), rec( content := "\n wi\ ll use ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 443038, root := ~, start := 443024, stop := 443037 ), rec( content := " to deco\ de the digraphs contained in ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 443090, root := ~, start := 443075, stop := 443089 ), rec( content := ".\n " , count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "P", next := 443100, root := ~, start := 443096, stop := 443099 ), rec( content := "\n\n \ The purpose of this function is to easily allow looping over digraphs\n enc\ oded in a file when loading all of the encoded digraphs would require\n too\ much memory. ", count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "P", next := 443278, root := ~, start := 443274, stop := 443277 ), rec( content := "\n\n \ To see what file types are available, see ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "Ref", next := 443353, root := ~, start := 443326, stop := 443352 ), rec( content := ".\n\n \ ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap\ > file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> Writ\ eDigraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\ \ngap> iter := IteratorFromDigraphFile(filename);\n\ngap> for x in i\ ter do od;\n", count := [ 9, 2, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "Example", next := 443702, root := ~, start := 443360, stop := 443701 ), rec( content := "\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Description", next := 443719, root := ~, start := 442563, stop := 443718 ) ], count := [ 9, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 443733, root := ~, start := 442446, stop := 443732 ), rec( content := "\n\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineEncoder" ), content := 0, count := [ 9, 2, 8, 2 ], mathmode := "Text", name := "Func", next := 443835, root := ~, start := 443754, stop := 443834 ), rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineDecoder" ), content := 0, count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 443919, root := ~, start := 443838, stop := 443918 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Returns", next := 443950, root := ~, start := 443922, stop := 443949 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se two functions return a function which encodes or decodes a digraph in\n \ a plain text format.", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "P", next := 444075, root := ~, start := 444071, stop := 444074 ), rec( content := "\n\n \ ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineEncoder", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 444115, root := ~, start := 444081, stop := 444114 ), rec( content := " returns\ a function which takes a single\n digraph as an argument. The function re\ turns a string describing the edges\n of that digraph; each edge is written\ as a pair of integers separated by the\n string ", count := [ 9, 2, 8, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter2", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 444345, root := ~, start := 444328, stop := 444344 ), rec( content := ", and th\ e edges themselves are separated by the\n string ", count := [ 9, 2, 8, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter1", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 444421, root := ~, start := 444404, stop := 444420 ), rec( content := ". ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 444458, root := ~, start := 444424, stop := 444457 ), rec( content := " returns\ the\n corresponding decoder function, which takes a string argument in thi\ s format\n and returns a digraph.", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "P", next := 444582, root := ~, start := 444578, stop := 444581 ), rec( content := "\n\n \ If only one delimiter is passed as an argument to\n ", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "A", next := 444676, root := ~, start := 444642, stop := 444675 ), rec( content := ", it wil\ l return a function which decodes\n a single edge, returning its contents a\ s a list of integers.", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "P", next := 444786, root := ~, start := 444782, stop := 444785 ), rec( content := "\n\n \ The argument ", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "A", next := 444818, root := ~, start := 444805, stop := 444817 ), rec( content := " should \ be an integer, which will describe a\n number to be added to each vertex be\ fore it is encoded, or after it is\n decoded. This may be used, for exampl\ e, to label vertices starting at 0\n instead of 1.", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 9 ], mathmode := "Text", name := "P", next := 445036, root := ~, start := 445032, stop := 445035 ), rec( content := "\n\n \ Note that the number of vertices of a digraph is not stored, and so vertices\n\ which are not connected to any edge may be lost.\n\n ", count := [ 9, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEncoder(\" \", \" \", -1);;\ngap> d\ ec := DigraphPlainTextLineDecoder(\" \", \" \", 1);;\ngap> enc(gr);\n\"0 1 0\ 2 1 0 2 0\"\ngap> dec(last);\n\ \n", count := [ 9, 2, 8, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 10 ], mathmode := "Text", name := "Example", next := 445503, root := ~, start := 445177, stop := 445502 ), rec( content := "\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 445520, root := ~, start := 443953, stop := 445519 ) ], count := [ 9, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 445534, root := ~, start := 443739, stop := 445533 ), rec( content := "\n\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TournamentLineD\ ecoder" ), content := 0, count := [ 9, 2, 9, 2 ], mathmode := "Text", name := "Oper", next := 445601, root := ~, start := 445555, stop := 445600 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Returns", next := 445633, root := ~, start := 445604, stop := 445632 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "A", next := 445693, root := ~, start := 445683, stop := 445692 ), rec( content := ", decode\ s it, and then returns the\n tournament [see ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Ref", next := 445774, root := ~, start := 445748, stop := 445773 ), rec( content := "] which \ it defines, according to\n the following rules. ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "P", next := 445836, root := ~, start := 445832, stop := 445835 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "A", next := 445881, root := ~, start := 445871, stop := 445880 ), rec( content := " represe\ nt the entries in the upper\n triangle of a tournament's adjacency matrix. \ The number of vertices\n ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "C", next := 446001, root := ~, start := 445993, stop := 446000 ), rec( content := " will be\ detected from the length of the string and will be as\n large as possible.\ ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "P", next := 446092, root := ~, start := 446088, stop := 446091 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 446160, root := ~, start := 446147, stop := 446159 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 446201, root := ~, start := 446188, stop := 446200 ), rec( content := " and so \ on until ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 446231, root := ~, start := 446218, stop := 446230 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 446290, root := ~, start := 446277, stop := 446289 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 446367, root := ~, start := 446352, stop := 446366 ), rec( content := ". ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "P", next := 446373, root := ~, start := 446369, stop := 446372 ), rec( content := "\n\n \ If a character of the string with corresponding edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 446445, root := ~, start := 446432, stop := 446444 ), rec( content := " is equa\ l\n to ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 446470, root := ~, start := 446462, stop := 446469 ), rec( content := ", then t\ he edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 446499, root := ~, start := 446486, stop := 446498 ), rec( content := " is pres\ ent in the tournament.\n Otherwise, the edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 446567, root := ~, start := 446554, stop := 446566 ), rec( content := " is pres\ ent instead. In this way, all the\n possible edges are encoded one-by-one.\ \n\n ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := TournamentLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1, 2 ] \ ]\n", count := [ 9, 2, 9, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 7 ], mathmode := "Text", name := "Example", next := 446842, root := ~, start := 446658, stop := 446841 ), rec( content := "\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Description", next := 446859, root := ~, start := 445636, stop := 446858 ) ], count := [ 9, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 446873, root := ~, start := 445540, stop := 446872 ), rec( content := "\n\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "AdjacencyMatrix\ UpperTriangleLineDecoder" ), content := 0, count := [ 9, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 446958, root := ~, start := 446894, stop := 446957 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 446990, root := ~, start := 446961, stop := 446989 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "A", next := 447050, root := ~, start := 447040, stop := 447049 ), rec( content := ", decode\ s it, and then returns the\n topologically sorted digraph [see ", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Ref", next := 447159, root := ~, start := 447123, stop := 447158 ), rec( content := "]\n w\ hich it defines, according to the following rules. ", count := [ 9, 2, 10, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "P", next := 447221, root := ~, start := 447217, stop := 447220 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "A", next := 447266, root := ~, start := 447256, stop := 447265 ), rec( content := " represe\ nt the entries in the upper\n triangle of a digraph's adjacency matrix. The\ number of vertices ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 447379, root := ~, start := 447371, stop := 447378 ), rec( content := "\n wi\ ll be detected from the length of the string and will be as large as\n poss\ ible. ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "P", next := 447474, root := ~, start := 447470, stop := 447473 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447542, root := ~, start := 447529, stop := 447541 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447583, root := ~, start := 447570, stop := 447582 ), rec( content := " and so \ on until ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447613, root := ~, start := 447600, stop := 447612 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447672, root := ~, start := 447659, stop := 447671 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447749, root := ~, start := 447734, stop := 447748 ), rec( content := ". If a \ character of the string with\n corresponding edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447822, root := ~, start := 447809, stop := 447821 ), rec( content := " is equa\ l to ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 447843, root := ~, start := 447835, stop := 447842 ), rec( content := ", then t\ his edge is\n present in the digraph. Otherwise, it is not present. In thi\ s way, all the\n possible edges are encoded one-by-one. ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "P", next := 447990, root := ~, start := 447986, stop := 447989 ), rec( content := "\n\n \ In particular, note that there exists no edge ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "C", next := 448055, root := ~, start := 448042, stop := 448054 ), rec( content := " if ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ \\leq\n i", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "M", next := 448080, root := ~, start := 448059, stop := 448079 ), rec( content := ". In or\ der words, the digraph will be topologically sorted.\n\n ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := AdjacencyMatrixUpperTriangleLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 4 \ ], [ ] ]\ngap> gr := AdjacencyMatrixUpperTriangleLineDecoder(\"111111x111\");\ \n\ngap> OutNeighbours(gr);\n[ [ 2\ , 3, 4, 5 ], [ 3, 4 ], [ 4, 5 ], [ 5 ], [ ] ]\n", count := [ 9, 2, 10, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 10, 7 ], mathmode := "Text", name := "Example", next := 448525, root := ~, start := 448146, stop := 448524 ), rec( content := "\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 448542, root := ~, start := 446993, stop := 448541 ) ], count := [ 9, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 448556, root := ~, start := 446879, stop := 448555 ), rec( content := "\n\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TCodeDecoder" ) , content := 0, count := [ 9, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 448614, root := ~, start := 448577, stop := 448613 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Returns", next := 448646, root := ~, start := 448617, stop := 448645 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "A", next := 448680, root := ~, start := 448670, stop := 448679 ), rec( content := " is a st\ ring consisting of at least two non-negative integers\n separated by spaces\ , then this function will attempt to return the digraph\n which it defines \ as a TCode string. ", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "P", next := 448865, root := ~, start := 448861, stop := 448864 ), rec( content := "\n\n \ The first integer of the string defines the number of vertices ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 448942, root := ~, start := 448934, stop := 448941 ), rec( content := " in\n \ the digraph, and the second defines the number of edges ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 449014, root := ~, start := 449006, stop := 449013 ), rec( content := ". The\n\ following ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 449044, root := ~, start := 449035, stop := 449043 ), rec( content := " integer\ s should be vertex numbers in the range ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 0 ..\n v-1]", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 449115, root := ~, start := 449092, stop := 449114 ), rec( content := ". These \ integers are read in pairs and define the digraph's\n edges. This function\ will return an error if ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "A", next := 449235, root := ~, start := 449225, stop := 449234 ), rec( content := " has few\ er than\n ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e+2", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 449266, root := ~, start := 449255, stop := 449265 ), rec( content := " entries\ . ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "P", next := 449280, root := ~, start := 449276, stop := 449279 ), rec( content := "\n\n \ Note that the vertex numbers will be incremented by 1 in the digraph\n retu\ rned. Hence the string fragment ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ 6", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "C", next := 449406, root := ~, start := 449396, stop := 449405 ), rec( content := " will de\ scribe the edge\n ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1,7]", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "C", next := 449446, root := ~, start := 449434, stop := 449445 ), rec( content := ".\n " , count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := TCodeDecoder(\"3 2 0 2 2 1\");\n\ngap> OutNeighbours(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeD\ ecoder(\"12 3 0 10 5 2 8 8\");\n\ \ngap> OutNeighbours(gr);\n[ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ]\ , [ 9 ], [ ], \n [ ], [ ] ]\n", count := [ 9, 2, 11, 6 ], name := "PCDATA" , root := ~ ) ], count := [ 9, 2, 11, 6 ], mathmode := "Text", name := "Example", next := 449814, root := ~, start := 449452, stop := 449813 ), rec( content := "\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 449831, root := ~, start := 448649, stop := 449830 ) ], count := [ 9, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 449845, root := ~, start := 448562, stop := 449844 ), rec( content := "\n\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlainTextString\ " ), content := 0, count := [ 9, 2, 12, 2 ], mathmode := "Text", name := "Oper", next := 449910, root := ~, start := 449866, stop := 449909 ), rec( attributes := rec( Arg := "s", Name := "DigraphFromPlai\ nTextString" ), content := 0, count := [ 9, 2, 12, 4 ], mathmode := "Text", name := "Oper", next := 449962, root := ~, start := 449913, stop := 449961 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Returns", next := 449993, root := ~, start := 449965, stop := 449992 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lainTextString", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 450036, root := ~, start := 450014, stop := 450035 ), rec( content := " takes a\ single digraph, and returns a string\n describing the edges of that digrap\ h. ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 450158, root := ~, start := 450125, stop := 450157 ), rec( content := "\n ta\ kes such a string and returns the digraph which it describes. Each edge\n \ is written as a pair of integers separated by a single space. The edges\n \ themselves are separated by a double space. Vertex numbers are reduced by\n \ 1 when they are encoded, so that vertices in the string are labelled\n st\ arting at 0.", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "P", next := 450489, root := ~, start := 450485, stop := 450488 ), rec( content := "\n\n \ Note that the number of vertices of a digraph is not stored, and so vertices\n\ which are not connected to any edge may be lost.", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "P", next := 450628, root := ~, start := 450624, stop := 450627 ), rec( content := "\n\n \ \n The operation ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "C", next := 450686, root := ~, start := 450653, stop := 450685 ), rec( content := "\n ta\ kes an optional first argument ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "Ref", next := 450754, root := ~, start := 450724, stop := 450753 ), rec( content := "\n or\ ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "Ref", next := 450794, root := ~, start := 450762, stop := 450793 ), rec( content := ", which \ specifies whether the output digraph shall\n be mutable or immutable. If no\ first argument is provided, then an immutable\n digraph is returned by def\ ault.\n\n ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> PlainTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFromP\ lainTextString(last);\n\n", count := [ 9, 2, 12, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 8 ], mathmode := "Text", name := "Example", next := 451215, root := ~, start := 450967, stop := 451214 ), rec( content := "\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Description", next := 451232, root := ~, start := 449996, stop := 451231 ) ], count := [ 9, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 451246, root := ~, start := 449851, stop := 451245 ), rec( content := "\n\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ h, delimiter, offset", Name := "WritePlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 2 ], mathmode := "Text", name := "Func", next := 451346, root := ~, start := 451267, stop := 451345 ), rec( attributes := rec( Arg := "filename, delimi\ ter, offset, ignore", Name := "ReadPlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 451426, root := ~, start := 451349, stop := 451425 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se functions write and read a single digraph in a human-readable plain\n te\ xt format as follows: each line contains a single edge, and each edge is\n \ written as a pair of integers separated by the string ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "A", next := 451674, root := ~, start := 451658, stop := 451673 ), rec( content := ".", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "P", next := 451679, root := ~, start := 451675, stop := 451678 ), rec( content := "\n\n \ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 451700, root := ~, start := 451685, stop := 451699 ), rec( content := " should \ be the name of a file which will be written to or\n read from, and ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 451790, root := ~, start := 451777, stop := 451789 ), rec( content := " should \ be an integer which is added to each\n vertex number as it is written or re\ ad. For example, if\n ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ ritePlainTextDigraph", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 451928, root := ~, start := 451900, stop := 451927 ), rec( content := " is call\ ed with ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 451957, root := ~, start := 451944, stop := 451956 ), rec( content := " ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 451967, root := ~, start := 451958, stop := 451966 ), rec( content := ", then\n\ the vertices will be numbered in the file starting from 0 instead of 1 -\n\ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadPlainTextDigraph", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 452082, root := ~, start := 452055, stop := 452081 ), rec( content := " would t\ hen need to be called with ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 452130, root := ~, start := 452117, stop := 452129 ), rec( content := "\n ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 452143, root := ~, start := 452135, stop := 452142 ), rec( content := " to conv\ ert back to the original graph.", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "P", next := 452186, root := ~, start := 452182, stop := 452185 ), rec( content := "\n\n \ ", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ gnore", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "A", next := 452205, root := ~, start := 452192, stop := 452204 ), rec( content := " should \ be a list of characters which will be ignored when\n reading the graph.\n \ ", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[1, 2, 3], [1, 1], [2]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out\ /plain.txt\");;\ngap> WritePlainTextDigraph(filename, gr, \",\", -1);\ngap> Re\ adPlainTextDigraph(filename, \",\", 1, ['/', '%']);\n\n", count := [ 9, 2, 13, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 8 ], mathmode := "Text", name := "Example", next := 452648, root := ~, start := 452291, stop := 452647 ), rec( content := "\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 452665, root := ~, start := 451429, stop := 452664 ) ], count := [ 9, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 452679, root := ~, start := 451252, stop := 452678 ), rec( content := "\n\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ h", Name := "WriteDIMACSDigraph" ), content := 0, count := [ 9, 2, 14, 2 ], mathmode := "Text", name := "Oper", next := 452757, root := ~, start := 452700, stop := 452756 ), rec( attributes := rec( Arg := "filename", Name := "ReadDIMACSDigra\ ph" ), content := 0, count := [ 9, 2, 14, 4 ], mathmode := "Text", name := "Oper", next := 452807, root := ~, start := 452760, stop := 452806 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations write or read the single symmetric digraph ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 452902, root := ~, start := 452888, stop := 452901 ), rec( content := "\n to\ or from a file in DIMACS format, as appropriate. The operation\n ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDIMACSDigraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "C", next := 453002, root := ~, start := 452977, stop := 453001 ), rec( content := " records\ the vertices and edges of ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 453051, root := ~, start := 453037, stop := 453050 ), rec( content := ".\n T\ he vertex labels of ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 453092, root := ~, start := 453078, stop := 453091 ), rec( content := " will be\ recorded only if they are\n integers. See ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Ref", next := 453177, root := ~, start := 453145, stop := 453176 ), rec( content := " and ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Ref", next := 453221, root := ~, start := 453182, stop := 453220 ), rec( content := ".", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "P", next := 453226, root := ~, start := 453222, stop := 453225 ), rec( content := "\n\n \ The first argument ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "A", next := 453266, root := ~, start := 453251, stop := 453265 ), rec( content := " should \ be the name of the file which will\n be written to or read from. A file ca\ n contain one symmetric digraph in\n DIMACS format. If ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "A", next := 453423, root := ~, start := 453408, stop := 453422 ), rec( content := " ends in\ one of ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 453449, root := ~, start := 453439, stop := 453448 ), rec( content := ", ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 453462, root := ~, start := 453451, stop := 453461 ), rec( content := ",\n o\ r ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 453481, root := ~, start := 453471, stop := 453480 ), rec( content := ", then t\ he file is compressed, or decompressed, appropriately.\n ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 7 ], mathmode := "Text", name := "P", next := 453552, root := ~, start := 453548, stop := 453551 ), rec( content := "\n\n \ The DIMACS format is described as follows. Each line in the DIMACS file has\n\ one of four types:\n ", count := [ 9, 2, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 14, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "c", count := [ 9, 2, 14, 9 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 9 ] , math\ mode := "Text", name := "C", next := 453720, root := ~, start := 453712, stop \ := 453719 ), rec( content := " and followed by any number of characters\n \ is a comment line, and is ignored.\n ", count := [ 9, 2, 14, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 9 ], name := "Ite\ m", next := 453818, root := ~, start := 453675, stop := 453817 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "p", count := [ 9, 2, 14, 11 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 11\ ], mathmode := "Text", name := "C", next := 453870, root := ~, start := 45386\ 2, stop := 453869 ), rec( content := " defines the numbers of vertices and edg\ es\n the digraph. This line has the format ", count := [ 9, 2, 14, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "p edge \n ", count :\ = [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 454020, root :=\ ~, start := 453960, stop := 454019 ), rec( content := ", where ", count := [ \ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "", count := [ 9, 2\ , 14, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 454054, root :\ = ~, start := 454028, stop := 454053 ), rec( content := " and\n ", coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2,\ 14, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 454090, root :\ = ~, start := 454067, stop := 454089 ), rec( content := " are replaced by the \ relevant integers. There\n must be exactly one such line in the file, a\ nd it must occur before any\n of the following kinds of line.", count :\ = [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 14, 12 ], mathmode := "Text" , name\ := "P", next := 454259, root := ~, start := 454255, stop := 454258 ), rec( cont\ ent := "\n\n Although it is required to be present, the value of\n \ ", count := [ 9, 2, 14, 12 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 14, 12 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 14, 12 ], mathmode := "Text", name := "C", next :\ = 454352, root := ~, start := 454329, stop := 454351 ), rec( content := " will\ be ignored. The correct number of edges\n will be deduced from the res\ t of the information in the file.\n ", count := [ 9, 2, 14, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 11 ], name := "Ite\ m", next := 454481, root := ~, start := 453825, stop := 454480 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line of the form ", count\ := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "e ", count := [ 9, 2, \ 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2\ , 14, 14 ], mathmode := "Text", name := "C", next := 454550, root := ~, start \ := 454522, stop := 454549 ), rec( content := ", where ", count := [ 9, 2, 14, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], mathmode\ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 1\ 4 ], mathmode := "Text", name := "C", next := 454574, root := ~, start := 4545\ 58, stop := 454573 ), rec( content := "\n and ", count := [ 9, 2, 14, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 14\ ], mathmode := "Text", name := "C", next := 454603, root := ~, start := 45458\ 7, stop := 454602 ), rec( content := " are integers in the range ", count := [ \ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "[1 ..\n ]\ ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "C", next := 45467\ 3, root := ~, start := 454630, stop := 454672 ), rec( content := ", specifies \ that there is a (symmetric) edge\n in the digraph between the vertices \ ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2, 14, 14\ ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ \ 9, 2, 14, 14 ], mathmode := "Text", name := "C", next := 454778, root := ~, star\ t := 454762, stop := 454777 ), rec( content := " and\n ", count := [ 9,\ 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], \ mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, \ 2, 14, 14 ], mathmode := "Text", name := "C", next := 454807, root := ~, start\ := 454791, stop := 454806 ), rec( content := ". A symmetric edge only needs t\ o be defined once; an\n additional line ", count := [ 9, 2, 14, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "e ", count := [ 9, 2, 14, 14 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], mathmode :=\ "Text", name := "C", next := 454912, root := ~, start := 454884, stop := 4549\ 11 ), rec( content := ", or ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "e \n ", count := [ 9, 2, 14, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], mathmode := "Text", name\ := "C", next := 454955, root := ~, start := 454917, stop := 454954 ), rec( cont\ ent := ", will be interpreted as an additional, multiple, edge.\n Loops\ are permitted.\n ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], name := "Item", next := 4550\ 53, root := ~, start := 454488, stop := 455052 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n A line of the form ", count := [ 9, 2, 14, 16 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "n \">\n https://gap-packages.github.io/grapeGrape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suiteedge-addition-planarity-suite<\ /Alt>\">\n https://\ pallini.di.uniroma1.it/nauty\">\n https://github.com/gap-packages/NautyTracesInterface<\ /URL>NautyTracesInterface\ \">\n \n https://gap-pack\ ages.github.io/io/\nIO\"\ >\n \n https://gap-pack\ ages.github.io/orb/\n Orb<\ /Alt>\">\n https://github.com/gap-packages/datastructuresdatastructures\">\n \n\ \n\n\n\n\n\n\n\ \n\n ]>\n\n\n\n &Digraphs;</Title\ >\n <Version>Version &VERSION;</Version>\n\n <Author>Jan De Beule\n <Emai\ l>jdebeule@cage.ugent.be</Email>\n <Homepage>http://homepages.vub.ac.be/~jd\ beule</Homepage>\n </Author>\n\n <Author>Julius Jonušas\n <Email>jul\ ius.jonusas@tuwien.ac.at</Email>\n <Homepage>http://julius.jonusas.work</Ho\ mepage>\n </Author>\n\n <Author>James D. Mitchell\n <Email>jdm3@st-and.ac\ .uk</Email>\n <Homepage>https://jdbm.me</Homepage>\n </Author>\n\n <Autho\ r>Michael C. Torpey\n <Email>mct25@st-andrews.ac.uk</Email>\n <Homepage>\ https://mtorpey.github.io</Homepage>\n </Author>\n\n <Author>Maria Tsalakou\ \n <Email>mt200@st-andrews.ac.uk</Email>\n <Homepage>https://mariatsalak\ ou.github.io</Homepage>\n </Author>\n\n <Author>Wilf A. Wilson\n <Email>g\ ap@wilf-wilson.net</Email>\n <Homepage>https://wilf.me</Homepage>\n </Auth\ or>\n\n <Author>Stuart Burrell</Author>\n\n <Author>Reinis Cirpons</Author>\ \n\n <Author>Luke Elliott</Author>\n\n <Author>Max Horn</Author>\n\n <Autho\ r>Christopher Jefferson</Author>\n\n <Author>Markus Pfeiffer</Author>\n\n <A\ uthor>Christopher Russell</Author>\n\n <Author>Finn Smith</Author>\n\n <Auth\ or>Murray White</Author>\n\n <Copyright>©right; ©RIGHTYEARS; by\n \ Jan De Beule, Julius Jonušas, James D. Mitchell, Michael\n Torpey, Wil\ f A. Wilson et al.<P/>\n\n &Digraphs; is free software; you can redistribut\ e it and/or modify it\n under the terms of the <URL Text=\"GNU General Publ\ ic License\">\n https://www.fsf.org/licenses/gpl.html</URL> as published by\ the Free Software\n Foundation; either version 3 of the License, or (at yo\ ur option) any later\n version.\n </Copyright>\n\n <Abstract>\n The &D\ igraphs; package is a &GAP; package containing methods for graphs,\n digrap\ hs, and multidigraphs.\n </Abstract>\n\n <Acknowledgements>\n We would li\ ke to thank Christopher Jefferson for his help in including \n &bliss; in &\ Digraphs;.\n\n This package's methods for computing digraph homomorphisms a\ re based on work\n by Max Neunhöffer, and independently Artur Schä\ ;fer.\n </Acknowledgements>\n\n</TitlePage>\n\n\n<TableOfContents/>\n\n<Body>\ \n <Chapter Label=\"The Digraphs package\">\n <Heading>\n The &Digraphs; \ package\n </Heading>\n <Index Key=\"Digraphs\">&Digraphs; package overview</\ Index>\n\n<Section><Heading>Introduction</Heading>\n\n This is the manual for\ version &VERSION; of the &Digraphs; package.\n This package was developed at\ the University of St Andrews by:\n <List>\n <Item>Jan De Beule,</Item>\n \ <Item>Julius Jonušas,</Item>\n <Item>James D. Mitchell,</Item>\n \ <Item>Michael C. Torpey,</Item>\n <Item>Maria Tsalakou, and</Item>\n <I\ tem>Wilf A. Wilson.</Item>\n </List>\n Additional contributions were made by\ :\n <List>\n <Item>Stuart Burrell,</Item>\n <Item>Reinis Cirpons,</Item\ >\n <Item>Luke Elliott,</Item>\n <Item>Max Horn,</Item>\n <Item>Chris\ topher Jefferson,</Item>\n <Item>Markus Pfeiffer,</Item>\n <Item>Christo\ pher Russell,</Item>\n <Item>Finn Smith, and</Item>\n <Item>Murray Whyte\ .</Item>\n </List>\n\n The &Digraphs; package contains a variety of methods \ for efficiently creating\n and storing mutable and immutable digraphs and com\ puting information about\n them. Full explanations of all the functions co\ ntained in the package are\n provided below. <P/>\n\n If the &Grape; packa\ ge is available, it will be loaded automatically.\n Digraphs created with the\ &Digraphs; package can be converted to &Grape;\n graphs with <Ref Oper=\"Gra\ ph\"/>, and conversely &Grape; graphs can be\n converted to &Digraphs; object\ s with <Ref Oper=\"Digraph\"/>. &Grape; is not\n required for &Digraphs; to \ run. <P/>\n\n The &bliss; tool <Cite Key=\"JK07\"/> is included in this packa\ ge. It\n is an open-source tool for computing automorphism groups and canoni\ cal forms\n of graphs, written by Tommi Junttila and Petteri Kaski. Several o\ f the methods\n in the &Digraphs; package rely on &bliss;. If the &NautyTrace\ sInterface;\n package for GAP is available then it is also possible to use &n\ auty; <Cite\n Key=\"MP14\"/> for computing automorphism groups and canonica\ l forms\n in &Digraphs;. See Section \n <Ref Sect=\"Isomorphisms and canonic\ al labellings\"/> for more details. \n <P/>\n\n The &edge-addition-planarit\ y-suite; is also included in &Digraphs;; see\n <Cite Key=\"BM04\"/>, <Cite Ke\ y=\"B06\"/>, <Cite Key=\"BM06\"/>, and <Cite\n Key=\"B12\"/> . The &edge-add\ ition-planarity-suite; is an open-source\n implementation of the edge additio\ n planar graph embedding algorithm and\n related algorithms by John M. Boyer.\ See Section <Ref Sect=\"Planarity\"/> for\n more details.\n <P/>\n\n From \ version 1.0.0 of this package, digraphs can be either mutable or immutable.\n \ Mutable digraphs can be changed in-place by many of the methods in the\n pac\ kage, which avoids unnecessary copying. Immutable digraphs cannot be\n change\ d in-place, but their advantage is that the value of an attribute of\n an imm\ utable digraph is only ever computed once. Mutable digraphs can be\n converte\ d into immutable digraphs in-place using <Ref Func=\"MakeImmutable\"\n BookNa\ me = \"ref\"/>. One of the motivations for introducing mutable\n digraphs in\ version 1.0.0 was that in practice the authors often wanted\n to create a di\ graph and immediately modify it (removing certain edges,\n loops, and so on).\ Before version 1.0.0, this involved copying the digraph\n several times, wit\ h each copy being discarded almost immediately.\n From version 1.0.0, this un\ necessary copying can be eliminated by\n first creating a mutable digraph, th\ en changing it in-place, and finally\n converting the mutable digraph to an \ immutable one in-place\n (if desirable).\n\n <Subsection Label=\"Definitions\ \"><Heading>Definitions</Heading>\n\n For the purposes of this package and \ its documentation, the following\n definitions apply: <P/>\n\n A <E>digr\ aph</E> <M>E=(E^0,E^1,r,s)</M>, also known as a <E>directed\n graph</E>, \ consists of a set of vertices <M>E^0</M> and a set of edges\n <M>E^1</M> to\ gether with functions <M>s, r: E^1 \\to E^0</M>, called the\n <E>source</E>\ and <E>range</E>, respectively.\n\n The source and range of an edge is res\ pectively the values of <M>s, r</M> at\n that edge.\n\n An edge is calle\ d a <E>loop</E> if its source and range are the same.\n\n A digraph is call\ ed a <E>multidigraph</E> if there exist two or more edges\n with the same s\ ource and the same range. <P/>\n\n A <E>directed walk</E> on a digraph is a\ sequence of alternating vertices\n and edges <M>(v_1, e_1, v_2, e_2, ..., \ e_{n-1}, v_n)</M> such that each edge\n <M>e_i</M> has source <M>v_i</M> an\ d range <M>v_{i+1}</M>. A <E>directed\n path</E> is a directed walk where\ no vertex (and hence no edge) is\n repeated. A <E>directed circuit</E> is\ a directed walk where <M>v_1 =\n v_n</M>, and a <E>directed cycle</E> is\ a directed circuit where where no\n vertex is repeated, except for <M>v_1 \ = v_n</M>. <P/>\n\n The <E>length</E> of a directed walk <M>(v_1, e_1, v_2\ , e_2, ..., e_{n-1},\n v_n)</M> is equal to <M>n-1</M>, the number of edg\ es it contains. A\n directed walk (or path) <M>(v_1, e_1, v_2, e_2, ..., e\ _{n-1}, v_n)</M> is\n sometimes called a directed walk (or path) <E>from ve\ rtex <M>v_1</M> to\n vertex <M>v_n</M></E>. A directed walk of zero leng\ th, i.e. a sequence\n <M>(v)</M> for some vertex <M>v</M>, is called <E>tri\ vial</E>. A trivial\n directed walk is considered to be both a circuit and \ a cycle, as is the\n empty directed walk <M>()</M>. A <E>simple circuit</E>\ is another name for a\n non-trivial and non-empty directed cycle.<P/>\n\n \ </Subsection>\n </Section>\n</Chapter>\n <!-- intro -->\n <Chapter Label=\"\ Installing Digraphs\">\n <Heading>Installing &Digraphs;</Heading>\n\n <Secti\ on Label=\"For those in a hurry\">\n <Heading>For those in a hurry</Heading\ >\n In this section we give a brief description of how to start using\n \ &Digraphs;.<P/>\n\n It is assumed that you have a working copy of &GAP; wit\ h version number\n &GAPVERS; or higher. The most up-to-date version of &GAP\ ; and instructions\n on how to install it can be obtained from the main &GA\ P; webpage\n <URL>https://www.gap-system.org</URL>.<P/>\n\n The followin\ g is a summary of the steps that should lead to a\n successful installation\ of &Digraphs;:\n\n <List>\n\n <Item>\n ensure that the &IO; pa\ ckage version &IOVERS; or higher is available.\n &IO; must be compiled \ before &Digraphs; can be loaded.\n </Item>\n\n <Item>\n ensur\ e that the &Orb; package version &ORBVERS; or higher is available.\n &O\ rb; has better performance when compiled, but although compilation is\n \ recommended, it is not required to be compiled for &Digraphs; to be\n \ loaded.\n </Item>\n\n <Item> ensure that the &datastructures; packag\ e version\n &DATASTRUCTURESVERS; or higher is available. \n </Item>\ \n\n <Item>\n <B>This step is optional:</B> certain functions in &\ Digraphs; require\n the &Grape; package to be available; see Section\n \ <Ref Subsect=\"The Grape package\"/> for full details. To use these\n \ functions make sure\n that the &Grape; package version &GRAPEVER\ S; or higher is available.\n If &Grape; is not available, then &Digraph\ s; can be used as normal with\n the exception that the functions listed\ in Subsection\n <Ref Subsect=\"The Grape package\"/> will not work.\n \ </Item>\n \n <Item>\n <B>This step is optional:</B> cert\ ain functions in &Digraphs; require\n the &NautyTracesInterface; pack\ age to be available.\n <!-- TODO \n see Section\n <Ref \ Subsect=\"The Grape package\"/> for full details. -->\n If you want to\ make use of these functions, please ensure that the\n &NautyTracesInte\ rface; package version &NAUTYTRACESINTERFACEVERS; or\n higher is availa\ ble. If &NautyTracesInterface; is not available,\n then &Digraphs; can\ be used as normal with the exception that \n functions whose names c\ ontain <Q>Nauty</Q> will not work.\n </Item>\n\n <Item>\n dow\ nload the package archive <F>&ARCHIVENAME;.tar.gz</F> from\n <URL Text=\ \"the Digraph package webpage\">https://digraphs.github.io/Digraphs/</URL>.\n \ </Item>\n\n <Item>\n unzip and untar the file, this should cr\ eate a directory called\n <F>&ARCHIVENAME;</F>.\n </Item>\n\n \ <Item>\n locate the <F>pkg</F> directory of your &GAP; directory, whic\ h contains\n the directories <F>lib</F>, <F>doc</F> and so on. Move the\ directory\n <F>&ARCHIVENAME;</F> into the <F>pkg</F> directory.\n \ </Item>\n\n <Item>\n it is necessary to compile the &Digraphs; pa\ ckage.\n Inside the <F>pkg/&ARCHIVENAME;</F> directory, type\n <\ Listing>\n<![CDATA[./configure\nmake]]></Listing>\n Further information\ about this step can be found in Section\n <Ref Sect=\"Compiling the ke\ rnel module\"/>.\n </Item>\n\n <Item>\n start &GAP; in the us\ ual way (i.e. type <C>gap</C> at the command line).\n </Item>\n\n <I\ tem>\n type <C>LoadPackage(\"digraphs\");</C>\n </Item>\n </Lis\ t>\n\n If you want to check that the package is working correctly, you\n \ should run some of the tests described in Section\n <Ref Sect=\"Testing yo\ ur installation\"/>.\n <Subsection Label=\"Configuration options\">\n \ <Heading>Configuration options</Heading>\n\nIn addition to the usual autoconf \ generated configuration flags, the following\nflags are provided.\n\n<Table Al\ ign=\"|l|l|\">\n <Caption>Configuration flags</Caption>\n <HorLine/>\n <Row>\ \n <Item>Option</Item><Item>Meaning</Item>\n </Row>\n <HorLine/>\n <HorLine\ />\n <Row>\n <Item><C>--enable-code-coverage</C></Item><Item> enable code co\ verage support</Item>\n </Row>\n <Row>\n <Item><C>--enable-compile-warnings<\ /C></Item><Item> enable compiler warnings</Item>\n </Row>\n <Row>\n <Item><C\ >--enable-debug</C></Item><Item> enable debug mode</Item>\n </Row>\n <Row>\n \ <Item><C>--with-external-bliss</C></Item><Item> use external &bliss;</Item>\n\ </Row>\n <Row>\n <Item><C>--with-external-planarity</C></Item><Item> use ex\ ternal &edge-addition-planarity-suite;</Item>\n </Row>\n <Row>\n <Item><C>--\ with-gaproot</C></Item><Item> specify root of GAP installation</Item>\n </Row>\ \n <Row>\n <Item><C>--without-intrinsics</C></Item><Item> do not use compile\ r intrinsics even if available</Item>\n </Row>\n <HorLine/>\n</Table>\n</Subse\ ction>\n </Section>\n\n <!--************************************************\ **********************-->\n <!--*********************************************\ *************************-->\n\n <Section Label=\"Optional package dependenci\ es\">\n <Heading>Optional package dependencies</Heading>\n\n <!--*******\ *************************************************************-->\n\n The &D\ igraphs; package is written in &GAP; and C code and requires the\n &IO; pac\ kage. The &IO; package is used to read and write transformations,\n partia\ l permutations, and bipartitions to a file. <P/>\n\n <!--******************\ **************************************************-->\n\n <Subsection Label\ =\"The Grape package\">\n <Heading>The Grape package</Heading>\n\n T\ he &Grape; package must be available for the following operations to be\n \ available:\n\n <List>\n <Item><Ref Oper=\"Graph\"/> with a digrap\ h argument</Item>\n <Item><Ref Attr=\"AsGraph\"/> with a digraph argume\ nt</Item>\n <Item><Ref Oper=\"Digraph\"/> with a &Grape; graph argument\ </Item>\n </List>\n\n If &Grape; is not available, then &Digraphs; c\ an be used as normal with\n the exception that the functions above will n\ ot work.\n </Subsection>\n </Section>\n\n <!--***************************\ *******************************************-->\n <!--************************\ **********************************************-->\n\n <Section Label=\"Compil\ ing the kernel module\">\n <Heading>Compiling the kernel module</Heading>\n\ The &Digraphs; package has a &GAP; kernel component in\n C which should\ be compiled. This component contains certain low-level\n functions require\ d by &Digraphs;.\n <P/>\n\n It is not possible to use the &Digraphs; pac\ kage without compiling it.<P/>\n\n To compile the kernel component inside t\ he <F>pkg/&ARCHIVENAME;</F>\n directory, type\n<Listing>\n<![CDATA[\n./conf\ igure\nmake]]></Listing>\n <P/>\n\n If you installed the package in anot\ her 'pkg' directory than the standard 'pkg'\n directory in your &GAP; insta\ llation, then you have to do two things. Firstly\n during compilation you h\ ave to use the option '--with-gaproot=PATH' of the\n 'configure' script whe\ re 'PATH' is a path to the main GAP root directory (if\n not given the defa\ ult '../..' is assumed).<P/>\n\n If you installed &GAP; on several architec\ tures, you must execute the\n configure/make step for each of the architect\ ures. You can either do this\n immediately after configuring and compiling \ GAP itself on this architecture, or\n alternatively set the environment\n \ variable 'CONFIGNAME' to the name of the configuration you used when compili\ ng\n GAP before running './configure'. Note however that your compiler cho\ ice and\n flags (environment variables 'CC' and 'CFLAGS') need to be chosen\ to match the\n setup of the original GAP compilation. For example you have\ to specify 32-bit\n or 64-bit mode correctly!\n </Section>\n\n <!--*****\ *****************************************************************-->\n <!--**\ ********************************************************************-->\n\n <\ Section Label=\"Rebuilding the documentation\">\n <Heading>Rebuilding the d\ ocumentation</Heading>\n The &Digraphs; package comes complete with pdf, ht\ ml, and text versions of\n the documentation. However, you might find it ne\ cessary, at some point, to\n rebuild the documentation. To rebuild the docu\ mentation, please use the\n function <Ref Func=\"DigraphsMakeDoc\"/>.\n\n \ <ManSection>\n <Func Name=\"DigraphsMakeDoc\" Arg=\"\"/>\n <Returns>Nothin\ g</Returns>\n <Description>\n This function should be called with no argum\ ent to compile the &Digraphs;\n documentation.\n </Description>\n</ManSect\ ion>\n\n </Section>\n\n <!--************************************************\ **********************-->\n <!--*********************************************\ *************************-->\n\n <Section Label=\"Testing your installation\"\ >\n <Heading>Testing your installation</Heading>\n\n In this section we \ describe how to test that &Digraphs; is working as\n intended. To test tha\ t &Digraphs; is installed correctly use\n <Ref Func=\"DigraphsTestInstall\"\ /> or for more extensive tests use\n <Ref Func=\"DigraphsTestStandard\"/>. \ <P/>\n\n If something goes wrong, then please review the instructions in Se\ ction <Ref\n Sect=\"For those in a hurry\"/> and ensure that &Digraphs; h\ as been\n properly installed. If you continue having problems, please use \ the <URL\n Text=\"issue tracker\">https://github.com/digraphs/Digraphs/is\ sues</URL>\n to report the issues you are having.\n\n <ManSection>\n <F\ unc Name=\"DigraphsTestInstall\" Arg=\"\"/>\n <Returns><K>true</K> or <K>fals\ e</K>.</Returns>\n <Description>\n This function can be called without arg\ uments to test your installation of\n &Digraphs; is working correctly. Thes\ e tests should take no more than a\n few seconds to complete. To test more\ comprehensively that\n &Digraphs; is working correctly, use <Ref Func=\"Di\ graphsTestStandard\"/>.\n </Description>\n</ManSection>\n\n <ManSection>\n\ <Func Name=\"DigraphsTestStandard\" Arg=\"\"/>\n <Returns><K>true</K> or <K\ >false</K>.</Returns>\n <Description>\n This function can be called withou\ t arguments to test all of the methods\n included in &Digraphs;.\n These\ tests should take less than a minute to complete. <P/>\n\n To quickly test\ that &Digraphs; is installed correctly use <Ref\n Func=\"DigraphsTestInsta\ ll\"/>. For a more thorough test, use <Ref\n Func=\"DigraphsTestExtreme\"/\ >.\n </Description>\n</ManSection>\n\n <ManSection>\n <Func Name=\"Digrap\ hsTestExtreme\" Arg=\"\"/>\n <Returns>Nothing.</Returns>\n <Description>\n \ This function should be called with no argument. It executes a series of\n \ very demanding tests, which measure the performance of a variety of\n fun\ ctions on large examples. These tests take a long time to\n complete, at l\ east several minutes. <P/>\n\n For these tests to complete, the digraphs li\ brary <C>digraphs-lib</C> must\n be downloaded and placed in the <C>digraph\ s</C> directory in a subfolder\n named <F>digraphs-lib</F>.\n This libra\ ry can be found on the &Digraphs; website.\n </Description>\n</ManSection>\n\ \n\n </Section>\n\n <!--****************************************************\ ******************-->\n <!--*************************************************\ *********************-->\n\n\n</Chapter>\n <!-- installing -->\n <Chapter Lab\ el=\"Creating digraphs\"><Heading>Creating digraphs</Heading>\n\n In this cha\ pter we describe how to create digraphs.<P/>\n\n <Section><Heading>Creating d\ igraphs</Heading>\n <ManSection>\n <Filt Name=\"IsDigraph\" Type='Category\ '/>\n <Description>\n Every digraph in &Digraphs; belongs to the cat\ egory\n <C>IsDigraph</C>. Some basic attributes and operations for digraph\ s\n are <Ref Attr=\"DigraphVertices\"/>, <Ref Attr=\"DigraphEdges\"/>, and\ \n <Ref Attr=\"OutNeighbours\"/>.\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Filt Name=\"IsMutableDigraph\" Type='Category'/>\n <Descrip\ tion>\n <C>IsMutableDigraph</C> is a synonym for <Ref Filt=\"IsDigraph\" />\ and <Ref\n Filt=\"IsMutable\" BookName=\"ref\" />. A mutable digraph may \ be changed\n in-place by methods in the &Digraphs; package, and is not attr\ ibute-storing\n – see <Ref Filt=\"IsAttributeStoringRep\" BookName=\"ref\ \" />.\n <P/>\n\n A mutable digraph may be converted into an immutable a\ ttribute-storing\n digraph by calling <Ref Func=\"MakeImmutable\" BookName=\ \"ref\"/> on the\n digraph.\n </Description>\n</ManSection>\n\n <ManS\ ection>\n <Filt Name=\"IsImmutableDigraph\" Type='Category'/>\n <Description\ >\n <C>IsImmutableDigraph</C> is a subcategory of <Ref Filt=\"IsDigraph\" /\ >.\n Digraphs that lie in <C>IsImmutableDigraph</C> are immutable and\n \ attribute-storing. In particular, they lie in\n <Ref Filt=\"IsAttributeStor\ ingRep\" BookName=\"ref\" />. <P/>\n\n A mutable digraph may be converted t\ o an immutable\n digraph that lies in the category <C>IsImmutableDigraph</C\ > by calling\n <Ref Func=\"MakeImmutable\" BookName=\"ref\"/> on the digrap\ h.<P/>\n\n The operation <Ref Oper=\"DigraphMutableCopy\"/> can be used\n \ to construct a mutable copy of an immutable digraph. It is however not\n \ possible to convert an immutable digraph into a mutable digraph in-place.\n <\ /Description>\n</ManSection>\n\n <ManSection>\n <Filt Name=\"IsCayleyDigra\ ph\" Type='Category'/>\n <Description>\n <C>IsCayleyDigraph</C> is a subca\ tegory of <C>IsDigraph</C>. Digraphs that\n are Cayley digraphs of a group \ and that are constructed by the operation\n <Ref Oper=\"CayleyDigraph\"/> a\ re constructed in this category, and are\n always immutable.\n </Descrip\ tion>\n</ManSection>\n\n <ManSection>\n <Filt Name=\"IsDigraphWithAdjacenc\ yFunction\" Type='Category'/>\n <Description>\n <C>IsDigraphWithAdjacencyF\ unction</C> is a subcategory of <C>IsDigraph</C>.\n Digraphs that are <E>cr\ eated</E> using an adjacency function are constructed\n in this category.\n\ </Description>\n</ManSection>\n\n <ManSection>\n <Var Name=\"DigraphByOu\ tNeighboursType\"/>\n <Fam Name=\"DigraphFamily\"/>\n <Description>\n The\ type of all digraphs is <C>DigraphByOutNeighboursType</C>.\n The family of\ all digraphs is <C>DigraphFamily</C>.\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Oper Name = \"Digraph\" Arg = \"[filt,] obj[, source, range]\ \"/>\n <Oper Name = \"Digraph\" Arg = \"[filt,] list, func\" \n Label = \"\ for a list and function\"/>\n <Oper Name = \"Digraph\" Arg = \"[filt,] G, lis\ t, act, adj\"\n Label = \"for a group, list, function, and function\"/>\n \ <Returns>A digraph.</Returns>\n <Description>\n If the optional first argu\ ment <A>filt</A> is present, then this should\n specify the category or rep\ resentation the digraph being created will\n belong to. For example, if <A>\ filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the digraph being crea\ ted will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImmutableDigraph\"/>\ , then the digraph will be immutable. \n If the optional first argument <A>\ filt</A> is not present, then <Ref\n Filt=\"IsImmutableDigraph\"/> is used \ by default.\n <List>\n <Mark>for a list (i.e. an adjacency list)</Mark\ >\n <Item>\n if <A>obj</A> is a list of lists of positive integers\ in\n the range from <C>1</C> to <C>Length(<A>obj</A>)</C>, then this f\ unction\n returns the digraph with vertices\n <M>E ^ 0 = </M><C>\ [1 .. Length(<A>obj</A>)]</C>,\n and edges corresponding to the entries\ of <A>obj</A>. <P/>\n\n More precisely, there is an edge from vertex <\ C>i</C> to <C>j</C> if and\n only if <C>j</C> is in <C><A>obj</A>[i]</C\ >; the source of this edge is\n <C>i</C> and the range is <C>j</C>. If \ <C>j</C> occurs in\n <C><A>obj</A>[i]</C> with multiplicity <C>k</C>, t\ hen there are <C>k</C>\n edges from <C>i</C> to <C>j</C>.\n </Item\ >\n\n <Mark>for three lists</Mark>\n <Item>\n if <A>obj</A> i\ s a duplicate-free list, and <A>source</A> and\n <A>range</A> are lists\ of equal length consisting of positive integers\n in the list <C>[1 ..\ Length(<A>obj</A>)]</C>, then this function\n returns a digraph with v\ ertices\n <M>E ^ 0 = </M><C>[1 .. Length(<A>obj</A>)]</C>, and\n \ <C>Length(<A>source</A>)</C> edges. For each <C>i</C> in\n <C>[1 .. L\ ength(<A>source</A>)]</C> there exists an edge with source\n vertex <C>\ source[i]</C> and range vertex <C>range[i]</C>.\n See <Ref Attr=\"Digra\ phSource\"/> and <Ref Attr=\"DigraphRange\"/>.\n <P/>\n\n The ve\ rtices of the digraph will be labelled by the elements of\n <A>obj</A>.\ \n </Item>\n\n <Mark>for an integer, and two lists</Mark>\n <It\ em>\n if <A>obj</A> is an integer, and <A>source</A> and\n <A>ra\ nge</A> are lists of equal length consisting of positive integers\n in \ the list <C>[1 .. <A>obj</A>]</C>, then this function\n returns a digra\ ph with vertices\n <M>E ^ 0 = </M><C>[1 .. <A>obj</A>]</C>, and\n \ <C>Length(<A>source</A>)</C> edges. For each <C>i</C> in\n <C>[1 .. \ Length(<A>source</A>)]</C> there exists an edge with source\n vertex <C\ >source[i]</C> and range vertex <C>range[i]</C>.\n See <Ref Attr=\"Digr\ aphSource\"/> and <Ref Attr=\"DigraphRange\"/>.\n </Item>\n\n <Mark>\ for a list and a function</Mark>\n <Item>\n if <A>list</A> is a li\ st and <A>func</A> is a function taking 2\n arguments that are elements\ of <A>list</A>, and <A>func</A> returns\n <K>true</K> or <K>false</K>,\ then this operation creates a digraph with\n vertices <C>[1 .. Length(\ <A>list</A>)]</C> and an edge from vertex\n <C>i</C> to vertex <C>j</C>\ if and only if\n <C><A>func</A>(<A>list</A>[i], <A>list</A>[j])</C> re\ turns <K>true</K>.\n </Item>\n\n <Mark>for a group, a list, and two \ functions</Mark>\n <Item>\n The arguments will be <A>G, list, act,\ adj</A>. <P/>\n\n Let <A>G</A> be a group acting on the objects in <A>\ list</A> via\n the action <A>act</A>, and let <A>adj</A> be a function \ taking\n two objects from <A>list</A> as arguments and returning <C>tru\ e</C> or\n <C>false</C>. The function <A>adj</A> will describe the adja\ cency\n between objects from <A>list</A>, which is invariant under the \ action of\n <A>G</A>. This variant of the constructor returns a digraph\ with\n vertices the objects of <A>list</A> and directed edges <C>[x, y\ ]</C>\n when <C>f(x, y)</C> is <C>true</C>. <P/>\n\n The action \ of the group <A>G</A> on the objects in <A>list</A> is stored\n in the \ attribute <Ref Attr=\"DigraphGroup\"/>, and is used to speed up\n opera\ tions like <Ref Attr=\"DigraphDiameter\"/>.\n </Item>\n\n <Mark>for \ a Grape package graph</Mark>\n <Item>\n if <A>obj</A> is a &Grape;\ package graph (i.e. a record for which the\n function <C>IsGraph</C> r\ eturns <K>true</K>), then this function returns\n a digraph isomorphic \ to <A>obj</A>.\n </Item>\n\n <Mark>for a binary relation</Mark>\n \ <Item>\n if <A>obj</A> is a binary relation on the points <C>[1 .. n\ ]</C> for\n some positive integer <M>n</M>, then this function returns \ the digraph\n defined by <A>obj</A>. Specifically, this function return\ s a digraph\n which has <M>n</M> vertices, and which has an edge with s\ ource <C>i</C>\n and range <C>j</C> if and only if <C>[i,j]</C> is a pa\ ir in\n the binary relation <A>obj</A>.\n </Item>\n </List>\n\n\ <Example><![CDATA[\ngap> gr := Digraph([\n> [2, 5, 8, 10], [2, 3, 4, 2, 5,\ 6, 8, 9, 10], [1],\n> [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\ \n> [1, 5, 9], [1, 2, 7, 8], [3, 5]]);\n<immutable multidigraph with 10 vertic\ es, 38 edges>\ngap> gr := Digraph([\"a\", \"b\", \"c\"], [\"a\"], [\"b\"]);\n<\ immutable digraph with 3 vertices, 1 edge>\ngap> gr := Digraph(5, [1, 2, 2, 4,\ 1, 1], [2, 3, 5, 5, 1, 1]);\n<immutable multidigraph with 5 vertices, 6 edges\ >\ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, \ y) return Intersection(x, y) = []; end);;\ngap> Digraph(Petersen);\n<immutable\ digraph with 10 vertices, 30 edges>\ngap> gr := Digraph([1 .. 10], ReturnTrue\ );\n<immutable digraph with 10 vertices, 100 edges>]]></Example>\n\n The \ next example illustrates the uses of the fourth and fifth variants of\n t\ his constructor. The resulting digraph is a strongly regular graph, and\n \ it is actually the point graph of the van Lint-Schrijver partial\n geome\ try, <Cite Key=\"vLS81\"/>. The algebraic description\n is taken from th\ e seminal paper of Calderbank and Kantor <Cite\n Key=\"CK86\"/>.\n<Exam\ ple><![CDATA[\ngap> f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, x -> Orde\ r(x) = 5);\nZ(3^4)^64\ngap> L := Union([Zero(f)], List(Group(gamma)));\n[ 0*Z(\ 3), Z(3)^0, Z(3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ]\ngap> omega := Union(\ List(L, x -> List(Difference(L, [x]), y -> x - y)));\n[ Z(3)^0, Z(3), Z(3^4)^5\ , Z(3^4)^7, Z(3^4)^8, Z(3^4)^13, Z(3^4)^15, \n Z(3^4)^16, Z(3^4)^21, Z(3^4)^2\ 3, Z(3^4)^24, Z(3^4)^29, Z(3^4)^31, \n Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4\ )^45, Z(3^4)^47, Z(3^4)^48, \n Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(\ 3^4)^63, Z(3^4)^64, \n Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 \ ]\ngap> adj := function(x, y)\n> return x - y in omega;\n> end;\nfunction( x\ , y ) ... end\ngap> digraph := Digraph(AsList(f), adj);\n<immutable digraph wi\ th 81 vertices, 2430 edges>\ngap> group := Group(Z(3));;\ngap> act := \\*;\n<O\ peration \"*\">\ngap> digraph := Digraph(group, List(f), act, adj);\n<immutabl\ e digraph with 81 vertices, 2430 edges>\n]]></Example>\n </Description>\n</Ma\ nSection>\n\n <ManSection>\n <Oper Name=\"DigraphByAdjacencyMatrix\" Arg=\ \"[filt, ]list\"/>\n <Returns>A digraph.</Returns>\n <Description>\n If t\ he optional first argument <A>filt</A> is present, then this should\n speci\ fy the category or representation the digraph being created will\n belong t\ o. For example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then \ the digraph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"\ IsImmutableDigraph\"/>, then the digraph will be immutable. \n If the optio\ nal first argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutab\ leDigraph\"/> is used by default.<P/>\n\n If <A>list</A> is the adjacency m\ atrix of a digraph in the sense of\n <Ref Attr=\"AdjacencyMatrix\"/>, then \ this operation returns the digraph\n which is defined by <A>list</A>. <P/>\ \n\n Alternatively, if <A>list</A> is a square boolean matrix, then this\n \ operation returns the digraph with <C>Length(</C><A>list</A><C>)</C>\n v\ ertices which has the edge <C>[i,j]</C> if and only if\n <A>list</A><C>[i][\ j]</C> is <K>true</K>. <P/>\n\n <Example><![CDATA[\ngap> DigraphByAdjacency\ Matrix([\n> [0, 1, 0, 2, 0],\n> [1, 1, 1, 0, 1],\n> [0, 3, 2, 1, 1],\n> [0, 0,\ 1, 0, 1],\n> [2, 0, 0, 0, 0]]);\n<immutable multidigraph with 5 vertices, 18 \ edges>\ngap> D := DigraphByAdjacencyMatrix([\n> [true, false, true],\n> [false\ , false, true],\n> [false, true, false]]);\n<immutable digraph with 3 vertices\ , 4 edges>\ngap> OutNeighbours(D);\n[ [ 1, 3 ], [ 3 ], [ 2 ] ]\ngap> D := Digr\ aphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, false, true],\n> [false, f\ alse, true],\n> [false, true, false]]);\n<mutable digraph with 3 vertices, 4 \ edges>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Oper Name=\"DigraphByEdges\" Arg=\"[filt, ]list[, n]\"/>\n <Returns>A digrap\ h.</Returns>\n <Description>\n If the optional first argument <A>filt</A> \ is present, then this should\n specify the category or representation the d\ igraph being created will\n belong to. For example, if <A>filt</A> is <Ref \ Filt=\"IsMutableDigraph\"/>,\n then the digraph being created will be mutab\ le, if <A>filt</A> is <Ref\n Filt=\"IsImmutableDigraph\"/>, then the digrap\ h will be immutable. \n If the optional first argument <A>filt</A> is not p\ resent, then <Ref\n Filt=\"IsImmutableDigraph\"/> is used by default.<P/>\n\ \n If <A>list</A> is list of pairs of positive integers, then this function\ \n returns the digraph with the minimum number of vertices <C>m</C> such th\ at\n its list equal <A>list</A>.<P/>\n\n If the optional second argument\ <A>n</A> is a positive integer with\n <C><A>n</A> >= m</C> (with <C>m</C> \ defined as above),\n then this function returns the digraph with <A>n</A> v\ ertices and list\n <A>list</A>. <P/>\n\n See <Ref Attr=\"DigraphEdges\"/\ >.\n\n <Example><![CDATA[\ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3],\ [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]]);\n<immutable dig\ raph with 6 vertices, 10 edges>\ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, \ 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n<immuta\ ble digraph with 12 vertices, 10 edges>\ngap> DigraphByEdges(IsMutableDigraph,\ \n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, \ 6], [6, 6]], 12);\n<mutable digraph with 12 vertices, 10 edges>\n]]></Example>\ \n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"EdgeOrbi\ tsDigraph\" Arg=\"G, edges[, n]\"/>\n <Returns>\n An immutable digraph.\n \ </Returns>\n <Description>\n If <A>G</A> is a permutation group, <A>edges\ </A> is an edge or list of\n edges, and <A>n</A> is a non-negative integer \ such that <A>G</A> fixes\n <C>[1 .. <A>n</A>]</C> setwise, then\n this o\ peration returns an immutable digraph with <A>n</A> vertices and the union\n \ of the orbits of the edges in <A> edges </A> under the action of the\n pe\ rmutation group <A>G</A>. An edge in this context is simply a pair of\n po\ sitive integers.\n <P/>\n\n If the optional third argument <A>n</A> is n\ ot present, then the largest\n moved point of the permutation group <A>G</A\ > is used by default.<P/>\n\n <Example><![CDATA[\ngap> digraph := EdgeOrbit\ sDigraph(Group((1, 3), (1, 2)(3, 4)),\n> [[1, \ 2], [4, 5]], 5);\n<immutable digraph with 5 vertices, 12 edges>\ngap> OutNeigh\ bours(digraph);\n[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ], [ ] ]\ \ngap> RepresentativeOutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ ] ]\n]]></Exam\ ple>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Digr\ aphByInNeighbours\" Arg=\"[filt, ]list\"/>\n <Oper Name=\"DigraphByInNeighbor\ s\" Arg=\"[filt, ]list\"/>\n <Returns>A digraph.</Returns>\n <Description>\n\ If the optional first argument <A>filt</A> is present, then this should\n s\ pecify the category or representation the digraph being created will\n belong\ to. For example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then \ the digraph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"Is\ ImmutableDigraph\"/>, then the digraph will be immutable. \n If the optional \ first argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutableDigr\ aph\"/> is used by default.<P/>\n\n If <A>list</A> is a list of lists of posi\ tive integers list the range\n <C>[1 .. Length(<A>list</A>)]</C>, then this f\ unction returns the\n digraph with vertices <M>E^0=</M><C>[1 .. Length(<A>lis\ t</A>)]</C>, and\n edges corresponding to the entries of <A>list</A>. More pr\ ecisely, there is an\n edge with source vertex <C>i</C> and range vertex <C>j\ </C> if <C>i</C> is list\n <C><A>list</A>[j]</C>. <P/>\n\n If <C>i</C> occur\ s list <C><A>list</A>[j]</C> with multiplicity <C>k</C>,\n then there are <C>\ k</C> multiple edges from <C>i</C> to <C>j</C>. <P/>\n\n See <Ref Attr=\"InNe\ ighbours\"/>.\n\n <Example><![CDATA[\ngap> D := DigraphByInNeighbours([\n> \ [2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10],\n> [1], [3, 5, 7, 8, 10], [2, 5, 7],\ [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9], [1, 2, 7, 8], [3, 5]]);\n<immutable d\ igraph with 10 vertices, 37 edges>\ngap> D := DigraphByInNeighbours([[2, 3, 2]\ , [1], [1, 2, 3]]);\n<immutable multidigraph with 3 vertices, 7 edges>\ngap> D\ := DigraphByInNeighbours(IsMutableDigraph, \n> \ [[2, 3, 2], [1], [1, 2, 3]]);\n<mutable multidigraph with 3 vertices, 7 edges>\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper \ Name=\"CayleyDigraph\" Arg=\"G[, gens]\"/>\n <Returns>An immutable digraph.</\ Returns>\n <Description>\n Let <A>G</A> be any group and let <A>gens</A> b\ e a list of elements of\n <A>G</A>. This operation returns an immutable dig\ raph that corresponds to\n the Cayley graph of <A>G</A> with respect\n <\ A>gens</A>. The vertices are the elements of <A>G</A>. There exists an edge\n \ from the vertex <C>u</C> to the vertex <C>v</C> if and only if there exists\ \n a generator <C>g</C> in <A>gens</A> such that <C>x * g = y</C>. <P/>\n\n\ <!-- FIXME What is the correspondence between vertices and elements? -->\n\ \n If the optional second argument <A>gens</A> is not present, then the\n \ generators of <A>G</A> are used by default.<P/>\n\n The digraph created b\ y this operation belongs to the category <Ref\n Filt=\"IsCayleyDigraph\"/\ >, the group <A>G</A> can be recovered from the\n digraph using <Ref Attr=\ \"GroupOfCayleyDigraph\"/>, and the generators\n <A>gens</A> can be obtaine\ d using <Ref Attr=\"GeneratorsOfCayleyDigraph\"/>.<P/>\n\n Note that this f\ unction can only return an immutable digraph.\n\n <Example><![CDATA[\ngap> \ G := DihedralGroup(8);\n<pc group of size 8 with 3 generators>\ngap> CayleyDig\ raph(G);\n<immutable digraph with 8 vertices, 24 edges>\ngap> G := DihedralGro\ up(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\n<imm\ utable digraph with 8 vertices, 16 edges>\ngap> digraph := CayleyDigraph(G, [(\ )]);\n<immutable digraph with 8 vertices, 8 edges>\ngap> GroupOfCayleyDigraph(\ digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]]]></Exam\ ple>\n </Description>\n</ManSection>\n\n </Section>\n\n <Section><Heading>C\ hanging representations</Heading>\n <ManSection>\n <Oper Name=\"AsBinaryRe\ lation\" Arg=\"digraph\"/>\n <Returns>A binary relation.</Returns>\n <Descri\ ption>\n If <A>digraph</A> is a digraph with a positive number of vertices\ \n <M>n</M>, and no multiple edges, then this operation returns a binary\n \ relation on the points <C>[1..n]</C>.\n The pair <C>[i,j]</C> is in the \ binary relation if and only if\n <C>[i,j]</C> is an edge in <A>digraph</A>.\ \n <P/>\n<Example><![CDATA[\ngap> D := Digraph([[3, 2], [1, 2], [2], [3, 4]\ ]);\n<immutable digraph with 4 vertices, 7 edges>\ngap> AsBinaryRelation(D);\n\ Binary Relation on 4 points\n]]></Example>\n </Description>\n</ManSection>\n\ \n <ManSection>\n <Oper Name=\"AsDigraph\" Arg=\"[filt, ]trans[, n]\"/>\n \ <Returns>A digraph, or <K>fail</K>.</Returns>\n <Description>\n If the op\ tional first argument <A>filt</A> is present, then this should\n specify th\ e category or representation the digraph being created will\n belong to. Fo\ r example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the d\ igraph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImm\ utableDigraph\"/>, then the digraph will be immutable. \n If the optional f\ irst argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutableDig\ raph\"/> is used by default.<P/>\n\n If <A>trans</A> is a transformation, a\ nd <A>n</A> is a non-negative integer\n such that the restriction of <A>tra\ ns</A> to <C>[1 .. <A>n</A>]</C> defines\n a transformation of <C>[1 .. <A>\ n</A>]</C>, then <C>AsDigraph</C>\n returns the functional digraph with <A>\ n</A> vertices defined by\n <A>trans</A>. See <Ref Prop=\"IsFunctionalDigra\ ph\"/>.\n <P/>\n\n Specifically, the digraph returned by <C>AsDigraph</C\ > has <A>n</A> edges:\n for each vertex <C>x</C> in <C>[1 .. <A>n</A>]</C>\ , there is a unique edge\n with source <C>x</C>; this edge has range <C>x^<\ A>trans</A></C>.\n <P/>\n\n If the optional second argument <A>n</A> is \ not supplied, then the degree of\n the transformation <A>trans</A> is used \ by default. If the restriction of\n <A>trans</A> to <C>[1 .. <A>n</A>]</C>\ does not define a transformation of\n <C>[1 .. <A>n</A>]</C>, then <C>AsDi\ graph(<A>trans</A>, <A>n</A>)</C>\n returns <K>fail</K>.\n\n <Example><!\ [CDATA[\ngap> f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTransform\ ation( [ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )\ngap> AsDigraph(f);\n<immutable fun\ ctional digraph with 10 vertices>\ngap> AsDigraph(f, 4);\n<immutable functiona\ l digraph with 4 vertices>\ngap> AsDigraph(f, 5);\nfail\ngap> b := BinaryRelat\ ionOnPoints(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation\ on 5 points\ngap> D := AsDigraph(b);\n<immutable digraph with 5 vertices, 11 \ edges>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Oper Name=\"Graph\" Arg=\"digraph\"/>\n <Returns>A &Grape; package graph.</R\ eturns>\n <Description>\n If <A>digraph</A> is a mutable or immutable digr\ aph without multiple edges, then this operation\n returns a &Grape; package\ graph that is isomorphic to <A>digraph</A>. <P/>\n\n If <A>digraph</A> is \ a multidigraph, then since &Grape; does not support\n multiple edges, the m\ ultiple edges will be reduced to a single edge in the\n result. In order wo\ rds, for a multidigraph this operation will return the\n same as <C>Graph(D\ igraphRemoveAllMultipleEdges(</C><A>digraph</A><C>))</C>.\n\n <Example><![C\ DATA[\ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function\ (x, y) return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nrec(\n\ adjacencies := [ [ 3, 5, 8 ] ],\n group := \n Group( [ ( 1, 2, 3, 5, 7)( \ 4, 6, 8, 9,10), ( 2, 4)( 6, 9)( 7,10) \n ] ),\n isGraph := true,\n names\ := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], \n [ 2, 4 ], [ 1,\ 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],\n order := 10,\n representatives := [ \ 1 ],\n schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digraph(Pe\ tersen);\n<immutable digraph with 10 vertices, 30 edges>\ngap> Graph(last) = P\ etersen;\ntrue]]></Example>\n </Description>\n</ManSection>\n\n <ManSectio\ n>\n <Attr Name=\"AsGraph\" Arg=\"digraph\"/>\n <Returns>A &Grape; package g\ raph.</Returns>\n <Description>\n If <A>digraph</A> is a digraph, then thi\ s method returns the same as\n <Ref Oper=\"Graph\"/>, except that if <A>dig\ raph</A> is immutable, then the\n result will be stored as a mutable attrib\ ute of <A>digraph</A>.\n In this latter case, when <C>AsGraph(</C><A>digrap\ h</A><C>)</C> is called\n subsequently, the same &GAP; object will be retur\ ned as before.\n\n <Example><![CDATA[\ngap> D := Digraph([[1, 2], [3], []])\ ;\n<immutable digraph with 3 vertices, 3 edges>\ngap> G := AsGraph(D);\nrec( a\ djacencies := [ [ 1, 2 ], [ 3 ], [ ] ], group := Group(()), \n isGraph := tr\ ue, names := [ 1 .. 3 ], order := 3, \n representatives := [ 1, 2, 3 ], schre\ ierVector := [ -1, -2, -3 ] )\n]]></Example>\n </Description>\n</ManSection>\ \n\n <ManSection>\n <Attr Name=\"AsTransformation\" Arg=\"digraph\"/>\n <\ Returns>A transformation, or <K>fail</K></Returns>\n <Description>\n If <A\ >digraph</A> is a functional digraph, then <C>AsTransformation</C>\n return\ s the transformation which is defined by <A>digraph</A>. See <Ref\n Prop\ =\"IsFunctionalDigraph\"/>. Otherwise,\n <C>AsTransformation(</C><A>digraph\ </A><C>)</C> returns <K>fail</K>. <P/>\n\n If <A>digraph</A> is a function\ al digraph with <M>n</M> vertices, then\n <C>AsTransformation(</C><A>digrap\ h</A><C>)</C> will return the\n transformation <C>f</C> of degree at most <\ M>n</M> where for each <M>1\n \\leq i \\leq n</M>, <C>i ^ f</C> is equal \ to the unique out-neighbour of\n vertex <C>i</C> in <A>digraph</A>.\n\n \ <Example><![CDATA[\ngap> D := Digraph([[1], [3], [2]]);\n<immutable digraph wi\ th 3 vertices, 3 edges>\ngap> AsTransformation(D);\nTransformation( [ 1, 3, 2 \ ] )\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 ver\ tices, 3 edges>\ngap> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] )\ngap\ > AsPermutation(last);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n<immuta\ ble digraph with 3 vertices, 2 edges>\ngap> AsTransformation(D);\nfail]]></Exa\ mple>\n </Description>\n</ManSection>\n\n </Section>\n\n <Section><Heading>\ New digraphs from old</Heading>\n <ManSection>\n <Oper Name=\"DigraphImmut\ ableCopy\" Arg=\"digraph\"/>\n <Oper Name=\"DigraphMutableCopy\" Arg=\"digrap\ h\"/>\n <Oper Name=\"DigraphCopySameMutability\" Arg=\"digraph\"/>\n <Oper N\ ame=\"DigraphCopy\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Des\ cription>\n Each of these operations returns a new copy of <A>digraph</A>,\ \n of the appropriate mutability, retaining none of the attributes or\n \ properties of <A>digraph</A>. <P/>\n\n <C>DigraphCopy</C> is a synonym for \ <C>DigraphCopySameMutability</C>.\n<Example><![CDATA[\ngap> D := CycleDigraph(\ 10);\n<immutable cycle digraph with 10 vertices>\ngap> DigraphCopy(D) = D;\ntr\ ue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\ngap> DigraphMutableCopy(D)\ ;\n<mutable digraph with 10 vertices, 10 edges>\n]]></Example>\n </Descriptio\ n>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphImmutableCopyIfIm\ mutable\" Arg=\"digraph\"/>\n <Oper Name=\"DigraphImmutableCopyIfMutable\" Ar\ g=\"digraph\"/>\n <Oper Name=\"DigraphMutableCopyIfMutable\" Arg=\"digraph\"/\ >\n <Oper Name=\"DigraphMutableCopyIfImmutable\" Arg=\"digraph\"/>\n <Return\ s>A digraph.</Returns>\n <Description>\n Each of these operations returns \ either the original argument\n <A>digraph</A>, or a new copy of <A>digraph<\ /A> of the appropriate\n mutability, according to the mutability of <A>digr\ aph</A>.\n<Example><![CDATA[\ngap> C := CycleDigraph(10);\n<immutable cycle di\ graph with 10 vertices>\ngap> D := DigraphImmutableCopyIfImmutable(C);\n<immut\ able digraph with 10 vertices, 10 edges>\ngap> IsIdenticalObj(C, D);\nfalse\ng\ ap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n<immutable cycl\ e digraph with 10 vertices>\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntr\ ue\ngap> D := DigraphMutableCopyIfMutable(C);\n<immutable cycle digraph with 1\ 0 vertices>\ngap> IsMutableDigraph(D);\nfalse\ngap> D := DigraphMutableCopyIfI\ mmutable(C);\n<mutable digraph with 10 vertices, 10 edges>\ngap> IsMutableDigr\ aph(D);\ntrue\ngap> C := CycleDigraph(IsMutableDigraph, 10);\n<mutable digraph\ with 10 vertices, 10 edges>\ngap> D := DigraphImmutableCopyIfImmutable(C);\n<\ mutable digraph with 10 vertices, 10 edges>\ngap> IsIdenticalObj(C, D);\ntrue\ \ngap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n<immutable d\ igraph with 10 vertices, 10 edges>\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C \ = D;\ntrue\ngap> D := DigraphMutableCopyIfMutable(C);\n<mutable digraph with 1\ 0 vertices, 10 edges>\ngap> IsMutableDigraph(D);\ntrue\ngap> D := DigraphMutab\ leCopyIfImmutable(C);\n<mutable digraph with 10 vertices, 10 edges>\ngap> IsId\ enticalObj(C, D);\ntrue\ngap> IsMutableDigraph(D);\ntrue]]></Example>\n </Des\ cription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"InducedSubdigraph\ \" Arg=\"digraph, verts\"/>\n <Returns>A digraph.</Returns>\n <Description>\ \n If <A>digraph</A> is a digraph, and <A>verts</A> is a subset of the\n \ vertices of <A>digraph</A>, then this operation returns a digraph\n constr\ ucted from <A>digraph</A> by retaining precisely those vertices in\n <A>ver\ ts</A>, and those edges whose source and range vertices are both\n containe\ d in <A>verts</A>. <P/>\n\n The vertices of the induced subdigraph are\n \ <C>[1..Length(verts)]</C> but the original vertex labels can be\n accessed\ via <Ref Oper=\"DigraphVertexLabels\"/>. <P/>\n\n If <A>digraph</A> belong\ s to <Ref Filt=\"IsMutableDigraph\"/>, then\n <A>digraph</A> is modified in\ place. If <A>digraph</A> belongs to <Ref\n Filt=\"IsImmutableDigraph\"/>\ , a new immutable digraph containing the\n appropriate\n vertices and ed\ ges is returned.<P/>\n\n <Example><![CDATA[\ngap> D := Digraph([[1, 1, 2, 3\ , 4, 4], [1, 3, 4], [3, 1], [1, 1]]);\n<immutable multidigraph with 4 vertices\ , 13 edges>\ngap> InducedSubdigraph(D, [1, 3, 4]);\n<immutable multidigraph wi\ th 3 vertices, 9 edges>\ngap> DigraphVertices(last);\n[ 1 .. 3 ]\ngap> D := Di\ graphMutableCopy(D);\n<mutable multidigraph with 4 vertices, 13 edges>\ngap> n\ ew := InducedSubdigraph(D, [1, 3, 4]);\n<mutable multidigraph with 3 vertices,\ 9 edges>\ngap> D = new;\ntrue\n]]></Example>\n </Description>\n</ManSection>\ \n\n <ManSection>\n <Oper Name=\"ReducedDigraph\" Arg=\"digraph\"/>\n <At\ tr Name=\"ReducedDigraphAttr\" Arg=\"digraph\"/>\n <Returns>A digraph.</Retur\ ns>\n <Description>\n This function returns a digraph isomorphic to the su\ bdigraph of\n <A>digraph</A> induced by the set of non-isolated vertices,\n\ i.e. the set of those vertices of <A>digraph</A> which are the source or\n\ range of some edge in <A>digraph</A>.\n See <Ref Oper=\"InducedSubdigra\ ph\"/>.\n <P/>\n\n The ordering of the remaining vertices of <A>digraph<\ /A> is preserved, as are\n the labels of the remaining vertices and edges;\ \n see <Ref Oper=\"DigraphVertexLabels\"/> and <Ref Oper=\"DigraphEdgeLabel\ s\"/>.\n This can allow one to match a vertex in the reduced digraph to the\ \n corresponding vertex in <A>digraph</A>.\n <P/>\n\n If <A>digraph</\ A> is immutable, then a new immutable digraph is returned.\n Otherwise, the\ isolated vertices of the mutable digraph <A>digraph</A> are\n removed in-p\ lace.\n <Example><![CDATA[\ngap> D := Digraph([[1, 2], [], [], [1, 4], []])\ ;\n<immutable digraph with 5 vertices, 4 edges>\ngap> R := ReducedDigraph(D);\ \n<immutable digraph with 3 vertices, 4 edges>\ngap> OutNeighbours(R);\n[ [ 1,\ 2 ], [ ], [ 1, 3 ] ]\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ],\ [ 4, 4 ] ]\ngap> DigraphEdges(R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]\ \ngap> DigraphVertexLabel(R, 3);\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap> D\ := Digraph(IsMutableDigraph, [[], [3], [2]]);\n<mutable digraph with 3 vertic\ es, 2 edges>\ngap> ReducedDigraph(D);\n<mutable digraph with 2 vertices, 2 edg\ es>\ngap> D;\n<mutable digraph with 2 vertices, 2 edges>\n]]></Example>\n </D\ escription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"MaximalSymmetri\ cSubdigraph\" Arg=\"digraph\"/>\n <Attr Name=\"MaximalSymmetricSubdigraphAttr\ \" Arg=\"digraph\"/>\n <Oper Name=\"MaximalSymmetricSubdigraphWithoutLoops\" \ Arg=\"digraph\"/>\n <Attr Name=\"MaximalSymmetricSubdigraphWithoutLoopsAttr\"\ Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Description>\n If <\ A>digraph</A> is a digraph, then <C>MaximalSymmetricSubdigraph</C>\n return\ s a symmetric digraph without multiple edges which has the same\n vertex se\ t as <A>digraph</A>, and whose edge list is formed from\n <A>digraph</A> by\ ignoring the multiplicity of edges, and by ignoring\n edges <C>[u,v]</C> f\ or which there does not exist an edge <C>[v,u]</C>.\n <P/>\n\n The digra\ ph returned by <C>MaximalSymmetricSubdigraphWithoutLoops</C> is\n the same,\ except that loops are removed.<P/>\n \n If <A>digraph</A> is immutable,\ then a new immutable digraph is returned.\n Otherwise, the mutable digraph\ <A>digraph</A> is changed in-place into such\n a digraph described above.<\ P/>\n\n See <Ref Prop=\"IsSymmetricDigraph\"/>, <Ref Prop=\"IsMultiDigraph\ \"/>, and\n <Ref Prop=\"DigraphHasLoops\"/> for more information.\n <Exa\ mple><![CDATA[\ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n<immutable \ multidigraph with 4 vertices, 7 edges>\ngap> not IsSymmetricDigraph(D) and IsM\ ultiDigraph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ \ 3, 1 ] ]\ngap> S := MaximalSymmetricSubdigraph(D);\n<immutable symmetric digra\ ph with 4 vertices, 4 edges>\ngap> IsSymmetricDigraph(S) and not IsMultiDigrap\ h(S);\ntrue\ngap> OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D :=\ CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges\ >\ngap> MaximalSymmetricSubdigraph(D);\n<mutable empty digraph with 3 vertices\ >\ngap> D;\n<mutable empty digraph with 3 vertices>\n]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"MaximalAntiSymmetric\ Subdigraph\" Arg=\"digraph\"/>\n <Attr Name=\"MaximalAntiSymmetricSubdigraphA\ ttr\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Description>\n \ If <A>digraph</A> is a digraph, then <C>MaximalAntiSymmetricSubdigraph</C>\n \ returns an anti-symmetric subdigraph of <A>digraph</A> formed by\n retai\ ning the vertices of <A>digraph</A>, discarding any duplicate edges,\n and \ discarding any edge <C>[i,j]</C> of <A>digraph</A> where <C>i > j</C>\n and\ the reverse edge <C>[j,i]</C> is an edge of <A>digraph</A>.\n In other wor\ ds, for every\n symmetric pair of edges <C>[i,j]</C> and <C>[j,i]</C> in <A\ >digraph</A>, where \n <C>i</C> and <C>j</C> are distinct, it discards the \ edge\n <M>[\\max(i,j),\\min(i,j)]</M>.\n <P/>\n\n If <A>digraph</A> i\ s immutable, then a new immutable digraph is returned.\n Otherwise, the mut\ able digraph <A>digraph</A> is changed in-place. <P/>\n\n See <Ref Prop=\"I\ sAntisymmetricDigraph\"/> for more information.\n <Example><![CDATA[\ngap> \ D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n<immutable multidigraph with 4 v\ ertices, 7 edges>\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph(D);\n\ true\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap> D \ := MaximalAntiSymmetricSubdigraph(D);\n<immutable antisymmetric digraph with 4\ vertices, 4 edges>\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigraph(D);\ \ntrue\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D := Digr\ aph(IsMutableDigraph, [[2], [1]]);\n<mutable digraph with 2 vertices, 2 edges>\ \ngap> MaximalAntiSymmetricSubdigraph(D);\n<mutable digraph with 2 vertices, 1\ edge>\ngap> D;\n<mutable digraph with 2 vertices, 1 edge>\n]]></Example>\n <\ /Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"UndirectedSpa\ nningForest\" Arg=\"digraph\"/>\n <Attr Name=\"UndirectedSpanningForestAttr\"\ Arg=\"digraph\"/>\n <Oper Name=\"UndirectedSpanningTree\" Arg=\"digraph\"/>\ \n <Attr Name=\"UndirectedSpanningTreeAttr\" Arg=\"digraph\"/>\n <Returns>A \ digraph, or <K>fail</K>.</Returns>\n <Description>\n If <A>digraph</A> is \ a digraph with at least one vertex, then\n <C>UndirectedSpanningForest</C> \ returns an undirected spanning forest of\n <A>digraph</A>, otherwise this a\ ttribute returns <K>fail</K>. See <Ref\n Oper=\"IsUndirectedSpanningFores\ t\" /> for the definition of an undirected\n spanning forest.<P/>\n\n If\ <A>digraph</A> is a digraph with at least one vertex and whose <Ref\n Op\ er=\"MaximalSymmetricSubdigraph\"/> is connected (see <Ref\n Prop=\"IsCon\ nectedDigraph\" />), then <C>UndirectedSpanningTree</C> returns\n an undire\ cted spanning tree of <A>digraph</A>, otherwise this attribute\n returns <K\ >fail</K>. See <Ref Oper=\"IsUndirectedSpanningTree\" /> for the\n definit\ ion of an undirected spanning tree.<P/>\n\n If <A>digraph</A> is immutable,\ then an immutable digraph is returned.\n Otherwise, the mutable digraph <A\ >digraph</A> is changed in-place into an\n undirected spanning tree of <A>d\ igraph</A>.<P/>\n\n Note that for an immutable digraph that has known undir\ ected spanning tree,\n the attribute <C>UndirectedSpanningTree</C> returns \ the same digraph as\n the attribute <C>UndirectedSpanningForest</C>.\n\n \ <Example><![CDATA[\ngap> D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);\n\ <immutable multidigraph with 4 vertices, 9 edges>\ngap> UndirectedSpanningTree\ (D);\nfail\ngap> forest := UndirectedSpanningForest(D);\n<immutable symmetric \ digraph with 4 vertices, 4 edges>\ngap> OutNeighbours(forest);\n[ [ 2 ], [ 1 ]\ , [ 4 ], [ 3 ] ]\ngap> IsUndirectedSpanningForest(D, forest);\ntrue\ngap> Digr\ aphConnectedComponents(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> DigraphCon\ nectedComponents(MaximalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ \ngap> UndirectedSpanningForest(MaximalSymmetricSubdigraph(D))\n> = forest;\nt\ rue\ngap> D := CompleteDigraph(4);\n<immutable complete digraph with 4 vertice\ s>\ngap> tree := UndirectedSpanningTree(D);\n<immutable symmetric digraph with\ 4 vertices, 6 edges>\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> tre\ e = UndirectedSpanningForest(D);\ntrue\ngap> UndirectedSpanningForest(EmptyDig\ raph(0));\nfail\ngap> D := PetersenGraph(IsMutableDigraph);\n<mutable digraph \ with 10 vertices, 30 edges>\ngap> UndirectedSpanningTree(D);\n<mutable digraph\ with 10 vertices, 18 edges>\ngap> D;\n<mutable digraph with 10 vertices, 18 e\ dges>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <\ Oper Name=\"QuotientDigraph\" Arg=\"digraph, p\"/>\n <Returns>A digraph.</Ret\ urns>\n <Description>\n If <A>digraph</A> is a digraph, and <A>p</A> is a \ partition of the vertices\n of <A>digraph</A>, then this operation returns \ a digraph constructed by\n amalgamating all vertices of <A>digraph</A> whic\ h lie in the same part of\n <A>p</A>.\n <P/>\n\n A partition of the v\ ertices of <A>digraph</A> is a list of non-empty\n disjoint lists, such tha\ t the union of all the sub-lists is equal\n to vertex set of <A>digraph</A>\ . In particular, each vertex must appear\n in precisely one sub-list.\n \ <P/>\n\n The vertices of <A>digraph</A> in part <C>i</C> of <A>p</A> will b\ ecome\n vertex <C>i</C> in the quotient, and there exists some edge in <A>d\ igraph</A>\n with source in part <C>i</C> and range in part <C>j</C> if and\ only if there\n is an edge from <C>i</C> to <C>j</C> in the quotient.\n \ In particular, this means that the quotient of a digraph has no multiple edge\ s.\n which was a change introduced in version 1.0.0 of the &Digraphs; packa\ ge.\n <P/>\n\n If <A>digraph</A> belongs to <Ref Filt=\"IsMutableDigraph\ \"/>, then\n <A>digraph</A> is modified in place. If <A>digraph</A> belongs\ to <Ref\n Filt=\"IsImmutableDigraph\"/>, a new immutable digraph with th\ e above\n properties is returned.<P/>\n\n <Example><![CDATA[\ngap> D := \ Digraph([[2, 1], [4], [1], [1, 3, 4]]);\n<immutable digraph with 4 vertices, 7\ edges>\ngap> DigraphVertices(D);\n[ 1 .. 4 ]\ngap> DigraphEdges(D);\n[ [ 1, 2\ ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], \n [ 4, 4 ] ]\ngap> p :\ = [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := QuotientDigraph\ (D, p);\n<immutable digraph with 3 vertices, 6 edges>\ngap> DigraphVertices(qu\ o);\n[ 1 .. 3 ]\ngap> DigraphEdges(quo);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 1 ], [ 2,\ 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> QuotientDigraph(EmptyDigraph(0), []);\n<immut\ able empty digraph with 0 vertices>\n]]></Example>\n </Description>\n</ManSec\ tion>\n\n <ManSection>\n <Oper Name=\"DigraphReverse\" Arg=\"digraph\"/>\n\ <Attr Name=\"DigraphReverseAttr\" Arg=\"digraph\"/>\n <Returns>A digraph.</\ Returns>\n <Description>\n The reverse of a digraph is the digraph formed \ by reversing the orientation\n of each of its edges, i.e. for every edge <C\ >[i, j]</C> of a digraph, the\n reverse contains the corresponding edge <C>\ [j, i]</C>.<P/>\n\n <C>DigraphReverse</C> returns the reverse of the digrap\ h <A>digraph</A>. If\n <A>digraph</A> is immutable, then a new immutable di\ graph is returned. Otherwise,\n the mutable digraph <A>digraph</A> is chang\ ed in-place into its reverse.\n <Example><![CDATA[\ngap> D := Digraph([[3],\ [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n<immutable digraph with 5 vertices, \ 11 edges>\ngap> DigraphReverse(D);\n<immutable digraph with 5 vertices, 11 edg\ es>\ngap> OutNeighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], [ 4 ], [\ 2, 5 ] ]\ngap> D := Digraph([[2, 4], [1], [4], [3, 4]]);\n<immutable digraph \ with 4 vertices, 6 edges>\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 4 ], [ 2, 1\ ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> DigraphEdges(DigraphReverse(D));\n[ [\ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> D := CycleDi\ graph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> \ OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> DigraphReverse(D);\n<mutable \ digraph with 3 vertices, 3 edges>\ngap> OutNeighbours(D);\n[ [ 3 ], [ 1 ], [ 2\ ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <O\ per Name=\"DigraphDual\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphDualAttr\" A\ rg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Description>\n The <E\ >dual</E> of <A>digraph</A> has the same vertices as <A>digraph</A>,\n and \ there is an edge in the dual from <C>i</C> to <C>j</C>\n whenever there is \ no edge from <C>i</C> to <C>j</C> in <A>digraph</A>.\n The <E>dual</E> is s\ ometimes called the <E>complement</E>.<P/>\n\n <C>DigraphDual</C> returns t\ he dual of the digraph <A>digraph</A>. If\n <A>digraph</A> is an immutable \ digraph, then a new immutable digraph is\n returned. Otherwise, the mutable\ digraph <A>digraph</A> is changed in-place\n into its dual.\n <Example>\ <![CDATA[\ngap> D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9], [], [\ ], []]);\n<immutable digraph with 9 vertices, 8 edges>\ngap> DigraphDual(D);\n\ <immutable digraph with 9 vertices, 73 edges>\ngap> D := CycleDigraph(IsMutabl\ eDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> DigraphDual(D)\ ;\n<mutable digraph with 3 vertices, 6 edges>\ngap> D;\n<mutable digraph with \ 3 vertices, 6 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n <M\ anSection>\n <Oper Name=\"DigraphSymmetricClosure\" Arg=\"digraph\"/>\n <Att\ r Name=\"DigraphSymmetricClosureAttr\" Arg=\"digraph\"/>\n <Returns>A digraph\ .</Returns>\n <Description>\n If <A>digraph</A> is a digraph, then this at\ tribute gives the minimal\n symmetric digraph which has the same vertices a\ nd contains all the edges of\n <A>digraph</A>.<P/>\n \n A digraph is \ <E>symmetric</E> if its adjacency matrix \n <Ref Attr=\"AdjacencyMatrix\"/>\ is symmetric. For a digraph with multiple\n edges this means that there ar\ e the same number of edges from a vertex\n <C>u</C> to a vertex <C>v</C> as\ there are from <C>v</C> to <C>u</C>; \n see <Ref Prop=\"IsSymmetricDigraph\ \"/>.<P/>\n\n If <A>digraph</A> is immutable, then a new immutable digraph \ is returned.\n Otherwise, the mutable digraph <A>digraph</A> is changed in-\ place into its\n symmetric closure.\n<Example><![CDATA[\ngap> D := Digraph(\ [[1, 2, 3], [2, 4], [1], [3, 4]]);\n<immutable digraph with 4 vertices, 8 edge\ s>\ngap> D := DigraphSymmetricClosure(D);\n<immutable symmetric digraph with 4\ vertices, 11 edges>\ngap> IsSymmetricDigraph(D);\ntrue\ngap> List(OutNeighbou\ rs(D), AsSet);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ]\ngap> D :=\ Digraph([[2, 2], [1]]);\n<immutable multidigraph with 2 vertices, 3 edges>\ng\ ap> D := DigraphSymmetricClosure(D);\n<immutable symmetric multidigraph with 2\ vertices, 4 edges>\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 1 ] ]\ngap> D :=\ CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges\ >\ngap> DigraphSymmetricClosure(D);\n<mutable digraph with 3 vertices, 6 edges\ >\ngap> D;\n<mutable digraph with 3 vertices, 6 edges>\n]]></Example>\n </Des\ cription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphTransitive\ Closure\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphTransitiveClosureAttr\" Arg\ =\"digraph\"/>\n <Oper Name=\"DigraphReflexiveTransitiveClosure\" Arg=\"digra\ ph\"/>\n <Attr Name=\"DigraphReflexiveTransitiveClosureAttr\" Arg=\"digraph\"\ />\n <Returns>A digraph.</Returns>\n <Description>\n If <A>digraph</A> is\ a digraph with no multiple edges, then these\n attributes return the (refl\ exive) transitive closure of <A>digraph</A>.<P/>\n\n A digraph is <E>reflex\ ive</E> if it has a loop at every vertex, and it is\n <E>transitive</E> if \ whenever <C>[i,j]</C> and <C>[j,k]</C> are edges of\n <A>digraph</A>, <C>[i\ ,k]</C> is also an edge. The <E>(reflexive)\n transitive closure</E> of \ a digraph <A>digraph</A> is the least\n (reflexive and) transitive digraph \ containing <A>digraph</A>.<P/>\n\n If <A>digraph</A> is immutable, then a n\ ew immutable digraph is returned.\n Otherwise, the mutable digraph <A>digra\ ph</A> is changed in-place into its\n (reflexive) transitive closure.<P/>\n\ \n Let <M>n</M> be the number of vertices of <A>digraph</A>, and let\n <\ M>m</M> be the number of edges. For an arbitrary digraph, these\n attribut\ es will use a version of the Floyd-Warshall algorithm, with\n complexity <M\ >O(n^3)</M>.\n\n However, for a topologically sortable digraph [see <Ref\n \ Attr=\"DigraphTopologicalSort\"/>], these attributes will use methods\n \ with complexity <M>O(m + n + m \\cdot n)</M> when this is faster.<P/>\n\n \ <Example><![CDATA[\ngap> D := DigraphFromDiSparse6String(\".H`eOWR`Ul^\");\n<\ immutable digraph with 9 vertices, 8 edges>\ngap> IsReflexiveDigraph(D) or IsT\ ransitiveDigraph(D);\nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1, 3 ], [ \ ], [ 5 ], [ ], [ 7, 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := DigraphTransiti\ veClosure(D);\n<immutable transitive digraph with 9 vertices, 18 edges>\ngap> \ OutNeighbours(T);\n[ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [ ], [\ 5 ], \n [ ], [ 7, 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphReflexiveTr\ ansitiveClosure(D);\n<immutable preorder digraph with 9 vertices, 27 edges>\ng\ ap> OutNeighbours(RT);\n[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7, 8, 9, 2\ ], [ 3 ], \n [ 5, 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]\ngap> D \ := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edg\ es>\ngap> DigraphReflexiveTransitiveClosure(D);\n<mutable digraph with 3 verti\ ces, 9 edges>\ngap> D;\n<mutable digraph with 3 vertices, 9 edges>\n]]></Examp\ le>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Digra\ phTransitiveReduction\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphTransitiveRed\ uctionAttr\" Arg=\"digraph\"/>\n <Oper Name=\"DigraphReflexiveTransitiveReduc\ tion\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphReflexiveTransitiveReductionAt\ tr\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Description>\n \ If <A>digraph</A> is a topologically sortable digraph\n [see <Ref Attr=\"Di\ graphTopologicalSort\"/>]\n with no multiple edges, then these operations \ return the\n (reflexive) transitive reduction of <A>digraph</A>.<P/>\n\n \ The (reflexive) transitive reduction of such a digraph is the unique\n lea\ st subgraph such that the (reflexive) transitive closure of the\n subgraph \ is equal to the (reflexive) transitive closure of <A>digraph</A>\n [see <Re\ f Oper=\"DigraphReflexiveTransitiveClosure\"/>].\n In order words, it is th\ e least subgraph of <A>digraph</A> which\n retains the same reachability as\ <A>digraph</A>.<P/>\n\n If <A>digraph</A> is immutable, then a new immutab\ le digraph is returned.\n Otherwise, the mutable digraph <A>digraph</A> is \ changed in-place into its\n (reflexive) transitive reduction.<P/>\n\n Le\ t <M>n</M> be the number of vertices of an arbitrary digraph, and let\n \ <M>m</M> be the number of edges.\n Then these operations\n use methods \ with complexity <M>O(m + n + m \\cdot n)</M>.\n <P/>\n <Example><![CDATA\ [\ngap> D := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(D);\ntrue\ \ngap> D1 := DigraphReflexiveTransitiveReduction(D);\n<immutable digraph with \ 3 vertices, 2 edges>\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeighbours(D1)\ ;\n[ [ 2 ], [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\n<immutab\ le digraph with 3 vertices, 4 edges>\ngap> DigraphHasLoops(D2);\ntrue\ngap> Ou\ tNeighbours(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveTransitiveC\ losure(D)\n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> DigraphTran\ sitiveClosure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D := Digraph(\ IsMutableDigraph, [[1], [1], [1, 2, 3]]);\n<mutable digraph with 3 vertices, 5\ edges>\ngap> DigraphReflexiveTransitiveReduction(D);\n<mutable digraph with 3\ vertices, 2 edges>\ngap> D;\n<mutable digraph with 3 vertices, 2 edges>\n]]><\ /Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\ \"DigraphAddVertex\" Arg=\"digraph[, label ]\"/>\n <Returns>A digraph.</Retur\ ns>\n <Description>\n The operation returns a digraph constructed from <A>\ digraph</A> by adding a\n single new vertex, and no new edges. <P/>\n\n \ If the optional second argument <A>label</A> is a &GAP; object,\n then the \ new vertex will be labelled <A>label</A>.\n <P/>\n\n If <A>digraph</A> b\ elongs to <Ref Filt=\"IsMutableDigraph\"/>, then the\n vertex is added dire\ ctly to <A>digraph</A>. If <A>digraph</A> belongs to\n <Ref Filt=\"IsImmuta\ bleDigraph\"/>, an immutable copy of <A>digraph</A> with\n the additional v\ ertex is returned.<P/>\n\n <Example><![CDATA[\ngap> D := CompleteDigraph(3)\ ;\n<immutable complete digraph with 3 vertices>\ngap> new := DigraphAddVertex(\ D);\n<immutable digraph with 4 vertices, 6 edges>\ngap> D = new;\nfalse\ngap> \ DigraphVertices(new);\n[ 1 .. 4 ]\ngap> new := DigraphAddVertex(D, Group([(1, \ 2)]));\n<immutable digraph with 4 vertices, 6 edges>\ngap> DigraphVertexLabels\ (new);\n[ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := CompleteBipartiteDigraph(IsMu\ tableDigraph, 2, 3);\n<mutable digraph with 5 vertices, 12 edges>\ngap> new :=\ DigraphAddVertex(D);\n<mutable digraph with 6 vertices, 12 edges>\ngap> D = n\ ew;\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\ \n <Oper Name=\"DigraphAddVertices\" Arg=\"digraph, m\"\n Label=\"for a di\ graph and an integer\"/>\n <Oper Name=\"DigraphAddVertices\" Arg=\"digraph, l\ abels\"\n Label=\"for a digraph and a list of labels\"/>\n <!-- The 3-argu\ ment version is deliberately undocumented. It is redundant,\n but it is \ included for backwards compatibility. -->\n <Returns>A digraph.</Returns>\n \ <Description>\n For a non-negative integer <A>m</A>, this operation returns\ a digraph\n constructed from <A>digraph</A> by adding <A>m</A> new vertice\ s.\n <P/>\n\n Otherwise, if <A>labels</A> is a list\n consisting of <\ C>k</C> &GAP; objects, then this operation returns a\n digraph constructed \ from <A>digraph</A> by adding <C>k</C> new vertices,\n which are labelled a\ ccording to this list.\n <P/>\n\n If <A>digraph</A> belongs to <Ref Filt\ =\"IsMutableDigraph\"/>, then the\n vertices are added directly to <A>digra\ ph</A>, which is changed in-place.\n If <A>digraph</A> belongs to <Ref Filt\ =\"IsImmutableDigraph\"/>, then\n <A>digraph</A> itself is returned if no v\ ertices are added\n (i.e. <C><A>m</A>=0</C> or <A>labels</A> is empty), oth\ erwise\n the result is a new immutable digraph.\n <P/>\n\n <Example><\ ![CDATA[\ngap> D := CompleteDigraph(3);\n<immutable complete digraph with 3 ve\ rtices>\ngap> new := DigraphAddVertices(D, 3);\n<immutable digraph with 6 vert\ ices, 6 edges>\ngap> DigraphVertices(new);\n[ 1 .. 6 ]\ngap> new := DigraphAdd\ Vertices(D, [Group([(1, 2)]), \"d\"]);\n<immutable digraph with 5 vertices, 6 \ edges>\ngap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]), \"d\" ]\n\ gap> DigraphAddVertices(D, 0) = D;\ntrue\ngap> D := CompleteBipartiteDigraph(I\ sMutableDigraph, 2, 3);\n<mutable digraph with 5 vertices, 12 edges>\ngap> new\ := DigraphAddVertices(D, 4);\n<mutable digraph with 9 vertices, 12 edges>\nga\ p> D = new;\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <ManS\ ection>\n <Oper Name=\"DigraphAddEdge\" Arg=\"digraph, edge\"\n Label=\"fo\ r a digraph and an edge\"/>\n <Oper Name=\"DigraphAddEdge\" Arg=\"digraph, sr\ c, ran\"\n Label=\"for a digraph, source, and range\"/>\n <Returns>A digra\ ph.</Returns>\n <Description>\n If <A>edge</A> is a pair of vertices of <A\ >digraph</A>, or <A>src</A> and\n <A>ran</A> are vertices of <A>digraph</A>\ ,\n then this operation returns a digraph constructed from <A>digraph</A>\n\ by adding a new edge with source <A>edge</A><C>[1]</C> [<A>src</A>] and ra\ nge\n <A>edge</A><C>[2]</C> [<A>ran</A>].\n <P/>\n \n If <A>digrap\ h</A> belongs to <Ref Filt=\"IsMutableDigraph\"/>, then the edge\n is added\ directly to <A>digraph</A>. If <A>digraph</A> belongs to <Ref\n Filt=\"I\ sImmutableDigraph\"/>, then an immutable copy of <A>digraph</A> with the\n \ additional edge is returned. <P/>\n\n <Example><![CDATA[\ngap> D1 := Digra\ ph([[2], [3], []]);\n<immutable digraph with 3 vertices, 2 edges>\ngap> Digrap\ hEdges(D1);\n[ [ 1, 2 ], [ 2, 3 ] ]\ngap> D2 := DigraphAddEdge(D1, [3, 1]);\n<\ immutable digraph with 3 vertices, 3 edges>\ngap> DigraphEdges(D2);\n[ [ 1, 2 \ ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D3 := DigraphAddEdge(D2, [2, 3]);\n<immutable mu\ ltidigraph with 3 vertices, 4 edges>\ngap> DigraphEdges(D3);\n[ [ 1, 2 ], [ 2,\ 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 4);\n<mut\ able digraph with 4 vertices, 4 edges>\ngap> new := DigraphAddEdge(D, [1, 3]);\ \n<mutable digraph with 4 vertices, 5 edges>\ngap> DigraphEdges(new);\n[ [ 1, \ 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ]\ngap> D = new;\ntrue\n]]></Examp\ le>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Digra\ phAddEdgeOrbit\" Arg=\"digraph, edge\"/>\n <Returns>\n A new digraph.\n <\ /Returns>\n <Description>\n This operation returns a new digraph with the \ same vertices and edges as\n <A>digraph</A> and with additional edges consi\ sting of the orbit of the\n edge <A>edge</A> under the action of the <Ref A\ ttr=\"DigraphGroup\"/> of\n <A>digraph</A>. If <A>edge</A> is already an e\ dge in <A>digraph</A>, then\n <A>digraph</A> is returned unchanged. The arg\ ument <A>digraph</A> must be\n an immutable digraph.\n <P/>\n\n An ed\ ge is simply a pair of vertices of <A>digraph</A>.\n\n <Example><![CDATA[\n\ gap> gr1 := CayleyDigraph(DihedralGroup(8));\n<immutable digraph with 8 vertic\ es, 24 edges>\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n<immutable digra\ ph with 8 vertices, 32 edges>\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ], \ [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4,\ 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 ]\ , [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\n\ gap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [\ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n [\ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5, \ 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ],\ [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := DigraphRe\ moveEdgeOrbit(gr2, [1, 8]);\n<immutable digraph with 8 vertices, 24 edges>\nga\ p> gr3 = gr1;\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <Ma\ nSection>\n <Oper Name=\"DigraphAddEdges\" Arg=\"digraph, edges\"/>\n <Retur\ ns>A digraph.</Returns>\n <Description>\n If <A>edges</A> is a (possibly e\ mpty) list of pairs of vertices of\n <A>digraph</A>, then this operation re\ turns a digraph constructed from\n <A>digraph</A> by adding the edges speci\ fied by <A>edges</A>. More\n precisely, for every <C>edge</C> in <A>edges<\ /A>, a new edge\n will be added with source <C>edge[1]</C> and range <C>edg\ es[2]</C>.\n <P/>\n\n If an edge is included in <A>edges</A> with multip\ licity <C>k</C>,\n then it will be added <C>k</C> times.\n\n If <A>digra\ ph</A> belongs to <Ref Filt=\"IsMutableDigraph\"/>, then the edges\n are ad\ ded directly to <A>digraph</A>. If <A>digraph</A> belongs to <Ref\n Filt=\ \"IsImmutableDigraph\"/>, then the result is returned as an immutable\n dig\ raph.\n <P/>\n\n <Example><![CDATA[\ngap> func := function(n)\n> local \ source, range, i;\n> source := [];\n> range := [];\n> for i in [1 .. n - 2\ ] do\n> Add(source, i);\n> Add(range, i + 1);\n> od;\n> return Digraph\ (n, source, range);\n> end;;\ngap> D := func(1024);\n<immutable digraph with 1\ 024 vertices, 1022 edges>\ngap> new := DigraphAddEdges(D,\n> [[1023, 1024], [1\ , 1024], [1023, 1024], [1024, 1]]);\n<immutable multidigraph with 1024 vertice\ s, 1026 edges>\ngap> D = new;\nfalse\ngap> D2 := DigraphMutableCopy(func(1024)\ );\n<mutable digraph with 1024 vertices, 1022 edges>\ngap> new := DigraphAddEd\ ges(D2,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n<mutable mult\ idigraph with 1024 vertices, 1026 edges>\ngap> D2 = new;\ntrue\n]]></Example>\ \n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphR\ emoveVertex\" Arg=\"digraph, v\"/>\n <Returns>A digraph.</Returns>\n <Descri\ ption>\n If <A>v</A> is a vertex of <A>digraph</A>, then this operation ret\ urns a\n digraph constructed from <A>digraph</A> by removing vertex <A>v</A\ >,\n along with any edge whose source or range vertex is <A>v</A>.<P/>\n\n \ If <A>digraph</A> has <C>n</C> vertices, then the vertices of the returned\ \n digraph are <C>[1..n-1]</C>, but the original labels can be\n accesse\ d via <Ref Oper=\"DigraphVertexLabels\"/>. <P/>\n\n If <A>digraph</A> belon\ gs to <Ref Filt=\"IsMutableDigraph\"/>, then the\n vertex is removed direct\ ly from <A>digraph</A>. If <A>digraph</A> belongs\n to <Ref Filt=\"IsImmuta\ bleDigraph\"/>, an immutable copy of <A>digraph</A>\n without the vertex is\ returned.<P/>\n\n <Example><![CDATA[\ngap> D := Digraph([\"a\", \"b\", \"c\ \"],\n> [\"a\", \"a\", \"b\", \"c\", \"c\"],\n> \ [\"b\", \"c\", \"a\", \"a\", \"c\"]);\n<immutable digraph with 3 vertices,\ 5 edges>\ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" ]\ngap> Digraph\ Edges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 3, 1 ], [ 3, 3 ] ]\ngap> new := D\ igraphRemoveVertex(D, 2);\n<immutable digraph with 2 vertices, 3 edges>\ngap> \ DigraphVertexLabels(new);\n[ \"a\", \"c\" ]\ngap> D := CycleDigraph(IsMutableD\ igraph, 5);\n<mutable digraph with 5 vertices, 5 edges>\ngap> new := DigraphRe\ moveVertex(D, 1);\n<mutable digraph with 4 vertices, 3 edges>\ngap> DigraphVer\ texLabels(D);\n[ 2, 3, 4, 5 ]\ngap> D = new;\ntrue\n]]></Example>\n </Descrip\ tion>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphRemoveVertices\ \" Arg=\"digraph, verts\"/>\n <Returns>A digraph.</Returns>\n <Description>\ \n If <A>verts</A> is a (possibly empty) duplicate-free list of vertices of\ \n <A>digraph</A>, then this operation returns a digraph constructed from\n\ <A>digraph</A> by removing every vertex in <A>verts</A>, along with any ed\ ge\n whose source or range vertex is in <A>verts</A>.<P/>\n\n If <A>digr\ aph</A> has <C>n</C> vertices, then the vertices of the new\n digraph are <\ C>[1 .. n-Length(<A>verts</A>)]</C>, but the\n original labels can be acces\ sed via <Ref Oper=\"DigraphVertexLabels\"/>. <P/>\n\n If <A>digraph</A> bel\ ongs to <Ref Filt=\"IsMutableDigraph\"/>, then the\n vertices are removed d\ irectly from <A>digraph</A>. If <A>digraph</A> belongs\n to <Ref Filt=\"IsI\ mmutableDigraph\"/>, an immutable copy of <A>digraph</A>\n without the vert\ ices is returned.<P/>\n\n <Example><![CDATA[\ngap> D := Digraph([[3], [1, 3\ , 5], [1], [1, 2, 4], [2, 3, 5]]);\n<immutable digraph with 5 vertices, 11 edg\ es>\ngap> SetDigraphVertexLabels(D, [\"a\", \"b\", \"c\", \"d\", \"e\"]);\ngap\ > new := DigraphRemoveVertices(D, [2, 4]);\n<immutable digraph with 3 vertices\ , 4 edges>\ngap> DigraphVertexLabels(new);\n[ \"a\", \"c\", \"e\" ]\ngap> D :=\ CycleDigraph(IsMutableDigraph, 5);\n<mutable digraph with 5 vertices, 5 edges\ >\ngap> new := DigraphRemoveVertices(D, [1, 3]);\n<mutable digraph with 3 vert\ ices, 1 edge>\ngap> DigraphVertexLabels(D);\n[ 2, 4, 5 ]\ngap> D = new;\ntrue\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper \ Name=\"DigraphRemoveEdge\" Arg=\"digraph, edge\"\n Label=\"for a digraph an\ d an edge\"/>\n <Oper Name=\"DigraphRemoveEdge\" Arg=\"digraph, src, ran\"\n \ Label=\"for a digraph, source, and range\"/>\n <Returns>A digraph.</Return\ s>\n <Description>\n If <A>digraph</A> is a digraph with no multiple edges\ and <A>edge</A> is a\n pair of vertices of <A>digraph</A>, or <A>src</A> a\ nd <A>ran</A> are\n vertices of <A>digraph</A>, then this operation returns\ a digraph\n constructed from <A>digraph</A> by removing the edge specified\ by\n <A>edge</A> or <A>[src, ran]</A>. <P/> \n\n If <A>digraph</A> bel\ ongs to <Ref\n Filt=\"IsMutableDigraph\"/>, then the edge is removed dire\ ctly from\n <A>digraph</A>. If <A>digraph</A> belongs to <Ref\n Filt=\ \"IsImmutableDigraph\"/>, an immutable copy of <A>digraph</A> without\n the\ edge is returned.\n <P/>\n\n Note that if <A>digraph</A> belongs to <Re\ f Filt=\"IsImmutableDigraph\"/>,\n then a new copy of <A>digraph</A> will b\ e returned even if <A>edge</A> or \n <A>[src, ran]</A> does not define an e\ dge of <A>digraph</A>.<P/>\n\n <Example><![CDATA[\ngap> D := CycleDigraph(2\ 50000);\n<immutable cycle digraph with 250000 vertices>\ngap> D := DigraphRemo\ veEdge(D, [250000, 1]);\n<immutable digraph with 250000 vertices, 249999 edges\ >\ngap> new := DigraphRemoveEdge(D, [25000, 2]);;\ngap> new = D;\ntrue\ngap> I\ sIdenticalObj(new, D);\nfalse\ngap> D := DigraphMutableCopy(D);;\ngap> new := \ DigraphRemoveEdge(D, 2500, 2);;\ngap> IsIdenticalObj(new, D);\ntrue]]>\n</Exam\ ple>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Digr\ aphRemoveEdgeOrbit\" Arg=\"digraph, edge\"/>\n <Returns>\n A new digraph.\ \n </Returns>\n <Description>\n This operation returns a new digraph with\ the same vertices as\n <A>digraph</A> and with the orbit of the edge <A>ed\ ge</A> (under the action\n of the <Ref Attr=\"DigraphGroup\"/> of <A>digrap\ h</A>) removed.\n If <A>edge</A> is not an edge in <A>digraph</A>, then <A>\ digraph</A> is\n returned unchanged. The argument <A>digraph</A> must be an\ immutable\n digraph.\n <P/>\n\n An edge is simply a pair of vertices\ of <A>digraph</A>.\n <Example><![CDATA[\ngap> gr1 := CayleyDigraph(Dihedra\ lGroup(8));\n<immutable digraph with 8 vertices, 24 edges>\ngap> gr2 := Digrap\ hAddEdgeOrbit(gr1, [1, 8]);\n<immutable digraph with 8 vertices, 32 edges>\nga\ p> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2\ , 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5\ , 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 \ ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\ngap> DigraphEdges(gr2);\n[ [ 1, 2\ ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], \ [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4,\ 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ]\ , [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n\ [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);\n<imm\ utable digraph with 8 vertices, 24 edges>\ngap> gr3 = gr1;\ntrue\n]]></Example\ >\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Digraph\ RemoveEdges\" Arg=\"digraph, edges\"/>\n <Returns>A digraph.</Returns>\n <De\ scription>\n If one of the following holds:\n <List>\n <Item>\n \ <A>digraph</A> is a digraph with no multiple edges, and\n <A>edges<\ /A> is a list of pairs of vertices of <A>digraph</A>, or\n </Item>\n \ <Item> <A>digraph</A> is a digraph and <A>edges</A> is an empty list </Item>\ \n </List>\n\n then this operation returns a digraph constructed from <A\ >digraph</A>\n by removing all of the edges specified by <A>edges</A>\n \ (see <Ref Oper=\"DigraphRemoveEdge\" Label=\"for a digraph and an edge\"/>).\n\ <P/>\n\n If <A>digraph</A> belongs to <Ref Filt=\"IsMutableDigraph\"/>,\ then the edge\n is removed directly from <A>digraph</A>. If <A>digraph</A>\ belongs to <Ref\n Filt=\"IsImmutableDigraph\"/>, the edge is removed fro\ m an immutable copy of\n <A>digraph</A> and this new digraph is returned.<P\ />\n\n Note that if <A>edges</A> is empty, then this operation\n will al\ ways return <A>digraph</A> rather than a copy. Also, if any element\n of <A\ >edges</A> is invalid (i.e. does not define an edge of <A>digraph</A>)\n th\ en that element will simply be ignored.\n\n <Example><![CDATA[\ngap> D := C\ ycleDigraph(250000);\n<immutable cycle digraph with 250000 vertices>\ngap> D :\ = DigraphRemoveEdges(D, [[250000, 1]]);\n<immutable digraph with 250000 vertic\ es, 249999 edges>\ngap> D := DigraphMutableCopy(D);\n<mutable digraph with 250\ 000 vertices, 249999 edges>\ngap> new := DigraphRemoveEdges(D, [[1, 2], [2, 3]\ , [3, 100]]);\n<mutable digraph with 250000 vertices, 249997 edges>\ngap> new \ = D;\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\ \n <Oper Name=\"DigraphRemoveLoops\" Arg=\"digraph\"/>\n <Attr Name=\"Digrap\ hRemoveLoopsAttr\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Desc\ ription>\n If <A>digraph</A> is a digraph, then this operation returns a\n \ digraph constructed from <A>digraph</A> by removing every loop. A loop\n \ is an edge with equal source and range.<P/>\n\n If <A>digraph</A> is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ removed from the mutable digraph <A>digraph</A>\n in-place.\n <Example>\ <![CDATA[\ngap> D := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);\ \n<immutable digraph with 5 vertices, 12 edges>\ngap> DigraphRemoveLoops(D);\n\ <immutable digraph with 5 vertices, 8 edges>\ngap> D := Digraph(IsMutableDigra\ ph, [[1, 2], [1]]);\n<mutable digraph with 2 vertices, 3 edges>\ngap> DigraphR\ emoveLoops(D);\n<mutable digraph with 2 vertices, 2 edges>\ngap> D;\n<mutable \ digraph with 2 vertices, 2 edges>\n]]></Example>\n </Description>\n</ManSecti\ on>\n\n <ManSection>\n <Oper Name=\"DigraphRemoveAllMultipleEdges\" Arg=\"\ digraph\"/>\n <Attr Name=\"DigraphRemoveAllMultipleEdgesAttr\" Arg=\"digraph\ \"/>\n <Returns>A digraph.</Returns>\n <Description>\n If <A>digraph</A> \ is a digraph, then this operation returns a\n digraph constructed from <A>\ digraph</A> by removing all multiple edges.\n The result is the largest sub\ digraph of <A>digraph</A> which does not\n contain multiple edges.<P/>\n\n \ If <A>digraph</A> is immutable, then a new immutable digraph is returned.\n\ Otherwise, the multiple edges of the mutable digraph <A>digraph</A> are\n \ removed in-place.\n <Example><![CDATA[\ngap> D1 := Digraph([[1, 2, 3, 2]\ , [1, 1, 3], [2, 2, 2]]);\n<immutable multidigraph with 3 vertices, 10 edges>\ \ngap> D2 := DigraphRemoveAllMultipleEdges(D1);\n<immutable digraph with 3 ver\ tices, 6 edges>\ngap> OutNeighbours(D2);\n[ [ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ]\nga\ p> D := Digraph(IsMutableDigraph, [[2, 2], [1]]);\n<mutable multidigraph with \ 2 vertices, 3 edges>\ngap> DigraphRemoveAllMultipleEdges(D);\n<mutable digraph\ with 2 vertices, 2 edges>\ngap> D;\n<mutable digraph with 2 vertices, 2 edges\ >\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper\ Name=\"DigraphReverseEdges\" Arg=\"digraph, edges\"\n Label=\"for a digrap\ h and a list of edges\"/>\n <Oper Name=\"DigraphReverseEdge\" Arg=\"digraph, \ edge\"\n Label=\"for a digraph and an edge\"/>\n <Oper Name=\"DigraphRever\ seEdge\" Arg=\"digraph, src, ran\"\n Label=\"for a digraph, source, and ran\ ge\"/>\n <Returns>A digraph.</Returns>\n <Description>\n If <A>digraph</A\ > is a digraph without multiple edges, and <A>edges</A> is a\n list of pair\ s of vertices of <A>digraph</A> (the entries of each pair\n corresponding t\ o the source and the range of an edge, respectively),\n then <C>DigraphReve\ rseEdges</C> returns a digraph constructed from\n <A>digraph</A> by reversi\ ng the orientation of every edge specified by\n <A>edges</A>. If only one e\ dge is to be reversed, then\n <C>DigraphReverseEdge</C> can be used instead\ . In this case, the second\n argument should just be a single vertex-pair, \ or the second and third\n arguments should be the source and range of an ed\ ge respectively. <P/>\n\n Note that even though <A>digraph</A> cannot have \ multiple edges, the\n output may have multiple edges. <P/>\n\n If <A>dig\ raph</A> belongs to <Ref Filt=\"IsMutableDigraph\"/>, then the edges\n are \ reversed in <A>digraph</A>. If <A>digraph</A> belongs to <Ref\n Filt=\"Is\ ImmutableDigraph\"/>, an immutable copy of <A>digraph</A> with the\n specif\ ied edges reversed is returned.<P/>\n <Example><![CDATA[\ngap> D := Digraph\ FromDiSparse6String(\".Tg?i@s?t_e?_qEsC\");\n<immutable digraph with 21 vertic\ es, 8 edges>\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ]\ , [ 7, 19 ], [ 9, 1 ], \n [ 11, 2 ], [ 21, 1 ] ]\ngap> new := DigraphReverseE\ dge(D, [7, 19]);\n<immutable digraph with 21 vertices, 8 edges>\ngap> DigraphE\ dges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, 1 ], [ 11, 2 ], \n\ [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := DigraphMutableCopy(new);;\ngap> new := Di\ graphReverseEdges(D2, [[19, 7]]);;\ngap> D2 = new;\ntrue\ngap> D = new;\ntrue\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Func \ Name=\"DigraphDisjointUnion\" Arg=\"D1, D2, ...\" Label=\"for an\n arbitrar\ y number of digraphs\"/>\n <Func Name=\"DigraphDisjointUnion\" Arg=\"list\" L\ abel=\"for a list of digraphs\"/>\n <Returns>A digraph.</Returns>\n <Descrip\ tion>\n In the first form, if <A>D1</A>, <A>D2</A>, etc. are digraphs, then\ \n <C>DigraphDisjointUnion</C> returns their disjoint union.\n\n In the \ second form, if <A>list</A> is a non-empty list of digraphs, then\n <C>Digr\ aphDisjointUnion</C> returns the disjoint union of the digraphs\n contained\ in the list. <P/>\n\n For a disjoint union of digraphs, the vertex set is\ the disjoint union of\n the vertex sets, and the edge list is the disjoint\ union of the edge lists.\n <P/>\n\n More specifically, for a collection\ of digraphs <A>D1</A>, <A>D2</A>,\n <C>...</C>, the disjoint union with ha\ ve\n <C>DigraphNrVertices(</C><A>D1</A><C>)</C> <C>+</C>\n <C>DigraphNrV\ ertices(</C><A>D2</A><C>)</C> <C>+</C> <C>...</C> vertices.\n The edges of \ <A>D1</A> will remain unchanged, whilst the edges of the\n <C>i</C>th digra\ ph, <A>D</A><C>[i]</C>, will be changed so that they\n belong to the vertic\ es of the disjoint union corresponding to\n <A>D</A><C>[i]</C>. In particu\ lar, the edges of <A>D</A><C>[i]</C> will\n have their source and range inc\ reased by\n <C>DigraphNrVertices(</C><A>D1</A><C>)</C> <C>+</C> <C>...</C> \ <C>+</C>\n <C>DigraphNrVertices(</C><A>D</A><C>[i-1])</C>.\n\n <P/>\n\n \ Note that previously set <Ref Oper=\"DigraphVertexLabels\"/> will be lost.<\ P/>\n\n If the first digraph <A>D1</A> [<A>list[1]</A>] belongs to <Ref\n \ Filt=\"IsMutableDigraph\"/>, then <A>D1</A> [<A>list[1]</A>] is modified in\ \n place to contain the appropriate vertices and edges. If <A>digraph</A> b\ elongs\n to <Ref Filt=\"IsImmutableDigraph\"/>, a new immutable digraph con\ taining the\n appropriate vertices and edges is returned.\n<Example><![CDAT\ A[\ngap> D1 := CycleDigraph(3);\n<immutable cycle digraph with 3 vertices>\nga\ p> OutNeighbours(D1);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> D2 := CompleteDigraph(3);\ \n<immutable complete digraph with 3 vertices>\ngap> OutNeighbours(D2);\n[ [ 2\ , 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> union := DigraphDisjointUnion(D1, D2);\n<imm\ utable digraph with 6 vertices, 9 edges>\ngap> OutNeighbours(union);\n[ [ 2 ],\ [ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ]\n]]></Example>\n </Description>\ \n</ManSection>\n\n <ManSection>\n <Func Name=\"DigraphEdgeUnion\" Arg=\"D\ 1, D2, ...\"\n Label=\"for a positive number of digraphs\"/>\n <Func Name=\ \"DigraphEdgeUnion\" Arg=\"list\" Label=\"for a list of digraphs\"/>\n <Retur\ ns>A digraph.</Returns>\n <Description>\n In the first form, if <A>D1</A>,\ <A>D2</A>, etc. are digraphs, then\n <C>DigraphEdgeUnion</C> returns their\ edge union.\n\n In the second form, if <A>list</A> is a non-empty list of \ digraphs, then\n <C>DigraphEdgeUnion</C> returns the edge union of the digr\ aphs contained\n in the list. <P/>\n\n The vertex set of the edge union\ of a collection of digraphs is the\n <E>union</E> of the vertex sets, whil\ st the edge list of the edge union is\n the <E>concatenation</E> of the edg\ e lists. The number of vertices of the\n edge union is equal to the <E>max\ imum</E> number of vertices of one of the\n digraphs, whilst the number of \ edges of the edge union will equal the\n <E>sum</E> of the number of edges \ of each digraph. <P/>\n\n Note that previously set <Ref Oper=\"DigraphVerte\ xLabels\"/> will be lost.\n <P/>\n\n If the first digraph <A>D1</A> [<A>\ list[1]</A>] belongs to <Ref\n Filt=\"IsMutableDigraph\"/>, then <A>D1</A> \ [<A>list[1]</A>] is modified in\n place to contain the appropriate vertices\ and edges. If <A>digraph</A> belongs\n to <Ref Filt=\"IsImmutableDigraph\"\ />, a new immutable digraph containing the\n appropriate vertices and edges\ is returned.\n<Example><![CDATA[\ngap> D := CycleDigraph(10);\n<immutable cyc\ le digraph with 10 vertices>\ngap> DigraphEdgeUnion(D, D);\n<immutable multidi\ graph with 10 vertices, 20 edges>\ngap> D1 := Digraph([[2], [1]]);\n<immutable\ digraph with 2 vertices, 2 edges>\ngap> D2 := Digraph([[2, 3], [2], [1]]);\n<\ immutable digraph with 3 vertices, 4 edges>\ngap> union := DigraphEdgeUnion(D1\ , D2);\n<immutable multidigraph with 3 vertices, 6 edges>\ngap> OutNeighbours(\ union);\n[ [ 2, 2, 3 ], [ 1, 2 ], [ 1 ] ]\ngap> union = DigraphByEdges(\n> Con\ catenation(DigraphEdges(D1), DigraphEdges(D2)));\ntrue\n]]></Example>\n </Des\ cription>\n</ManSection>\n\n <ManSection>\n <Func Name=\"DigraphJoin\" Arg\ =\"D1, D2, ...\"\n Label=\"for a positive number of digraphs\"/>\n <Func N\ ame=\"DigraphJoin\" Arg=\"list\" Label=\"for a list of digraphs\"/>\n <Return\ s>A digraph.</Returns>\n <Description>\n In the first form, if <A>D1</A>, \ <A>D2</A>, etc. are digraphs, then\n <C>DigraphJoin</C> returns their join.\ \n\n In the second form, if <A>list</A> is a non-empty list of digraphs,\n \ then <C>DigraphJoin</C> returns the join of the digraphs contained in the\n\ list. <P/>\n\n The join of a collection of digraphs <A>D1</A>, <A>D2</A\ >, <C>...</C> is\n formed by first taking the <Ref Func=\"DigraphDisjointUn\ ion\"\n Label=\"for a list of digraphs\"/> of the collection.\n\n In t\ he disjoint union, if <M>i \\neq j</M> then there are no edges between\n ve\ rtices corresponding to digraphs <A>D</A><C>[i]</C> and\n <A>D</A><C>[j]</C\ > in the collection; the join is created by including\n all such edges.<P/>\ \n\n For example, the join of two empty digraphs is a complete bipartite\n \ digraph.<P/>\n\n Note that previously set <Ref Oper=\"DigraphVertexLabel\ s\"/> will be lost.\n <P/>\n\n If the first digraph <A>D1</A> [<A>list[1\ ]</A>] belongs to <Ref\n Filt=\"IsMutableDigraph\"/>, then <A>D1</A> [<A>li\ st[1]</A>] is modified in\n place to contain the appropriate vertices and e\ dges. If <A>digraph</A> belongs\n to <Ref Filt=\"IsImmutableDigraph\"/>, a \ new immutable digraph containing the\n appropriate vertices and edges is re\ turned.\n<Example><![CDATA[\ngap> D := CompleteDigraph(3);\n<immutable complet\ e digraph with 3 vertices>\ngap> IsCompleteDigraph(DigraphJoin(D, D));\ntrue\n\ gap> D2 := CycleDigraph(3);\n<immutable cycle digraph with 3 vertices>\ngap> D\ igraphJoin(D, D2);\n<immutable digraph with 6 vertices, 27 edges>\n]]></Exampl\ e>\n </Description>\n</ManSection>\n\n <ManSection>\n <Func Name=\"Digrap\ hCartesianProduct\" Arg=\"gr1, gr2, ...\"\n Label=\"for a positive number o\ f digraphs\"/>\n <Func Name=\"DigraphCartesianProduct\" Arg=\"list\"\n Lab\ el=\"for a list of digraphs\"/>\n <Returns>A digraph.</Returns>\n <Descripti\ on>\n In the first form, if <A>gr1</A>, <A>gr2</A>, etc. are digraphs, then\ \n <C>DigraphCartesianProduct</C> returns a digraph isomorphic to their\n \ Cartesian product.\n <P/>\n\n In the second form, if <A>list</A> is a \ non-empty list of digraphs,\n then <C>DigraphCartesianProduct</C> returns a\ digraph isomorphic to the \n Cartesian product of the digraphs contained i\ n the list.\n <P/>\n \n Mathematically, the Cartesian product of two dig\ raphs <C>G</C>, <C>H</C>\n is a digraph with vertex set <C>Cartesian(Digrap\ hVertices(G),\n DigraphVertices(H))</C> such that there is an edge from <C>\ [u, u']</C> to\n <C>[v, v']</C> iff <C> u = v </C> and there is an edge fro\ m <C>u'</C> to\n <C>v'</C> in <C>H</C> or <C> u' = v'</C> and there is an e\ dge from\n <C>u</C> to <C>v</C> in <C>G</C>.\n <P/>\n\n Due to techni\ cal reasons, the digraph <C>D</C> returned by\n <C>DigraphCartesianProduct<\ /C> has vertex set\n <C>[1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]</\ C> instead, and the\n exact\n method of encoding pairs of vertices into\ integers is implementation\n specific. The original vertex pair can be so\ mewhat regained by using\n <Ref Attr=\"DigraphCartesianProductProjections\"\ />. In addition,\n <Ref Oper=\"DigraphVertexLabels\"/> are preserved: if ve\ rtex pair\n <C>[u,u']</C> was encoded as <C>i</C> then the vertex label of \ <C>i</C>\n will be the pair of vertex labels of <C>u</C> and <C>u'</C> i.e.\ \n <C>DigraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVer\ texLabel(H,u')]</C>.\n <P/>\n\n As the Cartesian product is associative,\ the Cartesian product of a collection \n of digraphs <A>gr1</A>, <A>gr2</A\ >, <C>...</C> is computed in the obvious \n fashion. \n <P/>\n\n<Example\ ><![CDATA[\ngap> gr := ChainDigraph(4);\n<immutable chain digraph with 4 verti\ ces>\ngap> gr2 := CycleDigraph(3);\n<immutable cycle digraph with 3 vertices>\ \ngap> gr3 := DigraphCartesianProduct(gr, gr2);\n<immutable digraph with 12 ve\ rtices, 21 edges>\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[2, 5], [3, 6], \ [4, 7], [8], \n> [6, 9], [7, 10], [8, 11], [12],\n> [10, 1],\ [11, 2], [12, 3], [4]]));\ntrue\n]]></Example>\n </Description>\n</ManSectio\ n>\n\n <ManSection>\n <Func Name=\"DigraphDirectProduct\" Arg=\"gr1, gr2, \ ...\"\n Label=\"for a positive number of digraphs\"/>\n <Func Name=\"Digra\ phDirectProduct\" Arg=\"list\"\n Label=\"for a list of digraphs\"/>\n <Ret\ urns>A digraph.</Returns>\n <Description>\n In the first form, if <A>gr1</\ A>, <A>gr2</A>, etc. are digraphs, then\n <C>DigraphDirectProduct</C> retur\ ns a digraph isomorphic to their direct\n product.\n <P/>\n\n In the \ second form, if <A>list</A> is a non-empty list of digraphs, then\n <C>Digr\ aphDirectProduct</C> returns a digraph isomorphic to the direct\n product o\ f the digraphs contained in the list.\n <P/>\n\n Mathematically, the dir\ ect product of two digraphs <C>G</C>, <C>H</C> is a\n digraph with vertex s\ et <C>Cartesian(DigraphVertices(G),\n DigraphVertices(H))</C> such that the\ re is an edge from <C>[u, u']</C>\n to <C>[v, v']</C> iff there is an edge \ from <C>u</C> to <C>v</C> in\n <C>G</C> and an edge from <C>u'</C> to <C>v'\ </C> in <C>H</C>. \n <P/> \n\n Due to technical reasons, the digraph <C>\ D</C> returned by\n <C>DigraphDirectProduct</C> has vertex set\n <C>[1 .\ . DigraphNrVertices(G)*DigraphNrVertices(H)]</C> instead, and the\n exact \ method of encoding pairs of vertices into integers is implementation\n spec\ ific. The original vertex pair can be somewhat regained by using\n <Ref At\ tr=\"DigraphDirectProductProjections\"/>. In addition\n <Ref Oper=\"Digraph\ VertexLabels\"/> are preserved: if vertex pair\n <C>[u,u']</C> was encoded \ as <C>i</C> then the vertex label of <C>i</C>\n will be the pair of vertex \ labels of <C>u</C> and <C>u'</C> i.e.\n <C>DigraphVertexLabel(D,i) = [Digra\ phVertexLabel(G,u),\n DigraphVertexLabel(H,u')]</C>.\n <P/>\n\n As th\ e direct product is associative, the direct product of a collection \n of d\ igraphs <A>gr1</A>, <A>gr2</A>, <C>...</C> is computed in the obvious \n fa\ shion. \n <P/>\n\n<Example><![CDATA[\ngap> gr := ChainDigraph(4);\n<immutab\ le chain digraph with 4 vertices>\ngap> gr2 := CycleDigraph(3);\n<immutable cy\ cle digraph with 3 vertices>\ngap> gr3 := DigraphDirectProduct(gr, gr2);\n<imm\ utable digraph with 12 vertices, 9 edges>\ngap> IsIsomorphicDigraph(gr3, \n> D\ igraph([[6], [7], [8], [], \n> [10], [11], [12], [],\n> [2],\ [3], [4], []]));\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Attr Name=\"DigraphCartesianProductProjections\" Arg=\"digra\ ph\"/>\n <Returns>A list of transformations.</Returns>\n <Description>\n \ If <A>digraph</A> is a Cartesian product digraph, \n <C>digraph = DigraphCa\ rtesianProduct(gr_1, gr_2, ... )</C>, then\n <C>DigraphCartesianProductProj\ ections</C> returns a list <C>proj</C>\n such that <C>proj[i]</C> is the pr\ ojection onto the <C>i</C>-th\n coordinate of the product.\n <P/>\n\n \ A projection is an idempotent endomorphism of <A>digraph</A>. If\n <C>gr1,\ gr2, ...</C> are all loopless digraphs, then the induced \n subdigraph of \ <A>digraph</A> on the image of <C>proj[i]</C> is \n isomorphic to <C>gr_i</\ C>.\n <P/>\n\n Currently this attribute is only set upon creating an imm\ utable \n digraph via <C>DigraphCartesianProduct</C> and there is no way \n\ of calculating it for other digraphs.\n <P/>\n\n For more informatio\ n see\n <Ref Func=\"DigraphCartesianProduct\" Label=\"for a list of digraph\ s\"/> \n\n <Example><![CDATA[\ngap> D := DigraphCartesianProduct(ChainDigra\ ph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigraphCartesianPro\ ductProjections(D);\ntrue\ngap> proj := DigraphCartesianProductProjections(D);\ ; Length(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\ngap> RankOfTransformati\ on(proj[3]);\n2\ngap> S := ImageSetOfTransformation(proj[2]);;\ngap> IsIsomorp\ hicDigraph(CycleDigraph(4), InducedSubdigraph(D, S));\ntrue\n]]></Example>\n \ </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphDirec\ tProductProjections\" Arg=\"digraph\"/>\n <Returns>A list of transformations.\ </Returns>\n <Description>\n If <A>digraph</A> is a direct product digraph\ , \n <C>digraph = DigraphDirectProduct(gr_1, gr_2, ... )</C>, then\n <C>\ DigraphDirectProductProjections</C> returns a list <C>proj</C>\n such that \ <C>proj[i]</C> is the projection onto the <C>i</C>-th\n coordinate of the p\ roduct.\n <P/>\n\n A projection is an idempotent endomorphism of <A>digr\ aph</A>. If\n <C>gr1, gr2, ...</C> are all loopless digraphs, then the imag\ e \n of <A>digraph</A> under <C>proj[i]</C> is isomorphic to <C>gr_i</C>.\n\ <P/>\n\n Currently this attribute is only set upon creating an immutabl\ e\n digraph via <C>DigraphDirectProduct</C> and there is no way of \n ca\ lculating it for other digraphs.\n <P/>\n\n For more information, see\n \ <Ref Func=\"DigraphDirectProduct\" Label=\"for a list of digraphs\"/> \n\n \ <Example><![CDATA[\ngap> D := DigraphDirectProduct(ChainDigraph(3), CycleDi\ graph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigraphDirectProductProjections(D\ );\ntrue\ngap> proj := DigraphDirectProductProjections(D);; Length(proj);\n3\n\ gap> IsIdempotent(proj[2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap\ > P := DigraphRemoveAllMultipleEdges(\n> ReducedDigraph(OnDigraphs(D, proj[2])\ ));; \ngap> IsIsomorphicDigraph(CycleDigraph(4), P);\ntrue\n]]></Example>\n <\ /Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"LineDigraph\"\ Arg=\"digraph\"/>\n <Oper Name=\"EdgeDigraph\" Arg=\"digraph\"/>\n <Returns\ >A digraph.</Returns>\n <Description>\n Given a digraph <A>digraph</A>, th\ e operation returns the digraph\n obtained by associating a vertex with eac\ h edge of <A>digraph</A>, and\n creating an edge from a vertex <K>v</K> to \ a vertex <K>u</K> if and only\n if the terminal vertex of the edge associat\ ed with <K>v</K> is the start\n vertex of the edge associated with <K>u</K>\ .<P/>\n\n Note that the returned digraph is always a new immutable digraph,\ and the\n argument <A>digraph</A> is never modified.\n <Example><![CDAT\ A[\ngap> LineDigraph(CompleteDigraph(3));\n<immutable digraph with 6 vertices,\ 12 edges>\ngap> LineDigraph(ChainDigraph(3));\n<immutable digraph with 2 vert\ ices, 1 edge>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSecti\ on>\n <Oper Name=\"LineUndirectedDigraph\" Arg=\"digraph\"/>\n <Oper Name=\"\ EdgeUndirectedDigraph\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n \ <Description>\n Given a symmetric digraph <A>digraph</A>, the operation ret\ urns the\n symmetric digraph obtained by associating a vertex with each edg\ e of\n <A>digraph</A>, ignoring directions and multiplicites, and adding an\ edge\n between two vertices if and only if the corresponding edges have a \ vertex in\n common.<P/>\n\n Note that the returned digraph is always a n\ ew immutable digraph, and the\n argument <A>digraph</A> is never modified.\ \n <Example><![CDATA[\ngap> LineUndirectedDigraph(CompleteDigraph(3));\n<im\ mutable digraph with 3 vertices, 6 edges>\ngap> LineUndirectedDigraph(DigraphS\ ymmetricClosure(ChainDigraph(3)));\n<immutable digraph with 2 vertices, 2 edge\ s>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Ope\ r Name=\"DoubleDigraph\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n \ <Description>\n Let <A>digraph</A> be a digraph with vertex set <C>V</C>. \ This function\n returns the double digraph of <A>digraph</A>. The vertex se\ t of the double\n digraph is the orginal vertex set together with a duplica\ te. The edges are\n <C>[u_1, v_2]</C> and <C>[u_2, v_1]</C> if and only if \ <C>[u, v]</C> is an\n edge in <A>digraph</A>, together with the original ed\ ges and their\n duplicates.<P/>\n\n If <A>digraph</A> is mutable, then <\ A>digraph</A> is modified in-place. If\n <A>digraph</A> is immutable, then \ a new immutable digraph constructed as\n described above is returned.\n \ <Example><![CDATA[\ngap> gamma := Digraph([[2], [3], [1]]);\n<immutable digrap\ h with 3 vertices, 3 edges>\ngap> DoubleDigraph(gamma);\n<immutable digraph wi\ th 6 vertices, 12 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Oper Name=\"BipartiteDoubleDigraph\" Arg=\"digraph\"/>\n <\ Returns>A digraph.</Returns>\n <Description>\n Let <A>digraph</A> be a dig\ raph with vertex set <C>V</C>. This function\n returns the bipartite double\ digraph of <A>digraph</A>. The vertex set of\n the double digraph is the o\ riginal vertex set together with a duplicate. The\n edges are <C>[u_1, v_2]\ </C> and <C>[u_2, v_1]</C> if and only if <C>[u,\n v]</C> is an edge in <\ A>digraph</A>. The resulting graph is bipartite,\n since the orignal edges \ are not included in the resulting digraph.<P/>\n\n If <A>digraph</A> is mut\ able, then <A>digraph</A> is modified in-place. If\n <A>digraph</A> is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n <Example><![CDATA[\ngap> gamma := Digraph([[2], [3], [1]]);\n<immu\ table digraph with 3 vertices, 3 edges>\ngap> BipartiteDoubleDigraph(gamma);\n\ <immutable digraph with 6 vertices, 6 edges>\n]]></Example>\n </Description>\ \n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphAddAllLoops\" Arg=\ \"digraph\"/>\n <Attr Name=\"DigraphAddAllLoopsAttr\" Arg=\"digraph\"/>\n <R\ eturns>A digraph.</Returns>\n <Description>\n For a digraph <A>digraph</A>\ this operation returns a new digraph constructed\n from <A>digraph</A>, su\ ch that a loop is added for every vertex which did not\n have a loop in <A>\ digraph</A>.<P/>\n\n If <A>digraph</A> is immutable, then a new immutable d\ igraph is returned.\n Otherwise, the loops are added to the loopless vertic\ es of the mutable digraph\n <A>digraph</A> in-place.\n <Example><![CDATA\ [\ngap> D := EmptyDigraph(13);\n<immutable empty digraph with 13 vertices>\nga\ p> D := DigraphAddAllLoops(D);\n<immutable reflexive digraph with 13 vertices,\ 13 edges>\ngap> OutNeighbours(D);\n[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ]\ , [ 7 ], [ 8 ], [ 9 ], \n [ 10 ], [ 11 ], [ 12 ], [ 13 ] ]\ngap> D := Digraph\ ([[1, 2, 3], [1, 3], [1]]);\n<immutable digraph with 3 vertices, 6 edges>\ngap\ > D := DigraphAddAllLoops(D);\n<immutable reflexive digraph with 3 vertices, 8\ edges>\ngap> OutNeighbours(D);\n[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ]\ngap> \ D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 e\ dges>\ngap> DigraphAddAllLoops(D);\n<mutable digraph with 3 vertices, 6 edges>\ \ngap> D;\n<mutable digraph with 3 vertices, 6 edges>\n]]></Example>\n </Desc\ ription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DistanceDigraph\" \ Arg=\"digraph, i\"\n Label=\"for digraph and int\"/>\n <Oper Name=\"Di\ stanceDigraph\" Arg=\"digraph, list\"\n Label=\"for digraph and list\"/\ >\n <Returns>A digraph.</Returns>\n <Description>\n The first argument is\ a digraph, the second argument is a non-negative\n integer or a list of po\ sitive integers. This operation returns a digraph on\n the same set of vert\ ices as <A>digraph</A>, with two vertices being adjacent\n if and only if t\ he distance between them in <A>digraph</A> equals <A>i</A>\n or is a number\ in <A>list</A>. See <Ref Oper=\"DigraphShortestDistance\"\n Label=\"for a\ digraph and two vertices\"/>. <P/>\n\n If <A>digraph</A> is mutable, then \ <A>digraph</A> is modified in-place. If\n <A>digraph</A> is immutable, then\ a new immutable digraph constructed as\n described above is returned.\n \ <Example><![CDATA[\ngap> digraph := DigraphFromSparse6String(\n> \":]n?AL`BC_\ DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfHMjKM\");\n<immutable symmetric digraph with \ 30 vertices, 90 edges>\ngap> DistanceDigraph(digraph, 1);\n<immutable digraph \ with 30 vertices, 90 edges>\ngap> DistanceDigraph(digraph, [1, 2]);\n<immutabl\ e digraph with 30 vertices, 270 edges>\n]]></Example>\n </Description>\n</Man\ Section>\n\n <ManSection>\n <Oper Name=\"DigraphClosure\" Arg=\"digraph, k\ \"/>\n <Returns>A digraph.</Returns>\n <Description>\n Given a symmetric \ loopless digraph with no multiple edges <A>digraph</A>,\n the <E><A>k</A>-c\ losure of <A>digraph</A></E> is defined to be the unique\n smallest symmetr\ ic loopless digraph <C>C</C> with no multiple edges on the\n vertices of <A\ >digraph</A> that contains all the edges of <A>digraph</A>\n and satsifies \ the property that the sum of the degrees of every two\n non-adjacenct verti\ ces in <C>C</C> is less than <A>k</A>. See <Ref\n Prop=\"IsSymmetricDigraph\ \"/>, <Ref Prop=\"DigraphHasLoops\"/>, <Ref\n Prop=\"IsMultiDigraph\"/>, an\ d <Ref Oper=\"OutDegreeOfVertex\"/>. <P/>\n\n The operation <C>DigraphClosu\ re</C> returns the <A>k</A>-closure of\n <A>digraph</A>.\n <Example><![C\ DATA[\ngap> D := CompleteDigraph(6);\n<immutable complete digraph with 6 verti\ ces>\ngap> D := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);\n<immutable digraph w\ ith 6 vertices, 28 edges>\ngap> closure := DigraphClosure(D, 6);\n<immutable d\ igraph with 6 vertices, 30 edges>\ngap> IsCompleteDigraph(closure);\ntrue\n]]>\ </Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\ \"DigraphMycielskian\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphMycielskianAtt\ r\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Description>\n I\ f <A>digraph</A> is a symmetric digraph, then\n <C>DigraphMycielskian</C> r\ eturns the Mycielskian of <A>digraph</A>.<P/>\n \n The Mycielskian of a \ symmetric digraph is a larger symmetric digraph\n constructed from it, whic\ h has a larger chromatic number. For further\n information, see <URL>https:\ //en.wikipedia.org/wiki/Mycielskian</URL>.<P/>\n\n If <A>digraph</A> is imm\ utable, then a new immutable digraph is returned.\n Otherwise, the mutable \ digraph <A>digraph</A> is changed in-place into\n its Mycielskian.<P/>\n\n \ <Example><![CDATA[\ngap> D := CycleDigraph(2);\n<immutable cycle digraph wi\ th 2 vertices>\ngap> ChromaticNumber(D);\n2\ngap> D := DigraphMycielskian(D);\ \n<immutable digraph with 5 vertices, 10 edges>\ngap> ChromaticNumber(D);\n3\n\ gap> D := DigraphMycielskian(D);\n<immutable digraph with 11 vertices, 40 edge\ s>\ngap> ChromaticNumber(D);\n4\ngap> D := CompleteBipartiteDigraph(IsMutable,\ 2, 3);\n<mutable digraph with 5 vertices, 12 edges>\ngap> DigraphMycielskian(\ D);\n<mutable digraph with 11 vertices, 46 edges>\ngap> D;\n<mutable digraph w\ ith 11 vertices, 46 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n\ </Section>\n\n <Section><Heading>Random digraphs</Heading>\n <ManSection\ >\n <Oper Name=\"RandomDigraph\" Arg=\"[filt, ]n[, p]\"/>\n <Returns>A digra\ ph.</Returns>\n <Description>\n If the optional first argument <A>filt</A>\ is present, then this should\n specify the category or representation the \ digraph being created will\n belong to. For example, if <A>filt</A> is <Ref\ Filt=\"IsMutableDigraph\"/>,\n then the digraph being created will be muta\ ble, if <A>filt</A> is <Ref\n Filt=\"IsImmutableDigraph\"/>, then the digra\ ph will be immutable. \n If the optional first argument <A>filt</A> is not \ present, then <Ref\n Filt=\"IsImmutableDigraph\"/> is used by default.<P/\ >\n\n If <A>n</A> is a positive integer, then this function returns a rando\ m\n digraph with <A>n</A> vertices and without multiple edges. The result\n\ may or may not have loops.<P/>\n\n If the optional second argument <A>p\ </A> is a float with value\n <M>0 \\leq </M> <A> p </A> <M> \\leq 1</M>, th\ en an edge will exist between each\n pair of vertices with probability appr\ oximately <A>p</A>.\n If <A>p</A> is not specified, then a random probabili\ ty will be assumed\n (chosen with uniform probability).\n <Log><![CDATA[\ \ngap> RandomDigraph(1000);\n<immutable digraph with 1000 vertices, 364444 edg\ es>\ngap> RandomDigraph(10000, 0.023);\n<immutable digraph with 10000 vertices\ , 2300438 edges>\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2);\n<mutable\ digraph with 1000 vertices, 499739 edges>\n]]></Log>\n </Description>\n</Man\ Section>\n\n <ManSection>\n <Oper Name=\"RandomMultiDigraph\" Arg=\"n[, m]\ \"/>\n <Returns>A digraph.</Returns>\n <Description>\n If <A>n</A> is a p\ ositive integer, then this function returns a random\n digraph with <A>n</A\ > vertices. If the optional second argument\n <A>m</A> is a positive intege\ r, then the digraph will have <A>m</A>\n edges. If <A>m</A> is not specifie\ d, then the number of edges will be\n chosen randomly (with uniform probabi\ lity) from the range\n <C>[1 .. </C><M>{n \\choose 2}</M><C>]</C>.\n <P/\ >\n\n The method used by this function chooses each edge from the set of\n \ all possible edges with uniform probability. No effort is made to avoid\n \ creating multiple edges, so it is possible (but not guaranteed) that the\n \ result will have multiple edges. The result may or may not have loops.<P/>\n\ <Log><![CDATA[\ngap> RandomMultiDigraph(1000);\n<immutable multidigraph wi\ th 1000 vertices, 216659 edges>\ngap> RandomMultiDigraph(1000, 950);\n<immutab\ le multidigraph with 1000 vertices, 950 edges>\n]]></Log>\n </Description>\n<\ /ManSection>\n\n <ManSection>\n <Oper Name=\"RandomTournament\" Arg=\"[fil\ t, ]n\"/>\n <Returns>A digraph.</Returns>\n <Description>\n If the option\ al first argument <A>filt</A> is present, then this should\n specify the ca\ tegory or representation the digraph being created will\n belong to. For ex\ ample, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the digra\ ph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImmutab\ leDigraph\"/>, then the digraph will be immutable. \n If the optional first\ argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutableDigra\ ph\"/> is used by default.<P/>\n\n If <A>n</A> is a non-negative integer, t\ his function returns a random\n tournament with <A>n</A> vertices. See <Ref\ Prop=\"IsTournament\"/>. <P/>\n <Log><![CDATA[\ngap> RandomTournament(10);\ \n<immutable tournament with 10 vertices>\ngap> RandomTournament(IsMutableDigr\ aph, 10);\n<mutable digraph with 1000 vertices, 500601 edges>\n]]></Log>\n </\ Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"RandomLattice\ \" Arg=\"n\"/>\n <Returns>A digraph.</Returns>\n <Description>\n If the o\ ptional first argument <A>filt</A> is present, then this should\n specify t\ he category or representation the digraph being created will\n belong to. F\ or example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the \ digraph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsIm\ mutableDigraph\"/>, then the digraph will be immutable. \n If the optional \ first argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutable\ Digraph\"/> is used by default.<P/>\n\n If <A>n</A> is a positive integer, \ this function return a random lattice\n with <C>m</C> vertices, where it is\ guaranteed that <C>m</C> is between\n <A>n</A> and <C>2 * <A>n</A></C>. Se\ e <Ref Prop=\"IsLatticeDigraph\"/>. <P/>\n <Log><![CDATA[\ngap> RandomLatti\ ce(10);\n<immutable lattice digraph with 10 vertices, 39 edges>\ngap> RandomLa\ ttice(IsMutableDigraph, 10);\n<mutable digraph with 12 vertices, 52 edges>\n]]\ ></Log>\n </Description>\n</ManSection>\n\n </Section>\n\n <Section><Headin\ g>Standard examples</Heading>\n <ManSection>\n <Oper Name=\"ChainDigraph\"\ Arg=\"[filt, ]n\"/>\n <Returns>A digraph.</Returns>\n <Description>\n If\ <A>n</A> is a positive integer, this function returns a chain\n with <A>n<\ /A> vertices and <C><A>n</A> - 1</C> edges.\n Specifically,\n for each v\ ertex <C>i</C> (with <C>i</C> < <C>n</C>), there is a directed\n edge wi\ th source <C>i</C> and range <C>i + 1</C>. <P/>\n\n If the optional first a\ rgument <A>filt</A> is present, then this should\n specify the category or \ representation the digraph being created will\n belong to. For example, if \ <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the digraph being c\ reated will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImmutableDigraph\ \"/>, then the digraph will be immutable. \n If the optional first argument\ <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutableDigraph\"/> is\ used by default.<P/>\n\n The <Ref Oper=\"DigraphReflexiveTransitiveClosure\ \"/> of a chain represents a\n total order.\n <Example><![CDATA[\ngap> C\ hainDigraph(42);\n<immutable chain digraph with 42 vertices>\ngap> ChainDigrap\ h(IsMutableDigraph, 10);\n<mutable digraph with 10 vertices, 9 edges>\n]]></Ex\ ample>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Co\ mpleteDigraph\" Arg=\"[filt, ]n\"/>\n <Returns>A digraph.</Returns>\n <Descr\ iption>\n If <A>n</A> is a non-negative integer, this function returns the \ complete\n digraph with <A>n</A> vertices. See <Ref Prop=\"IsCompleteDigrap\ h\"/>. <P/>\n\n If the optional first argument <A>filt</A> is present, then\ this should\n specify the category or representation the digraph being cre\ ated will\n belong to. For example, if <A>filt</A> is <Ref Filt=\"IsMutable\ Digraph\"/>,\n then the digraph being created will be mutable, if <A>filt</\ A> is <Ref\n Filt=\"IsImmutableDigraph\"/>, then the digraph will be immuta\ ble. \n If the optional first argument <A>filt</A> is not present, then <Re\ f\n Filt=\"IsImmutableDigraph\"/> is used by default.\n <Example><![CD\ ATA[\ngap> CompleteDigraph(20);\n<immutable complete digraph with 20 vertices>\ \ngap> CompleteDigraph(IsMutableDigraph, 10);\n<mutable digraph with 10 vertic\ es, 90 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSecti\ on>\n <Oper Name=\"CompleteBipartiteDigraph\" Arg=\"[filt, ]m, n\"/>\n <Retu\ rns>A digraph.</Returns>\n <Description>\n A complete bipartite digraph is\ a digraph whose vertices\n can be partitioned into two non-empty vertex se\ ts, such there exists\n a unique edge with source <C>i</C> and range <C>j</\ C> if and only if\n <C>i</C> and <C>j</C> lie in different vertex sets. <P/\ >\n\n If <A>m</A> and <A>n</A> are positive integers, this function\n re\ turns the complete bipartite digraph with vertex sets of sizes\n <A>m</A> (\ containing the vertices <C>[1 .. m]</C>) and\n <A>n</A> (containing the ver\ tices <C>[m + 1 .. m + n]</C>).\n <P/>\n\n If the optional first argumen\ t <A>filt</A> is present, then this should\n specify the category or repres\ entation the digraph being created will\n belong to. For example, if <A>fil\ t</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the digraph being created\ will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImmutableDigraph\"/>, t\ hen the digraph will be immutable. \n If the optional first argument <A>fil\ t</A> is not present, then <Ref\n Filt=\"IsImmutableDigraph\"/> is used b\ y default.\n\n <Example><![CDATA[\ngap> CompleteBipartiteDigraph(2, 3);\n<i\ mmutable complete bipartite digraph with bicomponent sizes 2 and 3>\ngap> Comp\ leteBipartiteDigraph(IsMutableDigraph, 3, 2);\n<mutable digraph with 5 vertice\ s, 12 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSect\ ion>\n <Oper Name=\"CompleteMultipartiteDigraph\" Arg = \"[filt, ]orders\"/\ >\n <Returns>A digraph.</Returns>\n <Description>\n For a list <A>o\ rders</A> of <C>n</C> positive integers,\n this function returns the digr\ aph containing <C>n</C> independent sets\n of vertices of orders <C>[<A>l\ </A>[1] .. <A>l</A>[n]]</C>. Moreover,\n each vertex is adjacent to every\ other not contained in the\n same independent set.\n <P/>\n\n \ If the optional first argument <A>filt</A> is present, then this should\n \ specify the category or representation the digraph being created will\n \ belong to. For example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n \ then the digraph being created will be mutable, if <A>filt</A> is <Ref\n \ Filt=\"IsImmutableDigraph\"/>, then the digraph will be immutable. \n \ If the optional first argument <A>filt</A> is not present, then <Ref\n \ Filt=\"IsImmutableDigraph\"/> is used by default.\n <Example><![CDATA[\ \ngap> CompleteMultipartiteDigraph([5, 4, 2]);\n<immutable complete multiparti\ te digraph with 11 vertices, 76 edges>\ngap> CompleteMultipartiteDigraph(IsMut\ ableDigraph, [5, 4, 2]);\n<mutable digraph with 11 vertices, 76 edges>\n]]></E\ xample>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"C\ ycleDigraph\" Arg=\"[filt, ]n\"/>\n <Returns>A digraph.</Returns>\n <Descrip\ tion>\n If <A>n</A> is a positive integer, this function returns a <E>cycle\ </E>\n digraph with <A>n</A> vertices and <A>n</A> edges. Specifically,\n \ for each vertex <C>i</C> (with <C>i</C> < <C>n</C>), there is a directed\ \n edge with source <C>i</C> and range <C>i + 1</C>. In addition, there is\ \n an edge with source <C>n</C> and range <C>1</C>. <P/>\n\n If the opti\ onal first argument <A>filt</A> is present, then this should\n specify the \ category or representation the digraph being created will\n belong to. For \ example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the dig\ raph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImmut\ ableDigraph\"/>, then the digraph will be immutable. \n If the optional fir\ st argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutableDigra\ ph\"/> is used by default.\n <Example><![CDATA[\ngap> CycleDigraph(1);\n<immu\ table digraph with 1 vertex, 1 edge>\ngap> CycleDigraph(123);\n<immutable cycl\ e digraph with 123 vertices>\ngap> CycleDigraph(IsMutableDigraph, 10);\n<mutab\ le digraph with 10 vertices, 10 edges>\n]]></Example>\n </Description>\n</Man\ Section>\n\n <ManSection>\n <Oper Name=\"EmptyDigraph\" Arg=\"[filt, ]n\"/\ >\n <Oper Name=\"NullDigraph\" Arg=\"[filt, ]n\"/>\n <Returns>A digraph.</Re\ turns>\n <Description>\n If <A>n</A> is a non-negative integer, this funct\ ion returns the\n <E>empty</E> or <E>null</E> digraph with <A>n</A> vertice\ s. An empty\n digraph is one with no edges.\n <P/>\n \n If the o\ ptional first argument <A>filt</A> is present, then this should\n specify t\ he category or representation the digraph being created will\n belong to. F\ or example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the \ digraph being created will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsIm\ mutableDigraph\"/>, then the digraph will be immutable. \n If the optional \ first argument <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutable\ Digraph\"/> is used by default.<P/>\n\n <C>NullDigraph</C> is a synonym for\ <C>EmptyDigraph</C>.\n\n <Example><![CDATA[\ngap> EmptyDigraph(20);\n<immu\ table empty digraph with 20 vertices>\ngap> NullDigraph(10);\n<immutable empty\ digraph with 10 vertices>\ngap> EmptyDigraph(IsMutableDigraph, 10);\n<mutable\ empty digraph with 10 vertices>]]></Example>\n </Description>\n</ManSection>\ \n\n <ManSection>\n <Oper Name=\"JohnsonDigraph\" Arg=\"[filt, ]n, k\"/>\n\ <Returns>A digraph.</Returns>\n <Description>\n If <A>n</A> and <A>k</A>\ are non-negative integers, then this operation\n returns a symmetric digra\ ph which corresponds to the undirected <E>Johnson\n graph</E> <M>J(n, k)</M\ >. <P/>\n\n If the optional first argument <A>filt</A> is present, then thi\ s should\n specify the category or representation the digraph being created\ will\n belong to. For example, if <A>filt</A> is <Ref Filt=\"IsMutableDigr\ aph\"/>,\n then the digraph being created will be mutable, if <A>filt</A> i\ s <Ref\n Filt=\"IsImmutableDigraph\"/>, then the digraph will be immutable.\ \n If the optional first argument <A>filt</A> is not present, then <Ref\n \ Filt=\"IsImmutableDigraph\"/> is used by default.<P/>\n\n The <E>Johns\ on graph</E> <M>J(n, k)</M> has vertices given by all the\n <A>k</A>-subset\ s of the range <C>[1 .. <A>n</A>]</C>, and two vertices are\n connected b\ y an edge iff their intersection has size <M><A>k</A> - 1</M>.\n\n <Example\ ><![CDATA[\ngap> gr := JohnsonDigraph(3, 1);\n<immutable symmetric digraph wit\ h 3 vertices, 6 edges>\ngap> OutNeighbours(gr);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 \ ] ]\ngap> gr := JohnsonDigraph(4, 2);\n<immutable symmetric digraph with 6 ver\ tices, 24 edges>\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [\ 1, 2, 5, 6 ], [ 1, 2, 5, 6 ], \n [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ]\ngap> John\ sonDigraph(1, 0);\n<immutable empty digraph with 1 vertex>\ngap> JohnsonDigrap\ h(IsMutableDigraph, 1, 0);\n<mutable empty digraph with 1 vertex>\n]]></Exampl\ e>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Peters\ enGraph\" Arg=\"[filt]\"/>\n <Returns>A digraph.</Returns>\n <Description>\n\ \n From <URL>https://en.wikipedia.org/wiki/Petersen_graph</URL>:<P/>\n\n \ <Q>The Petersen graph is an undirected graph with 10 vertices and 15 edges.\n\ It is a small graph that serves as a useful example and counterexample for\ \n many problems in graph theory. The Petersen graph is named after Julius\ \n Petersen, who in 1898 constructed it to be the smallest bridgeless cubic\ \n graph with no three-edge-coloring.</Q><P/>\n \n If the optional fi\ rst argument <A>filt</A> is present, then this should\n specify the categor\ y or representation the digraph being created will\n belong to. For example\ , if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"/>,\n then the digraph be\ ing created will be mutable, if <A>filt</A> is <Ref\n Filt=\"IsImmutableDig\ raph\"/>, then the digraph will be immutable. \n If the optional first argu\ ment <A>filt</A> is not present, then <Ref\n Filt=\"IsImmutableDigraph\"/\ > is used by default.<P/>\n\n See also <Ref Oper=\"GeneralisedPetersenGraph\ \"/>.\n\n<Example><![CDATA[\ngap> ChromaticNumber(PetersenGraph());\n3\ngap> P\ etersenGraph(IsMutableDigraph);\n<mutable digraph with 10 vertices, 30 edges>\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper \ Name=\"GeneralisedPetersenGraph\" Arg=\"[filt, ]n, k\"/>\n <Returns>A digraph\ .</Returns>\n <Description>\n If <A>n</A> is a positive integer and <A>k</\ A> is a non-negative\n integer less than <C><A>n</A> / 2</C>, then this ope\ ration returns\n the <E>generalised Petersen graph</E> <M>GPG(<A>n</A>, k)<\ /M>. <P/>\n\n From\n <URL>https://en.wikipedia.org/wiki/Generalized_Pete\ rsen_graph</URL>:\n <P/>\n\n <Q>The generalized Petersen graphs are a fa\ mily of cubic\n graphs formed by connecting the vertices of a regular polyg\ on to the\n corresponding vertices of a star polygon. They include the Pete\ rsen graph\n and generalize one of the ways of constructing the Petersen gr\ aph. The\n generalized Petersen graph family was introduced in 1950 by H. S\ . M.\n Coxeter and was given its name in 1969 by Mark Watkins.</Q>\n <P/\ >\n\n If the optional first argument <A>filt</A> is present, then this shou\ ld\n specify the category or representation the digraph being created will\ \n belong to. For example, if <A>filt</A> is <Ref Filt=\"IsMutableDigraph\"\ />,\n then the digraph being created will be mutable, if <A>filt</A> is <Re\ f\n Filt=\"IsImmutableDigraph\"/>, then the digraph will be immutable. \n \ If the optional first argument <A>filt</A> is not present, then <Ref\n \ Filt=\"IsImmutableDigraph\"/> is used by default.<P/>\n\n See also <Ref Ope\ r=\"PetersenGraph\"/>.\n\n<Example><![CDATA[\ngap> GeneralisedPetersenGraph(7,\ 2);\n<immutable symmetric digraph with 14 vertices, 42 edges>\ngap> Generalis\ edPetersenGraph(40, 1);\n<immutable symmetric digraph with 80 vertices, 240 ed\ ges>\ngap> D := GeneralisedPetersenGraph(5, 2);\n<immutable symmetric digraph \ with 10 vertices, 30 edges>\ngap> IsIsomorphicDigraph(D, PetersenGraph());\ntr\ ue\ngap> GeneralisedPetersenGraph(IsMutableDigraph, 9, 4);\n<mutable digraph w\ ith 18 vertices, 54 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n\ </Section>\n\n</Chapter>\n <!-- creating -->\n <Chapter Label=\"Operators\"\ ><Heading>Operators</Heading>\n\n <Section><Heading>Operators for digraphs</H\ eading>\n <List>\n <Mark><C><A>digraph1</A> = <A>digraph2</A></C></Mark\ >\n <Item>\n <Index Key=\"=\"><C>=</C> (for digraphs)</Index>\n \ returns <K>true</K> if <A>digraph1</A> and <A>digraph2</A> have the same\n \ vertices, and <C>DigraphEdges(<A>digraph1</A>) =\n DigraphEdges(<\ A>digraph2</A>)</C>, up to some re-ordering of the edge\n lists. <P/>\n\ \n Note that this operator does not compare the vertex labels\n of\ <A>digraph1</A> and <A>digraph2</A>.\n </Item>\n\n <Mark><C><A>digrap\ h1</A> < <A>digraph2</A></C></Mark>\n <Item>\n <Index Key=\"<\ \"><C><</C> (for digraphs)</Index>\n\n This operator returns <K>true<\ /K> if one of the following holds:\n\n <List>\n <Item>\n \ The number <M>n_1</M> of vertices in <A>digraph1</A> is less than\n \ the number <M>n_2</M> of vertices in <A>digraph2</A>;\n </Item>\n \ <Item>\n <M>n_1 = n_2</M>, and the number <M>m_1</M> of edges\ in\n <A>digraph1</A> is less than the number <M>m_2</M> of edges in\ \n <A>digraph2</A>;\n </Item>\n <Item>\n <\ M>n_1 = n_2</M>, <M>m_1 = m_2</M>, and\n <C>DigraphEdges(<A>digraph1\ </A>)</C> is less than\n <C>DigraphEdges(<A>digraph2</A>)</C> after \ having both of these sets\n have been sorted with respect to the lex\ icographical order.\n </Item>\n </List>\n </Item>\n </List\ >\n\n <ManSection>\n <Oper Name=\"IsSubdigraph\" Arg=\"super, sub\"/>\n <R\ eturns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n If <A>supe\ r</A> and <A>sub</A> are digraphs, then this operation returns\n <K>true</K\ > if <A>sub</A> is a subdigraph of <A>super</A>, and <K>false</K>\n if it i\ s not. <P/>\n\n A digraph <A>sub</A> is a <E>subdigraph</E> of a digraph <A\ >super</A> if\n <A>sub</A> and <A>super</A> share the same number of vertic\ es, and the\n collection of edges of <A>super</A> (including repeats) conta\ ins the\n collection of edges of <A>sub</A> (including repeats). <P/>\n\n \ In other words, <A>sub</A> is a subdigraph of <A>super</A> if and only if\n\ <C>DigraphNrVertices(<A>sub</A>) = DigraphNrVertices(<A>super</A>)</C>, an\ d\n for each pair of vertices <C>i</C> and <C>j</C>, there are at least as \ many\n edges of the form <C>[i, j]</C> in <A>super</A> as there are in <A>s\ ub</A>.\n <P/>\n\n <Example><![CDATA[\ngap> g := Digraph([[2, 3], [1], [\ 2, 3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> h := Digraph([[2,\ 3], [], [2]]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> IsSubdigra\ ph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse\ngap> IsSubdigraph(CompleteDi\ graph(4), CycleDigraph(4));\ntrue\ngap> IsSubdigraph(CycleDigraph(4), ChainDig\ raph(4));\ntrue\ngap> g := Digraph([[2, 2], [1]]);\n<immutable multidigraph wi\ th 2 vertices, 3 edges>\ngap> h := Digraph([[2], [1]]);\n<immutable digraph wi\ th 2 vertices, 2 edges>\ngap> IsSubdigraph(g, h);\ntrue\ngap> IsSubdigraph(h, \ g);\nfalse]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Oper Name=\"IsUndirectedSpanningTree\" Arg=\"super, sub\"/>\n <Oper Name=\"\ IsUndirectedSpanningForest\" Arg=\"super, sub\"/>\n <Returns><K>true</K> or <\ K>false</K>.</Returns>\n <Description>\n The operation <C>IsUndirectedSpan\ ningTree</C> returns <K>true</K> if the\n digraph <A>sub</A> is an undirect\ ed spanning tree of the digraph\n <A>super</A>, and the operation <C>IsUndi\ rectedSpanningForest</C> returns\n <K>true</K> if the digraph <A>sub</A> is\ an undirected spanning forest of\n the digraph <A>super</A>. <P/>\n\n A\ n <E>undirected spanning tree</E> of a digraph <A>super</A> is a subdigraph\n \ of <A>super</A> that is an undirected tree (see <Ref Oper=\"IsSubdigraph\" \ />\n and <Ref Prop=\"IsUndirectedTree\" />). Note that a digraph whose <Ref\ \n Oper=\"MaximalSymmetricSubdigraph\"/> is not connected has no undirect\ ed\n spanning trees (see <Ref Prop=\"IsConnectedDigraph\"/>). <P/>\n\n A\ n <E>undirected spanning forest</E> of a digraph <A>super</A> is a\n subdig\ raph of <A>super</A> that is an undirected forest (see <Ref\n Oper=\"IsSu\ bdigraph\"/> and <Ref Prop=\"IsUndirectedForest\"/>), and is not\n containe\ d in any larger such subdigraph of <A>super</A>. Equivalently, an\n undire\ cted spanning forest is a subdigraph of <A>super</A> whose connected\n comp\ onents coincide with those of the <Ref\n Oper=\"MaximalSymmetricSubdigrap\ h\"/> of <A>super</A> (see <Ref\n Attr=\"DigraphConnectedComponents\"/>).\ <P/>\n\n Note that an undirected spanning tree is an undirected spanning f\ orest that\n is connected.\n\n <Example><![CDATA[\ngap> D := CompleteDig\ raph(4);\n<immutable complete digraph with 4 vertices>\ngap> tree := Digraph([\ [3], [4], [1, 4], [2, 3]]);\n<immutable digraph with 4 vertices, 6 edges>\ngap\ > IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntrue\ngap> IsUndirectedSp\ anningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(4);\n<immutable empty \ digraph with 4 vertices>\ngap> IsSubdigraph(D, forest) and IsUndirectedForest(\ forest);\ntrue\ngap> IsUndirectedSpanningForest(D, forest);\nfalse\ngap> IsSub\ digraph(tree, forest);\ntrue\ngap> D := DigraphDisjointUnion(CycleDigraph(2), \ CycleDigraph(2));\n<immutable digraph with 4 vertices, 4 edges>\ngap> IsUndire\ ctedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUndirectedSpanningForest\ (D, D);\ntrue]]></Example>\n </Description>\n</ManSection>\n\n </Section>\n<\ /Chapter>\n <!-- operators/hashs -->\n <Chapter Label=\"Attributes and operat\ ions\"><Heading>Attributes and operations</Heading>\n <Section><Heading>Verti\ ces and edges</Heading>\n <ManSection>\n <Attr Name=\"DigraphVertices\" Ar\ g=\"digraph\"/>\n <Returns>A list of integers.</Returns>\n <Description>\n \ Returns the vertices of the digraph <A>digraph</A>. <P/>\n\n Note that th\ e vertices of a digraph are always a range of\n positive integers from <C>1\ </C> to the number of vertices of the graph.\n <Example><![CDATA[\ngap> gr \ := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \"b\"],\n\ > [\"b\", \"c\", \"a\"]);\n<immutable digraph with 3 vertices\ , 3 edges>\ngap> DigraphVertices(gr);\n[ 1 .. 3 ]\ngap> gr := Digraph([1, 2, 3\ , 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> [2, 7, 5,\ 3, 7]);\n<immutable digraph with 6 vertices, 5 edges>\ngap> DigraphVertices(g\ r);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 100 ]\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 \ edges>\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\n]]></Example>\n </Description>\ \n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphNrVertices\" Arg=\"\ digraph\"/>\n <Returns>An integer.</Returns>\n <Description>\n Returns th\ e number of vertices of the digraph <A>digraph</A>. <P/>\n\n <Example><![CD\ ATA[\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"\ b\", \"b\"],\n> [\"b\", \"c\", \"a\"]);\n<immutable digraph w\ ith 3 vertices, 3 edges>\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Digraph([\ 1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> [2\ , 7, 5, 3, 7]);\n<immutable digraph with 6 vertices, 5 edges>\ngap> DigraphNrV\ ertices(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(100));\n100\ngap> D := C\ ycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ \ngap> DigraphNrVertices(D);\n3\n]]></Example>\n </Description>\n</ManSection\ >\n\n <ManSection>\n <Attr Name=\"DigraphEdges\" Arg=\"digraph\"/>\n <Ret\ urns>A list of lists.</Returns>\n <Description>\n <C>DigraphEdges</C> retu\ rns a list of edges of the digraph <A>digraph</A>,\n where each edge is a p\ air of elements of <Ref Attr=\"DigraphVertices\"/> of\n the form <C>[source\ ,range]</C>.\n <P/>\n\n The entries of <C>DigraphEdges(</C><A>digraph</A\ ><C>)</C> are in one-to-one\n correspondence with the edges of <A>digraph</\ A>. Hence\n <C>DigraphEdges(</C><A>digraph</A><C>)</C> is duplicate-free i\ f and only if\n <A>digraph</A> contains no multiple edges. <P/>\n\n The \ entries of <C>DigraphEdges</C> are guaranteed to be sorted by their\n first\ component (i.e. by the source of each edge), but they are not\n necessaril\ y then sorted by the second component.\n <Example><![CDATA[\ngap> gr := Dig\ raphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n<immutable multidigraph with 5 v\ ertices, 16 edges>\ngap> edges := ShallowCopy(DigraphEdges(gr));; Sort(edges);\ \ngap> edges;\n[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 1 ], \ \n [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3, 4 ], [ 3, 5 ], \n [ 4, 2 ], \ [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ <mutable digraph with 3 vertices, 3 edges>\ngap> DigraphEdges(D);\n[ [ 1, 2 ],\ [ 2, 3 ], [ 3, 1 ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <\ ManSection>\n <Attr Name=\"DigraphNrEdges\" Arg=\"digraph\"/>\n <Returns>An \ integer.</Returns>\n <Description>\n This function returns the number of e\ dges of the digraph <A>digraph</A>.\n <Example><![CDATA[\ngap> gr := Digrap\ h([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;\ngap> Digraph\ NrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n<immutab\ le multidigraph with 3 vertices, 3 edges>\ngap> DigraphNrEdges(gr);\n3\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 ed\ ges>\ngap> DigraphNrEdges(D);\n3\n]]></Example>\n </Description>\n</ManSectio\ n>\n\n <ManSection>\n <Attr Name=\"DigraphNrLoops\" Arg=\"digraph\"/>\n <\ Returns>An integer.</Returns>\n <Description>\n This function returns the \ number of loops of the digraph <A>digraph</A>. See <Ref Prop=\"DigraphHasLoops\ \"/>. <P/>\n <Example><![CDATA[\ngap> D := Digraph([[2, 3], [1, 4], [3, 3, \ 5], [], [2, 5]]);\n<immutable multidigraph with 5 vertices, 9 edges>\ngap> Dig\ raphNrLoops(D);\n3\ngap> D := EmptyDigraph(5);\n<immutable empty digraph with \ 5 vertices>\ngap> DigraphNrLoops(D);\n0\ngap> D := CompleteDigraph(5);\n<immut\ able complete digraph with 5 vertices>\ngap> DigraphNrLoops(D);\n0\n]]></Examp\ le>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"Digra\ phSinks\" Arg=\"digraph\"/>\n <Returns>A list of vertices.</Returns>\n <Desc\ ription>\n This function returns a list of the sinks of the digraph\n <A\ >digraph</A>.\n A sink of a digraph is a vertex with out-degree zero.\n \ See <Ref Oper=\"OutDegreeOfVertex\"/>.\n <Example><![CDATA[\ngap> gr := Dig\ raph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n<immutable multidigraph with\ 5 vertices, 9 edges>\ngap> DigraphSinks(gr);\n[ 3, 5 ]\ngap> D := CycleDigrap\ h(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> Digr\ aphSinks(D);\n[ ]\n]]></Example>\n </Description>\n</ManSection>\n\n <Man\ Section>\n <Attr Name=\"DigraphSources\" Arg=\"digraph\"/>\n <Returns>A list\ of vertices.</Returns>\n <Description>\n This function returns an immutab\ le list of the sources of the digraph\n <A>digraph</A>.\n A source of a \ digraph is a vertex with in-degree zero.\n See <Ref Oper=\"InDegreeOfVertex\ \"/>.\n <Example><![CDATA[\ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, \ 2, 5, 3], []]);\n<immutable multidigraph with 5 vertices, 9 edges>\ngap> Digra\ phSources(gr);\n[ 1, 4 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<mutab\ le digraph with 3 vertices, 3 edges>\ngap> DigraphSources(D);\n[ ]\n]]></Exam\ ple>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"Digr\ aphTopologicalSort\" Arg=\"digraph\"/>\n <Returns>A list of positive integers\ , or <K>fail</K>.</Returns>\n <Description>\n If <A>digraph</A> is a digra\ ph whose only directed cycles are loops, then\n <C>DigraphTopologicalSort</\ C> returns the vertices of <A>digraph</A> ordered\n so that every edge's so\ urce appears no earlier in the list than its range.\n If the digraph <A>dig\ raph</A> contains directed cycles of length greater\n than <M>1</M>, then t\ his operation returns <K>fail</K>.\n <P/>\n\n See Section <Ref Subsect=\ \"Definitions\" Style=\"Number\" /> for the definition\n of a directed cycl\ e, and the definition of a loop.\n\n <P/>\n\n The method used for this a\ ttribute has complexity <M>O(m+n)</M> where\n <M>m</M> is the number of edg\ es (counting multiple edges as one) and\n <M>n</M> is the number of vertice\ s in the digraph. <P/>\n <Example><![CDATA[\ngap> D := Digraph([\n> [2, 3],\ [], [4, 6], [5], [], [7, 8, 9], [], [], []]);\n<immutable digraph with 9 vert\ ices, 8 edges>\ngap> DigraphTopologicalSort(D);\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ \ngap> D := Digraph(IsMutableDigraph, [[2, 3], [3], [4], []]);\n<mutable digra\ ph with 4 vertices, 4 edges>\ngap> DigraphTopologicalSort(D);\n[ 4, 3, 2, 1 ]\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper \ Name = \"DigraphVertexLabel\" Arg=\"digraph, i\"/>\n <Oper Name =\"SetDigraph\ VertexLabel\" Arg=\"digraph, i, obj\"/>\n <Description>\n If <A>digraph</A\ > is a digraph, then the first operation returns the label\n of the vertex \ <A>i</A>. The second operation can be used to set the label\n of the vertex\ <A>i</A> in <A>digraph</A> to the arbitrary &GAP; object\n <A>obj</A>. <P\ />\n\n The label of a vertex can be changed an arbitrary number of times. I\ f no\n label has been set for the vertex <A>i</A>, then the default value i\ s\n <A>i</A>. <P/>\n\n If <A>digraph</A> is a digraph created from a rec\ ord with a component\n <C>DigraphVertices</C>, then the labels of the verti\ ces are set to\n the value of this component.<P/>\n\n Induced subdigraph\ s, and some other operations which create new digraphs from\n old ones, inh\ erit their labels from their parents.\n\n <Example><![CDATA[\ngap> D := Dig\ raphFromDigraph6String(\"&DHUEe_\");\n<immutable digraph with 5 vertices, 11 e\ dges>\ngap> DigraphVertexLabel(D, 3);\n3\ngap> D := Digraph([\"a\", \"b\", \"c\ \"], [], []);\n<immutable empty digraph with 3 vertices>\ngap> DigraphVertexLa\ bel(D, 2);\n\"b\"\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap> DigraphVerte\ xLabel(D, 2);\n\"d\"\ngap> D := InducedSubdigraph(D, [1, 2]);\n<immutable empt\ y digraph with 2 vertices>\ngap> DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := D\ igraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n<mutable empty digra\ ph with 3 vertices>\ngap> DigraphVertexLabel(D, 1);\n\"e\"\ngap> SetDigraphVer\ texLabel(D, 1, \"h\");\ngap> DigraphVertexLabel(D, 1);\n\"h\"\ngap> InducedSub\ digraph(D, [1, 2]);\n<mutable empty digraph with 2 vertices>\ngap> DigraphVert\ exLabel(D, 1);\n\"h\"\n]]></Example>\n </Description>\n</ManSection>\n\n <\ ManSection>\n <Oper Name=\"DigraphVertexLabels\" Arg=\"digraph\"/>\n <Oper N\ ame=\"SetDigraphVertexLabels\" Arg=\"digraph, list\"/>\n <Description>\n I\ f <A>digraph</A> is a digraph, then <C>DigraphVertexLabels</C> returns a\n \ copy of the labels of the vertices in <A>digraph</A>.\n <C>SetDigraphVertex\ Labels</C> can be used to set the labels of the vertices\n in <A>digraph</A\ > to the list of\n arbitrary &GAP; objects <A>list</A>, which must be of th\ e same length\n as the number of vertices of <A>digraph</A>. <P/>\n\n If\ the list <A>list</A> is immutable, then the vertex labels are set to a\n m\ utable copy of <A>list</A>. Otherwise, the labels are set to exactly\n <A>l\ ist</A>. <P/>\n\n The label of a vertex can be changed an arbitrary number \ of times. If no\n label has been set for the vertex <A>i</A>, then the defa\ ult value is\n <A>i</A>. <P/>\n\n If <A>digraph</A> is a digraph created\ from a record with a component\n <C>DigraphVertices</C>, then the labels o\ f the vertices are set to the\n value of this component. As in the above, i\ f the component is immutable\n then the digraph's vertex labels are set to \ a mutable copy of\n <C>DigraphVertices</C>. Otherwise, they are set to exac\ tly\n <C>DigraphVertices</C>. <P/>\n\n Induced subdigraphs, and other op\ erations which create new digraphs from\n old ones, inherit their labels fr\ om their parents.\n\n <Example><![CDATA[\ngap> D := DigraphFromDigraph6Stri\ ng(\"&DHUEe_\");\n<immutable digraph with 5 vertices, 11 edges>\ngap> DigraphV\ ertexLabels(D);\n[ 1 .. 5 ]\ngap> D := Digraph([\"a\", \"b\", \"c\"], [], []);\ \n<immutable empty digraph with 3 vertices>\ngap> DigraphVertexLabels(D);\n[ \ \"a\", \"b\", \"c\" ]\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap> DigraphV\ ertexLabels(D);\n[ \"a\", \"d\", \"c\" ]\ngap> D := InducedSubdigraph(D, [1, 3\ ]);\n<immutable empty digraph with 2 vertices>\ngap> DigraphVertexLabels(D);\n\ [ \"a\", \"c\" ]\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [\ ], []);\n<mutable empty digraph with 3 vertices>\ngap> SetDigraphVertexLabels(\ D, [\"h\", \"i\", \"j\"]);\ngap> DigraphVertexLabels(D);\n[ \"h\", \"i\", \"j\ \" ]\ngap> InducedSubdigraph(D, [1, 3]);\n<mutable empty digraph with 2 vertic\ es>\ngap> DigraphVertexLabels(D);\n[ \"h\", \"j\" ]\n]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Oper Name = \"DigraphEdgeLabel\"\ Arg=\"digraph, i, j\"/>\n <Oper Name = \"SetDigraphEdgeLabel\" Arg=\"digraph\ , i, j, obj\"/>\n <Description>\n If <A>digraph</A> is a digraph without m\ ultiple edges, then the first\n operation returns the label of the edge fro\ m vertex <A>i</A> to vertex\n <A>j</A>. The second operation can be used t\ o set the label of the edge\n between vertex <A>i</A> and vertex <A>j</A> t\ o the arbitrary &GAP; object\n <A>obj</A>. <P/>\n\n The label of an edge\ can be changed an arbitrary number of times. If no\n label has been set fo\ r the edge, then the default value is <A>1</A>. <P/>\n\n Induced subdigraph\ s, and some other operations which create new digraphs from\n old ones, inh\ erit their edge labels from their parents.\n\n See also <Ref Oper=\"Digraph\ EdgeLabels\"/>.\n\n <Example><![CDATA[\ngap> D := DigraphFromDigraph6String\ (\"&DHUEe_\");\n<immutable digraph with 5 vertices, 11 edges>\ngap> DigraphEdg\ eLabel(D, 3, 1);\n1\ngap> SetDigraphEdgeLabel(D, 2, 5, [42]);\ngap> DigraphEdg\ eLabel(D, 2, 5);\n[ 42 ]\ngap> D := InducedSubdigraph(D, [2, 5]);\n<immutable \ digraph with 2 vertices, 3 edges>\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 42 ]\nga\ p> D := ChainDigraph(IsMutableDigraph, 5);\n<mutable digraph with 5 vertices, \ 4 edges>\ngap> DigraphEdgeLabel(D, 2, 3);\n1\ngap> SetDigraphEdgeLabel(D, 4, 5\ , [1729]);\ngap> DigraphEdgeLabel(D, 4, 5);\n[ 1729 ]\ngap> InducedSubdigraph(\ D, [4, 5]);\n<mutable digraph with 2 vertices, 1 edge>\ngap> DigraphEdgeLabel(\ D, 1, 2);\n[ 1729 ]\n]]></Example>\n </Description>\n</ManSection>\n\n <Ma\ nSection>\n <Oper Name=\"DigraphEdgeLabels\" Arg=\"digraph\"/>\n <Oper Name=\ \"SetDigraphEdgeLabels\" Arg=\"digraph, labels\"\n Label=\"for a digraph an\ d a list of lists\"/>\n <Oper Name=\"SetDigraphEdgeLabels\" Arg=\"digraph, fu\ nc\"\n Label=\"for a digraph and a function\"/>\n <Description>\n If <A\ >digraph</A> is a digraph without multiple edges, then\n <C>DigraphEdgeLabe\ ls</C> returns a copy of the labels of the edges\n in <A>digraph</A> as a l\ ist of lists <C>labels</C> such that\n <C>labels[i][j]</C> is the label on \ the edge from vertex <C>i</C>\n to vertex <C>OutNeighbours(digraph)[i][j]</\ C>.\n\n <C>SetDigraphEdgeLabels</C> can be used to set the labels of the ed\ ges in\n <A>digraph</A> without multiple edges to the list <A>labels</A> of\ lists of\n arbitrary &GAP; objects such that <C>list[i][j]</C> is the labe\ l on the edge\n from vertex <C>i</C> to the vertex <C>OutNeighbours(digraph\ >[i][j]</C>.\n\n Alternatively <C>SetDigraphEdgeLabels</C> can be called wi\ th binary function\n <A>func</A> that as its second argument that when pass\ ed two vertices <C>i</C>\n and <C>j</C> returns the label for the edge betw\ een vertex <C>i</C> and vertex\n <C>j</C>. <P/>\n\n The label of an edge\ can be changed an arbitrary number of times. If no\n label has been set fo\ r an edge, then the default value is <C>1</C>. <P/>\n\n Induced subdigraphs\ , and some other operations which create new digraphs\n from old ones, inhe\ rit their labels from their parents.\n\n <Example><![CDATA[\ngap> D := Digr\ aphFromDigraph6String(\"&DHUEe_\");\n<immutable digraph with 5 vertices, 11 ed\ ges>\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ 1, 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ \ 1, 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 1, \"d\");\ngap> DigraphEdgeLabels\ (D);\n[ [ 1 ], [ \"d\", 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> D := I\ nducedSubdigraph(D, [1, 2, 3]);\n<immutable digraph with 3 vertices, 4 edges>\ \ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1 ], [ 1 ] ]\ngap> OutNeighbou\ rs(D);\n[ [ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := CompleteBipartiteDigraph(IsMutab\ leDigraph, 2, 3);\n<mutable digraph with 5 vertices, 12 edges>\ngap> DigraphEd\ geLabels(D);\n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap>\ SetDigraphEdgeLabel(D, 2, 4, \"a\");\ngap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1\ ], [ 1, \"a\", 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> InducedSubdigraph(D,\ [1, 2, 3, 4]);\n<mutable digraph with 4 vertices, 8 edges>\ngap> DigraphEdgeL\ abels(D);\n[ [ 1, 1 ], [ 1, \"a\" ], [ 1, 1 ], [ 1, 1 ] ]\ngap> OutNeighbors(D\ );\n[ [ 3, 4 ], [ 3, 4 ], [ 1, 2 ], [ 1, 2 ] ]\n]]></Example>\n </Description\ >\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphInEdges\" Arg=\"di\ graph, vertex\"/>\n <Returns>A list of edges.</Returns>\n <Description>\n \ <C>DigraphInEdges</C> returns the list of all edges of <A>digraph</A>\n wh\ ich have <A>vertex</A> as their range.\n <Example><![CDATA[\ngap> D := Digr\ aph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n<immutable multidigraph with 4 vertice\ s, 8 edges>\ngap> DigraphInEdges(D, 2);\n[ [ 1, 2 ], [ 1, 2 ] ]\n]]></Example>\ \n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphO\ utEdges\" Arg=\"digraph, vertex\"/>\n <Returns>A list of edges.</Returns>\n \ <Description>\n <C>DigraphOutEdges</C> returns the list of all edges of <A>\ digraph</A>\n which have <A>vertex</A> as their source.\n <Example><![CD\ ATA[\ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n<immutable multidi\ graph with 4 vertices, 8 edges>\ngap> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2,\ 3 ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Oper Name=\"IsDigraphEdge\" Arg=\"digraph, list\"\n Label=\"for digrap\ h and list\"/>\n <Oper Name=\"IsDigraphEdge\" Arg=\"digraph, u, v\"\n \ Label=\"for digraph and two pos ints\"/>\n <Returns><K>true</K> or <K>false</\ K>.</Returns>\n <Description>\n In the first form, this function returns <\ K>true</K> if and only if the list\n <A>list</A> specifies an edge in the d\ igraph <A>digraph</A>. Specifically,\n this operation returns <K>true</K> \ if <A>list</A> is a pair of positive\n integers where <A>list</A><C>[1]</C>\ is the source and <A>list</A><C>[2]</C>\n is the range of an edge in <A>di\ graph</A>, and <K>false</K> otherwise. <P/>\n\n The second form simply retu\ rns <K>true</K> if <C>[<A>u</A>, <A>v</A>]</C> is\n an edge in <A>digraph</\ A>, and <K>false</K> otherwise.\n\n<Example><![CDATA[\ngap> D := Digraph([[2, \ 2], [6], [], [3], [], [1]]);\n<immutable multidigraph with 6 vertices, 5 edges\ >\ngap> IsDigraphEdge(D, [1, 1]);\nfalse\ngap> IsDigraphEdge(D, [1, 2]);\ntrue\ \ngap> IsDigraphEdge(D, [1, 8]);\nfalse\n]]></Example>\n </Description>\n</Ma\ nSection>\n\n <ManSection>\n <Oper Name=\"IsMatching\" Arg=\"digraph, list\ \"/>\n <Oper Name=\"IsMaximalMatching\" Arg=\"digraph, list\"/>\n <Oper Name\ =\"IsMaximumMatching\" Arg=\"digraph, list\"/>\n <Oper Name=\"IsPerfectMatchi\ ng\" Arg=\"digraph, list\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns\ >\n <Description>\n If <A>digraph</A> is a digraph and <A>list</A> is a li\ st of pairs of\n vertices of <A>digraph</A>, then <C>IsMatching</C> returns\ <K>true</K> if\n <A>list</A> is a matching of <A>digraph</A>. The operati\ on\n <C>IsMaximalMatching</C> returns <K>true</K> if <A>list</A> is a maxim\ al \n matching, <C>IsMaximumMatching</C> returns <K>true</K> if <A>list</A>\ is a\n maximum matching and <C>IsPerfectMatching</C> returns <K>true</K> i\ f \n <A>list</A> is a perfect, matching of <A>digraph</A>, respectively. \ \n Otherwise, each of these operations return <K>false</K>.\n <P/>\n\n \ A <E>matching</E> <C>M</C> of a digraph <A>digraph</A> is a subset of the\n \ edges of <A>digraph</A>, i.e. <C>DigraphEdges(<A>digraph</A>)</C>, such\n \ that no pair of distinct edges in <C>M</C> are incident to the same vertex\n\ of <A>digraph</A>. Note that this definition allows a matching to contain\ \n loops. See <Ref Prop=\"DigraphHasLoops\" />. The matching <C>M</C> is\ \n <E>maximal</E> if it is contained in no larger matching of the digraph,\ \n is <E>maximum</E> if it has the greatest cardinality among all matchings\ and\n is <E>perfect</E> if every vertex of the digraph is incident to an e\ dge in\n the matching. Every maximum or perfect matching is maximal. Note,\ however,\n that not every perfect matching of digraphs with loops is maxim\ um.\n\n <Example><![CDATA[\ngap> D := Digraph([[1, 2], [1, 2], [2, 3, 4], [\ 3, 5], [1]]);\n<immutable digraph with 5 vertices, 10 edges>\ngap> IsMatching(\ D, [[2, 1], [3, 2]]);\nfalse\ngap> edges := [[3, 2]];;\ngap> IsMatching(D, edg\ es);\ntrue\ngap> IsMaximalMatching(D, edges);\nfalse\ngap> edges := [[2, 1], [\ 3, 4]];;\ngap> IsMaximalMatching(D, edges);\ntrue\ngap> IsPerfectMatching(D, e\ dges);\nfalse\ngap> edges := [[1, 2], [3, 3], [4, 5]];;\ngap> IsPerfectMatchin\ g(D, edges);\ntrue\ngap> IsMaximumMatching(D, edges);\nfalse\ngap> edges := [[\ 1, 1], [2, 2], [3, 3], [4, 5]];;\ngap> IsMaximumMatching(D, edges);\ntrue\n]]>\ </Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\ \"DigraphMaximalMatching\" Arg=\"digraph\"/>\n <Returns>A list of pairs of ve\ rtices.</Returns>\n <Description>\n This function returns a maximal matchi\ ng of the digraph <A>digraph</A>.\n <P/>\n\n For the definition of a max\ imal matching, see <Ref Oper=\"IsMaximalMatching\"/>. \n\n <Example><![CDAT\ A[\ngap> D := DigraphFromDiSparse6String(\".IeAoXCJU@|SHAe?d\");\n<immutable d\ igraph with 10 vertices, 13 edges>\ngap> M := DigraphMaximalMatching(D);; IsMa\ ximalMatching(D, M);\ntrue\ngap> D := RandomDigraph(100);;\ngap> IsMaximalMatc\ hing(D, DigraphMaximalMatching(D));\ntrue\ngap> D := GeneralisedPetersenGraph(\ IsMutableDigraph, 9, 2);\n<mutable digraph with 18 vertices, 54 edges>\ngap> I\ sMaximalMatching(D, DigraphMaximalMatching(D));\ntrue\n]]></Example>\n </Desc\ ription>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphMaximumMatc\ hing\" Arg=\"digraph\"/>\n <Returns>A list of pairs of vertices.</Returns>\n \ <Description>\n This function returns a maximum matching of the digraph <A\ >digraph</A>.\n <P/>\n\n For the definition of a maximum matching, see <\ Ref Oper=\"IsMaximumMatching\"/>.\n If <A>digraph</A> is bipartite (see <Re\ f Oper=\"IsBipartiteDigraph\"/>), then \n the algorithm used has complexity\ <C>O(m*sqrt(n))</C>. Otherwise for general \n graphs the complexity is <C>\ O(m*n*log(n))</C>. Here <C>n</C> is the number of vertices \n and <C>m</C> \ is the number of edges.\n\n <Example><![CDATA[\ngap> D := DigraphFromDigrap\ h6String(\"&I@EA_A?AdDp[_c??OO\");\n<immutable digraph with 10 vertices, 23 ed\ ges>\ngap> M := DigraphMaximumMatching(D);; IsMaximalMatching(D, M);\ntrue\nga\ p> Length(M);\n5\ngap> D := Digraph([[5, 6, 7, 8], [6, 7, 8], [7, 8], [8], \n>\ [], [], [], []]);;\ngap> M := DigraphMaximumMatching(D);\n[ \ [ 1, 5 ], [ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D := GeneralisedPetersenGraph(I\ sMutableDigraph, 9, 2);\n<mutable digraph with 18 vertices, 54 edges>\ngap> M \ := DigraphMaximumMatching(D);;\ngap> IsMaximalMatching(D, M);\ntrue\ngap> Leng\ th(M);\n9\n]]></Example>\n </Description>\n</ManSection>\n\n </Section>\n\n \ <Section><Heading>Neighbours and degree</Heading>\n <ManSection>\n <Attr \ Name=\"AdjacencyMatrix\" Arg=\"digraph\"/>\n <Oper Name=\"AdjacencyMatrixMuta\ bleCopy\" Arg=\"digraph\"/>\n <Returns>A square matrix of non-negative intege\ rs.</Returns>\n <Description>\n This function returns the adjacency matrix\ <C>mat</C> of the digraph\n <A>digraph</A>.\n The value of the matrix e\ ntry <C>mat[i][j]</C> is the number of edges\n in <A>digraph</A> with sourc\ e <C>i</C> and range <C>j</C>. If <A>digraph</A>\n has no vertices, then th\ e empty list is returned. <P/>\n\n The function <C>AdjacencyMatrix</C> retu\ rns an immutable list of\n lists, whereas the function <C>AdjacencyMatrixMu\ tableCopy</C> returns a copy\n of <C>AdjacencyMatrix</C> that is a mutable \ list of mutable lists. <P/>\n\n <Example><![CDATA[\ngap> gr := Digraph([\n>\ [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3, 9], [3, 3], [3, 5, 6,\ 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],\n> [2, 3, 4,\ 6, 7, 10]]);\n<immutable multidigraph with 10 vertices, 44 edges>\ngap> mat :\ = AdjacencyMatrix(gr);;\ngap> Display(mat);\n[ [ 0, 3, 0, 0, 0, 0, 0, \ 0, 0, 0 ],\n [ 1, 0, 1, 0, 0, 1, 0, 1, 1, 1 ],\n [ 0, 0, 0, \ 1, 0, 1, 0, 1, 0, 0 ],\n [ 1, 1, 1, 0, 0, 0, 0, 0, 1, 0 ],\n \ [ 0, 0, 2, 0, 0, 0, 0, 0, 0, 0 ],\n [ 0, 0, 1, 0, 1, 1, 0, \ 0, 0, 1 ],\n [ 1, 1, 0, 0, 0, 0, 1, 0, 0, 0 ],\n [ 1, 1, 1, \ 0, 1, 1, 0, 0, 0, 2 ],\n [ 1, 0, 1, 1, 1, 0, 0, 1, 0, 1 ],\n\ [ 0, 1, 1, 1, 0, 1, 1, 0, 0, 1 ] ]\ngap> D := CycleDigraph(IsMutab\ leDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> Display(Adjac\ encyMatrix(D));\n[ [ 0, 1, 0 ],\n [ 0, 0, 1 ],\n [ 1, 0, 0 ] ]\n]]>\ </Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\ \"CharacteristicPolynomial\" Arg=\"digraph\"/>\n <Returns>A polynomial with i\ nteger coefficients.</Returns>\n <Description>\n This function returns the\ characteristic polynomial of the digraph\n <A>digraph</A>. That is it ret\ urns the characteristic polynomial\n of the adjacency matrix of the digra\ ph <A>digraph</A>\n <Example><![CDATA[\ngap> D := Digraph([\n> [2, 2, 2], [\ 1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, \ 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\ \n<immutable multidigraph with 10 vertices, 44 edges>\ngap> CharacteristicPoly\ nomial(D);\nx_1^10-3*x_1^9-7*x_1^8-x_1^7+14*x_1^6+x_1^5-26*x_1^4+51*x_1^3-10*x\ _1^2\\\n+18*x_1-30\ngap> D := CompleteDigraph(5);\n<immutable complete digraph\ with 5 vertices>\ngap> CharacteristicPolynomial(D);\nx_1^5-10*x_1^3-20*x_1^2-\ 15*x_1-4\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with \ 3 vertices, 3 edges>\ngap> CharacteristicPolynomial(D);\nx_1^3-1\n]]></Example\ >\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"Boolean\ AdjacencyMatrix\" Arg=\"digraph\"/>\n <Oper Name=\"BooleanAdjacencyMatrixMuta\ bleCopy\" Arg=\"digraph\"/>\n <Returns>A square matrix of booleans.</Returns>\ \n <Description>\n If <A>digraph</A> is a digraph with a positive number o\ f vertices\n <C>n</C>, then <C>BooleanAdjacencyMatrix(</C><A>digraph</A><C>\ )</C>\n returns the boolean adjacency matrix <C>mat</C> of <A>digraph</A>. \ The\n value of the matrix entry <C>mat[j][i]</C> is <K>true</K> if and onl\ y if\n there exists an edge in <A>digraph</A> with source <C>j</C> and rang\ e\n <C>i</C>. If <A>digraph</A> has no vertices, then the empty list is\n \ returned. <P/>\n\n Note that the boolean adjacency matrix loses informat\ ion about multiple\n edges. <P/>\n\n The attribute <C>BooleanAdjacencyM\ atrix</C> returns an immutable list of\n immutable lists, whereas the funct\ ion\n <C>BooleanAdjacencyMatrixMutableCopy</C> returns a copy of the\n <\ C>BooleanAdjacencyMatrix</C> that is a mutable list of mutable lists. <P/>\n \ <Example><![CDATA[\ngap> gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);\n<\ immutable digraph with 4 vertices, 8 edges>\ngap> PrintArray(BooleanAdjacencyM\ atrix(gr));\n[ [ false, false, true, true ],\n [ false, true, true\ , false ],\n [ true, true, false, true ],\n [ false, false, false\ , true ] ]\ngap> gr := CycleDigraph(4);;\ngap> PrintArray(BooleanAdjacencyMa\ trix(gr));\n[ [ false, true, false, false ],\n [ false, false, true,\ false ],\n [ false, false, false, true ],\n [ true, false, false,\ false ] ]\ngap> BooleanAdjacencyMatrix(EmptyDigraph(0));\n[ ]\ngap> D := Cy\ cleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\n\ gap> PrintArray(BooleanAdjacencyMatrix(D));\n[ [ false, true, false ],\n \ [ false, false, true ],\n [ true, false, false ] ]\n]]></Example>\n \ </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphAdjac\ encyFunction\" Arg=\"digraph\"/>\n <Returns>A function.</Returns>\n <Descrip\ tion>\n If <A>digraph</A> is a digraph, then <C>DigraphAdjacencyFunction</C\ > returns\n a function which takes two integer parameters <C>x, y</C> and r\ eturns\n <K>true</K> if there exists an edge from vertex <C>x</C> to vertex\ <C>y</C>\n in <A>digraph</A> and <K>false</K> if not.\n\n <Example><![C\ DATA[\ngap> digraph := Digraph([[1, 2], [3], []]);\n<immutable digraph with 3 \ vertices, 3 edges>\ngap> foo := DigraphAdjacencyFunction(digraph);\nfunction( \ u, v ) ... end\ngap> foo(1, 1);\ntrue\ngap> foo(1, 2);\ntrue\ngap> foo(1, 3);\ \nfalse\ngap> foo(3, 1);\nfalse\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\ \"]);\n<immutable multidigraph with 3 vertices, 3 edges>\ngap> foo := DigraphA\ djacencyFunction(gr);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> f\ oo(3, 2);\nfalse\ngap> foo(3, 1);\nfalse\ngap> D := CycleDigraph(IsMutableDigr\ aph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> foo := DigraphAdjac\ encyFunction(D);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(2,\ 1);\nfalse\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection\ >\n <Attr Name=\"DigraphRange\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphSour\ ce\" Arg=\"digraph\"/>\n <Returns>A list of positive integers.</Returns>\n <\ Description>\n <C>DigraphRange</C> and <C>DigraphSource</C> return the rang\ e and source of\n the digraph <A>digraph</A>. More precisely, position <C>i\ </C> in\n <C>DigraphRange(<A>digraph</A>)</C> is the range of the <C>i</C>t\ h edge of\n <A>digraph</A>.\n\n <Example><![CDATA[\ngap> gr := Digraph([\ \n> [2, 1, 3, 5], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);\n<immutable digraph \ with 5 vertices, 14 edges>\ngap> DigraphRange(gr);\n[ 2, 1, 3, 5, 1, 3, 4, 2, \ 3, 2, 1, 2, 3, 4 ]\ngap> DigraphSource(gr);\n[ 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5\ , 5, 5, 5 ]\ngap> DigraphEdges(gr);\n[ [ 1, 2 ], [ 1, 1 ], [ 1, 3 ], [ 1, 5 ],\ [ 2, 1 ], [ 2, 3 ], \n [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ 4, 2 ], [ 5, 1 ], [ 5\ , 2 ], \n [ 5, 3 ], [ 5, 4 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\ \n<mutable digraph with 3 vertices, 3 edges>\ngap> DigraphRange(D);\n[ 2, 3, 1\ ]\ngap> DigraphSource(D);\n[ 1, 2, 3 ]\n]]></Example>\n </Description>\n</Ma\ nSection>\n\n <ManSection>\n <Attr Name=\"OutNeighbours\" Arg=\"digraph\"/\ >\n <Attr Name=\"OutNeighbors\" Arg=\"digraph\"/>\n <Oper Name=\"OutNeighbou\ rsMutableCopy\" Arg=\"digraph\"/>\n <Oper Name=\"OutNeighborsMutableCopy\" Ar\ g=\"digraph\"/>\n <Returns>The adjacencies of a digraph.</Returns>\n <Descri\ ption>\n This function returns the list <C>out</C> of out-neighbours of eac\ h vertex\n of the digraph <A>digraph</A>.\n More specifically, a vertex \ <C>j</C> appears in <C>out[i]</C> each time\n there exists an edge with sou\ rce <C>i</C> and range <C>j</C> in\n <A>digraph</A>. <P/>\n\n The functi\ on <C>OutNeighbours</C> returns an immutable list of\n lists, whereas the f\ unction <C>OutNeighboursMutableCopy</C> returns a copy\n of <C>OutNeighbour\ s</C> which is a mutable list of mutable lists. <P/>\n\n <Example><![CDATA[\ \ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\",\ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n<immutable digraph with \ 3 vertices, 3 edges>\ngap> OutNeighbours(gr);\n[ [ 2 ], [ 1, 3 ], [ ] ]\ngap>\ gr := Digraph([[1, 2, 3], [2, 1], [3]]);\n<immutable digraph with 3 vertices,\ 6 edges>\ngap> OutNeighbours(gr);\n[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]\ngap> gr \ := DigraphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n\ <immutable multidigraph with 3 vertices, 7 edges>\ngap> OutNeighbours(gr);\n[ \ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutNeighboursMutableCopy(gr);\n[ [ 1, \ 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<m\ utable digraph with 3 vertices, 3 edges>\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3\ ], [ 1 ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection\ >\n <Attr Name=\"InNeighbours\" Arg=\"digraph\"/>\n <Attr Name=\"InNeighbors\ \" Arg=\"digraph\"/>\n <Oper Name=\"InNeighboursMutableCopy\" Arg=\"digraph\"\ />\n <Oper Name=\"InNeighborsMutableCopy\" Arg=\"digraph\"/>\n <Returns>A li\ st of lists of vertices.</Returns>\n <Description>\n This function returns\ the list <C>inn</C> of in-neighbours of each vertex\n of the digraph <A>di\ graph</A>.\n More specifically, a vertex <C>j</C> appears in <C>inn[i]</C> \ each time\n there exists an edge with source <C>j</C> and range <C>i</C> in\ \n <A>digraph</A>. <P/>\n\n The function <C>InNeighbours</C> returns an \ immutable list of\n lists, whereas the function <C>InNeighboursMutableCopy<\ /C> returns a copy\n of <C>InNeighbours</C> which is a mutable list of muta\ ble lists. <P/>\n\n Note that each entry of <C>inn</C> is sorted into ascen\ ding order. <P/>\n\n <Example><![CDATA[\ngap> gr := Digraph([\"a\", \"b\", \ \"c\"],\n> [\"a\", \"b\", \"b\"],\n> [\"b\",\ \"a\", \"c\"]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> InNeighbo\ urs(gr);\n[ [ 2 ], [ 1 ], [ 2 ] ]\ngap> gr := Digraph([[1, 2, 3], [2, 1], [3]]\ );\n<immutable digraph with 3 vertices, 6 edges>\ngap> InNeighbours(gr);\n[ [ \ 1, 2 ], [ 1, 2 ], [ 1, 3 ] ]\ngap> gr := DigraphByAdjacencyMatrix([\n> [1, 2,\ 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n<immutable multidigraph with 3 vertices,\ 7 edges>\ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> I\ nNeighboursMutableCopy(gr);\n[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> D := Cy\ cleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\n\ gap> InNeighbours(D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n]]></Example>\n </Description\ >\n</ManSection>\n\n <ManSection>\n <Attr Name=\"OutDegrees\" Arg=\"digrap\ h\"/>\n <Attr Name=\"OutDegreeSequence\" Arg=\"digraph\"/>\n <Attr Name=\"Ou\ tDegreeSet\" Arg=\"digraph\"/>\n <Returns>A list of non-negative integers.</R\ eturns>\n <Description>\n\n Given a digraph <A>digraph</A> with <M>n</M> v\ ertices, the function\n <C>OutDegrees</C> returns an immutable list <C>out<\ /C> of length <M>n</M>, such that\n for a vertex <C>i</C> in <A>digraph</A>\ , the value of <C>out[i]</C> is the\n out-degree of vertex <C>i</C>.\n S\ ee <Ref Oper=\"OutDegreeOfVertex\"/>. <P/>\n\n The function <C>OutDegreeSeq\ uence</C> returns the same list,\n after it has been sorted into non-increa\ sing order. <P/>\n\n The function <C>OutDegreeSet</C> returns the same list\ , sorted into\n increasing order with duplicate entries removed. <P/>\n\n \ <Example><![CDATA[\ngap> D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);\n<imm\ utable multidigraph with 4 vertices, 6 edges>\ngap> OutDegrees(D);\n[ 4, 0, 2,\ 0 ]\ngap> OutDegreeSequence(D);\n[ 4, 2, 0, 0 ]\ngap> OutDegreeSet(D);\n[ 0, \ 2, 4 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 \ vertices, 3 edges>\ngap> OutDegrees(D);\n[ 1, 1, 1 ]\n]]></Example>\n </Descr\ iption>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"InDegrees\" Arg=\"d\ igraph\"/>\n <Attr Name=\"InDegreeSequence\" Arg=\"digraph\"/>\n <Attr Name=\ \"InDegreeSet\" Arg=\"digraph\"/>\n <Returns>A list of non-negative integers.\ </Returns>\n <Description>\n\n Given a digraph <A>digraph</A> with <M>n</M\ > vertices, the function\n <C>InDegrees</C> returns an immutable list <C>in\ n</C> of length <M>n</M>, such that\n for a vertex <C>i</C> in <A>digraph</\ A>, the value of <C>inn[i]</C> is\n the in-degree of vertex <C>i</C>.\n \ See <Ref Oper=\"InDegreeOfVertex\"/>. <P/>\n\n The function <C>InDegreeSequ\ ence</C> returns the same list,\n after it has been sorted into non-increas\ ing order. <P/>\n\n The function <C>InDegreeSet</C> returns the same list, \ sorted into\n increasing order with duplicate entries removed. <P/>\n\n \ <Example><![CDATA[\ngap> D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);\n\ <immutable multidigraph with 4 vertices, 8 edges>\ngap> InDegrees(D);\n[ 2, 3,\ 1, 2 ]\ngap> InDegreeSequence(D);\n[ 3, 2, 2, 1 ]\ngap> InDegreeSet(D);\n[ 1,\ 2, 3 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3\ vertices, 3 edges>\ngap> InDegrees(D);\n[ 1, 1, 1 ]\n]]></Example>\n </Descr\ iption>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"OutDegreeOfVertex\"\ Arg=\"digraph, vertex\"/>\n <Returns>The non-negative integer.</Returns>\n \ <Description>\n This operation returns the out-degree of the vertex <A>vert\ ex</A> in the\n digraph <A>digraph</A>.\n The out-degree of <A>vertex</A\ > is the number of edges in <A>digraph</A>\n whose source is <A>vertex</A>.\ \n <P/>\n <Example><![CDATA[\ngap> D := Digraph([\n> [2, 2, 1], [1, 4], \ [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]]);\n<immutable multidigraph with 4 vertice\ s, 16 edges>\ngap> OutDegreeOfVertex(D, 1);\n3\ngap> OutDegreeOfVertex(D, 2);\ \n2\ngap> OutDegreeOfVertex(D, 3);\n4\ngap> OutDegreeOfVertex(D, 4);\n7\n]]></\ Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"\ OutNeighboursOfVertex\" Arg=\"digraph, vertex\"/>\n <Oper Name=\"OutNeighbors\ OfVertex\" Arg=\"digraph, vertex\"/>\n <Returns>A list of vertices.</Returns>\ \n <Description>\n\n This operation returns the list <C>out</C> of vertice\ s of the digraph\n <A>digraph</A>.\n A vertex <C>i</C> appears in the li\ st <C>out</C> each time there exists an\n edge with source <A>vertex</A> a\ nd range <C>i</C> in <A>digraph</A>; in\n particular, this means that <C>ou\ t</C> may contain duplicates.<P/>\n <Example><![CDATA[\ngap> D := Digraph([\ \n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]]);\n<immutable mult\ idigraph with 4 vertices, 16 edges>\ngap> OutNeighboursOfVertex(D, 1);\n[ 2, 2\ , 3 ]\ngap> OutNeighboursOfVertex(D, 3);\n[ 2, 2, 3 ]\n]]></Example>\n </Desc\ ription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"InDegreeOfVertex\"\ Arg=\"digraph, vertex\"/>\n <Returns>A non-negative integer.</Returns>\n <D\ escription>\n This operation returns the in-degree of the vertex <A>vertex<\ /A> in the\n digraph <A>digraph</A>.\n The in-degree of <A>vertex</A> is\ the number of edges in <A>digraph</A>\n whose range is <A>vertex</A>.\n \ <Example><![CDATA[\ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [\ 1, 1, 2, 2, 1, 2, 2]]);\n<immutable multidigraph with 4 vertices, 16 edges>\ng\ ap> InDegreeOfVertex(D, 1);\n5\ngap> InDegreeOfVertex(D, 2);\n9\ngap> InDegree\ OfVertex(D, 3);\n0\ngap> InDegreeOfVertex(D, 4);\n2\n]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"InNeighboursOfVertex\ \" Arg=\"digraph, vertex\"/>\n <Oper Name=\"InNeighborsOfVertex\" Arg=\"digra\ ph, vertex\"/>\n <Returns>A list of postitive vertices.</Returns>\n <Descrip\ tion>\n\n This operation returns the list <C>inn</C> of vertices of the dig\ raph\n <A>digraph</A>.\n A vertex <C>i</C> appears in the list <C>inn</C\ > each time there exists an\n edge with source <C>i</C> and range <A>verte\ x</A> in <A>digraph</A>; in\n particular, this means that <C>inn</C> may co\ ntain duplicates. <P/>\n\n <Example><![CDATA[\ngap> D := Digraph([\n> [2, 2\ , 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]]);\n<immutable multidigraph \ with 4 vertices, 16 edges>\ngap> InNeighboursOfVertex(D, 1);\n[ 2, 4, 4, 4 ]\n\ gap> InNeighboursOfVertex(D, 2);\n[ 1, 1, 3, 3, 4, 4, 4, 4 ]\n]]></Example>\n \ </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphLoop\ s\" Arg=\"digraph\"/>\n <Returns>A list of vertices.</Returns>\n <Descriptio\ n>\n If <A>digraph</A> is a digraph, then <C>DigraphLoops</C> returns the l\ ist\n consisting of the <Ref Attr=\"DigraphVertices\"/> of <A>digraph</A>\n\ at which there is a loop. See <Ref Prop=\"DigraphHasLoops\"/>. <P/>\n\n \ <Example><![CDATA[\ngap> D := Digraph([[2], [3], []]);\n<immutable digraph wi\ th 3 vertices, 2 edges>\ngap> DigraphHasLoops(D);\nfalse\ngap> DigraphLoops(D)\ ;\n[ ]\ngap> D := Digraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);\n<immutable\ digraph with 5 vertices, 9 edges>\ngap> DigraphLoops(D);\n[ 3, 4 ]\ngap> D :=\ Digraph(IsMutableDigraph, [[1], [1]]);\n<mutable digraph with 2 vertices, 2 e\ dges>\ngap> DigraphLoops(D);\n[ 1 ]\n]]></Example>\n </Description>\n</ManSec\ tion>\n\n <ManSection>\n <Oper Name=\"PartialOrderDigraphMeetOfVertices\"\ \n Arg=\"digraph, u, v\"/>\n <Oper Name=\"PartialOrderDigraphJoinOfVertice\ s\"\n Arg=\"digraph, u, v\"/>\n <Returns>A positive integer or <K>fail</K>\ </Returns>\n <Description>\n If the first argument is a partial order di\ graph <Ref\n Prop=\"IsPartialOrderDigraph\"/> then these operations return \ the meet, or\n the join, of the two input vertices. If the meet (or join) i\ s does not\n exist then <K>fail</K> is returned. The meet (or join) is guar\ anteed to\n exist when the first argument satisfies\n <Ref Prop=\"IsMeet\ SemilatticeDigraph\"/> (or\n <Ref Prop=\"IsJoinSemilatticeDigraph\"/>) - s\ ee the documentation for these\n properties for the definition of the meet \ (or the join).\n\n <Example><![CDATA[\ngap> D := Digraph([[1], [1, 2], [1, \ 3], [1, 2, 3, 4]]);\n<immutable digraph with 4 vertices, 9 edges>\ngap> Partia\ lOrderDigraphMeetOfVertices(D, 2, 3);\n4\ngap> PartialOrderDigraphJoinOfVertic\ es(D, 2, 3);\n1\ngap> PartialOrderDigraphMeetOfVertices(D, 1, 2);\n2\ngap> Par\ tialOrderDigraphJoinOfVertices(D, 3, 4);\n3\ngap> D := Digraph([[1], [2], [1, \ 2, 3], [1, 2, 4]]);\n<immutable digraph with 4 vertices, 8 edges>\ngap> Partia\ lOrderDigraphMeetOfVertices(D, 3, 4);\nfail\ngap> PartialOrderDigraphJoinOfVer\ tices(D, 3, 4);\nfail\ngap> PartialOrderDigraphMeetOfVertices(D, 1, 2);\nfail\ \ngap> PartialOrderDigraphJoinOfVertices(D, 1, 2);\nfail\n]]></Example>\n </D\ escription>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DegreeMatrix\" \ Arg=\"digraph\"/>\n <Returns>A square matrix of non-negative integers.</Retur\ ns>\n <Description>\n This function returns the outdegree matrix <C>mat</C\ > of the digraph\n <A>digraph</A>. The value of the <C>i</C>th diagonal mat\ rix entry is the\n outdegree of the vertex <C>i</C> in <A>digraph</A>. All \ off-diagonal entries \n are <C>0</C>.\n If <A>digraph</A> has no vertice\ s, then the empty list is returned. <P/>\n\n See <Ref Attr=\"OutDegrees\"/>\ for more information.\n <Example><![CDATA[\ngap> D := Digraph([[1, 2, 3], \ [4], [1, 3, 4], []]);\n<immutable digraph with 4 vertices, 7 edges>\ngap> Prin\ tArray(DegreeMatrix(D));\n[ [ 3, 0, 0, 0 ],\n [ 0, 1, 0, 0 ],\n [ 0\ , 0, 3, 0 ],\n [ 0, 0, 0, 0 ] ]\ngap> D := CycleDigraph(5);;\ngap> Pri\ ntArray(DegreeMatrix(D));\n[ [ 1, 0, 0, 0, 0 ],\n [ 0, 1, 0, 0, 0 ]\ ,\n [ 0, 0, 1, 0, 0 ],\n [ 0, 0, 0, 1, 0 ],\n [ 0, 0, 0, 0, \ 1 ] ]\ngap> DegreeMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMut\ ableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> Display(Deg\ reeMatrix(D));\n[ [ 1, 0, 0 ],\n [ 0, 1, 0 ],\n [ 0, 0, 1 ] ]\n]]><\ /Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\ \"LaplacianMatrix\" Arg=\"digraph\"/>\n <Returns>A square matrix of integers.\ </Returns>\n <Description>\n This function returns the outdegree Laplacian\ matrix <C>mat</C> of the\n digraph <A>digraph</A>. The outdegree Laplacian\ matrix is defined as\n <C>DegreeMatrix(digraph) - AdjacencyMatrix(digraph)\ </C>. If\n <A>digraph</A> has no vertices, then the empty list is returned\ . <P/>\n\n See <Ref Attr=\"DegreeMatrix\"/> and <Ref Attr=\"AdjacencyMatrix\ \"/> for more\n information.\n <Example><![CDATA[\ngap> gr := Digraph([[\ 1, 2, 3], [4], [1, 3, 4], []]);\n<immutable digraph with 4 vertices, 7 edges>\ \ngap> PrintArray(LaplacianMatrix(gr));\n[ [ 2, -1, -1, 0 ],\n [ 0, \ 1, 0, -1 ],\n [ -1, 0, 2, -1 ],\n [ 0, 0, 0, 0 ] ]\ngap> L\ aplacianMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableDigrap\ h, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> Display(LaplacianMatr\ ix(D));\n[ [ 1, -1, 0 ],\n [ 0, 1, -1 ],\n [ -1, 0, 1 ] ]\n]]\ ></Example>\n </Description>\n</ManSection>\n\n </Section>\n\n <Section><He\ ading>Reachability and connectivity</Heading>\n <ManSection>\n <Attr Name=\ \"DigraphDiameter\" Arg=\"digraph\"/>\n <Returns>An integer or <C>fail</C>.</\ Returns>\n <Description>\n This function returns the diameter of the digra\ ph <A>digraph</A>.\n <P/>\n\n If a digraph <A>digraph</A> is strongly co\ nnected and has at least 1\n vertex, then the <E>diameter</E> is the maximu\ m shortest distance between\n any pair of distinct vertices. Otherwise then\ the diameter of\n <A>digraph</A> is undefined, and this function returns t\ he value\n <C>fail</C>. <P/>\n\n See <Ref Attr=\"DigraphShortestDistance\ s\"/>. <P/>\n <Example><![CDATA[\ngap> D := Digraph([[2], [3], [4, 5], [5],\ [1, 2, 3, 4, 5]]);\n<immutable digraph with 5 vertices, 10 edges>\ngap> Digra\ phDiameter(D);\n3\ngap> D := ChainDigraph(2);\n<immutable chain digraph with 2\ vertices>\ngap> DigraphDiameter(D);\nfail\ngap> IsStronglyConnectedDigraph(D)\ ;\nfalse\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 6, 2);\n<mutabl\ e digraph with 12 vertices, 36 edges>\ngap> DigraphDiameter(D);\n4\n]]></Examp\ le>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Digra\ phShortestDistance\" Label=\"for a digraph and two vertices\"\n Arg=\"d\ igraph, u, v\"/>\n <Oper Name=\"DigraphShortestDistance\" Label=\"for a digra\ ph and a list\"\n Arg=\"digraph, list\"/>\n <Oper Name=\"DigraphShorte\ stDistance\" Label=\"for a digraph, a list, and a list\"\n Arg=\"digrap\ h, list1, list2\"/>\n <Returns>An integer or <K>fail</K></Returns>\n <Descri\ ption>\n If there is a directed path in the digraph <A>digraph</A> between \ vertex\n <A>u</A> and vertex <A>v</A>, then this operation returns the leng\ th of the\n shortest such directed path. If no such directed path exists, \ then this\n operation returns <K>fail</K>. See Section <Ref Subsect=\"Defin\ itions\"\n Style=\"Number\" /> for the definition of a directed path. <P/\ >\n\n If the second form is used, then <A>list</A> should be a list of leng\ th two,\n containing two positive integers which correspond to the vertices\ <A>u</A>\n and <A>v</A>. <P/>\n\n Note that as usual, a vertex is consi\ dered to be at distance 0 from\n itself .<P/>\n\n If the third form is u\ sed, then <A>list1</A> and <A>list2</A> are both lists\n of vertices. The s\ hortest directed path between <A>list1</A> and\n <A>list2</A> is then the l\ ength of the shortest directed path which starts\n with a vertex in <A>list\ 1</A> and terminates at a vertex in <A>list2</A>, if\n such directed path e\ xists. If <A>list1</A> and <A>list2</A> have non-empty\n intersection, the \ operation returns <C>0</C>.<P/>\n <Example><![CDATA[\ngap> D := Digraph([[2\ ], [3], [1, 4], [1, 3], [5]]);\n<immutable digraph with 5 vertices, 7 edges>\n\ gap> DigraphShortestDistance(D, 1, 3);\n2\ngap> DigraphShortestDistance(D, [3,\ 3]);\n0\ngap> DigraphShortestDistance(D, 5, 2);\nfail\ngap> DigraphShortestDi\ stance(D, [1, 2], [4, 5]);\n2\ngap> DigraphShortestDistance(D, [1, 3], [3, 5])\ ;\n0\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <A\ ttr Name=\"DigraphShortestDistances\" Arg=\"digraph\"/>\n <Returns>A square m\ atrix.</Returns>\n <Description>\n If <A>digraph</A> is a digraph with <M>\ n</M> vertices, then this\n function returns an <M>n \\times n</M> matrix <\ C>mat</C>, where each entry is\n either a non-negative integer, or <K>fail<\ /K>. If <M>n = 0</M>, then an\n empty list is returned. <P/>\n\n If the\ re is a directed path from vertex <C>i</C> to vertex <C>j</C>, then\n the v\ alue of <C>mat[i][j]</C> is the length of the shortest such directed\n path\ . If no such directed path exists, then the value of <C>mat[i][j]</C> is\n \ <C>fail</C>. We use the convention that the distance from every vertex to\n \ itself is <C>0</C>, i.e. <C>mat[i][i] = 0</C> for all vertices <C>i</C>.\n \ <P/>\n\n The method used in this function is a version of the Floyd-Warsh\ all\n algorithm, and has complexity <M>O(n^3)</M>.\n\n <Example><![CDATA\ [\ngap> D := Digraph([[1, 2], [3], [1, 2], [4]]);\n<immutable digraph with 4 v\ ertices, 6 edges>\ngap> mat := DigraphShortestDistances(D);;\ngap> PrintArray(\ mat);\n[ [ 0, 1, 2, fail ],\n [ 2, 0, 1, fail ],\n \ [ 1, 1, 0, fail ],\n [ fail, fail, fail, 0 ] ]\ngap> D :\ = CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with 3 vertices, 3 edge\ s>\ngap> DigraphShortestDistances(D);\n[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ]\ ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Ope\ r Name=\"DigraphLongestDistanceFromVertex\" Arg=\"digraph, v\"/>\n <Returns>A\ n integer, or <K>infinity</K>.</Returns>\n <Description>\n If <A>digraph</\ A> is a digraph and <A>v</A> is a vertex in <A>digraph</A>,\n then this ope\ ration returns the length of the longest directed walk in\n <A>digraph</A> \ which begins at vertex <A>v</A>. See Section <Ref\n Subsect=\"Definition\ s\" Style=\"Number\" /> for the definitions of directed\n walk, directed cy\ cle, and loop.\n <P/>\n\n <List>\n <Item>\n If there exists \ a directed walk starting at vertex <A>v</A>\n which traverses a loop or\ a directed cycle,\n then we consider there to be a walk of infinite le\ ngth from <A>v</A>\n (realised by repeatedly traversing the loop/direct\ ed cycle),\n and so the result is <K>infinity</K>.\n To disallow\ walks using loops, try using\n <Ref Oper=\"DigraphRemoveLoops\"/>:<P/>\ \n\n <C>DigraphLongestDistanceFromVertex(DigraphRemoveLoops(<A>digraph<\ /A>,<A>v</A>))</C>.\n </Item>\n\n <Item>\n Otherwise, if all \ directed walks\n starting at vertex <A>v</A> have finite length,\n \ then the length of the longest such walk is returned.\n </Item>\n <\ /List>\n\n Note that the result is <C>0</C> if and only if <A>v</A> is a si\ nk of\n <A>digraph</A>. See <Ref Attr=\"DigraphSinks\"/>.\n\n<Example><![C\ DATA[\ngap> D := Digraph([[2], [3, 4], [], [5], [], [6]]);\n<immutable digraph\ with 6 vertices, 5 edges>\ngap> DigraphLongestDistanceFromVertex(D, 1);\n3\ng\ ap> DigraphLongestDistanceFromVertex(D, 3);\n0\ngap> 3 in DigraphSinks(D);\ntr\ ue\ngap> DigraphLongestDistanceFromVertex(D, 6);\ninfinity\n]]></Example>\n <\ /Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphDistan\ ceSet\" Label=\"for a digraph, a pos int, and an int\" Arg=\"digraph, vertex, \ distance\"/>\n <Oper Name=\"DigraphDistanceSet\" Label=\"for a digraph, a pos\ int, and a list\" Arg=\"digraph, vertex, distances\"/>\n <Returns>A list of \ vertices</Returns>\n <Description>\n This operation returns the list of al\ l vertices in digraph <A>digraph</A>\n such that the shortest distance to a\ vertex <A>vertex</A> is\n <A>distance</A> or is in the list <A>distances</\ A>. <P/>\n\n <A>digraph</A> should be a digraph, <A>vertex</A> should be a \ positive\n integer, <A>distance</A> should be a non-negative integer, and\n\ <A>distances</A> should be a list of non-negative integers.\n\n <Exampl\ e><![CDATA[\ngap> D := Digraph([[2], [3], [1, 4], [1, 3]]);\n<immutable digrap\ h with 4 vertices, 6 edges>\ngap> DigraphDistanceSet(D, 2, [1, 2]);\n[ 3, 1, 4\ ]\ngap> DigraphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ngap> DigraphDistanceSet(D, 2,\ 0);\n[ 2 ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection\ >\n <Attr Name=\"DigraphGirth\" Arg=\"digraph\"/>\n <Returns>An integer, or \ <K>infinity</K>.</Returns>\n <Description>\n This attribute returns the <E\ >girth</E> of the digraph <A>digraph</A>.\n The <E>girth</E> of a digraph i\ s the length of its shortest simple circuit.\n See Section <Ref Subsect=\"D\ efinitions\" Style=\"Number\" /> for the definitions\n of simple circuit, d\ irected cycle, and loop.\n <P/>\n\n If <A>digraph</A> has no directed cy\ cles, then this function will return\n <K>infinity</K>. If <A>digraph</A> \ contains a loop, then this function will\n return <C>1</C>.\n <P/>\n\n \ In the worst case, the method used in this function is a version of the\n \ Floyd-Warshall algorithm, and has complexity <C>O(<A>n</A> ^ 3)</C>, where\n \ <A>n</A> is the number of vertices in <A>digraph</A>. If the digraph has\n\ known automorphisms [see <Ref Attr=\"DigraphGroup\"/>], then the performan\ ce\n is likely to be better.\n <P/>\n\n For symmetric digraphs, see a\ lso <Ref Attr=\"DigraphUndirectedGirth\"/>. <P/>\n <Example><![CDATA[\ngap>\ D := Digraph([[1], [1]]);\n<immutable digraph with 2 vertices, 2 edges>\ngap>\ DigraphGirth(D);\n1\ngap> D := Digraph([[2, 3], [3], [4], []]);\n<immutable d\ igraph with 4 vertices, 4 edges>\ngap> DigraphGirth(D);\ninfinity\ngap> D := D\ igraph([[2, 3], [3], [4], [1]]);\n<immutable digraph with 4 vertices, 5 edges>\ \ngap> DigraphGirth(D);\n3\ngap> D := GeneralisedPetersenGraph(IsMutableDigrap\ h, 6, 2);\n<mutable digraph with 12 vertices, 36 edges>\ngap> DigraphGirth(D);\ \n2\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <At\ tr Name=\"DigraphOddGirth\" Arg=\"digraph\"/>\n <Returns>An integer, or <K>in\ finity</K>.</Returns>\n <Description>\n This attribute returns the <E>odd \ girth</E> of the digraph <A>digraph</A>.\n The <E>odd girth</E> of a digrap\ h is the length of its shortest simple\n circuit of odd length. See Sectio\ n <Ref Subsect=\"Definitions\"\n Style=\"Number\" /> for the definitions of\ simple circuit, directed\n cycle, and loop.\n <P/>\n\n If <A>digraph\ </A> has no directed cycles of odd length, then this function\n will return\ <K>infinity</K>, even if <A>digraph</A> has a directed cycle\n of even len\ gth. If <A>digraph</A> contains a loop, then this function\n will return <\ C>1</C>.\n <P/>\n\n See also <Ref Attr=\"DigraphGirth\"/>. <P/>\n <Ex\ ample><![CDATA[\ngap> D := Digraph([[2], [3, 1], [1]]);\n<immutable digraph wi\ th 3 vertices, 4 edges>\ngap> DigraphOddGirth(D);\n3\ngap> D := CycleDigraph(4\ );\n<immutable cycle digraph with 4 vertices>\ngap> DigraphOddGirth(D);\ninfin\ ity\ngap> D := Digraph([[2], [3], [], [3], [4]]);\n<immutable digraph with 5 v\ ertices, 4 edges>\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := CycleDigraph(\ IsMutableDigraph, 4);\n<mutable digraph with 4 vertices, 4 edges>\ngap> Digrap\ hDisjointUnion(D, CycleDigraph(5));\n<mutable digraph with 9 vertices, 9 edges\ >\ngap> DigraphOddGirth(D);\n5\n]]></Example>\n </Description>\n</ManSection>\ \n\n <ManSection>\n <Attr Name=\"DigraphUndirectedGirth\" Arg=\"digraph\"/\ >\n <Returns>An integer or <K>infinity</K>.</Returns>\n <Description>\n I\ f <A>digraph</A> is a symmetric digraph, then this function returns the\n g\ irth of <A>digraph</A> when treated as an undirected graph (i.e. each pair\n \ of edges <M>[i, j]</M> and <M>[j, i]</M> is treated as a single edge between\ \n <M>i</M> and <M>j</M>). <P/>\n\n The <E>girth</E> of an undirected gr\ aph is the length of its shortest simple\n cycle, i.e. the shortest non-tri\ vial path starting and ending at the same\n vertex and passing through no v\ ertex or edge more than once. <P/>\n\n If <A>digraph</A> has no cycles, the\ n this function will return\n <K>infinity</K>. <P/>\n <Example><![CDATA[\ \ngap> D := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);\n<immutable digraph wit\ h 4 vertices, 8 edges>\ngap> DigraphUndirectedGirth(D);\n4\ngap> D := Digraph(\ [[2], [1, 3], [2]]);\n<immutable digraph with 3 vertices, 4 edges>\ngap> Digra\ phUndirectedGirth(D);\ninfinity\ngap> D := Digraph([[1], [], [4], [3]]);\n<imm\ utable digraph with 4 vertices, 3 edges>\ngap> DigraphUndirectedGirth(D);\n1\n\ gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n<mutable digraph \ with 18 vertices, 54 edges>\ngap> DigraphUndirectedGirth(D);\n5\n]]></Example>\ \n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphC\ onnectedComponents\" Arg=\"digraph\"/>\n <Attr Name=\"DigraphNrConnectedCompo\ nents\" Arg=\"digraph\"/>\n <Returns>A record.</Returns>\n <Description>\n \ This function returns the record <C>wcc</C> corresponding to the weakly\n \ connected components of the digraph <A>digraph</A>. Two vertices of\n <A>\ digraph</A> are in the same weakly connected component whenever they are\n \ equal, or there exists a directed path (ignoring the orientation of edges)\n \ between them. More formally, two vertices are in the same weakly connected\ \n component of <A>digraph</A> if and only if they are in the same strongly\ \n connected component (see <Ref Attr=\"DigraphStronglyConnectedComponents\ \"/>)\n of the <Ref Oper=\"DigraphSymmetricClosure\"/> of <A>digraph</A>. \ <P/>\n\n The set of weakly connected components is a partition of\n the \ vertex set of <A>digraph</A>. <P/>\n\n The record <C>wcc</C> has 2 componen\ ts: <C>comps</C> and <C>id</C>.\n The component <C>comps</C> is a list of t\ he weakly connected components\n of <A>digraph</A> (each of which is a lis\ t of vertices).\n The component <C>id</C> is a list such that the vertex <C\ >i</C> is an\n element of the weakly connected component <C>comps[id[i]]</C\ >. <P/>\n\n The method used in this function has complexity <M>O(m+n)</M>,\ where\n <M>m</M> is the number of edges and\n <M>n</M> is the number of\ vertices in the digraph. <P/>\n\n <C>DigraphNrConnectedComponents(<A>digra\ ph</A>)</C> is simply a shortcut\n for <C>Length(DigraphConnectedComponents\ (<A>digraph</A>).comps)</C>,\n and is no more efficient.\n\n <Example><!\ [CDATA[\ngap> gr := Digraph([[2], [3, 1], []]);\n<immutable digraph with 3 ver\ tices, 3 edges>\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ [ 1, 2,\ 3 ] ], id := [ 1, 1, 1 ] )\ngap> gr := Digraph([[1], [1, 2], []]);\n<immutabl\ e digraph with 3 vertices, 3 edges>\ngap> DigraphConnectedComponents(gr);\nrec\ ( comps := [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )\ngap> DigraphNrConnectedCo\ mponents(gr);\n2\ngap> gr := EmptyDigraph(0);\n<immutable empty digraph with 0\ vertices>\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ ], id := [ \ ] )\ngap> D := CycleDigraph(IsMutableDigraph, 2);\n<mutable digraph with 2 ve\ rtices, 2 edges>\ngap> G := CycleDigraph(3);\n<immutable cycle digraph with 3 \ vertices>\ngap> DigraphDisjointUnion(D, G);\n<mutable digraph with 5 vertices,\ 5 edges>\ngap> DigraphConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3,\ 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n]]></Example>\n </Description>\n</ManSe\ ction>\n\n <ManSection>\n <Oper Name=\"DigraphConnectedComponent\" Arg=\"d\ igraph, vertex\"/>\n <Returns>A list of vertices.</Returns>\n <Description>\ \n If <A>vertex</A> is a vertex in the digraph <A>digraph</A>, then\n th\ is operation returns the connected component of <A>vertex</A>\n in <A>digra\ ph</A>.\n See <Ref Attr=\"DigraphConnectedComponents\"/> for more informati\ on.\n <Example><![CDATA[\ngap> D := Digraph([[3], [2], [1, 2], [4]]);\n<imm\ utable digraph with 4 vertices, 5 edges>\ngap> DigraphConnectedComponent(D, 3)\ ;\n[ 1, 2, 3 ]\ngap> DigraphConnectedComponent(D, 2);\n[ 1, 2, 3 ]\ngap> Digra\ phConnectedComponent(D, 4);\n[ 4 ]\n]]></Example>\n </Description>\n</ManSect\ ion>\n\n <ManSection>\n <Attr Name=\"DigraphStronglyConnectedComponents\" \ Arg=\"digraph\"/>\n <Attr Name=\"DigraphNrStronglyConnectedComponents\" Arg=\ \"digraph\"/>\n <Returns>A record.</Returns>\n <Description>\n This funct\ ion returns the record <C>scc</C> corresponding to the strongly\n connected\ components of the digraph <A>digraph</A>. Two vertices of\n <A>digraph</A\ > are in the same strongly connected component whenever they\n are equal, o\ r there is a directed path from each vertex to the other. The\n set of str\ ongly connected components is a partition of the vertex set of\n <A>digraph\ </A>.\n <P/>\n\n The record <C>scc</C> has 2 components: <C>comps</C> an\ d <C>id</C>.\n The component <C>comps</C> is a list of the strongly connect\ ed components\n of <A>digraph</A> (each of which is a list of vertices).\n \ The component <C>id</C> is a list such that the vertex <C>i</C> is an\n \ element of the strongly connected component <C>comps[id[i]]</C>. <P/>\n\n T\ he method used in this function is a non-recursive version of Gabow's\n Alg\ orithm <Cite Key=\"Gab00\"/> and has complexity <M>O(m+n)</M> where\n <M>m<\ /M> is the number of edges (counting multiple edges as one) and\n <M>n</M> \ is the number of vertices in the digraph. <P/>\n\n <C>DigraphNrStronglyConn\ ectedComponents(<A>digraph</A>)</C> is simply a\n shortcut for\n <C>Leng\ th(DigraphStronglyConnectedComponents(<A>digraph</A>).comps)</C>,\n and is \ no more efficient.\n\n <Example><![CDATA[\ngap> gr := Digraph([[2], [3, 1],\ []]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> DigraphStronglyConn\ ectedComponents(gr);\nrec( comps := [ [ 3 ], [ 1, 2 ] ], id := [ 2, 2, 1 ] )\n\ gap> DigraphNrStronglyConnectedComponents(gr);\n2\ngap> D := DigraphDisjointUn\ ion(CycleDigraph(4), CycleDigraph(5));\n<immutable digraph with 9 vertices, 9 \ edges>\ngap> DigraphStronglyConnectedComponents(D);\nrec( comps := [ [ 1, 2, 3\ , 4 ], [ 5, 6, 7, 8, 9 ] ], \n id := [ 1, 1, 1, 1, 2, 2, 2, 2, 2 ] )\ngap> Di\ graphNrStronglyConnectedComponents(D);\n2\ngap> D := CycleDigraph(IsMutableDig\ raph, 2);\n<mutable digraph with 2 vertices, 2 edges>\ngap> G := CycleDigraph(\ 3);\n<immutable cycle digraph with 3 vertices>\ngap> DigraphDisjointUnion(D, G\ );\n<mutable digraph with 5 vertices, 5 edges>\ngap> DigraphStronglyConnectedC\ omponents(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ], id := [ 1, 1, 2, 2, 2 \ ] )\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Op\ er Name=\"DigraphStronglyConnectedComponent\" Arg=\"digraph, vertex\"/>\n <Re\ turns>A list of vertices.</Returns>\n <Description>\n If <A>vertex</A> is \ a vertex in the digraph <A>digraph</A>, then\n this operation returns the s\ trongly connected component of <A>vertex</A>\n in <A>digraph</A>.\n See \ <Ref Attr=\"DigraphStronglyConnectedComponents\"/> for more information.\n \ <Example><![CDATA[\ngap> D := Digraph([[3], [2], [1, 2], [3]]);\n<immutable di\ graph with 4 vertices, 5 edges>\ngap> DigraphStronglyConnectedComponent(D, 3);\ \n[ 1, 3 ]\ngap> DigraphStronglyConnectedComponent(D, 2);\n[ 2 ]\ngap> Digraph\ StronglyConnectedComponent(D, 4);\n[ 4 ]\n]]></Example>\n </Description>\n</M\ anSection>\n\n <ManSection>\n <Attr Name=\"DigraphBicomponents\" Arg=\"dig\ raph\"/>\n <Returns>A pair of lists of vertices, or <K>fail</K>.</Returns>\n \ <Description>\n If <A>digraph</A> is a bipartite digraph, i.e. if it satis\ fies <Ref\n Prop=\"IsBipartiteDigraph\"/>, then <C>DigraphBicomponents</C\ > returns a\n pair of bicomponents of <A>digraph</A>. Otherwise,\n <C>Di\ graphBicomponents</C> returns <K>fail</K>.<P/>\n\n For a bipartite digraph,\ the vertices can be partitioned into two non-empty\n sets such that the so\ urce and range of any edge are in distinct sets. The\n parts of this partit\ ion are called <E>bicomponents</E> of <A>digraph</A>.\n Equivalently, a pai\ r of bicomponents of <A>digraph</A> consists of the\n color-classes of a 2-\ coloring of <A>digraph</A>. <P/>\n\n For a bipartite digraph with at least \ 3 vertices, there is a unique pair of\n bicomponents of bipartite if and on\ ly if the digraph is connected. See <Ref\n Prop=\"IsConnectedDigraph\"/> \ for more information.\n <Example><![CDATA[\ngap> D := CycleDigraph(3);\n<im\ mutable cycle digraph with 3 vertices>\ngap> DigraphBicomponents(D);\nfail\nga\ p> D := ChainDigraph(5);\n<immutable chain digraph with 5 vertices>\ngap> Digr\ aphBicomponents(D);\n[ [ 1, 3, 5 ], [ 2, 4 ] ]\ngap> D := Digraph([[5], [1, 4]\ , [5], [5], []]);\n<immutable digraph with 5 vertices, 5 edges>\ngap> DigraphB\ icomponents(D);\n[ [ 1, 3, 4 ], [ 2, 5 ] ]\ngap> D := CompleteBipartiteDigraph\ (IsMutableDigraph, 2, 3);\n<mutable digraph with 5 vertices, 12 edges>\ngap> D\ igraphBicomponents(D);\n[ [ 1, 2 ], [ 3, 4, 5 ] ]\n]]></Example>\n </Descript\ ion>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"ArticulationPoints\" A\ rg=\"digraph\"/>\n <Returns>A list of vertices.</Returns>\n <Description>\n \ A connected digraph is <E>biconnected</E> if it is still connected (in the\ \n sense of <Ref Prop=\"IsConnectedDigraph\"/>) when any vertex is removed.\ \n If the digraph <A>digraph</A> is not biconnected but is connected, then\ any\n vertex <C>v</C> of <A>digraph</A> whose removal makes the resulting \ digraph\n disconnected is called an <E>articulation point</E>.<P/>\n\n <\ C>ArticulationPoints</C> returns a list of the articulation points of\n <A>\ digraph</A>, if any, and, in particular, returns the empty list if\n <A>dig\ raph</A> is not connected. <P/>\n\n Multiple edges are ignored by this meth\ od. <P/>\n\n The method used in this operation has complexity <M>O(m+n)</M>\ where\n <M>m</M> is the number of edges and <M>n</M> is the number of vert\ ices in\n the digraph.<P/>\n \n If <A>D</A> has a bridge (see <Ref \ Attr=\"Bridges\"/>), then a node incident\n to the bridge is an articulatio\ n point if and only if it has degree at\n least <M>2</M>. It follows that \ if <A>D</A> has a bridge and at least\n <M>3</M> nodes, then at least one o\ f the nodes incident to the bridge is\n an articulation point. The convers\ e does not hold, there are digraphs\n with articulation points, but no brid\ ges.\n <P/>\n\n See also <Ref Prop=\"IsBiconnectedDigraph\"/> and \n \ <Ref Prop=\"IsBridgelessDigraph\"/>.\n<Example><![CDATA[\ngap> ArticulationPoi\ nts(CycleDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [3, 5], [4], [2], [6],\ [1], []]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap> ArticulationPoints(C\ hainDigraph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(NullDigraph(5));\n[ ]\ \ngap> D := ChainDigraph(IsMutableDigraph, 4);\n<mutable digraph with 4 vertic\ es, 3 edges>\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"Bridges\" Arg=\"D\"/\ >\n <Returns>A (possibly empty) list of edges.</Returns>\n <Description>\n \ A connected digraph is <E>2-edge-connected</E> if it is still connected (in\ \n the sense of <Ref Prop=\"IsConnectedDigraph\"/>) when any edge is\n r\ emoved. If the digraph <A>D</A> is not 2-edge-connected but is\n connected\ , then any edge <C>[u, v]</C> of <A>D</A> whose removal\n makes the resulti\ ng digraph disconnected is called a <E>bridge</E>.<P/>\n\n <C>Bridges</C> r\ eturns a list of the bridges of <A>D</A>, if any, and, in\n particular, ret\ urns the empty list if <A>D</A> is not connected. <P/>\n\n Multiple edges a\ re ignored by this method. <P/>\n\n The method used in this operation has c\ omplexity <M>O(m+n)</M> where\n <M>m</M> is the number of edges and <M>n</M\ > is the number of vertices in\n the digraph.\n <P/>\n\n If <A>D</A> \ has a bridge, then a node incident to the bridge is an\n articulation point\ (see <Ref Attr=\"ArticulationPoints\"/>) if and only if\n it has degree at\ least <M>2</M>. It follows that if <A>D</A> has a\n bridge and at least <\ M>3</M> nodes, then at least one of the nodes\n incident to the bridge is a\ n articulation point. The converse does not\n hold, there are digraphs with\ articulation points, but no bridges.\n <P/>\n\n See also <Ref Prop=\"Is\ BiconnectedDigraph\"/> and \n <Ref Prop=\"IsBridgelessDigraph\"/>.\n<Exampl\ e><![CDATA[\ngap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]])\ ;\n<immutable digraph with 5 vertices, 12 edges>\ngap> Bridges(D);\n[ ]\ngap>\ D := Digraph([[2], [3], [4], [2]]);\n<immutable digraph with 4 vertices, 4 ed\ ges>\ngap> Bridges(D);\n[ [ 1, 2 ] ]\ngap> Bridges(ChainDigraph(2));\n[ [ 1, 2\ ] ]\ngap> ArticulationPoints(ChainDigraph(2));\n[ ]\n]]></Example>\n </Desc\ ription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"StrongOrientation\ \" Arg=\"D\"/>\n <Attr Name=\"StrongOrientationAttr\" Arg=\"D\"/>\n <Returns\ >A digraph or <K>fail</K>.</Returns>\n <Description>\n A <E>strong orienta\ tion</E> of a connected symmetric digraph <A>D</A> (if\n it exists) is a st\ rongly connected subdigraph <C>C</C> of <A>D</A> such\n that for every edge\ <C>[u, v]</C> of <A>D</A> either <C>[u, v]</C> or\n <C>[v, u]</C> is an ed\ ge of <C>C</C> but not both. Robbin's Theorem states\n that a digraph admi\ ts a strong orientation if and only if it is\n bridgeless (see <Ref Prop=\"\ IsBridgelessDigraph\"/>). \n <P/>\n\n This operation returns a strong or\ ientation of the digraph <A>D</A> if \n <A>D</A> is symmetric and <A>D</A> \ admits a strong orientation. If <A>D</A>\n is symmetric but does not admit \ a strong orientation, then <K>fail</K> is\n returned. If <A>D</A> is not sy\ mmetric, then an error is given. \n <P/>\n\n If <A>D</A> is immutable, <\ C>StrongOrientation(<A>D</A>)</C> returns an\n immutable digraph, and if <A\ >D</A> is mutable, then\n <C>StrongOrientation(<A>D</A>)</C> returns a muta\ ble digraph. <P/>\n\n The method used in this operation has complexity <M>O\ (m+n)</M> where\n <M>m</M> is the number of edges and <M>n</M> is the numbe\ r of vertices in\n the digraph.\n<Example><![CDATA[\ngap> StrongOrientation\ (DigraphSymmetricClosure(CycleDigraph(5))) \n> = CycleDigraph(5);\ntrue\ngap> \ D := DigraphSymmetricClosure(Digraph(\n> [[2, 7], [3, 5], [4], [2], [6], [1], \ []]));;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> StrongOrientation(D);\nfail\ \ngap> StrongOrientation(NullDigraph(0));\n<immutable empty digraph with 0 ver\ tices>\ngap> StrongOrientation(DigraphDisjointUnion(CompleteDigraph(3), \n> \ CompleteDigraph(3)));\nfail\n]]></Exam\ ple>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"Digr\ aphPeriod\" Arg=\"digraph\"/>\n <Returns>An integer.</Returns>\n <Descriptio\ n>\n This function returns the period of the digraph <A>digraph</A>.<P/>\n\ \n If a digraph <A>digraph</A> has at least one directed cycle, then the pe\ riod\n is the greatest positive integer which divides the lengths of all di\ rected\n cycles of <A>digraph</A>. If <A>digraph</A> has no directed cycle\ s, then\n this function returns <M>0</M>. See Section <Ref Subsect=\"Defin\ itions\"\n Style=\"Number\" /> for the definition of a directed cycle. <P\ />\n\n A digraph with a period of <M>1</M> is said to be <E>aperiodic</E>. \ See\n <Ref Prop=\"IsAperiodicDigraph\"/>. <P/>\n <Example><![CDATA[\nga\ p> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n<immutable multidigraph w\ ith 6 vertices, 7 edges>\ngap> DigraphPeriod(D);\n6\ngap> D := Digraph([[2], [\ 3, 5], [4], [5], [1, 2]]);\n<immutable digraph with 5 vertices, 7 edges>\ngap>\ DigraphPeriod(D);\n1\ngap> D := ChainDigraph(2);\n<immutable chain digraph wi\ th 2 vertices>\ngap> DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\nga\ p> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n<mutable digraph wi\ th 10 vertices, 30 edges>\ngap> DigraphPeriod(D);\n1\n]]></Example>\n </Descr\ iption>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphFloydWarshal\ l\" Arg=\"digraph, func, nopath, edge\"/>\n <Returns>A matrix.</Returns>\n <\ Description>\n If <A>digraph</A> is a digraph with <M>n</M> vertices, then\ \n this operation returns an <M>n \\times n</M> matrix <C>mat</C> containin\ g\n the output of a generalised version of the Floyd-Warshall algorithm,\n \ applied to <A>digraph</A>. <P/>\n\n The operation <C>DigraphFloydWarshal\ l</C> is customised by the arguments\n <A>func</A>, <A>nopath</A>, and <A>e\ dge</A>.\n The arguments <A>nopath</A> and <A>edge</A> can be arbitrary &GA\ P; objects.\n The argument <A>func</A> must be a function which accepts 4 a\ rguments:\n the matrix <C>mat</C>, followed by 3 postive integers. The func\ tion\n <A>func</A> is where the work to calculate the desired outcome must \ be\n performed. <P/>\n\n This method initialises the matrix <C>mat</C> b\ y setting entry\n <C>mat[i][j]</C> to equal <A>edge</A> if there is an edge\ with source\n <C>i</C> and range <C>j</C>, and by setting entry <C>mat[i][\ j]</C> to equal\n <A>nopath</A> otherwise.\n The final part of <C>Digrap\ hFloydWarshall</C> then calls the function\n <A>func</A> inside three neste\ d for loops, over the vertices of\n <A>digraph</A>: <P/>\n\n<Listing><![CDA\ TA[\nfor i in DigraphsVertices(digraph) do\n for j in DigraphsVertices(digrap\ h) do\n for k in DigraphsVertices(digraph) do\n func(mat, i, j, k);\n \ od;\n od;\nod;\n]]></Listing>\n\n The matrix <C>mat</C> is then returne\ d as the result. An example of using\n <C>DigraphFloydWarshall</C> to calcu\ late the shortest (non-zero) distances\n between the vertices of a digraph \ is shown below: <P/>\n\n <Example><![CDATA[\ngap> D := DigraphFromDigraph6S\ tring(\"&EAHQeDB\");\n<immutable digraph with 6 vertices, 12 edges>\ngap> func\ := function(mat, i, j, k)\n> if mat[i][k] <> -1 and mat[k][j] <> -1 then\n>\ if (mat[i][j] = -1) or (mat[i][j] > mat[i][k] + mat[k][j]) then\n> \ mat[i][j] := mat[i][k] + mat[k][j];\n> fi;\n> fi;\n> end;\nfunction( mat\ , i, j, k ) ... end\ngap> shortest_distances := DigraphFloydWarshall(D, func, \ -1, 1);;\ngap> Display(shortest_distances);\n[ [ 3, -1, -1, 2, 1, 2 \ ],\n [ 4, 2, 1, 3, 2, 1 ],\n [ 3, 1, 2, 2, 1, 2 ],\n\ [ 1, -1, -1, 1, 1, 2 ],\n [ 2, -1, -1, 1, 2, 1 ],\n [ \ 3, -1, -1, 2, 1, 1 ] ]\n]]></Example>\n </Description>\n</ManSectio\ n>\n\n <ManSection>\n <Oper Name=\"IsReachable\" Arg=\"digraph, u, v\"/>\n\ <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This o\ peration returns <K>true</K> if there exists a non-trivial directed\n walk \ from vertex <A>u</A> to vertex <A>v</A> in the digraph <A>digraph</A>,\n an\ d <K>false</K> if there does not exist such a directed walk. See Section\n \ <Ref Subsect=\"Definitions\" Style=\"Number\" /> for the definition of a\n \ non-trivial directed walk.\n <P/>\n\n The method for <C>IsReachable</C>\ has worst case complexity of <M>O(m +\n n)</M> where <M>m</M> is the num\ ber of edges and <M>n</M> the number of\n vertices in <A>digraph</A>.\n\n<E\ xample><![CDATA[\ngap> D := Digraph([[2], [3], [2, 3]]);\n<immutable digraph w\ ith 3 vertices, 4 edges>\ngap> IsReachable(D, 1, 3);\ntrue\ngap> IsReachable(D\ , 2, 1);\nfalse\ngap> IsReachable(D, 3, 3);\ntrue\ngap> IsReachable(D, 1, 1);\ \nfalse\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Oper Name=\"VerticesReachableFrom\" Arg=\"digraph, root\"/>\n <Returns>A li\ st.</Returns>\n <Description>\n This operation returns a list of the verti\ ces <A>v</A>, for which there exists\n a non-trivial directed walk from ve\ rtex <A>root</A> to vertex <A>v</A> in the digraph\n <A>digraph</A>. See Se\ ction <Ref Subsect=\"Definitions\" Style=\"Number\" />\n for the definition\ of a non-trivial directed walk.\n <P/>\n\n The method for <C>VerticesRe\ achableFrom</C> has worst case complexity of <M>O(m +\n n)</M> where <M>m\ </M> is the number of edges and <M>n</M> the number of\n vertices in <A>dig\ raph</A>.\n\n<Example><![CDATA[\ngap> D := CompleteDigraph(5);\n<immutable com\ plete digraph with 5 vertices>\ngap> VerticesReachableFrom(D, 1);\n[ 2, 1, 3, \ 4, 5 ]\ngap> VerticesReachableFrom(D, 3);\n[ 1, 2, 3, 4, 5 ]\ngap> D := EmptyD\ igraph(5);\n<immutable empty digraph with 5 vertices>\ngap> VerticesReachableF\ rom(D, 1);\n[ ]\ngap> VerticesReachableFrom(D, 3);\n[ ]\ngap> D := CycleDigr\ aph(4);\n<immutable cycle digraph with 4 vertices>\ngap> VerticesReachableFrom\ (D, 1);\n[ 2, 3, 4, 1 ]\ngap> VerticesReachableFrom(D, 3);\n[ 4, 1, 2, 3 ]\nga\ p> D := ChainDigraph(5);\n<immutable chain digraph with 5 vertices>\ngap> Vert\ icesReachableFrom(D, 1);\n[ 2, 3, 4, 5 ]\ngap> VerticesReachableFrom(D, 3);\n[\ 4, 5 ]\ngap> VerticesReachableFrom(D, 5);\n[ ]\n]]></Example>\n </Descripti\ on>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphPath\" Arg=\"dig\ raph, u, v\"/>\n <Returns>A pair of lists, or <K>fail</K>.</Returns>\n <Desc\ ription>\n If there exists a non-trivial directed path (or a non-trivial cy\ cle, in the\n case that <A>u</A> <C>=</C> <A>v</A>) from vertex <A>u</A> to\ vertex\n <A>v</A> in the digraph <A>digraph</A>, then this operation retur\ ns such a\n directed path (or directed cycle). Otherwise, this operation re\ turns\n <K>fail</K>. See Section <Ref Subsect=\"Definitions\" Style=\"Text\ \"/> for the\n definition of a directed path and a directed cycle.\n <P/\ >\n\n A directed path (or directed cycle) of non-zero length <C>n-1</C>,\n \ <M>(v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)</M>,\n is represented by a pa\ ir of lists <C>[v,a]</C> as follows:\n\n <List>\n <Item>\n <C>v\ </C> is the list <M>[v_1, v_2, ..., v_n]</M>.\n </Item>\n <Item>\n \ <C>a</C> is the list of positive integers <M>[a_1, a_2, ..., a_{n-1}]</M\ >\n where for each each <M>i < n</M>, <M>a_i</M> is the position of\ \n <M>v_{i+1}</M> in\n <C>OutNeighboursOfVertex(</C><A>digraph</\ A><C>,</C><M>v_i</M><C>)</C>\n corresponding to the edge <M>e_i</M>. T\ his can be useful if the\n position of a vertex in a list of out-neigho\ urs is significant, for\n example in orbit digraphs.\n </Item>\n \ </List>\n\n The method for <C>DigraphPath</C> has worst case complexity o\ f <M>O(m +\n n)</M> where <M>m</M> is the number of edges and <M>n</M> th\ e number of\n vertices in <A>digraph</A>.\n\n<Example><![CDATA[\ngap> D := \ Digraph([[2], [3], [2, 3]]);\n<immutable digraph with 3 vertices, 4 edges>\nga\ p> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ngap> DigraphPath(D, 2, 1)\ ;\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ 2 ] ]\ngap> DigraphPath(D, \ 1, 1);\nfail\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSectio\ n>\n <Oper Name=\"DigraphShortestPath\" Arg=\"digraph, u, v\"/>\n <Returns>A\ pair of lists, or <K>fail</K>.</Returns>\n <Description>\n Returns the sh\ ortest directed path in the digraph digraph from the vertex\n <A>u</A> to t\ he vertex <A>v</A>, if such a path exists. If <C><A>u</A> =\n <A>v</A></C>,\ then the shortest non-trivial cycle is returned, again, if it\n exists. Ot\ herwise, this operation returns <K>fail</K>. See Section\n <Ref Subsect=\"\ Definitions\" Style=\"Text\"/> for the definition of a directed\n path and \ a directed cycle.\n <P/>\n\n See <Ref Oper=\"DigraphPath\"/> for details\ on the output.\n\n The method for <C>DigraphShortestPath</C> has worst cas\ e complexity of\n <M>O(m + n)</M> where <M>m</M> is the number of edges and\ <M>n</M> the\n number of vertices in <A>digraph</A>.\n\n<Example><![CDATA[\ \ngap> D := Digraph([[1, 2], [3], [2, 4], [1], [2, 4]]);\n<immutable digraph w\ ith 5 vertices, 8 edges>\ngap> DigraphShortestPath(D, 5, 1);\n[ [ 5, 4, 1 ], [\ 2, 1 ] ]\ngap> DigraphShortestPath(D, 3, 3);\n[ [ 3, 2, 3 ], [ 1, 1 ] ]\ngap>\ DigraphShortestPath(D, 5, 5);\nfail\ngap> DigraphShortestPath(D, 1, 1);\n[ [ \ 1, 1 ], [ 1 ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSec\ tion>\n <Oper Name=\"IteratorOfPaths\" Arg=\"digraph, u, v\"/>\n <Returns>An\ iterator.</Returns>\n <Description>\n If <A>digraph</A> is a digraph or a\ list of adjacencies which defines a\n digraph - see <Ref Attr=\"OutNeighbo\ urs\"/> - then this operation returns an\n iterator of the non-trivial dire\ cted paths (or directed cycles, in the case\n that <A>u</A> <C>=</C> <A>v</\ A>) in <A>digraph</A> from the vertex <A>u</A>\n to the vertex <A>v</A>.\n \ <P/>\n\n See <Ref Oper=\"DigraphPath\"/> for more information about the \ repesentation\n of a directed path or directed cycle which is used, and see\ <Ref\n Sect=\"Iterators\" BookName=\"ref\"/> for more information about \ iterators.\n See Section <Ref Subsect=\"Definitions\" Style=\"Text\"/> for \ the definition of\n a directed path and a directed cycle.\n <P/>\n\n<Exa\ mple><![CDATA[\ngap> D := Digraph([[1, 4, 4, 2], [3, 5], [2, 3], [1, 2], [4]])\ ;\n<immutable multidigraph with 5 vertices, 11 edges>\ngap> iter := IteratorOf\ Paths(D, 1, 4);\n<iterator>\ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 2 ] ]\nga\ p> NextIterator(iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> NextIterator(iter);\n[ [ 1, \ 2, 5, 4 ], [ 4, 2, 1 ] ]\ngap> IsDoneIterator(iter);\ntrue\ngap> iter := Itera\ torOfPaths(D, 4, 3);\n<iterator>\ngap> NextIterator(iter);\n[ [ 4, 1, 2, 3 ], \ [ 1, 4, 1 ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSecti\ on>\n <Attr Name=\"DigraphAllSimpleCircuits\" Arg=\"digraph\"/>\n <Returns>A\ list of lists of vertices.</Returns>\n <Description>\n If <A>digraph</A> \ is a digraph, then <C>DigraphAllSimpleCircuits</C>\n returns a list of the \ <E>simple circuits</E> in <A>digraph</A>. <P/>\n\n See Section <Ref Subsect\ =\"Definitions\" Style=\"Number\" /> for the definition\n of a simple circu\ it, and related notions. Note that a loop is a simple\n circuit. <P/>\n\n \ For a digraph without multiple edges, a simple circuit is uniquely\n dete\ rmined by its subsequence of vertices. However this is not the case for\n a\ multidigraph. The attribute <C>DigraphAllSimpleCircuits</C> ignores\n mul\ tiple edges, and identifies a simple circuit using only its subsequence\n o\ f vertices. For example, although the simple circuits <M>(v, e, v)</M> and\n \ <M>(v, e', v)</M> (for distinct edges <M>e</M> and <M>e'</M>) are\n mathe\ matically distinct, <C>DigraphAllSimpleCircuits</C> considers them to\n be \ the same. <P/>\n\n With this approach, a directed circuit of length <C>n</C\ > can be determined\n by a list of its first <C>n</C> vertices. Thus a simp\ le circuit <M>(v_1,\n e_1, v_2, e_2, ..., e_{n-1}, v_n, e_{n+1}, v_1)</M>\ can be represented as\n the list <M>[v_1, \\ldots, v_n]</M>, or any cyclic\ permutation thereof. For\n each simple circuit of <A>digraph</A>,\n <C\ >DigraphAllSimpleCircuits(<A>digraph</A>)</C> includes precisely one such\n \ list to represent the circuit. <P/>\n\n <Example><![CDATA[\ngap> D := Dig\ raph([[], [3], [2, 4], [5, 4], [4]]);\n<immutable digraph with 5 vertices, 6 e\ dges>\ngap> DigraphAllSimpleCircuits(D);\n[ [ 4 ], [ 4, 5 ], [ 2, 3 ] ]\ngap> \ D := ChainDigraph(10);;\ngap> DigraphAllSimpleCircuits(D);\n[ ]\ngap> D := Di\ graph([[3], [1], [1]]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> Di\ graphAllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D := Digraph([[1, 1]]);\n<immut\ able multidigraph with 1 vertex, 2 edges>\ngap> DigraphAllSimpleCircuits(D);\n\ [ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<mutable digraph with\ 3 vertices, 3 edges>\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 2, 3 ] ]\n]]>\ </Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\ \"DigraphLongestSimpleCircuit\" Arg=\"digraph\"/>\n <Returns>A list of vertic\ es, or <K>fail</K>.</Returns>\n <Description>\n If <A>digraph</A> is a dig\ raph, then <C>DigraphLongestSimpleCircuit</C>\n returns the longest <E>simp\ le circuit</E> in <A>digraph</A>. See Section\n <Ref Subsect=\"Definitions\ \" Style=\"Number\" /> for the definition of simple\n circuit, and the defi\ nition of length for a simple circuit.<P/>\n\n This attribute computes\n \ <C>DigraphAllSimpleCircuits(</C><A>digraph</A><C>)</C> to find all the\n s\ imple circuits of <A>digraph</A>, and returns one of maximal length. A\n s\ imple circuit is represented as a list of vertices, in the same way as\n de\ scribed in <Ref Attr=\"DigraphAllSimpleCircuits\"/>.<P/>\n\n If <A>digraph<\ /A> has no simple circuits, then this attribute returns\n <K>fail</K>. If \ <A>digraph</A> has multiple simple circuits of maximal\n length, then this \ attribute returns one of them.<P/>\n\n <Example><![CDATA[\ngap> D := Digrap\ h([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSimpleCircuit(D);\n[ 4\ , 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSimpleCircuit(D);\nfai\ l\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> DigraphLongestSimpleC\ ircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, \ 4, 1);\n<mutable digraph with 8 vertices, 24 edges>\ngap> DigraphLongestSimple\ Circuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n]]></Example>\n </Description>\n</Ma\ nSection>\n\n <ManSection>\n <Oper Name=\"DigraphLayers\" Arg=\"digraph, v\ ertex\"/>\n <Returns>A list.</Returns>\n <Description>\n This operation r\ eturns a list <K>list</K> such that <K>list[i]</K> is the\n list of vertice\ s whose minimum distance from the vertex <A>vertex</A> in\n <A>digraph</A> \ is <K>i - 1</K>. Vertex <A>vertex</A> is\n assumed to be at distance <K>0</\ K> from itself.\n <Example><![CDATA[\ngap> D := CompleteDigraph(4);;\ngap> \ DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4 ] ]\n]]></Example>\n </Description>\ \n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphDegeneracy\" Arg=\"\ digraph\"/>\n <Returns>A non-negative integer, or <K>fail</K>.</Returns>\n <\ Description>\n If <A>digraph</A> is a symmetric digraph without multiple ed\ ges - see\n <Ref Prop=\"IsSymmetricDigraph\"/> and <Ref Prop=\"IsMultiDigra\ ph\"/> - then\n this attribute returns the degeneracy of <A>digraph</A>. <P\ />\n\n The degeneracy of a digraph is the least integer <C>k</C> such\n \ that every induced of <A>digraph</A> contains a vertex whose number of\n ne\ ighbours (excluding itself) is at most <C>k</C>. Note that this means\n tha\ t loops are ignored.<P/>\n\n If <A>digraph</A> is not symmetric or has mult\ iple edges then this\n attribute returns <K>fail</K>.\n <Example><![CDAT\ A[\ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegenera\ cy(D);\n1\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracy(D);\n4\ngap>\ D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n<immutable digra\ ph with 6 vertices, 10 edges>\ngap> DigraphDegeneracy(D);\n1\ngap> D := Genera\ lisedPetersenGraph(IsMutableDigraph, 10, 3);\n<mutable digraph with 20 vertice\ s, 60 edges>\ngap> DigraphDegeneracy(D);\n3\n]]></Example>\n </Description>\n\ </ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphDegeneracyOrdering\" \ Arg=\"digraph\"/>\n <Returns>A list of integers, or <K>fail</K>.</Returns>\n \ <Description>\n If <A>digraph</A> is a digraph for which\n <C>DigraphDe\ generacy(</C><A>digraph</A><C>)</C> is a non-negative integer\n <C>k</C> - \ see <Ref Attr=\"DigraphDegeneracy\"/> - then this attribute\n returns a deg\ eneracy ordering of the vertices of the vertices of\n <A>digraph</A>.<P/>\n\ \n A degeneracy ordering of <A>digraph</A> is a list <C>ordering</C> of the\ \n vertices of <A>digraph</A> ordered such that for any\n position <C>i<\ /C> of the list, the vertex <C>ordering[i]</C> has at most\n <C>k</C> neigh\ bours in later position of the list.<P/>\n\n If <C>DigraphDegeneracy(</C><A\ >digraph</A><C>)</C> returns <K>fail</K>,\n then this attribute returns <K>\ fail</K>.\n <Example><![CDATA[\ngap> D := DigraphSymmetricClosure(ChainDigr\ aph(5));;\ngap> DigraphDegeneracyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := Co\ mpleteDigraph(5);;\ngap> DigraphDegeneracyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap\ > D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n<immutable digr\ aph with 6 vertices, 10 edges>\ngap> DigraphDegeneracyOrdering(D);\n[ 1, 6, 5,\ 2, 4, 3 ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 3, 1);\n<muta\ ble digraph with 6 vertices, 18 edges>\ngap> DigraphDegeneracyOrdering(D);\n[ \ 6, 5, 4, 1, 3, 2 ]\n]]></Example>\n </Description>\n</ManSection>\n\n <Man\ Section>\n <Attr Name=\"HamiltonianPath\" Arg=\"digraph\"/>\n <Returns>A lis\ t or <K>fail</K>.</Returns>\n <Description>\n Returns a Hamiltonian path i\ f one exists, <K>fail</K> if not.<P/>\n\n A <E>Hamiltonian path</E> of a di\ graph with <C>n</C> vertices is directed\n cycle of length <C>n</C>. If <A>\ digraph</A> is a digraph that contains a\n Hamiltonian path, then this func\ tion returns one, described in the form\n used by <Ref Attr=\"DigraphAllSim\ pleCircuits\"/>. Note if <A>digraph</A> has\n 0 or 1 vertices, then <C>Hami\ ltonianPath</C> returns <C>[]</C> or\n <C>[1]</C>, respectively.<P/>\n\n \ The method used in this attribute has the same worst case complexity as\n \ <Ref Oper=\"DigraphMonomorphism\"/>.\n\n <Example><![CDATA[\ngap> D := Digr\ aph([[]]);\n<immutable empty digraph with 1 vertex>\ngap> HamiltonianPath(D);\ \n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n<immutable digraph with 2 vertices, 2\ edges>\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D := Digraph([[3], [], [2]])\ ;\n<immutable digraph with 3 vertices, 2 edges>\ngap> HamiltonianPath(D);\nfai\ l\ngap> D := Digraph([[2], [3], [1]]);\n<immutable digraph with 3 vertices, 3 \ edges>\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := GeneralisedPetersenGr\ aph(IsMutableDigraph, 5, 2);\n<mutable digraph with 10 vertices, 30 edges>\nga\ p> HamiltonianPath(D);\nfail\n]]></Example>\n </Description>\n</ManSection>\n\ \n <ManSection>\n <Attr Name=\"NrSpanningTrees\" Arg=\"digraph\"/>\n <Ret\ urns>An integer.</Returns>\n <Description>\n Returns the number of spannin\ g trees of the symmetric digraph\n <A>digraph</A>.\n <C>NrSpanningTrees<\ /C> will return an error if <A>digraph</A> is not a\n symmetric digraph. <P\ /> \n\n See <Ref Prop=\"IsSymmetricDigraph\"/> and\n <Ref Oper=\"IsUndir\ ectedSpanningTree\" /> for more information. \n\n <Example><![CDATA[\ngap> \ D := CompleteDigraph(5);\n<immutable complete digraph with 5 vertices>\ngap> N\ rSpanningTrees(D);\n125\ngap> D := DigraphSymmetricClosure(CycleDigraph(24));;\ \ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(EmptyDigraph(0));\n0\ngap\ > D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n<mutable digraph wit\ h 18 vertices, 54 edges>\ngap> NrSpanningTrees(D);\n1134225\n]]></Example>\n \ </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphDijks\ tra\" Arg=\"digraph, source, target\" Label=\"for a source and target\"/>\n <\ Oper Name=\"DigraphDijkstra\" Arg=\"digraph, source\" Label=\"for a source\"/>\ \n <Returns>Two lists.</Returns>\n <Description>\n If <A>digraph</A> is a d\ igraph and <A>source</A> and <A>target</A> are\n vertices of <A>digraph</A>,\ then <C>DigraphDijkstra</C> calculates the\n length of the shortest path fro\ m <A>source</A> to <A>target</A> and returns\n two lists. Each element of the\ first list is the distance of the\n corresponding element from <A>source</A>\ . If a vertex was not visited in\n the process of calculating the shortest d\ istance to <A>target</A> or if\n there is no path connecting that vertex with\ <A>source</A>, then\n the corresponding distance is <K>infinity</K>. Each e\ lement of the\n second list gives the previous vertex in the shortest path\n \ from <A>source</A> to the corresponding vertex. For\n <A>source</A> and for\ any vertices that remained unvisited this\n will be <C>-1</C>.<P/>\n\n \ If the optional second argument <A>target</A> is not present, then\n <C>Dig\ raphDijkstra</C> returns the shortest path from <A>source</A> to\n every ve\ rtex that is reachable from <A>source</A>.\n\n <Example><![CDATA[\ngap> mat :\ = [[0, 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1 ], [ 0, 0, 1 ], [ 0, 0, 0 ] \ ]\ngap> D := DigraphByAdjacencyMatrix(mat);\n<immutable digraph with 3 vertice\ s, 3 edges>\ngap> DigraphDijkstra(D, 2, 3);\n[ [ infinity, 0, 1 ], [ -1, -1, 2\ ] ]\ngap> DigraphDijkstra(D, 1, 3);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\ngap> Digr\ aphDijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n]]></Example>\n</Descrip\ tion>\n</ManSection>\n\n </Section>\n\n <Section><Heading>Cayley graphs of g\ roups</Heading>\n <ManSection>\n <Attr Name=\"GroupOfCayleyDigraph\" Arg=\ \"digraph\"/>\n <Attr Name=\"SemigroupOfCayleyDigraph\" Arg=\"digraph\"/>\n \ <Returns>A group or semigroup.</Returns>\n <Description>\n If <A>digraph</\ A> is an immutable Cayley graph of a group <C>G</C> and\n <A>digraph</A> be\ longs to the category <Ref Filt=\"IsCayleyDigraph\"/>, then\n <C>GroupOfCay\ leyDigraph</C> returns <C>G</C>.\n <P/>\n\n If <A>digraph</A> is a Cayle\ y graph of a semigroup <C>S</C> and\n <A>digraph</A> belongs to the categor\ y <Ref Filt=\"IsCayleyDigraph\"/>, then\n <C>SemigroupOfCayleyDigraph</C> r\ eturns <C>S</C>.\n <P/>\n\n See also <Ref Attr=\"GeneratorsOfCayleyDigra\ ph\"/>.\n <Example><![CDATA[\ngap> G := DihedralGroup(IsPermGroup, 8);\nGro\ up([ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyDigraph(G);\n<immutable digrap\ h with 8 vertices, 16 edges>\ngap> GroupOfCayleyDigraph(digraph) = G;\ntrue\n]\ ]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Nam\ e=\"GeneratorsOfCayleyDigraph\" Arg=\"digraph\"/>\n <Returns>A list of genera\ tors.</Returns>\n <Description>\n If <A>digraph</A> is an immutable Cayley\ graph of a group or semigroup with\n respect to a set of generators <C>gen\ s</C> and <A>digraph</A> belongs to\n the category <Ref Filt=\"IsCayleyDigr\ aph\"/>, then\n <C>GeneratorsOfCayleyDigraph</C> return the list of generat\ ors <C>gens</C>\n over which <A>digraph</A> is defined.\n <P/>\n\n Se\ e also <Ref Attr=\"GroupOfCayleyDigraph\"/> \n or <Ref Attr=\"SemigroupOfCa\ yleyDigraph\"/>.\n <Example><![CDATA[\ngap> G := DihedralGroup(IsPermGroup,\ 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyDigraph(G);\n<immutab\ le digraph with 8 vertices, 16 edges>\ngap> GeneratorsOfCayleyDigraph(digraph)\ = GeneratorsOfGroup(G);\ntrue\ngap> digraph := CayleyDigraph(G, [()]);\n<immu\ table digraph with 8 vertices, 8 edges>\ngap> GeneratorsOfCayleyDigraph(digrap\ h) = [()];\ntrue]]></Example>\n </Description>\n</ManSection>\n\n </Section>\ \n\n <Section><Heading>Associated semigroups</Heading>\n <ManSection>\n \ <Oper Name=\"AsSemigroup\" Arg=\"filt, digraph\"\n Label=\"for a filte\ r and a digraph\"/>\n <Oper Name=\"AsMonoid\" Arg=\"filt, digraph\"/>\n \ <Returns>A semilattice of partial perms.</Returns>\n <Description>\n T\ he operation <C>AsSemigroup</C> requires that <A>filt</A> be equal to\n <\ Ref Filt=\"IsPartialPermSemigroup\" BookName=\"ref\"/>.\n If <A>digraph</\ A> is a\n <Ref Prop=\"IsJoinSemilatticeDigraph\"/> or <Ref Prop=\"IsLatti\ ceDigraph\"/>\n then <C>AsSemigroup</C> returns a semigroup of partial pe\ rms which is\n isomorphic to the semigroup whose elements are the vertice\ s of\n <A>digraph</A> with the binary operation\n <Ref Oper=\"Partia\ lOrderDigraphJoinOfVertices\"/>.\n If <A>digraph</A>\n satisfies <R\ ef Prop=\"IsMeetSemilatticeDigraph\"/> but not\n <Ref Prop=\"IsJoinSemila\ tticeDigraph\"/> then\n <C>AsSemigroup</C> returns a semigroup of partial\ perms which is\n isomorphic to the semigroup whose elements are the vert\ ices of\n <A>digraph</A> with the binary operation\n <Ref Oper=\"Par\ tialOrderDigraphMeetOfVertices\"/>.\n <P/>\n\n The operation <C>AsMo\ noid</C> behaves similarly to <C>AsSemigroup</C>\n except that <A>filt</A\ > may also be equal to\n <Ref Filt=\"IsPartialPermMonoid\" BookName=\"ref\ \"/>,\n <A>digraph</A> must satisfy <Ref Prop=\"IsLatticeDigraph\"/>, and\ the\n output satisfies <Ref Filt=\"IsMonoid\" BookName=\"ref\"/>. <P/>\n\ \n The output of both of these operations is guaranteed to be of minimal\ \n degree (see <Ref Attr=\"DegreeOfPartialPermSemigroup\" BookName=\"ref\ \"/>).\n Furthermore the <Ref Attr=\"GeneratorsOfSemigroup\" BookName=\"r\ ef\"/> of\n the output is guaranteed to be the unique generating set of m\ inimal size.\n <Example><![CDATA[\ngap> di := Digraph([[1], [1, 2], [1, 3\ ], [1, 4], [1, 2, 3, 5]]);\n<immutable digraph with 5 vertices, 11 edges>\ngap\ > S := AsSemigroup(IsPartialPermSemigroup, di);\n<partial perm semigroup of ra\ nk 3 with 4 generators>\ngap> ForAll(Elements(S), IsIdempotent);\ntrue\ngap> I\ sInverseSemigroup(S);\ntrue\ngap> Size(S);\n5\ngap> di := Digraph([[1], [1, 2]\ , [1, 2, 3]]);\n<immutable digraph with 3 vertices, 6 edges>\ngap> M := AsMono\ id(IsPartialPermMonoid, di);\n<partial perm monoid of rank 2 with 3 generators\ >\ngap> Size(M);\n3]]></Example>\n </Description>\n </ManSection>\n\n \ <ManSection>\n <Oper Name=\"AsSemigroup\" Arg=\"filt, Y, gps, homs\"\n \ Label=\"for a filter, semilattice digraph, and two lists\"/>\n <Returns> \ A Clifford semigroup of partial perms. </Returns>\n <Description>\n Th\ e operation <C>AsSemigroup</C> requires that <A>filt</A> be equal to\n <R\ ef Filt=\"IsPartialPermSemigroup\" BookName=\"ref\"/>.\n If <A>Y</A> is a\ <Ref Prop=\"IsJoinSemilatticeDigraph\"/> or\n <Ref Prop=\"IsMeetSemilatt\ iceDigraph\"/>,\n <A>gps</A> is a list of groups corresponding to each ve\ rtex, and\n <A>homs</A> is a list containing for each edge <C>(i, j)</C> \ in\n the transitive reduction of <A>digraph</A> a triple <C>[i, j, hom]</\ C>\n where <C>hom</C> is a group homomorphism from <C>gps[i]</C> to\n \ <C>gps[j]</C>, and the diagram of homomorphisms commutes, then\n <C>AsS\ emigroup</C> returns a semigroup of partial perms which is\n isomorphic t\ o the strong semilattice of groups <M>S[Y; gps; homs]</M>.\n <Example><![\ CDATA[\ngap> G1 := AlternatingGroup(4);;\ngap> G2 := SymmetricGroup(2);;\ngap>\ G3 := SymmetricGroup(3);;\ngap> gr := Digraph([[1, 3], [2, 3], [3]]);;\ngap> \ sgn := function(x)\n> if SignPerm(x) = 1 then\n> return ();\n> fi;\n> return (\ 1, 2);\n> end;;\ngap> hom13 := GroupHomomorphismByFunction(G1, G3, sgn);;\ngap\ > hom23 := GroupHomomorphismByFunction(G2, G3, sgn);;\ngap> T := AsSemigroup(I\ sPartialPermSemigroup,\n> gr,\n> [G1, G2, G3], [[1, 3, hom13], [2, 3, hom23]])\ ;;\ngap> Size(T);\n20\ngap> D := GreensDClasses(T);;\ngap> List(D, x -> Size(x\ ));\n[ 6, 12, 2 ]\n]]></Example>\n </Description>\n </ManSection>\n\n </S\ ection>\n\n <Section><Heading>Planarity</Heading>\n <ManSection>\n <Attr \ Name=\"KuratowskiPlanarSubdigraph\" Arg=\"digraph\"/>\n <Returns>A list or <K\ >fail</K>.</Returns>\n <Description>\n <C>KuratowskiPlanarSubdigraph</C> r\ eturns the immutable list of\n lists of out-neighbours of a (not necessaril\ y induced) subdigraph of the\n digraph <A>digraph</A> that witnesses the fa\ ct that <A>digraph</A> is not\n planar, or <K>fail</K> if <A>digraph</A> is\ planar. In other words,\n <C>KuratowskiPlanarSubdigraph</C> returns the o\ ut-neighbours of a\n subdigraph of <A>digraph</A> that is homeomorphic to t\ he complete graph\n with <C>5</C> vertices, or to the complete bipartite gr\ aph with vertex sets\n of sizes <C>3</C> and <C>3</C>. <P/>\n \n The \ directions and multiplicities of any edges in <A>digraph</A> are\n ignored \ when considering whether or not <A>digraph</A> is planar. <P/>\n\n See also\ <Ref Prop=\"IsPlanarDigraph\"/>\n and <Ref Attr=\"SubdigraphHomeomorphicTo\ K33\"/>.\n <P/>\n \n This method uses the reference implementation in\ \n &edge-addition-planarity-suite; by John Boyer of the algorithms describe\ d\n in <Cite Key=\"BM06\"/>.\n\n<Example><![CDATA[\ngap> D := Digraph([[3, \ 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2\ , 11], [1, 2], [5, 9]]);\n<immutable digraph with 11 vertices, 25 edges>\ngap>\ KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := Digraph([[2, 4, 7, 9, 10], [1\ , 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4\ , 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], \ [3, 5]]);\n<immutable digraph with 10 vertices, 50 edges>\ngap> IsPlanarDigrap\ h(D);\nfalse\ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ]\ , [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\ngap> D := Digra\ ph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [\ 4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n<mutable digraph with 11 ve\ rtices, 25 edges>\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := Digrap\ h(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5,\ 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, \ 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n<mutable digraph with 10 \ vertices, 50 edges>\ngap> IsPlanarDigraph(D);\nfalse\ngap> KuratowskiPlanarSub\ digraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ \ 7, 9, 3 ], [ ], [ ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Attr Name=\"KuratowskiOuterPlanarSubdigraph\" Arg=\"digraph\ \"/>\n <Returns>A list or <K>fail</K>.</Returns>\n <Description>\n <C>Kur\ atowskiOuterPlanarSubdigraph</C> returns the immutable list of\n immutable \ lists of out-neighbours of a (not necessarily induced)\n subdigraph of the \ digraph <A>digraph</A> that witnesses the fact\n that <A>digraph</A> is not\ outer planar, or <K>fail</K> if\n <A>digraph</A> is outer planar. In othe\ r words,\n <C>KuratowskiOuterPlanarSubdigraph</C> returns the out-neighbour\ s of a\n subdigraph of <A>digraph</A> that is homeomorphic to the complete \ graph\n with <C>4</C> vertices, or to the complete bipartite graph with ver\ tex sets\n of sizes <C>2</C> and <C>3</C>. <P/>\n \n The directions a\ nd multiplicities of any edges in <A>digraph</A> are\n ignored when conside\ ring whether or not <A>digraph</A> is outer planar. \n <P/>\n\n See also\ \n <Ref Prop=\"IsOuterPlanarDigraph\"/>,\n <Ref Attr=\"SubdigraphHomeom\ orphicToK4\"/>, and \n <Ref Attr=\"SubdigraphHomeomorphicToK23\"/>.\n <P\ />\n\n This method uses the reference implementation in\n &edge-addition\ -planarity-suite; by John Boyer of the algorithms described\n in <Cite Key=\ \"BM06\"/>.\n\n<Example><![CDATA[\ngap> D := Digraph([[3, 5, 10], [8, 9, 10], \ [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9\ ]]);\n<immutable digraph with 11 vertices, 25 edges>\ngap> KuratowskiOuterPlan\ arSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [\ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph([[2, 4, 7, 9, 10], [1,\ 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4,\ 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [\ 3, 5]]);\n<immutable digraph with 10 vertices, 50 edges>\ngap> IsOuterPlanarDi\ graph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ], [ ], [ ]\ , [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]\ngap> D := Digrap\ h(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n<mutable digraph with 11 ver\ tices, 25 edges>\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ \ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ \ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10\ ], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n\ > [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n<mutabl\ e digraph with 10 vertices, 50 edges>\ngap> IsOuterPlanarDigraph(D);\nfalse\ng\ ap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [\ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]]]></Example>\n </Description>\n</Ma\ nSection>\n\n <ManSection>\n <Attr Name=\"PlanarEmbedding\" Arg=\"digraph\ \"/>\n <Returns>A list or <K>fail</K>.</Returns>\n <Description>\n If <A>\ digraph</A> is a planar digraph, then \n <C>PlanarEmbedding</C> returns the\ immutable list of lists of\n out-neighbours of a subdigraph of <A>digraph<\ /A> such that each vertex's\n neighbours are given in clockwise order. If <\ A>digraph</A> is not planar,\n then <K>fail</K> is returned. <P/>\n \n \ The directions and multiplicities of any edges in <A>digraph</A> are\n ig\ nored by <C>PlanarEmbedding</C>.\n <P/>\n\n See also \n <Ref Prop=\"I\ sPlanarDigraph\"/>.\n <P/>\n\n This method uses the reference implementa\ tion in\n &edge-addition-planarity-suite; by John Boyer of the algorithms d\ escribed\n in <Cite Key=\"BM06\"/>.\n\n<Example><![CDATA[\ngap> D := Digrap\ h([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6, 8], [2,\ 7], [2, 11], [1, 2], [5, 9]]);\n<immutable digraph with 11 vertices, 25 edges\ >\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11,\ 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph([[2, 4, \ 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, \ 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, \ 5, 6, 7, 8], [3, 5]]);\n<immutable digraph with 10 vertices, 50 edges>\ngap> P\ lanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8,\ 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2\ ], [5, 9]]);\n<mutable digraph with 11 vertices, 25 edges>\ngap> PlanarEmbeddi\ ng(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11, 7 ], [ 7 ], [ 8 ], \ \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7,\ 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, \ 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5\ , 6, 7, 8], [3, 5]]);\n<mutable digraph with 10 vertices, 50 edges>\ngap> Plan\ arEmbedding(D);\nfail\n]]></Example>\n </Description>\n</ManSection>\n\n <\ ManSection>\n <Attr Name=\"OuterPlanarEmbedding\" Arg=\"digraph\"/>\n <Retur\ ns>A list or <K>fail</K>.</Returns>\n <Description>\n If <A>digraph</A> is\ an outer planar digraph, then\n <C>OuterPlanarEmbedding</C> returns the im\ mutable list of lists\n of out-neighbours of a subdigraph of <A>digraph</A>\ such that each\n vertex's neighbours are given in clockwise order. If <A>d\ igraph</A> is not\n outer planar, then <K>fail</K> is returned. <P/>\n \ \n The directions and multiplicities of any edges in <A>digraph</A> are\n \ ignored by <C>OuterPlanarEmbedding</C>.\n <P/>\n\n See also <Ref Prop=\ \"IsOuterPlanarDigraph\"/>.\n <P/>\n\n This method uses the reference im\ plementation in\n &edge-addition-planarity-suite; by John Boyer of the algo\ rithms described\n in <Cite Key=\"BM06\"/>.\n\n<Example><![CDATA[\ngap> D :\ = Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6,\ 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n<immutable digraph with 11 vertices, \ 25 edges>\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph([[2, 4, 7, 9\ , 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 1\ 0], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6\ , 7, 8], [3, 5]]);\n<immutable digraph with 10 vertices, 50 edges>\ngap> Outer\ PlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteBipartiteDigraph(\ 2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigrap\ h, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2,\ 7], [2, 11], [1, 2], [5, 9]]);\n<mutable digraph with 11 vertices, 25 edges>\ \ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigraph, [[2\ , 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4,\ 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [\ 2, 3, 5, 6, 7, 8], [3, 5]]);\n<mutable digraph with 10 vertices, 50 edges>\nga\ p> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteBipartite\ Digraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\n]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"SubdigraphHomeomorph\ icToK23\" Arg=\"digraph\"/>\n <Attr Name=\"SubdigraphHomeomorphicToK33\" Arg=\ \"digraph\"/>\n <Attr Name=\"SubdigraphHomeomorphicToK4\" Arg=\"digraph\"/>\n\ <Returns>A list or <K>fail</K>.</Returns>\n <Description>\n These attrib\ utes return the immutable list of lists of\n out-neighbours of a subdigraph\ of the digraph <A>digraph</A> which is\n homeomorphic to one of the follow\ ing:\n the complete bipartite graph with vertex sets of sizes <C>2</C> and\ \n <C>3</C>; the complete bipartite graph with vertex sets of sizes <C>3</C\ >\n and <C>3</C>; or the complete graph with <C>4</C> vertices. If\n <A>\ digraph</A> has no such subdigraphs, then <K>fail</K> is returned. \n <P/>\ \n\n See also <Ref Prop=\"IsPlanarDigraph\"/> and <Ref\n Prop=\"IsOute\ rPlanarDigraph\"/> for more details.<P/>\n\n This method uses the reference\ implementation in\n &edge-addition-planarity-suite; by John Boyer of the a\ lgorithms described\n in <Cite Key=\"BM06\"/>.\n\n<Example><![CDATA[\ngap> \ D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 7, 11], \n> [4, 7], \ [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n<immutable digraph with 11 vertice\ s, 25 edges>\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 \ ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap>\ SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], \ [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph([[3, 5\ , 10], [8, 9, 10], [1, 4], [3, 6], [1, 11], \n> [4, 7], [6, 8], [2, 7], [2, 1\ 1], [1, 2], [5, 9]]);\n<immutable digraph with 11 vertices, 24 edges>\ngap> Su\ bdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23(D);\n[ [ \ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ]\ , [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> SubdigraphHom\ eomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK33(Comple\ teDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipartiteDigrap\ h(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ] ]\ngap> S\ ubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\ngap> D := Digraph(IsMuta\ bleDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6\ , 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n<mutable digraph with 11 vertices, 2\ 5 edges>\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [\ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> Sub\ digraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11\ ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableD\ igraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 11], [4, 7], [6, 8], [\ 2, 7], [2, 11], [1, 2], [5, 9]]);\n<mutable digraph with 11 vertices, 24 edges\ >\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23\ (D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ \ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> Sub\ digraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicTo\ K33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipar\ titeDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ]\ ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\n]]></Example>\ \n </Description>\n</ManSection>\n\n </Section>\n\n</Chapter>\n <!-- attribu\ tes, operations -->\n <Chapter Label=\"Properties of digraphs\"><Heading>Prop\ erties of digraphs</Heading>\n\n <Section><Heading>Edge properties</Heading>\ \n <ManSection>\n <Prop Name=\"DigraphHasLoops\" Arg=\"digraph\"/>\n <Ret\ urns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n Returns <K>t\ rue</K> if the digraph <A>digraph</A> has loops, and\n <K>false</K> if it d\ oes not. A loop is an edge with equal source and range.\n <P/>\n \n \ If the argument <A>digraph</A> is mutable, then the return value of\n this\ property is recomputed every time it is called. \n\n<Example><![CDATA[\ngap> \ D := Digraph([[1, 2], [2]]);\n<immutable digraph with 2 vertices, 3 edges>\nga\ p> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ngap> DigraphHasLoops(D)\ ;\ntrue\ngap> D := Digraph([[2, 3], [1], [2]]);\n<immutable digraph with 3 ver\ tices, 4 edges>\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 3, 2\ ] ]\ngap> DigraphHasLoops(D);\nfalse\ngap> D := CompleteDigraph(IsMutableDigr\ aph, 4);\n<mutable digraph with 4 vertices, 12 edges>\ngap> DigraphHasLoops(D)\ ;\nfalse]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Prop Name=\"IsAntiSymmetricDigraph\" Arg=\"digraph\"/>\n <Prop Name=\"IsAnti\ symmetricDigraph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.<\ /Returns>\n <Description>\n This property is <K>true</K> if the digraph <A\ >digraph</A>\n is antisymmetric, and <K>false</K> if it is not.\n <P/>\n\ \n A digraph is <E>antisymmetric</E> if whenever there is an edge with sour\ ce\n <C>u</C> and range <C>v</C>, and an edge with source <C>v</C> and rang\ e\n <C>u</C>, then the vertices <C>u</C> and <C>v</C> are equal.\n <P/>\ \n \n If the argument <A>digraph</A> is mutable, then the return value\ of\n this property is recomputed every time it is called. \n\n <Example\ ><![CDATA[\ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n<immutable digraph wi\ th 3 vertices, 5 edges>\ngap> IsAntisymmetricDigraph(gr1);\nfalse\ngap> Digrap\ hEdges(gr1){[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2 := Digraph([[1, 2], [3,\ 3], [1]]);\n<immutable multidigraph with 3 vertices, 5 edges>\ngap> IsAntisym\ metricDigraph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ [ 1, 1 ], [ 1, 2 ], [ 2,\ 3 ], [ 2, 3 ], [ 3, 1 ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n\ <ManSection>\n <Prop Name=\"IsBipartiteDigraph\" Arg=\"digraph\"/>\n <Re\ turns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This property\ is <K>true</K> if the digraph <A>digraph</A> is bipartite, and\n <K>false</K\ > if it is not. A digraph is bipartite if and only if the\n vertices of <A>d\ igraph</A> can be partitioned into two non-empty sets such\n that the source \ and range of any edge of <A>digraph</A> lie in distinct sets.\n Equivalently,\ a digraph is bipartite if and only if it is 2-colorable; see\n <Ref Attr=\"D\ igraphGreedyColouring\" Label=\"for a digraph\"/>. <P/>\n\n See also <Ref Att\ r=\"DigraphBicomponents\"/>.\n <P/>\n \n If the argument <A>digraph\ </A> is mutable, then the return value of\n this property is recomputed eve\ ry time it is called. \n <Example><![CDATA[\ngap> D := ChainDigraph(4);\n<i\ mmutable chain digraph with 4 vertices>\ngap> IsBipartiteDigraph(D);\ntrue\nga\ p> D := CycleDigraph(3);\n<immutable cycle digraph with 3 vertices>\ngap> IsBi\ partiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph\ , 5, 4);\n<mutable digraph with 9 vertices, 40 edges>\ngap> IsBipartiteDigraph\ (D);\ntrue]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n\ <Prop Name=\"IsCompleteBipartiteDigraph\" Arg=\"digraph\"/>\n <Returns><K>t\ rue</K> or <K>false</K>.</Returns>\n <Description>\n Returns <K>true</K> i\ f the digraph <A>digraph</A> is a complete bipartite\n digraph, and <K>fals\ e</K> if it is not. <P/>\n\n A digraph is a <E>complete bipartite digraph</\ E> if it is bipartite, see\n <Ref Prop=\"IsBipartiteDigraph\"/>, and there \ exists a unique edge with\n source <C>i</C> and range <C>j</C> if and only \ if <C>i</C> and <C>j</C> lie\n in different bicomponents of <A>digraph</A>,\ see <Ref\n Attr=\"DigraphBicomponents\"/>. <P/>\n\n Equivalently, a b\ ipartite digraph with bicomponents of size <M>m</M> and\n <M>n</M> is compl\ ete precisely when it has <M>2mn</M> edges, none of which\n are multiple ed\ ges. <P/>\n\n See also <Ref Oper=\"CompleteBipartiteDigraph\"/>.\n <P/>\ \n \n If the argument <A>digraph</A> is mutable, then the return value\ of\n this property is recomputed every time it is called. \n\n <Example\ ><![CDATA[\ngap> D := CycleDigraph(2);\n<immutable cycle digraph with 2 vertic\ es>\ngap> IsCompleteBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(4);\n<i\ mmutable cycle digraph with 4 vertices>\ngap> IsBipartiteDigraph(D);\ntrue\nga\ p> IsCompleteBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(I\ sMutableDigraph, 5, 4);\n<mutable digraph with 9 vertices, 40 edges>\ngap> IsC\ ompleteBipartiteDigraph(D);\ntrue]]></Example>\n </Description>\n</ManSection\ >\n\n <ManSection>\n <Prop Name=\"IsCompleteDigraph\" Arg=\"digraph\"/>\n \ <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n Returns\ <K>true</K> if the digraph <A>digraph</A> is complete, and\n <K>false</K> \ if it is not. <P/>\n\n A digraph is <E>complete</E> if it has no loops, and\ for all\n <E>distinct</E> vertices <C>i</C> and <C>j</C>,\n there is ex\ actly one edge with source <C>i</C> and range <C>j</C>.\n\n Equivalently, a\ digraph with <M>n</M> vertices is complete precisely when\n it has <M>n(n \ - 1)</M> edges, no loops, and no multiple edges.\n <P/>\n \n If the\ argument <A>digraph</A> is mutable, then the return value of\n this proper\ ty is recomputed every time it is called. \n\n <Example><![CDATA[\ngap> D :\ = Digraph([[2, 3], [1, 3], [1, 2]]);\n<immutable digraph with 3 vertices, 6 ed\ ges>\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Digraph([[2, 2], [1]]);\n<im\ mutable multidigraph with 2 vertices, 3 edges>\ngap> IsCompleteDigraph(D);\nfa\ lse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n<mutable dig\ raph with 9 vertices, 40 edges>\ngap> IsCompleteDigraph(D);\nfalse]]></Example\ >\n </Description>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsCompl\ eteMultipartiteDigraph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false\ </K>.</Returns>\n <Description>\n This property returns <K>true</K> if <A>\ digraph</A> is a complete\n multipartite digraph, and <K>false</K> if not. \ <P/>\n\n A digraph is a <E>complete multipartite digraph</E> if and only if\ \n its vertices can be partitioned into at least two maximal independent se\ ts,\n where every possible edge between these independent sets occurs in th\ e\n digraph exactly once.\n <P/>\n \n If the argument <A>digraph\ </A> is mutable, then the return value of\n this property is recomputed eve\ ry time it is called. \n\n <Example><![CDATA[\ngap> D := CompleteMultiparti\ teDigraph([2, 4, 6]);\n<immutable complete multipartite digraph with 12 vertic\ es, 88 edges>\ngap> IsCompleteMultipartiteDigraph(D);\ntrue\ngap> D := Complet\ eBipartiteDigraph(IsMutableDigraph, 5, 4);\n<mutable digraph with 9 vertices, \ 40 edges>\ngap> IsCompleteMultipartiteDigraph(D);\ntrue]]></Example>\n </Desc\ ription>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsEmptyDigraph\" A\ rg=\"digraph\"/>\n <Prop Name=\"IsNullDigraph\" Arg=\"digraph\"/>\n <Returns\ ><K>true</K> or <K>false</K>.</Returns>\n <Description>\n Returns <K>true<\ /K> if the digraph <A>digraph</A> is empty, and\n <K>false</K> if it is not\ . A digraph is <E>empty</E> if it has no\n edges.<P/>\n\n <C>IsNullDigra\ ph</C> is a synonym for <C>IsEmptyDigraph</C>.\n <P/>\n \n If the a\ rgument <A>digraph</A> is mutable, then the return value of\n this property\ is recomputed every time it is called. \n\n <Example><![CDATA[\ngap> D := \ Digraph([[], []]);\n<immutable empty digraph with 2 vertices>\ngap> IsEmptyDig\ raph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D := Digraph([[], [1]]);\n<\ immutable digraph with 2 vertices, 1 edge>\ngap> IsEmptyDigraph(D);\nfalse\nga\ p> IsNullDigraph(D);\nfalse]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Prop Name=\"IsEquivalenceDigraph\" Arg=\"digraph\"/>\n <R\ eturns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n A digraph \ is an equivalence digraph if and only if the digraph satisifies\n all of <R\ ef Prop=\"IsReflexiveDigraph\"/>,\n <Ref Prop=\"IsSymmetricDigraph\"/> and \ <Ref Prop=\"IsTransitiveDigraph\"/>.\n A partial order <A>digraph</A> corre\ sponds to an equivalence relation.\n <P/>\n \n If the argument <A>d\ igraph</A> is mutable, then the return value of\n this property is recomput\ ed every time it is called. \n\n<Example><![CDATA[\ngap> D := Digraph([[1, 3],\ [2], [1, 3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> IsEquivale\ nceDigraph(D);\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <M\ anSection>\n <Prop Name=\"IsFunctionalDigraph\" Arg=\"digraph\"/>\n <Returns\ ><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This property is\ <K>true</K> if the digraph <A>digraph</A> is\n functional. <P/>\n\n A d\ igraph is <E>functional</E> if every vertex is the source of a\n unique edg\ e.\n <P/>\n \n If the argument <A>digraph</A> is mutable, then the \ return value of\n this property is recomputed every time it is called. \n \ <Example><![CDATA[\ngap> gr1 := Digraph([[3], [2], [2], [1], [6], [5]]);\n<i\ mmutable digraph with 6 vertices, 6 edges>\ngap> IsFunctionalDigraph(gr1);\ntr\ ue\ngap> gr2 := Digraph([[1, 2], [1]]);\n<immutable digraph with 2 vertices, 3\ edges>\ngap> IsFunctionalDigraph(gr2);\nfalse\ngap> gr3 := Digraph(3, [1, 2, \ 3], [2, 3, 1]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> IsFunction\ alDigraph(gr3);\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <\ ManSection>\n <Prop Name=\"IsMultiDigraph\" Arg=\"digraph\"/>\n <Returns><K>\ true</K> or <K>false</K>.</Returns>\n <Description>\n A <E>multidigraph</E\ > is one that has at least two\n edges with equal source and range.<P/>\n\n\ If the argument <A>digraph</A> is mutable, then the return value of\n t\ his property is recomputed every time it is called. \n\n <Example><![CDATA[\ \ngap> D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \"b\", \"b\"], [\"b\", \"c\ \", \"a\"]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> IsMultiDigrap\ h(D);\nfalse\ngap> D := DigraphFromDigraph6String(\"&Bug\");\n<immutable digra\ ph with 3 vertices, 6 edges>\ngap> IsDuplicateFree(DigraphEdges(D));\ntrue\nga\ p> IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1, 2, 3, 2], [2, 1], [3]]);\ \n<immutable multidigraph with 3 vertices, 7 edges>\ngap> IsDuplicateFree(Digr\ aphEdges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\ngap> D := DigraphMutableC\ opy(D); \n<mutable multidigraph with 3 vertices, 7 edges>\ngap> IsMultiDigraph\ (D);\ntrue]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n\ <Prop Name=\"IsReflexiveDigraph\" Arg=\"digraph\"/>\n <Returns><K>true</K> \ or <K>false</K>.</Returns>\n <Description>\n This property is <K>true</K> \ if the digraph <A>digraph</A> is\n reflexive, and <K>false</K> if it is not\ .\n A digraph is <E>reflexive</E> if it has a loop at every vertex. <P/>\n \ <P/>\n \n If the argument <A>digraph</A> is mutable, then the retur\ n value of\n this property is recomputed every time it is called. \n <Ex\ ample><![CDATA[\ngap> D := Digraph([[1, 2], [2]]);\n<immutable digraph with 2 \ vertices, 3 edges>\ngap> IsReflexiveDigraph(D);\ntrue\ngap> D := Digraph([[3, \ 1], [4, 2], [3], [2, 1]]);\n<immutable digraph with 4 vertices, 7 edges>\ngap>\ IsReflexiveDigraph(D);\nfalse\n]]></Example>\n </Description>\n</ManSection>\ \n\n <ManSection>\n <Prop Name=\"IsSymmetricDigraph\" Arg=\"digraph\"/>\n \ <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This pr\ operty is <K>true</K> if the digraph <A>digraph</A>\n is symmetric, and <K>\ false</K> if it is not.<P/>\n\n A <E>symmetric digraph</E> is one where for\ each non-loop edge, having\n source <C>u</C> and range <C>v</C>, there is \ a corresponding edge with\n source <C>v</C> and range <C>u</C>. If there a\ re <C>n</C> edges with\n source <C>u</C> and range <C>v</C>, then there mus\ t be precisely <C>n</C>\n edges with source <C>v</C> and range <C>u</C>. I\ n other words, a symmetric\n digraph has a symmetric adjacency matrix <Ref \ Attr=\"AdjacencyMatrix\"/>.\n <P/>\n \n If the argument <A>digraph<\ /A> is mutable, then the return value of\n this property is recomputed ever\ y time it is called. \n <Example><![CDATA[\ngap> gr1 := Digraph([[2], [1, 3\ ], [2, 3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> IsSymmetricDi\ graph(gr1);\ntrue\ngap> adj1 := AdjacencyMatrix(gr1);;\ngap> Display(adj1);\n[\ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ] ]\ngap> adj1 = Transpose\ dMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(gr1);\ntrue\ngap> gr2 := Digraph(\ [[2, 3], [1, 3], [2, 3]]);\n<immutable digraph with 3 vertices, 6 edges>\ngap>\ IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 := AdjacencyMatrix(gr2);;\ngap> Di\ splay(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ] ]\ngap> a\ dj2 = TransposedMat(adj2);\nfalse\n]]></Example>\n </Description>\n</ManSecti\ on>\n\n <ManSection>\n <Prop Name=\"IsTournament\" Arg=\"digraph\"/>\n <R\ eturns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This prope\ rty is <K>true</K> if the digraph <A>digraph</A> is a tournament,\n and <K>\ false</K> if it is not. <P/>\n\n A tournament is an orientation of a comple\ te (undirected) graph.\n Specifically, a tournament is a digraph which has \ a unique directed edge\n (of some orientation) between any pair of distinct\ vertices, and no loops.\n <P/>\n \n If the argument <A>digraph</A>\ is mutable, then the return value of\n this property is recomputed every t\ ime it is called. \n <Example><![CDATA[\ngap> D := Digraph([[2, 3, 4], [3, \ 4], [4], []]);\n<immutable digraph with 4 vertices, 6 edges>\ngap> IsTournamen\ t(D);\ntrue\ngap> D := Digraph([[2], [1], [3]]);\n<immutable digraph with 3 ve\ rtices, 3 edges>\ngap> IsTournament(D);\nfalse\ngap> D := CycleDigraph(IsMutab\ leDigraph, 3);\n<mutable digraph with 3 vertices, 3 edges>\ngap> IsTournament(\ D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\n<mutable digraph with 3 vertices,\ 2 edges>\ngap> IsTournament(D);\nfalse\n]]></Example>\n </Description>\n</Ma\ nSection>\n\n <ManSection>\n <Prop Name=\"IsTransitiveDigraph\" Arg=\"digr\ aph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n \ This property is <K>true</K> if the digraph <A>digraph</A>\n is transiti\ ve, and <K>false</K> if it is not.\n\n A digraph is <E>transitive</E> if wh\ enever <C>[ i, j ]</C> and\n <C>[ j, k ]</C> are edges of the digraph, then\ <C>[ i, k ]</C> is also an\n edge of the digraph. <P/>\n\n Let <M>n</M>\ be the number of vertices of an arbitrary digraph, and let\n <M>m</M> \ be the number of edges.\n For general digraphs, the methods used for this p\ roperty use a version\n of the Floyd-Warshall algorithm, and have complexit\ y <M>O(n^3)</M>.\n\n However for digraphs which are topologically sortable\ \n [<Ref Attr=\"DigraphTopologicalSort\"/>], then methods with\n complex\ ity <M>O(m + n + m \\cdot n)</M> will be used when appropriate.\n <P/>\n \ \n If the argument <A>digraph</A> is mutable, then the return value of\n\ this property is recomputed every time it is called. \n <P/>\n <Exam\ ple><![CDATA[\ngap> D := Digraph([[1, 2], [3], [3]]);\n<immutable digraph with\ 3 vertices, 4 edges>\ngap> IsTransitiveDigraph(D);\nfalse\ngap> gr2 := Digrap\ h([[1, 2, 3], [3], [3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> \ IsTransitiveDigraph(gr2);\ntrue\ngap> gr2 = DigraphTransitiveClosure(D);\ntrue\ \ngap> gr3 := Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n<immutable multidigraph w\ ith 3 vertices, 7 edges>\ngap> IsTransitiveDigraph(gr3);\ntrue\n]]></Example>\ \n </Description>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsPreord\ erDigraph\" Arg=\"digraph\"/>\n <Prop Name=\"IsQuasiorderDigraph\" Arg=\"digr\ aph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n \ A digraph is a preorder digraph if and only if the digraph satisifies both\ \n <Ref Prop=\"IsReflexiveDigraph\"/> and <Ref Prop=\"IsTransitiveDigraph\"\ />.\n A preorder digraph (or quasiorder digraph) <A>digraph</A> corresponds\ to\n the preorder relation <M>\\leq</M> defined by <M>x \\leq y</M> if and\ only\n if <C>[x, y]</C> is an edge of <A>digraph</A>.\n <P/>\n \n \ If the argument <A>digraph</A> is mutable, then the return value of\n th\ is property is recomputed every time it is called. \n\n<Example><![CDATA[\ngap\ > D := Digraph([[1], [2, 3], [2, 3]]);\n<immutable digraph with 3 vertices, 5 \ edges>\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[1 .. 4], [1 .. 4\ ], [1 .. 4], [1 .. 4]]);\n<immutable digraph with 4 vertices, 16 edges>\ngap> \ IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], [3], [4], [5], [1]]);\n<i\ mmutable digraph with 5 vertices, 5 edges>\ngap> IsPreorderDigraph(D);\nfalse\ \ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n<immutable digraph with 3 vertice\ s, 5 edges>\ngap> IsQuasiorderDigraph(D);\nfalse\n]]></Example>\n </Descripti\ on>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsPartialOrderDigraph\"\ Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Desc\ ription>\n A digraph is a partial order digraph if and only if the digraph \ satisifies\n all of <Ref Prop=\"IsReflexiveDigraph\"/>,\n <Ref Prop=\"Is\ AntisymmetricDigraph\"/> and <Ref Prop=\"IsTransitiveDigraph\"/>.\n A parti\ al order <A>digraph</A> corresponds\n to the partial order relation <M>\\le\ q</M> defined by <M>x \\leq y</M> if and\n only if <C>[x, y]</C> is an edge\ of <A>digraph</A>.\n <P/>\n \n If the argument <A>digraph</A> is m\ utable, then the return value of\n this property is recomputed every time i\ t is called. \n\n<Example><![CDATA[\ngap> D := Digraph([[1, 3], [2, 3], [3]]);\ \n<immutable digraph with 3 vertices, 5 edges>\ngap> IsPartialOrderDigraph(D);\ \ntrue\ngap> D := CycleDigraph(5);\n<immutable cycle digraph with 5 vertices>\ \ngap> IsPartialOrderDigraph(D);\nfalse\ngap> D := Digraph([[1, 1], [1, 1, 2],\ [3], [3, 3, 4, 4]]);\n<immutable multidigraph with 4 vertices, 10 edges>\ngap\ > IsPartialOrderDigraph(D);\ntrue\n]]></Example>\n </Description>\n</ManSecti\ on>\n\n <ManSection>\n <Prop Name=\"IsMeetSemilatticeDigraph\" Arg=\"digra\ ph\"/>\n <Prop Name=\"IsJoinSemilatticeDigraph\" Arg=\"digraph\"/>\n <Prop N\ ame=\"IsLatticeDigraph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false\ </K>.</Returns>\n <Description>\n <C>IsMeetSemilatticeDigraph</C> returns \ <K>true</K> if the digraph\n <A>digraph</A> is a meet semilattice; <C>IsJoi\ nSemilatticeDigraph</C>\n returns <K>true</K> if the digraph <A>digraph</A>\ is a join semilattice;\n and <C>IsLatticeDigraph</C> returns <K>true</K> \ if the digraph\n <A>digraph</A> is both a meet and a join semilattice.\n \ <P/>\n\n For a partial order digraph <Ref Prop=\"IsPartialOrderDigraph\"/>\ the\n corresponding partial order is the relation <M>\\leq</M>, defined by\ \n <M>x \\leq y</M> if and only if <C>[x, y]</C> is an edge.\n A digraph\ is a <E>meet semilattice</E> if it is a partial order and every\n pair of \ vertices has a greatest lower bound (meet) with respect to the\n aforementi\ oned relation. A <E>join semilattice</E> is a partial order where\n every p\ air of vertices has a least upper bound (join) with respect to\n the relati\ on.\n <P/>\n \n If the argument <A>digraph</A> is mutable, then the\ return value of\n this property is recomputed every time it is called. \n\ \n <Example><![CDATA[\ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n<immutabl\ e digraph with 3 vertices, 5 edges>\ngap> IsMeetSemilatticeDigraph(D);\nfalse\ \ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ng\ ap> D := Digraph([[1], [2], [1 .. 3]]);\n<immutable digraph with 3 vertices, 5\ edges>\ngap> IsJoinSemilatticeDigraph(D);\nfalse\ngap> IsMeetSemilatticeDigra\ ph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D := Digraph([[1 .. 4], [\ 2, 4], [3, 4], [4]]);\n<immutable digraph with 4 vertices, 9 edges>\ngap> IsMe\ etSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> \ IsLatticeDigraph(D);\ntrue\ngap> D := Digraph([[1, 1, 1], [1, 1, 2, 2],\n> \ [1, 3, 3], [1, 2, 3, 3, 4]]);\n<immutable multidigraph with 4 v\ ertices, 15 edges>\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemila\ tticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\n]]></Example>\n </De\ scription>\n</ManSection>\n\n </Section>\n\n <Section><Heading>Regularity</H\ eading>\n <ManSection>\n <Prop Name=\"IsInRegularDigraph\" Arg=\"digraph\"\ />\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n Th\ is property is <K>true</K> if there is an integer <C>n</C> such that\n for \ every vertex <C>v</C> of digraph <A>digraph</A> there are exactly\n <C>n</C\ > edges terminating in <C>v</C>.\n\n See also <Ref Prop=\"IsOutRegularDigra\ ph\"/> and\n <Ref Prop=\"IsRegularDigraph\"/>.\n <P/>\n \n If th\ e argument <A>digraph</A> is mutable, then the return value of\n this prope\ rty is recomputed every time it is called. \n <Example><![CDATA[\ngap> IsIn\ RegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInRegularDigraph(ChainDigrap\ h(4));\nfalse\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSecti\ on>\n <Prop Name=\"IsOutRegularDigraph\" Arg=\"digraph\"/>\n <Returns><K>tru\ e</K> or <K>false</K>.</Returns>\n <Description>\n This property is <K>tru\ e</K> if there is an integer <C>n</C> such that\n for every vertex <C>v</C>\ of digraph <A>digraph</A> there are exactly\n <C>n</C> edges starting at <\ C>v</C>.\n <P/>\n\n See also <Ref Prop=\"IsInRegularDigraph\"/> and\n \ <Ref Prop=\"IsRegularDigraph\"/>.\n <P/>\n \n If the argument <A>d\ igraph</A> is mutable, then the return value of\n this property is recomput\ ed every time it is called. \n <Example><![CDATA[\ngap> IsOutRegularDigraph\ (CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph(ChainDigraph(4));\nfalse\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Prop \ Name=\"IsRegularDigraph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>fals\ e</K>.</Returns>\n <Description>\n This property is <K>true</K> if there i\ s an integer <C>n</C> such that\n for every vertex <C>v</C> of digraph <A>d\ igraph</A> there are exactly\n <C>n</C> edges starting and terminating at <\ C>v</C>. In other words,\n the property is <K>true</K> if <A>digraph</A> is\ both in-regular and\n and out-regular.\n\n See also <Ref Prop=\"IsInReg\ ularDigraph\"/> and\n <Ref Prop=\"IsOutRegularDigraph\"/>.\n <P/>\n \ \n If the argument <A>digraph</A> is mutable, then the return value of\n \ this property is recomputed every time it is called. \n <Example><![CDATA\ [\ngap> IsRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsRegularDigraph(Cha\ inDigraph(4));\nfalse\n]]></Example>\n </Description>\n</ManSection>\n\n <\ ManSection>\n <Prop Name=\"IsDistanceRegularDigraph\" Arg=\"digraph\"/>\n <R\ eturns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n If <A>digr\ aph</A> is a connected symmetric graph, this property returns\n <K>true</K>\ if for any two vertices <C>u</C> and <C>v</C> of <A>digraph</A>\n and any \ two integers <C>i</C> and <C>j</C> between <C>0</C> and the\n diameter of <\ A>digraph</A>, the number of vertices at distance <C>i</C>\n from <C>u</C> \ and distance <C>j</C> from <C>v</C> depends only on\n <C>i</C>, <C>j</C>, a\ nd the distance between vertices <C>u</C> and\n <C>v</C>.<P/>\n\n Altern\ atively, a distance regular graph is a graph for which there exist\n intege\ rs <C>b_i</C>, <C>c_i</C>, and <C>i</C> such that for any two\n vertices <C\ >u</C>, <C>v</C> in <A>digraph</A> which are distance <C>i</C>\n apart, the\ re are exactly <C>b_i</C> neighbors of <C>v</C> which are at\n distance <C>\ i - 1</C> away from <C>u</C>, and <C>c_i</C> neighbors of\n <C>v</C> which \ are at distance <C>i + 1</C> away from <C>u</C>. This\n definition is used \ to check whether <A>digraph</A> is distance regular.<P/>\n\n In the case wh\ ere <A>digraph</A> is not symmetric or not connected, the\n property is <K>\ false</K>.\n <P/>\n \n If the argument <A>digraph</A> is mutable, t\ hen the return value of\n this property is recomputed every time it is call\ ed. \n\n <Example><![CDATA[\ngap> D := DigraphSymmetricClosure(ChainDigraph\ (5));;\ngap> IsDistanceRegularDigraph(D);\nfalse\ngap> D := Digraph([[2, 3, 4]\ , [1, 3, 4], [1, 2, 4], [1, 2, 3]]);\n<immutable digraph with 4 vertices, 12 e\ dges>\ngap> IsDistanceRegularDigraph(D);\ntrue\n]]></Example>\n </Description\ >\n</ManSection>\n\n </Section>\n\n <Section><Heading>Connectivity and cycle\ s</Heading>\n <ManSection>\n <Prop Name=\"IsAcyclicDigraph\" Arg=\"digraph\ \"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n \ This property is <K>true</K> if the digraph <A>digraph</A> is acyclic, and\n \ <K>false</K> if it is not. A digraph is <E>acyclic</E> if every directed\n \ cycle on the digraph is trivial. See Section <Ref Subsect=\"Definitions\"\n \ Style=\"Number\" /> for the definition of a directed cycle, and of a triv\ ial\n directed cycle.<P/>\n\n The method used in this operation has comp\ lexity <M>O(m+n)</M> where\n <M>m</M> is the number of edges (counting mult\ iple edges as one) and\n <M>n</M> is the number of vertices in the digraph.\ <P/>\n\n If the argument <A>digraph</A> is mutable, then the return value \ of\n this property is recomputed every time it is called. \n\n <Example>\ <![CDATA[\ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> func\ tion(x, y)\n> return IsEmpty(Intersection(x, y));\n> end);;\ngap> D := Digra\ ph(Petersen);\n<immutable digraph with 10 vertices, 30 edges>\ngap> IsAcyclicD\ igraph(D);\nfalse\ngap> D := DigraphFromDiSparse6String(\n> \".b_OGCIDBaPGkULE\ bQHCeRIdrHcuZMfRyDAbPhTi|zF\");\n<immutable digraph with 35 vertices, 34 edges\ >\ngap> IsAcyclicDigraph(D);\ntrue\ngap> IsAcyclicDigraph(ChainDigraph(10));\n\ true\ngap> D := CompleteDigraph(IsMutableDigraph, 4);\n<mutable digraph with 4\ vertices, 12 edges>\ngap> IsAcyclicDigraph(D);\nfalse\ngap> IsAcyclicDigraph(\ CycleDigraph(10));\nfalse]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Prop Name=\"IsChainDigraph\" Arg=\"digraph\"/>\n <Returns><\ K>true</K> or <K>false</K>.</Returns>\n <Description>\n <C>IsChainDigraph<\ /C> returns <K>true</K> if the digraph <A>digraph</A> is\n isomorphic to th\ e chain digraph with the same number of vertices as\n <A>digraph</A>, and <\ K>false</K> if it is not; see <Ref\n Oper=\"ChainDigraph\"/>.<P/>\n\n \ A digraph is a <E>chain</E> if and only if it is a directed tree, in which\n \ every vertex has out degree at most one; see <Ref Prop=\"IsDirectedTree\"/>\ \n and <Ref Attr=\"OutDegrees\"/>.\n <P/>\n \n If the argument <\ A>digraph</A> is mutable, then the return value of\n this property is recom\ puted every time it is called. \n\n<Example><![CDATA[\ngap> D := Digraph([[1, \ 3], [2, 3], [3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> IsChain\ Digraph(D);\nfalse\ngap> D := ChainDigraph(5);\n<immutable chain digraph with \ 5 vertices>\ngap> IsChainDigraph(D);\ntrue\ngap> D := DigraphReverse(D);\n<imm\ utable digraph with 5 vertices, 4 edges>\ngap> IsChainDigraph(D);\ntrue\ngap> \ D := ChainDigraph(IsMutableDigraph, 5);\n<mutable digraph with 5 vertices, 4 e\ dges>\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphReverse(D);\n<mutable digrap\ h with 5 vertices, 4 edges>\ngap> IsChainDigraph(D);\ntrue]]></Example>\n </D\ escription>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsConnectedDigr\ aph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n \ <Description>\n This property is <K>true</K> if the digraph <A>digraph</A>\ \n is weakly connected and <K>false</K> if it is not. A digraph\n <A>di\ graph</A> is <E>weakly connected</E> if it is possible to travel\n from any\ vertex to any other vertex by traversing edges in either\n direction (poss\ ibly against the orientation of some of them). <P/>\n\n The method used in \ this function has complexity <M>O(m)</M> if the\n digraph's <Ref Attr=\"Dig\ raphSource\"/> attribute is set, otherwise it has\n complexity <M>O(m+n)</M\ > (where\n <M>m</M> is the number of edges and\n <M>n</M> is the number \ of vertices of the digraph).\n <P/>\n \n If the argument <A>digraph\ </A> is mutable, then the return value of\n this property is recomputed eve\ ry time it is called. \n\n <Example><![CDATA[\ngap> D := Digraph([[2], [3],\ []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D := Digraph([[1, 3], [4], [3\ ], []]);;\ngap> IsConnectedDigraph(D);\nfalse\ngap> D := Digraph(IsMutableDigr\ aph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D := Digraph(I\ sMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\ ]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Prop Na\ me=\"IsBiconnectedDigraph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>fa\ lse</K>.</Returns>\n <Description>\n A connected digraph is <E>biconnected\ </E> if it is still connected (in the\n sense of <Ref Prop=\"IsConnectedDig\ raph\"/>) when any vertex is removed. \n If <A>D</A> has at least 3 verti\ ces, then <C>IsBiconnectedDigraph</C>\n implies <Ref Prop=\"IsBridgelessD\ igraph\"/>;\n see <Ref Attr=\"ArticulationPoints\"/> or <Ref Attr=\"Bridges\ \"/> for a more\n detailed explanation.\n <P/>\n\n <C>IsBiconnectedDi\ graph</C> returns <K>true</K> if the digraph\n <A>digraph</A> is biconnecte\ d, and <K>false</K> if it is not. In\n particular, <C>IsBiconnectedDigraph<\ /C> returns <K>false</K> if\n <A>digraph</A> is not connected. <P/>\n \ \n If the argument <A>digraph</A> is mutable, then the return value of\n \ this property is recomputed every time it is called. <P/>\n\n Multiple edg\ es are ignored by this method. <P/>\n\n The method used in this operation h\ as complexity <M>O(m+n)</M> where\n <M>m</M> is the number of edges and <M>\ n</M> is the number of vertices in\n the digraph.\n <P/>\n\n See also\ <Ref Attr=\"Bridges\"/>, <Ref Attr=\"ArticulationPoints\"/>, and \n <Ref P\ rop=\"IsBridgelessDigraph\"/>.\n <P/>\n \n<Example><![CDATA[\ngap> IsB\ iconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBiconnectedD\ igraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3,\ 3, 4, 4]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := CompleteBiparti\ teDigraph(IsMutableDigraph, 5, 4);\n<mutable digraph with 9 vertices, 40 edges\ >\ngap> IsBiconnectedDigraph(D);\ntrue]]></Example>\n </Description>\n</ManSe\ ction>\n\n <ManSection>\n <Prop Name=\"IsBridgelessDigraph\" Arg=\"D\"/>\n\ <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n A conn\ ected digraph is <E>bridgeless</E> if it is still connected (in the\n sense\ of <Ref Prop=\"IsConnectedDigraph\"/>) when any edge is removed. \n If <A>\ D</A> has at least 3 vertices, then <Ref\n Prop=\"IsBiconnectedDigraph\"/> \ implies <C>IsBridgelessDigraph</C>;\n see <Ref Attr=\"ArticulationPoints\"/\ > or <Ref Attr=\"Bridges\"/> for a more\n detailed explanation.\n <P/>\n\ \n <C>IsBridgelessDigraph</C> returns <K>true</K> if the digraph\n <A>D<\ /A> is bridgeless, and <K>false</K> if it is not. In\n particular, <C>IsBri\ dgelessDigraph</C> returns <K>false</K> if\n <A>D</A> is not connected. <P/\ >\n \n If the argument <A>D</A> is mutable, then the return value of\n\ this property is recomputed every time it is called. <P/>\n\n Multiple \ edges are ignored by this method. <P/>\n\n The method used in this operatio\ n has complexity <M>O(m+n)</M> where\n <M>m</M> is the number of edges and \ <M>n</M> is the number of vertices in\n the digraph.\n <P/>\n\n See a\ lso <Ref Attr=\"Bridges\"/>, <Ref Attr=\"ArticulationPoints\"/>, and \n <Re\ f Prop=\"IsBiconnectedDigraph\"/>.\n \n<Example><![CDATA[\ngap> IsBridgel\ essDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBridgelessDigraph(C\ ycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4\ ]]);;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph\ (IsMutableDigraph, 5, 4);\n<mutable digraph with 9 vertices, 40 edges>\ngap> I\ sBridgelessDigraph(D);\ntrue\ngap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4],\ [2, 3], [1, 2]]);\n<immutable digraph with 5 vertices, 12 edges>\ngap> IsBrid\ gelessDigraph(D);\ntrue\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := Digra\ ph([[2], [3], [4], [2]]);\n<immutable digraph with 4 vertices, 4 edges>\ngap> \ IsBridgelessDigraph(D);\nfalse\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> IsB\ ridgelessDigraph(ChainDigraph(2));\nfalse\ngap> IsBiconnectedDigraph(ChainDigr\ aph(2));\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSect\ ion>\n <Prop Name=\"IsStronglyConnectedDigraph\" Arg=\"digraph\"/>\n <Return\ s><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This property i\ s <K>true</K> if the digraph <A>digraph</A> is strongly\n connected and <K>\ false</K> if it is not. <P/>\n\n A digraph <A>digraph</A> is <E>strongly co\ nnected</E> if there is a directed\n path from every vertex to every other \ vertex. See Section <Ref\n Subsect=\"Definitions\" Style=\"Number\" /> f\ or the definition of a directed\n path. <P/>\n\n The method used in t\ his operation is based on Gabow's Algorithm <Cite\n Key=\"Gab00\"/> and h\ as complexity <M>O(m+n)</M>, where <M>m</M> is\n the number of edges (count\ ing multiple edges as one) and <M>n</M> is the\n number of vertices in the \ digraph.\n <P/>\n \n If the argument <A>digraph</A> is mutable, the\ n the return value of\n this property is recomputed every time it is called\ . \n\n <Example><![CDATA[\ngap> D := CycleDigraph(250000);\n<immutable cycl\ e digraph with 250000 vertices>\ngap> IsStronglyConnectedDigraph(D);\ntrue\nga\ p> D := DigraphRemoveEdges(D, [[250000, 1]]);\n<immutable digraph with 250000 \ vertices, 249999 edges>\ngap> IsStronglyConnectedDigraph(D);\nfalse\ngap> D :=\ CycleDigraph(IsMutableDigraph, 250000);\n<mutable digraph with 250000 vertice\ s, 250000 edges>\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> DigraphRemov\ eEdge(D, [250000, 1]);\n<mutable digraph with 250000 vertices, 249999 edges>\n\ gap> IsStronglyConnectedDigraph(D);\nfalse\n]]></Example>\n </Description>\n<\ /ManSection>\n\n <ManSection>\n <Prop Name=\"IsAperiodicDigraph\" Arg=\"di\ graph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\ \n This property is <K>true</K> if the digraph <A>digraph</A>\n is aperi\ odic, i.e. if its <Ref Attr = \"DigraphPeriod\"/> is equal to 1.\n Otherwis\ e, the property is <K>false</K>.\n <P/>\n \n If the argument <A>dig\ raph</A> is mutable, then the return value of\n this property is recomputed\ every time it is called. \n <Example><![CDATA[\ngap> D := Digraph([[6], [1\ ], [2], [3], [4, 4], [5]]);\n<immutable multidigraph with 6 vertices, 7 edges>\ \ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Digraph([[2], [3, 5], [4], [5]\ , [1, 2]]);\n<immutable digraph with 5 vertices, 7 edges>\ngap> IsAperiodicDig\ raph(D);\ntrue\ngap> D := Digraph(IsMutableDigraph, [[2], [3, 5], [4], [5], [1\ , 2]]);\n<mutable digraph with 5 vertices, 7 edges>\ngap> IsAperiodicDigraph(D\ );\ntrue]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Prop Name=\"IsDirectedTree\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>\ false</K>.</Returns>\n <Description>\n Returns <K>true</K> if the digraph \ <A>digraph</A> is a directed tree, and\n <K>false</K> if it is not. <P/>\n\ \n A <E>directed tree</E> is an acyclic digraph with precisely 1 source,\n \ such that no two vertices share an out-neighbour.\n Note the empty digra\ ph is not considered a directed\n tree as it has no source. <P/>\n\n See\ also <Ref Attr=\"DigraphSources\"/>.\n <P/>\n \n If the argument <\ A>digraph</A> is mutable, then the return value of\n this property is recom\ puted every time it is called. \n\n <Example><![CDATA[\ngap> D := Digraph([\ [], [2]]);\n<immutable digraph with 2 vertices, 1 edge>\ngap> IsDirectedTree(D\ );\nfalse\ngap> D := Digraph([[3], [3], []]);\n<immutable digraph with 3 verti\ ces, 2 edges>\ngap> IsDirectedTree(D);\nfalse\ngap> D := Digraph([[2], [3], []\ ]);\n<immutable digraph with 3 vertices, 2 edges>\ngap> IsDirectedTree(D);\ntr\ ue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [], []]);\n<immutable digraph \ with 6 vertices, 5 edges>\ngap> IsDirectedTree(D);\ntrue\n]]></Example>\n </D\ escription>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsUndirectedTre\ e\" Arg=\"digraph\"/>\n <Prop Name=\"IsUndirectedForest\" Arg=\"digraph\"/>\n\ <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n The pr\ operty <C>IsUndirectedTree</C> returns <K>true</K> if the digraph\n <A>digr\ aph</A> is an undirected tree, and the property\n <C>IsUndirectedForest</C>\ returns <K>true</K> if <A>digraph</A> is an\n undirected forest; otherwise\ , these properties return <K>false</K>. <P/>\n\n An <E>undirected tree</E> \ is a symmetric digraph without loops, in which for\n any pair of distinct v\ ertices <C>u</C> and <C>v</C>, there is exactly one\n directed path from <C\ >u</C> to <C>v</C>. See <Ref\n Prop=\"IsSymmetricDigraph\"/> and <Ref Pro\ p=\"DigraphHasLoops\"/>, and see\n Section <Ref Subsect=\"Definitions\" Sty\ le=\"Number\" /> for the definition of\n directed path. This definition imp\ lies that an undirected tree has\n no multiple edges. <P/>\n\n An <E>und\ irected forest</E> is a digraph, each of whose connected components\n is an\ undirected tree. In other words, an undirected forest is isomorphic to\n a\ disjoint union of undirected trees. See <Ref\n Attr=\"DigraphConnectedCo\ mponents\" /> and <Ref Func=\"DigraphDisjointUnion\"\n Label=\"for a list\ of digraphs\" />. In particular, every\n undirected tree is an undirected \ forest. <P/>\n\n Please note that the digraph with zero vertices is conside\ red to be neither\n an undirected tree nor an undirected forest.\n <P/>\ \n \n If the argument <A>digraph</A> is mutable, then the return value\ of\n this property is recomputed every time it is called. \n\n <Example\ ><![CDATA[\ngap> D := Digraph([[3], [3], [1, 2]]);\n<immutable digraph with 3 \ vertices, 4 edges>\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmetricDigraph(D\ ) and not DigraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], [1, 4], [3],\ [2]]);\n<immutable digraph with 5 vertices, 6 edges>\ngap> IsConnectedDigraph\ (D);\nfalse\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D);\ntr\ ue\ngap> D := Digraph([[1, 2], [1], [2]]);\n<immutable digraph with 3 vertices\ , 4 edges>\ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfalse\ngap> Is\ SymmetricDigraph(D) or not DigraphHasLoops(D);\nfalse]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsEulerianDigraph\" \ Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Descr\ iption>\n This property returns true if the digraph <A>digraph</A> is Euler\ ian.\n <P/>\n\n A connected digraph is called <E>Eulerian</E> if there e\ xists a directed\n circuit on the digraph which includes every edge exactly\ once. See\n Section <Ref Subsect=\"Definitions\" Style=\"Number\" /> for \ the definition of\n a directed circuit. Note that the empty digraph with at\ most one vertex is\n considered to be Eulerian.\n <P/>\n \n If \ the argument <A>digraph</A> is mutable, then the return value of\n this pro\ perty is recomputed every time it is called. \n\n <Example><![CDATA[\ngap> \ D := Digraph([[]]);\n<immutable empty digraph with 1 vertex>\ngap> IsEulerianD\ igraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n<immutable digraph with 2 ver\ tices, 1 edge>\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph([[3], [],\ [2]]);\n<immutable digraph with 3 vertices, 2 edges>\ngap> IsEulerianDigraph(\ D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n<immutable digraph with 3 ver\ tices, 3 edges>\ngap> IsEulerianDigraph(D);\ntrue\n]]></Example>\n </Descript\ ion>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsHamiltonianDigraph\"\ Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Desc\ ription>\n If <A>digraph</A> is Hamiltonian, then this property returns\n \ <K>true</K>, and <K>false</K> if it is not. <P/>\n\n A digraph with <C>n<\ /C> vertices is <E>Hamiltonian</E> if it has a\n directed cycle of length <\ C>n</C>. See Section <Ref Subsect=\"Definitions\"\n Style=\"Number\" /> fo\ r the definition of a directed cycle.\n Note the empty digraphs on 0 and 1 \ vertices are considered to be\n Hamiltonian.<P/>\n\n The method used in \ this operation has the worst case complexity as\n <Ref Oper=\"DigraphMonomo\ rphism\"/>.\n <P/>\n \n If the argument <A>digraph</A> is mutable, \ then the return value of\n this property is recomputed every time it is cal\ led. \n\n <Example><![CDATA[\ngap> g := Digraph([[]]);\n<immutable empty di\ graph with 1 vertex>\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g := Digraph([\ [2], [1]]);\n<immutable digraph with 2 vertices, 2 edges>\ngap> IsHamiltonianD\ igraph(g);\ntrue\ngap> g := Digraph([[3], [], [2]]);\n<immutable digraph with \ 3 vertices, 2 edges>\ngap> IsHamiltonianDigraph(g);\nfalse\ngap> g := Digraph(\ [[2], [3], [1]]);\n<immutable digraph with 3 vertices, 3 edges>\ngap> IsHamilt\ onianDigraph(g);\ntrue\n]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Prop Name=\"IsCycleDigraph\" Arg=\"digraph\"/>\n <Returns><K\ >true</K> or <K>false</K>.</Returns>\n <Description>\n <C>IsCycleDigraph</\ C> returns <K>true</K> if the digraph <A>digraph</A> is\n isomorphic to the\ cycle digraph with the same number of vertices as\n <A>digraph</A>, and <K\ >false</K> if it is not; see <Ref\n Oper=\"CycleDigraph\"/>.<P/>\n\n A\ digraph is a <E>cycle</E> if and only if it is strongly connected and has\n \ the same number of edges as vertices. \n <P/>\n \n If the argumen\ t <A>digraph</A> is mutable, then the return value of\n this property is re\ computed every time it is called. \n\n<Example><![CDATA[\ngap> D := Digraph([[\ 1, 3], [2, 3], [3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> IsCy\ cleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\n<immutable cycle digraph wi\ th 5 vertices>\ngap> IsCycleDigraph(D);\ntrue\ngap> D := OnDigraphs(D, (1, 2, \ 3));\n<immutable digraph with 5 vertices, 5 edges>\ngap> D = CycleDigraph(5);\ \nfalse\ngap> IsCycleDigraph(D);\ntrue]]></Example>\n </Description>\n</ManSe\ ction>\n\n </Section>\n \n <Section Label=\"Planarity\"><Heading>Planarity<\ /Heading>\n <ManSection>\n <Prop Name=\"IsPlanarDigraph\" Arg=\"digraph\"/\ >\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n A <\ E>planar</E> digraph is a digraph that can be embedded in the plane in\n su\ ch a way that its edges do not intersect. A digraph is planar if and only\n \ if it does not have a subdigraph that is homeomorphic to either the\n comp\ lete graph on <C>5</C> vertices or the complete bipartite graph with\n vert\ ex sets of sizes <C>3</C> and <C>3</C>.\n <P/>\n \n <C>IsPlanarDigrap\ h</C> returns <K>true</K> if the digraph <A>digraph</A> is\n planar and <K>\ false</K> if it is not. The directions and multiplicities of\n any edges in\ <A>digraph</A> are ignored by <C>IsPlanarDigraph</C>.\n <P/> \n\n See a\ lso <Ref Prop=\"IsOuterPlanarDigraph\"/>.\n <P/>\n\n This method uses th\ e reference implementation in\n &edge-addition-planarity-suite; by John Boy\ er of the algorithms described\n in <Cite Key=\"BM06\"/>.\n\n <Example><\ ![CDATA[\ngap> IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigrap\ h(CompleteDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2\ , 3));\ntrue\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\nga\ p> IsPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanar\ Digraph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(Co\ mpleteBipartiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(C\ ompleteBipartiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n]]></Example>\n </D\ escription>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsOuterPlanarDi\ graph\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n\ <Description>\n An <E>outer planar</E> digraph is a digraph that can be e\ mbedded in the\n plane in such a way that its edges do not intersect, and a\ ll vertices\n belong to the unbounded face of the embedding. A digraph is \ outer planar\n if and only if it does not have a subdigraph that is homeomo\ rphic to either\n the complete graph on <C>4</C> vertices or the complete b\ ipartite graph\n with vertex sets of sizes <C>2</C> and <C>3</C>.\n <P/>\ \n \n <C>IsOuterPlanarDigraph</C> returns <K>true</K> if the digraph\n \ <A>digraph</A> is outer planar and <K>false</K> if it is not. The\n direc\ tions and multiplicities of any edges in <A>digraph</A> are ignored by\n <C\ >IsPlanarDigraph</C>. <P/>\n\n See also <Ref Prop=\"IsPlanarDigraph\"/>.\n\ \n This method uses the reference implementation in\n &edge-addition-pla\ narity-suite; by John Boyer of the algorithms described\n in <Cite Key=\"BM\ 06\"/>.\n\n<Example><![CDATA[\ngap> IsOuterPlanarDigraph(CompleteDigraph(4));\ \nfalse\ngap> IsOuterPlanarDigraph(CompleteDigraph(5));\nfalse\ngap> IsOuterPl\ anarDigraph(CompleteBipartiteDigraph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph\ (CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigra\ ph(10));\ntrue\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4)\ );\nfalse\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 5));\nf\ alse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterP\ lanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleD\ igraph(IsMutableDigraph, 10));\ntrue\n\n]]></Example>\n </Description>\n</Man\ Section>\n\n </Section>\n\n <Section><Heading>Homomorphisms and transformati\ ons</Heading>\n <ManSection>\n <Prop Name=\"IsDigraphCore\" Arg=\"digraph\ \"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n \ This property returns <K>true</K> if <A>digraph</A> is a core,\n and <K>fal\ se</K> if it is not.<P/>\n\n A digraph <C>D</C> is a <E>core</E> if and onl\ y if it has no proper\n subdigraphs <C>A</C> such that there exists a homom\ orphism from <C>D</C>\n to <C>A</C>. In other words, a digraph <C>D</C> is \ a core if and only if\n every endomorphism on <C>D</C> is an automorphism o\ n <C>D</C>.\n <P/>\n \n If the argument <A>digraph</A> is mutable, \ then the return value of\n this property is recomputed every time it is cal\ led. \n\n <Example><![CDATA[\ngap> D := CompleteDigraph(6);\n<immutable com\ plete digraph with 6 vertices>\ngap> IsDigraphCore(D);\ntrue\ngap> D := Digrap\ hSymmetricClosure(CycleDigraph(6));\n<immutable symmetric digraph with 6 verti\ ces, 12 edges>\ngap> DigraphHomomorphism(D, CompleteDigraph(2));\nTransformati\ on( [ 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigraphCore(D);\nfalse\n]]></Example>\n </\ Description>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsEdgeTransiti\ ve\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <\ Description>\n If <A>digraph</A> is a digraph without multiple edges, then\ \n <C>IsEdgeTransitive</C> returns <K>true</K> if <A>digraph</A>\n is ed\ ge transitive, and <K>false</K> otherwise. A digraph is\n <E>edge transitiv\ e</E> if its automorphism group acts\n transitively on its edges (via the a\ ction\n <Ref Func=\"OnPairs\" BookName=\"ref\"/>).\n <P/>\n \n I\ f the argument <A>digraph</A> is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n <Example><![CDATA[\ngap\ > IsEdgeTransitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(ChainDigr\ aph(3));\nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\nError,\ the argument <D> must be a digraph with no multiple edges,\n]]></Example>\n \ </Description>\n</ManSection>\n\n <ManSection>\n <Prop Name=\"IsVertexTran\ sitive\" Arg=\"digraph\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\ \n <Description>\n If <A>digraph</A> is a digraph, then <C>IsVertexTransit\ ive</C> returns\n <K>true</K> if <A>digraph</A> is vertex transitive, and <\ K>false</K>\n otherwise. A digraph is <E>vertex transitive</E> if its autom\ orphism group\n acts transitively on its vertices.\n <P/>\n \n I\ f the argument <A>digraph</A> is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n <Example><![CDATA[\ngap\ > IsVertexTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitive(Chain\ Digraph(3));\nfalse\n]]></Example>\n </Description>\n</ManSection>\n\n </Sec\ tion>\n\n \n</Chapter>\n <!-- properties -->\n <Chapter Label=\"Homomorphism\ s\"><Heading>Homomorphisms</Heading>\n <!--Let $E_a=(E_a^0,E_a^1,\\ra_a,\\so_\ a)$ and $E_b=(E_b^0,E_b^1,\\ra_b,\\so_b)$ be two\ngraphs, and let $\\phi_0 : E\ _a^0 \\to E_b^0$ and $\\phi_1 : E_a^1 \\to E_b^1$ be\nfunctions. Then the pair\ $\\phi = (\\phi_0, \\phi_1)$ is a \\emph{graph homomorphism\nfrom} $E_a$ \\em\ ph{to} $E_b$ if $\\phi_0(\\so_a(e)) = \\so_b(\\phi_1(e))$ and\n$\\phi_0(\\ra_a\ (e)) = \\ra_b(\\phi_1(e))$ for every $e \\in E_a^1$. If $\\phi_0$ and\n$\\phi_\ 1$ are in addition bijective, then $\\phi$ is a \\emph{graph isomorphism\nfrom\ } $E_a$ \\emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are\n\\emph\ {isomorphic} and write $E_a \\cong E_b$.-->\n\n <Section><Heading>Acting on d\ igraphs</Heading>\n <ManSection>\n <Oper Name=\"OnDigraphs\" Arg=\"digraph\ , perm\"\n Label=\"for a digraph and a perm\"/>\n <Oper Name=\"OnDigraphs\ \" Arg=\"digraph, trans\"\n Label=\"for a digraph and a transformation\"/>\ \n <Returns>A digraph.</Returns>\n <Description>\n If <A>digraph</A> is a\ digraph, and the second argument <A>perm</A> is a\n <E>permutation</E> of \ the vertices of <A>digraph</A>, then this operation\n returns a digraph con\ structed by relabelling the vertices of\n <A>digraph</A> according to <A>pe\ rm</A>. Note that for an automorphism\n <C>f</C> of a digraph, we have <C>\ OnDigraphs(<A>digraph</A>, f) =\n </C><A>digraph</A>.\n <P/>\n\n If t\ he second argument is a <E>transformation</E> <A>trans</A> of the\n vertice\ s of <A>digraph</A>, then this operation returns a digraph\n constructed by\ transforming the source and range of each edge according to\n <A>trans</A>\ . Thus a vertex which does not appear in the image of\n <A>trans</A> will b\ e isolated in the returned digraph, and the returned\n digraph may contain \ multiple edges, even if <A>digraph</A> does not.\n If <A>trans</A> is mathe\ matically a permutation, then the result coincides\n with <C>OnDigraphs(<A>\ digraph</A>, AsPermutation(<A>trans</A>))</C>.\n <P/>\n\n The <Ref Oper=\ \"DigraphVertexLabels\"/> of <A>digraph</A> will not be retained\n in the r\ eturned digraph. <P/>\n\n If <A>digraph</A> belongs to <Ref Filt=\"IsMutabl\ eDigraph\"/>, then\n relabelling of the vertices is performed directly on <\ A>digraph</A>.\n If <A>digraph</A> belongs to <Ref Filt=\"IsImmutableDigrap\ h\"/>, an immutable\n copy of <A>digraph</A> with the vertices relabelled i\ s returned.<P/>\n\n <Example><![CDATA[\ngap> D := Digraph([[3], [1, 3, 5], \ [1], [1, 2, 4], [2, 3, 5]]);\n<immutable digraph with 5 vertices, 11 edges>\ng\ ap> new := OnDigraphs(D, (1, 2));\n<immutable digraph with 5 vertices, 11 edge\ s>\ngap> OutNeighbours(new);\n[ [ 2, 3, 5 ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3\ , 5 ] ]\ngap> D := Digraph([[2], [], [2]]);\n<immutable digraph with 3 vertice\ s, 2 edges>\ngap> t := Transformation([1, 2, 1]);;\ngap> new := OnDigraphs(D, \ t);\n<immutable multidigraph with 3 vertices, 2 edges>\ngap> OutNeighbours(new\ );\n[ [ 2, 2 ], [ ], [ ] ]\ngap> ForAll(DigraphEdges(D),\n> e -> IsDigraphE\ dge(new, [e[1] ^ t, e[2] ^ t]));\ntrue\n]]></Example>\n </Description>\n</Man\ Section>\n\n <ManSection>\n <Oper Name=\"OnMultiDigraphs\" Arg=\"digraph, \ pair\"/>\n <Oper Name=\"OnMultiDigraphs\" Arg=\"digraph, perm1, perm2\"\n \ Label=\"for a digraph, perm, and perm\"/>\n <Returns>A digraph.</Returns>\n \ <Description>\n If <A>digraph</A> is a digraph, and <A>pair</A> is a pair c\ onsisting of a\n permutation of the vertices and a permutation of the edges\ of\n <A>digraph</A>, then this operation returns a digraph\n constructe\ d by relabelling the vertices and edges of <A>digraph</A>\n according to <A\ >perm[1]</A> and <A>perm[2]</A>, respectively. <P/>\n\n In its second form,\ <C>OnMultiDigraphs</C> returns a digraph with vertices\n and edges permute\ d by <A>perm1</A> and <A>perm2</A>, respectively. <P/>\n\n Note that <C>OnD\ igraphs(<A>digraph</A>,\n perm)=OnMultiDigraphs(<A>digraph</A>, [perm, ()\ ])</C> where <C>perm</C> is\n a permutation of the vertices of <A>digraph</\ A>. If you are only interested\n in the action of a permutation on the vert\ ices of a digraph, then you can\n use <C>OnDigraphs</C> instead of <C>OnMul\ tiDigraphs</C>.\n\n If <A>digraph</A> belongs to <Ref Filt=\"IsMutableDigra\ ph\"/>, then\n relabelling of the vertices is performed directly on <A>digr\ aph</A>.\n If <A>digraph</A> belongs to <Ref Filt=\"IsImmutableDigraph\"/>,\ an immutable\n copy of <A>digraph</A> with the vertices relabelled is retu\ rned.<P/>\n\n <Example><![CDATA[\ngap> D1 := Digraph([\n> [3, 6, 3], [], [3\ ], [9, 10], [9], [], [], [10, 4, 10], [], []]);\n<immutable multidigraph with\ 10 vertices, 10 edges>\ngap> p := BlissCanonicalLabelling(D1);\n[ (1,7)(3,6)(\ 4,10)(5,9), () ]\ngap> D2 := OnMultiDigraphs(D1, p);\n<immutable multidigraph \ with 10 vertices, 10 edges>\ngap> OutNeighbours(D2);\n[ [ ], [ ], [ ], [ ]\ , [ ], [ 6 ], [ 6, 3, 6 ], [ 4, 10, 4 ], \n [ 5 ], [ 5, 4 ] ]]]></Example>\n\ </Description>\n</ManSection>\n\n </Section>\n\n <Section Label=\"Isomorph\ isms and canonical labellings\">\n <Heading>Isomorphisms and canonical labe\ llings</Heading>\n From version 0.11.0 of &Digraphs; it is possible to use \ either &bliss; or\n &nauty; (via &NautyTracesInterface;) to calculate canon\ ical labellings and\n automorphism groups of digraphs; see <Cite Key=\"JK07\ \"/> and <Cite\n Key=\"MP14\"/> for more details about &bliss; and &nauty\ ;,\n respectively. \n\n <ManSection>\n <Func Name=\"DigraphsUseNauty\" \ Arg=\"\"/>\n <Func Name=\"DigraphsUseBliss\" Arg=\"\"/>\n <Returns>Nothing.<\ /Returns>\n <Description>\n These functions can be used to specify whether\ &nauty; or &bliss; should be\n used by default by &Digraphs;. If &NautyTra\ cesInterface; is not available,\n then these functions do nothing. Otherwis\ e, by calling\n <C>DigraphsUseNauty</C> subsequent computations will defaul\ t to using\n &nauty; rather than &bliss;, where possible.\n <P/>\n\n \ You can call these functions at any point in a &GAP; session, as many\n tim\ es as you like, it is guaranteed that existing digraphs remain\n valid, and\ that comparison of existing digraphs and newly created digraphs\n via\n \ <Ref Oper=\"IsIsomorphicDigraph\" Label=\"for digraphs\"/>,\n <Ref Oper=\"\ IsIsomorphicDigraph\"\n Label=\"for digraphs and homogeneous lists\"/>,\n\ <Ref Oper=\"IsomorphismDigraphs\" Label=\"for digraphs\"/>, and\n <Ref \ Oper=\"IsomorphismDigraphs\"\n Label=\"for digraphs and homogeneous lists\ \"/> are also valid.<P/>\n\n It is also possible to compute the automorphis\ m group of a specific digraph\n using both &nauty; and &bliss; using <Ref A\ ttr=\"NautyAutomorphismGroup\"/>\n and <Ref Attr=\"BlissAutomorphismGroup\"\ Label=\"for a digraph\"/>, respectively.\n </Description>\n</ManSection>\n\n\ <ManSection>\n <Attr Name=\"AutomorphismGroup\" Label=\"for a digraph\" A\ rg=\"digraph\"/>\n <Returns>A permutation group.</Returns>\n <Description>\n\ If <A>digraph</A> is a digraph, then this attribute contains the group of\ \n automorphisms of <A>digraph</A>. An <E>automorphism</E> of <A>digraph</\ A>\n is an isomorphism from <A>digraph</A> to itself. See <Ref\n Oper=\ \"IsomorphismDigraphs\" Label=\"for digraphs\" /> for more information\n ab\ out isomorphisms of digraphs. <P/>\n\n If <A>digraph</A> is not a multidigr\ aph then the automorphism group is\n returned as a group of permutations on\ the set of vertices of\n <A>digraph</A>. <P/>\n\n If <A>digraph</A> is \ a multidigraph then the automorphism group is returned\n as the direct prod\ uct of a group of permutations on the set of vertices of\n <A>digraph</A> w\ ith a group of permutations on the set of edges of\n <A>digraph</A>. These \ groups can be accessed using <Ref Oper=\"Projection\"\n Label=\"for a dom\ ain and a positive integer\" BookName=\"ref\"/> on the\n returned group.<P/\ >\n\n By default, the automorphism group is found using &bliss; by Tommi Ju\ nttila\n and Petteri Kaski. If &NautyTracesInterface; is available, then &n\ auty; by\n Brendan Mckay and Adolfo Piperno can be used instead; see\n <\ Ref Attr=\"BlissAutomorphismGroup\" Label=\"for a digraph\"/>, \n <Ref Attr\ =\"NautyAutomorphismGroup\"/>, <Ref Func=\"DigraphsUseBliss\"/>,\n and <Ref\ Func=\"DigraphsUseNauty\"/>.\n <P/>\n\n If the argument <A>digraph</A> \ is mutable, then the return value of\n this attribute is recomputed every t\ ime it is called. \n\n <Example><![CDATA[\ngap> johnson := DigraphFromGraph\ 6String(\"E}lw\");\n<immutable symmetric digraph with 6 vertices, 24 edges>\ng\ ap> G := AutomorphismGroup(johnson);\nGroup([ (3,4), (2,3)(4,5), (1,2)(5,6) ])\ \ngap> cycle := CycleDigraph(9);\n<immutable cycle digraph with 9 vertices>\ng\ ap> G := AutomorphismGroup(cycle);\nGroup([ (1,2,3,4,5,6,7,8,9) ])\ngap> IsCyc\ lic(G) and Size(G) = 9;\ntrue]]></Example>\n </Description>\n </ManSection\ >\n\n <ManSection>\n <Attr Name=\"BlissAutomorphismGroup\" Arg=\"digraph\"\ Label=\"for a digraph\"/>\n <Oper Name=\"BlissAutomorphismGroup\" Arg=\"digr\ aph, vertex_colours\" \n Label=\"for a digraph and homogenous list\"/>\n <\ Oper Name=\"BlissAutomorphismGroup\" Arg=\"digraph, vertex_colours, edge_colou\ rs\" \n Label=\"for a digraph, homogenous list, and list\"/>\n <Returns>A \ permutation group.</Returns>\n <Description>\n If <A>digraph</A> is a digr\ aph, then this attribute contains the group of\n automorphisms of <A>digrap\ h</A> as calculated using &bliss; by Tommi Junttila\n and Petteri Kaski. <P\ />\n\n The attribute <Ref Attr=\"AutomorphismGroup\" Label=\"for a digraph\ \"/> and\n operation <Ref Oper=\"AutomorphismGroup\" \n Label=\"for a \ digraph and a homogeneous list\"/> returns\n the value of either <C>BlissAu\ tomorphismGroup</C> or <Ref\n Attr=\"NautyAutomorphismGroup\"/>. These gr\ oups are, of course, equal but\n their generating sets may differ. <P/>\n \ The attribute <Ref Attr=\"AutomorphismGroup\" \n Label=\"for a digraph,\ homogeneous list, and list\"/> returns the value of\n <C>BlissAutomorphism\ Group</C> as it is not implemented for &nauty; The \n requirements for the \ optional arguments <A>vertex_colours</A> and \n <A>edge_colours</A> are doc\ umented in <Ref Attr=\"AutomorphismGroup\" \n Label=\"for a digraph, homo\ geneous list, and list\"/>.\n\n See also <Ref Func=\"DigraphsUseBliss\"/>, \ and <Ref Func=\"DigraphsUseNauty\"/>.\n <P/>\n\n If the argument <A>digr\ aph</A> is mutable, then the return value of\n this attribute is recomputed\ every time it is called. \n\n <Example><![CDATA[\ngap> G := BlissAutomorph\ ismGroup(JohnsonDigraph(5, 2));;\ngap> IsSymmetricGroup(G);\ntrue\ngap> Size(G\ );\n120]]></Example>\n </Description>\n </ManSection>\n\n <ManSection>\ \n <Attr Name=\"NautyAutomorphismGroup\" Arg=\"digraph[, vert_colours]\"/>\n \ <Returns>A permutation group.</Returns>\n <Description>\n If <A>digraph</\ A> is a digraph, then this attribute contains the group of\n automorphisms \ of <A>digraph</A> as calculated using &nauty; by\n Brendan Mckay and Adolfo\ Piperno via &NautyTracesInterface;.\n\n <!-- FIXME be more explicit about \ what the second arg is -->\n\n The attribute <Ref Attr=\"AutomorphismGroup\ \" Label=\"for a digraph\"/> and\n operation <Ref Oper=\"AutomorphismGroup\ \" \n Label=\"for a digraph and a homogeneous list\"/> returns\n the v\ alue of either <C>NautyAutomorphismGroup</C> or <Ref\n Attr=\"BlissAutom\ orphismGroup\" Label=\"for a digraph\"/>.\n These groups are, of course, eq\ ual but their generating sets may differ.<P/>\n\n See also <Ref Func=\"Digr\ aphsUseBliss\"/>, and <Ref Func=\"DigraphsUseNauty\"/>.\n <P/>\n\n If th\ e argument <A>digraph</A> is mutable, then the return value of\n this attri\ bute is recomputed every time it is called. \n\n <Log><![CDATA[\ngap> Nauty\ AutomorphismGroup(JohnsonDigraph(5, 2));\nGroup([ (3,4)(6,7)(8,9), (2,3)(5,6)(\ 9,10), (2,5)(3,6)(4,7),\n (1,2)(6,8)(7,9) ])]]></Log>\n </Description>\n <\ /ManSection>\n\n <ManSection>\n <Oper Name=\"AutomorphismGroup\" Label=\"f\ or a digraph and a homogeneous list\"\n Arg=\"digraph, vert_colours\"/>\n \ <Returns>A permutation group.</Returns>\n <Description>\n\n This operation\ computes the automorphism group of a vertex-coloured digraph.\n A vertex-c\ oloured digraph can be specified by its underlying digraph\n <A>digraph</A>\ and its colouring <A>vert_colours</A>. Let <C>n</C> be the\n number of ver\ tices of <A>digraph</A>. The colouring <A>vert_colours</A> may\n have one o\ f the following two forms:\n\n <List>\n <Item>\n a list of <C>n\ </C> integers, where <A>vert_colours</A><C>[i]</C> is the\n colour of v\ ertex <C>i</C>, using the colours <C>[1 .. m]</C> for some\n <C>m <=\ n</C>; or\n </Item>\n <Item>\n a list of non-empty disjoint \ lists whose union is\n <C>DigraphVertices(<A>digraph</A>)</C>, such tha\ t\n <A>vert_colours</A><C>[i]</C> is the list of all vertices with colo\ ur\n <C>i</C>.\n </Item>\n </List>\n\n The <E>automorphism g\ roup</E> of a coloured digraph <A>digraph</A> with\n colouring <A>vert_colo\ urs</A> is the group consisting of its automorphisms; an\n <E>automorphism<\ /E> of <A>digraph</A> is an isomorphism of coloured\n digraphs from <A>digr\ aph</A> to itself. This group is equal to the\n subgroup of <C>Automorphis\ mGroup(<A>digraph</A>)</C> consisting of those\n automorphisms that preserv\ e the colouring specified by <A>vert_colours</A>. See\n <Ref Attr=\"Automor\ phismGroup\" Label=\"for a digraph\" />, and see <Ref\n Oper=\"Isomorphis\ mDigraphs\" Label=\"for digraphs and homogeneous lists\" />\n for more info\ rmation about isomorphisms of coloured digraphs. <P/>\n\n If <A>digraph</A>\ is not a multidigraph then the automorphism group is\n returned as a group\ of permutations on the set of vertices of\n <A>digraph</A>. <P/>\n\n If\ <A>digraph</A> is a multidigraph then the automorphism group is returned\n \ as the direct product of a group of permutations on the set of vertices of\n \ <A>digraph</A> with a group of permutations on the set of edges of\n <A>\ digraph</A>. These groups can be accessed using <Ref Oper=\"Projection\"\n \ Label=\"for a domain and a positive integer\" BookName=\"ref\"/> on the\n \ returned group.<P/>\n\n By default, the automorphism group is found using \ &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterface; is\ available, then &nauty; by\n Brendan Mckay and Adolfo Piperno can be used \ instead; see\n <Ref Attr=\"BlissAutomorphismGroup\"\n Label=\"for a di\ graph and homogenous list\"/>,\n <Ref Attr=\"NautyAutomorphismGroup\"/>, <R\ ef Func=\"DigraphsUseBliss\"/>,\n and <Ref Func=\"DigraphsUseNauty\"/>.\n\n\ <Example><![CDATA[\ngap> cycle := CycleDigraph(9);\n<immutable cycle digraph\ with 9 vertices>\ngap> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and \ Size(G) = 9;\ntrue\ngap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> \ H := AutomorphismGroup(cycle, colours);;\ngap> Size(H);\n3\ngap> H = Automorph\ ismGroup(cycle, [1, 2, 3, 1, 2, 3, 1, 2, 3]);\ntrue\ngap> H = SubgroupByProper\ ty(G, p -> OnTuplesSets(colours, p) = colours);\ntrue\ngap> IsTrivial(Automorp\ hismGroup(cycle, [1, 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"AutomorphismGroup\"\ \n Label=\"for a digraph, homogeneous list, and list\"\n Arg=\"digraph, \ vert_colours, edge_colours\"/>\n <Returns>A permutation group.</Returns>\n <\ Description>\n\n This operation computes the automorphism group of a vertex\ - and/or\n edge-coloured digraph.\n A coloured digraph can be specified \ by its underlying digraph\n <A>digraph</A> and colourings <A>vert_colours</\ A>, <A>edge_colours</A>.\n Let <C>n</C> be the number of vertices of <A>dig\ raph</A>. The colourings\n must have the following forms:\n\n <List>\n \ <Item>\n <A>vert_colours</A> must be <K>fail</K> or a list of <C>n<\ /C> integers,\n where <A>vert_colours</A><C>[i]</C> is the colour of ve\ rtex <C>i</C>,\n using the colours <C>[1 .. m]</C> for some <C>m <= \ n</C>;\n </Item>\n <Item>\n <A>edge_colours</A> must be <K>fa\ il</K> or a list of <C>n</C> lists of\n integers of the same shape as <\ C>OutNeighbours(digraph)</C>, where\n <A>edge_colours</A><C>[i][j]</C> \ is the colour of the edge\n <C>OutNeighbours(digraph)[i][j]</C>, using \ the colours <C>[1 .. k]</C>\n for some <C>k <= n</C>;\n </Item>\ \n </List>\n\n Giving <A>vert_colours</A> [<A>edge_colours</A>] as <C>fa\ il</C> is\n equivalent to setting all vertices [edges] to be the same colou\ r. <P/>\n\n Unlike <Ref Attr=\"AutomorphismGroup\" Label=\"for a digraph\"/\ >, it is\n possible to obtain the automorphism group of an edge-coloured mu\ ltidigraph\n (see <Ref Prop=\"IsMultiDigraph\" />) when no two edges share \ the same\n source, range, and colour.\n\n The <E>automorphism group</E> \ of a vertex/edge-coloured digraph\n <A>digraph</A> with colouring <A>c</A> \ is the group consisting of its\n vertex/edge-colour preserving automorphism\ s; an\n <E>automorphism</E> of <A>digraph</A> is an isomorphism of vertex/e\ dge-coloured\n digraphs from <A>digraph</A> to itself. This group is equal\ to the\n subgroup of <C>AutomorphismGroup(<A>digraph</A>)</C> consisting o\ f those\n automorphisms that preserve the colouring specified by <A>colours\ </A>. See\n <Ref Attr=\"AutomorphismGroup\" Label=\"for a digraph\" />, and\ see <Ref\n Oper=\"IsomorphismDigraphs\" Label=\"for digraphs and homogen\ eous lists\" />\n for more information about isomorphisms of coloured digra\ phs. <P/>\n\n If <A>digraph</A> is not a multidigraph then the automorphism\ group is\n returned as a group of permutations on the set of vertices of\n\ <A>digraph</A>. <P/>\n\n If <A>digraph</A> is a multidigraph then the a\ utomorphism group is returned\n as the direct product of a group of permuta\ tions on the set of vertices of\n <A>digraph</A> with a group of permutatio\ ns on the set of edges of\n <A>digraph</A>. These groups can be accessed us\ ing <Ref Oper=\"Projection\"\n Label=\"for a domain and a positive intege\ r\" BookName=\"ref\"/> on the\n returned group.<P/>\n\n By default, the \ automorphism group is found using &bliss; by Tommi Junttila\n and Petteri K\ aski. If &NautyTracesInterface; is available, then &nauty; by\n Brendan Mck\ ay and Adolfo Piperno can be used instead; see\n <Ref Attr=\"BlissAutomorp\ hismGroup\"\n Label=\"for a digraph, homogenous list, and list\"/>,\n \ <Ref Attr=\"NautyAutomorphismGroup\"/>, <Ref Func=\"DigraphsUseBliss\"/>, an\ d\n <Ref Func=\"DigraphsUseNauty\"/>.\n\n <Example><![CDATA[\ngap> cycle \ := CycleDigraph(12);\n<immutable cycle digraph with 12 vertices>\ngap> vert_co\ lours := List([1 .. 12], x -> x mod 3 + 1);;\ngap> edge_colours := List([1 .. \ 12], x -> [x mod 2 + 1]);;\ngap> Size(AutomorphismGroup(cycle));\n12\ngap> Siz\ e(AutomorphismGroup(cycle, vert_colours));\n4\ngap> Size(AutomorphismGroup(cyc\ le, fail, edge_colours));\n6\ngap> Size(AutomorphismGroup(cycle, vert_colours,\ edge_colours));\n2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> vert_colours, \ List([1 .. 12], x -> [x mod 4 + 1])));\ntrue\n]]></Example>\n </Description>\ \n</ManSection>\n\n <ManSection>\n <Attr Name=\"BlissCanonicalLabelling\" \ Label=\"for a digraph\" Arg=\"digraph\"/>\n <Attr Name=\"NautyCanonicalLabell\ ing\" Label=\"for a digraph\" Arg=\"digraph\"/>\n <Returns>A permutation, or \ a list of two permutations.</Returns>\n <Description>\n A function <M>\\rh\ o</M> that maps a digraph to a digraph is a <E>canonical\n representative\ map</E> if the following two conditions hold for all\n digraphs <M>G</M> a\ nd <M>H</M>: <P/>\n\n <List>\n <Item>\n <M>\\rho(G)</M> and <M>\ G</M> are isomorphic as digraphs; and\n </Item>\n <Item>\n <M\ >\\rho(G)=\\rho(H)</M> if and only if <M>G</M> and <M>H</M> are\n isomo\ rphic as digraphs.\n </Item>\n </List>\n\n A <E>canonical labelling\ </E> of a digraph <M>G</M> (under <M>\\rho</M>) is an\n isomorphism of <M>G\ </M> onto its <E>canonical representative</E>,\n <M>\\rho(G)</M>. See <Ref\ Oper=\"IsomorphismDigraphs\" Label=\"for digraphs\" />\n for more informat\ ion about isomorphisms of digraphs. <P/>\n\n <C>BlissCanonicalLabelling</C>\ returns a canonical labelling of the digraph\n <A>digraph</A> found using \ &bliss; by Tommi Junttila and Petteri Kaski.\n <C>NautyCanonicalLabelling</\ C> returns a canonical labelling of the digraph\n <A>digraph</A> found usin\ g &nauty; by Brendan McKay and Adolfo Piperno.\n Note that the canonical la\ bellings returned by &bliss; and &nauty; are not\n usually the same (and ma\ y depend of the version used).<P/>\n\n <C>BlissCanonicalLabelling</C> can o\ nly be computed if <A>digraph</A> has\n no multiple edges; see <Ref Prop=\"\ IsMultiDigraph\" />. <P/>\n\n <Example><![CDATA[\ngap> digraph1 := DigraphF\ romDiSparse6String(\".ImNS_AiB?qRN\");\n<immutable digraph with 10 vertices, 8\ edges>\ngap> BlissCanonicalLabelling(digraph1);\n(1,9,5,7)(3,6,4,10)\ngap> p \ := (1, 2, 7, 5)(3, 9)(6, 10, 8);;\ngap> digraph2 := OnDigraphs(digraph1, p);\n\ <immutable digraph with 10 vertices, 8 edges>\ngap> digraph1 = digraph2;\nfals\ e\ngap> OnDigraphs(digraph1, BlissCanonicalLabelling(digraph1)) =\n> OnDigr\ aphs(digraph2, BlissCanonicalLabelling(digraph2));\ntrue]]></Example>\n </Des\ cription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"BlissCanonicalLab\ elling\" Label=\"for a digraph and a list\"\n Arg=\"digraph, colours\"/>\n \ <Oper Name=\"NautyCanonicalLabelling\" Label=\"for a digraph and a list\"\n \ Arg=\"digraph, colours\"/>\n <Returns>A permutation.</Returns>\n <Descript\ ion>\n A function <M>\\rho</M> that maps a coloured digraph to a coloured d\ igraph is\n a <E>canonical representative map</E> if the following two cond\ itions hold\n for all coloured digraphs <M>G</M> and <M>H</M>:\n\n <List\ >\n <Item>\n <M>\\rho(G)</M> and <M>G</M> are isomorphic as colour\ ed digraphs; and\n </Item>\n <Item>\n <M>\\rho(G)=\\rho(H)</M\ > if and only if <M>G</M> and <M>H</M> are\n isomorphic as coloured dig\ raphs.\n </Item>\n </List>\n\n A <E>canonical labelling</E> of a co\ loured digraph <M>G</M> (under\n <M>\\rho</M>) is an isomorphism of <M>G</M\ > onto its <E>canonical\n representative</E>, <M>\\rho(G)</M>. See <Ref \ Oper=\"IsomorphismDigraphs\"\n Label=\"for digraphs and homogeneous lists\ \" /> for more information about\n isomorphisms of coloured digraphs. <P/>\ \n\n A coloured digraph can be specified by its underlying digraph <A>digra\ ph</A>\n and its colouring <A>colours</A>. Let <C>n</C> be the number of v\ ertices of\n <A>digraph</A>. The colouring <A>colours</A> may have one of t\ he following\n two forms: <P/>\n\n <List>\n <Item>\n a list \ of <C>n</C> integers, where <A>colours</A><C>[i]</C> is the\n colour of\ vertex <C>i</C>, using the colours <C>[1 .. m]</C> for some\n <C>m <\ ;= n</C>; or\n </Item>\n <Item>\n a list of non-empty disjoin\ t lists whose union is\n <C>DigraphVertices(<A>digraph</A>)</C>, such t\ hat\n <A>colours</A><C>[i]</C> is the list of all vertices with colour\ \n <C>i</C>.\n </Item>\n </List>\n\n If <A>digraph</A> and <\ A>colours</A> together form a coloured digraph,\n <C>BlissCanonicalLabellin\ g</C> returns a canonical labelling of the digraph\n <A>digraph</A> found u\ sing &bliss; by Tommi Junttila and Petteri Kaski.\n Similarly, <C>NautyCano\ nicalLabelling</C> returns a canonical labelling of\n the digraph <A>digrap\ h</A> found using &nauty; by Brendan McKay and Adolfo\n Piperno. Note that\ the canonical labellings returned by &bliss; and\n &nauty; are not usually\ the same (and may depend of the version used).<P/>\n\n <C>BlissCanonicalLa\ belling</C> can only be computed if <A>digraph</A> has\n no multiple edges;\ see <Ref Prop=\"IsMultiDigraph\" />. The canonical\n labelling of <A>digra\ ph</A> is given as a permutation of its vertices.\n The canonical represent\ ative of <A>digraph</A> can be created from\n <A>digraph</A> and its canoni\ cal labelling <C>p</C> by using the\n operation <Ref Oper=\"OnDigraphs\" La\ bel=\"for a digraph and a perm\" />:\n <Log>gap> OnDigraphs(digraph, p);</L\ og>\n\n The colouring of the canonical representative can easily be\n co\ nstructed. A vertex <C>v</C> (in <A>digraph</A>) has colour <C>i</C> if\n a\ nd only if the vertex <C>v ^ p</C> (in the canonical representative) has\n \ colour <C>i</C>, where <C>p</C> is the permutation of the canonical\n label\ ling that acts on the vertices of <A>digraph</A>. In particular, if\n <A>co\ lours</A> has the first form that is described above, then the\n colouring \ of the canonical representative is given by:\n\n <Log>gap> List(DigraphVert\ ices(digraph), i -> colours[i / p]);</Log>\n\n On the other hand, if <A>col\ ours</A> has the second form above, then the\n canonical representative has\ colouring:\n\n <Log>gap> OnTuplesSets(colours, p);</Log>\n\n <P/>\n\n \ <Example><![CDATA[\ngap> digraph := DigraphFromDiSparse6String(\".ImNS_AiB?q\ RN\");\n<immutable digraph with 10 vertices, 8 edges>\ngap> colours := [[1, 2,\ 8, 9, 10], [3, 4, 5, 6, 7]];;\ngap> p := BlissCanonicalLabelling(digraph, col\ ours);\n(1,5,8,4,10,3,9)(6,7)\ngap> OnDigraphs(digraph, p);\n<immutable digrap\ h with 10 vertices, 8 edges>\ngap> OnTuplesSets(colours, p);\n[ [ 1, 2, 3, 4, \ 5 ], [ 6, 7, 8, 9, 10 ] ]\ngap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ng\ ap> p := BlissCanonicalLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\nga\ p> OnDigraphs(digraph, p);\n<immutable digraph with 10 vertices, 8 edges>\ngap\ > List(DigraphVertices(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2,\ 2, 3, 3 ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n\ <Attr Name=\"BlissCanonicalDigraph\" Arg=\"digraph\"/>\n <Attr Name=\"Nauty\ CanonicalDigraph\" Arg=\"digraph\"/>\n <Returns>A digraph.</Returns>\n <Desc\ ription>\n The attribute <C>BlissCanonicalLabelling</C> returns the canonic\ al\n representative found by applying <Ref Attr=\"BlissCanonicalLabelling\"\ \n Label=\"for a digraph\"/>. The digraph returned is canonical in the se\ nse\n that\n\n <List>\n <Item>\n <C>BlissCanonicalDigraph(<A\ >digraph</A>)</C> and <A>digraph</A> are\n isomorphic as digraphs; and\ \n </Item>\n <Item>\n If <C>gr</C> is any digraph then <C>Bli\ ssCanonicalDigraph(gr)</C>\n and <C>BlissCanonicalDigraph(<A>digraph</A\ >)</C> are equal if and only\n if <C>gr</C> and <A>digraph</A> are isom\ orphic as digraphs.\n </Item>\n </List>\n\n Analogously, the attrib\ ute <C>NautyCanonicalLabelling</C> returns the\n canonical representative f\ ound by applying <Ref\n Attr=\"NautyCanonicalLabelling\" Label=\"for a di\ graph\"/>.\n <P/>\n\n If the argument <A>digraph</A> is mutable, then th\ e return value of\n this attribute is recomputed every time it is called. \ \n\n <Example><![CDATA[\ngap> digraph := Digraph([[1], [2, 3], [3], [1, 2, \ 3]]);\n<immutable digraph with 4 vertices, 7 edges>\ngap> canon := BlissCanoni\ calDigraph(digraph);\n<immutable digraph with 4 vertices, 7 edges>\ngap> OutNe\ ighbours(canon);\n[ [ 1 ], [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]\n]]></Example>\n </\ Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"DigraphGroup\"\ Arg=\"digraph\"/>\n <Returns>A permutation group.</Returns>\n <Description>\ \n If <A>digraph</A> is immutable and was created knowing a subgroup of\n \ its automorphism group, then this group is stored in the attribute\n <C>D\ igraphGroup</C>. If <A>digraph</A> is mutable, or was not created\n knowing\ a subgroup of its automorphism group, then <C>DigraphGroup</C>\n returns t\ he entire automorphism group of <A>digraph</A>. \n Note that if <A>digra\ ph</A> is mutable, then the automorphism group is\n recomputed every time t\ his function is called.\n <P/>\n\n Note that certain other constructor o\ perations such as <Ref\n Oper=\"CayleyDigraph\"/>, <Ref Oper=\"Bipartite\ DoubleDigraph\"/>, and <Ref\n Oper=\"DoubleDigraph\"/>, may not require a\ group as one of the arguments,\n but use the standard constructor method u\ sing a group, and hence set the\n <C>DigraphGroup</C> attribute for the res\ ulting digraph.\n\n <Example><![CDATA[\ngap> n := 4;;\ngap> adj := function\ (x, y)\n> return (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);\n> \ end;;\ngap> group := CyclicGroup(IsPermGroup, n);\nGroup([ (1,2,3,4) ])\ngap\ > D := Digraph(IsMutableDigraph, group, [1 .. n], \\^, adj);\n<mutable digraph\ with 4 vertices, 8 edges>\ngap> HasDigraphGroup(D);\nfalse\ngap> DigraphGroup\ (D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> AutomorphismGroup(D);\nGroup([ (2,4),\ (1,2)(3,4) ])\ngap> D := Digraph(group, [1 .. n], \\^, adj);\n<immutable digr\ aph with 4 vertices, 8 edges>\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGro\ up(D);\nGroup([ (1,2,3,4) ])\ngap> D := DoubleDigraph(D);\n<immutable digraph \ with 8 vertices, 32 edges>\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGroup(\ D);\nGroup([ (1,2,3,4)(5,6,7,8), (1,5)(2,6)(3,7)(4,8) ])\ngap> AutomorphismGro\ up(D) =\n> Group([(6, 8), (5, 7), (4, 6), (3, 5), (2, 4),\n> (1, 2)(3, \ 4)(5, 6)(7, 8)]);\ntrue\ngap> D := Digraph([[2, 3], [], []]);\n<immutable digr\ aph with 3 vertices, 2 edges>\ngap> HasDigraphGroup(D);\nfalse\ngap> HasAutomo\ rphismGroup(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (2,3) ])\ngap> HasAutom\ orphismGroup(D);\ntrue\ngap> group := DihedralGroup(8);\n<pc group of size 8 w\ ith 3 generators>\ngap> D := CayleyDigraph(group);\n<immutable digraph with 8 \ vertices, 24 edges>\ngap> HasDigraphGroup(D);\ntrue\ngap> GeneratorsOfGroup(Di\ graphGroup(D));\n[ (1,2)(3,5)(4,6)(7,8), (1,7,4,3)(2,5,6,8), (1,4)(2,6)(3,7)(5\ ,8) ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <At\ tr Name=\"DigraphOrbits\" Arg=\"digraph\"/>\n <Returns>\n An immutable lis\ t of lists of integers.\n </Returns>\n <Description>\n <C>DigraphOrbits</\ C> returns the orbits of the action of the\n <Ref Attr=\"DigraphGroup\"/> o\ n the set of vertices of <A>digraph</A>.\n\n <Example><![CDATA[\ngap> G := \ Group([(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := EdgeOrbitsDigr\ aph(G, [1, 2]);\n<immutable digraph with 9 vertices, 6 edges>\ngap> DigraphOrb\ its(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := DigraphMutableCo\ py(D);\n<mutable digraph with 9 vertices, 6 edges>\ngap> DigraphOrbits(D);\n[ \ [ 1, 2, 3 ], [ 4, 5, 6, 7, 8, 9 ] ]\n]]></Example>\n </Description>\n</ManSec\ tion>\n\n <ManSection>\n <Attr Name=\"DigraphOrbitReps\" Arg=\"digraph\"/>\ \n <Returns>\n An immutable list of integers.\n </Returns>\n <Descriptio\ n>\n <C>DigraphOrbitReps</C> returns a list of orbit representatives\n o\ f the action of the <Ref Attr=\"DigraphGroup\"/> on the set of vertices of\n \ <A>digraph</A>.\n\n <Example><![CDATA[\ngap> D := CayleyDigraph(Alternati\ ngGroup(4));\n<immutable digraph with 12 vertices, 24 edges>\ngap> DigraphOrbi\ tReps(D);\n[ 1 ]\ngap> D := DigraphMutableCopy(D);\n<mutable digraph with 12 v\ ertices, 24 edges>\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphFromDig\ raph6String(\"&IGO??S?`?_@?a?CK?O\");\n<immutable digraph with 10 vertices, 14\ edges>\ngap> DigraphOrbitReps(D);\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> Di\ graphOrbitReps(DigraphMutableCopy(D));\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n]]>\ </Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\ \"DigraphSchreierVector\" Arg=\"digraph\"/>\n <Returns>\n An immutable lis\ t of integers.\n </Returns>\n <Description>\n <C>DigraphSchreierVector</C\ > returns the so-called <E>Schreier vector</E>\n of the action of the <Ref \ Attr=\"DigraphGroup\"/> on the set of vertices of\n <A>digraph</A>. The Sch\ reier vector is a list <C>sch</C> of integers with\n length <C>DigraphNrVer\ tices(<A>digraph</A>)</C> where:\n <List>\n <Mark><C>sch[i] < 0:</C\ ></Mark>\n <Item>\n implies that <C>i</C> is an orbit representati\ ve and\n <C>DigraphOrbitReps(<A>digraph</A>)[-sch[i]] = i</C>.\n <\ /Item>\n\n <Mark><C>sch[i] > 0:</C></Mark>\n <Item>\n impl\ ies that <C>i / gens[sch[i]]</C>\n is one step closer to the root (or r\ epresentative) of the tree, where\n <C>gens</C> is the generators of <C\ >DigraphGroup(<A>digraph</A>)</C>.\n </Item>\n </List>\n\n <Example\ ><![CDATA[\ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y)\ mod n) = 1) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGr\ oup(IsPermGroup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph\ , group, [1 .. n], \\^, adj);\n<mutable digraph with 4 vertices, 8 edges>\ngap\ > sch := DigraphSchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(A\ lternatingGroup(4));\n<immutable digraph with 12 vertices, 24 edges>\ngap> sch\ := DigraphSchreierVector(D);\n[ -1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1 ]\ngap> D\ igraphOrbitReps(D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n\ [ (1,7,5)(2,10,9)(3,4,11)(6,8,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12) ]\ngap> 10 \ / gens[sch[10]];\n2\ngap> 7 / gens[sch[7]];\n1\ngap> 5 / gens[sch[5]];\n7]]></\ Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"\ DigraphStabilizer\" Arg=\"digraph, v\"/>\n <Returns>\n A permutation group\ .\n </Returns>\n <Description>\n <C>DigraphStabilizer</C> returns the sta\ bilizer of the vertex <A>v</A>\n under of the action of the <Ref Attr=\"Dig\ raphGroup\"/> on the set of\n vertices of <A>digraph</A>.\n\n <Example><\ ![CDATA[\ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n<immutable \ digraph with 8 vertices, 24 edges>\ngap> DigraphStabilizer(D, 8);\nGroup(())\n\ gap> DigraphStabilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n<m\ utable digraph with 8 vertices, 24 edges>\ngap> DigraphStabilizer(D, 8);\nGrou\ p(())\ngap> DigraphStabilizer(D, 2);\nGroup(())\n]]></Example>\n </Descriptio\ n>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"IsIsomorphicDigraph\" La\ bel=\"for digraphs\" Arg=\"digraph1, digraph2\"/>\n <Returns><K>true</K> or <\ K>false</K>.</Returns>\n <Description>\n This operation returns <K>true</K\ > if there exists an isomorphism from the\n digraph <A>digraph1</A> to the \ digraph <A>digraph2</A>. See <Ref\n Oper=\"IsomorphismDigraphs\" Label=\ \"for digraphs\" /> for more information\n about isomorphisms of digraphs. \ <P/>\n\n By default, an isomorphism is found using the canonical labellings\ of the\n digraphs obtained from &bliss; by Tommi Junttila\n and Petteri\ Kaski. If &NautyTracesInterface; is available, then &nauty; by\n Brendan M\ ckay and Adolfo Piperno can be used instead; see\n <Ref Func=\"DigraphsUseB\ liss\"/>, and <Ref Func=\"DigraphsUseNauty\"/>.\n\n <Example><![CDATA[\ngap\ > digraph1 := CycleDigraph(4);\n<immutable cycle digraph with 4 vertices>\ngap\ > digraph2 := CycleDigraph(5);\n<immutable cycle digraph with 5 vertices>\ngap\ > IsIsomorphicDigraph(digraph1, digraph2);\nfalse\ngap> digraph2 := DigraphRev\ erse(digraph1);\n<immutable digraph with 4 vertices, 4 edges>\ngap> IsIsomorph\ icDigraph(digraph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], []]);\ \n<immutable digraph with 3 vertices, 1 edge>\ngap> digraph2 := Digraph([[], [\ ], [2]]);\n<immutable digraph with 3 vertices, 1 edge>\ngap> IsIsomorphicDigra\ ph(digraph1, digraph2);\ntrue]]></Example>\n </Description>\n</ManSection>\n\ \n <ManSection>\n <Oper Name=\"IsIsomorphicDigraph\" Label=\"for digraphs \ and homogeneous lists\"\n Arg=\"digraph1, digraph2, colours1, colours2\"/>\ \n <Returns><K>true</K> or <K>false</K>.</Returns>\n <Description>\n This\ operation tests for isomorphism of coloured digraphs. A coloured\n digrap\ h can be specified by its underlying digraph <A>digraph1</A> and its\n colo\ uring <A>colours1</A>. Let <C>n</C> be the number of vertices of\n <A>digr\ aph1</A>. The colouring <A>colours1</A> may have one of the following\n two\ forms:\n\n <List>\n <Item>\n a list of <C>n</C> integers, wher\ e <A>colours</A><C>[i]</C> is the\n colour of vertex <C>i</C>, using th\ e colours <C>[1 .. m]</C> for some\n <C>m <= n</C>; or\n </Item\ >\n <Item>\n a list of non-empty disjoint lists whose union is\n \ <C>DigraphVertices(<A>digraph</A>)</C>, such that\n <A>colours</A\ ><C>[i]</C> is the list of all vertices with colour\n <C>i</C>.\n \ </Item>\n </List>\n\n If <A>digraph1</A> and <A>digraph2</A> are digraph\ s without multiple edges,\n and <A>colours1</A> and <A>colours2</A> are col\ ourings of <A>digraph1</A>\n and <A>digraph2</A>, respectively, then this o\ peration returns <K>true</K>\n if there exists an isomorphism between these\ two coloured digraphs. See\n <Ref Oper=\"IsomorphismDigraphs\" Label=\"fo\ r digraphs and homogeneous lists\"\n /> for more information about isomor\ phisms of coloured digraphs. <P/>\n\n By default, an isomorphism is found u\ sing the canonical labellings of the\n digraphs obtained from &bliss; by To\ mmi Junttila\n and Petteri Kaski. If &NautyTracesInterface; is available, t\ hen &nauty; by\n Brendan Mckay and Adolfo Piperno can be used instead; see\ \n <Ref Func=\"DigraphsUseBliss\"/>, and <Ref Func=\"DigraphsUseNauty\"/>.\ \n\n <Example><![CDATA[\ngap> digraph1 := ChainDigraph(4);\n<immutable chai\ n digraph with 4 vertices>\ngap> digraph2 := ChainDigraph(3);\n<immutable chai\ n digraph with 3 vertices>\ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [\ [1, 4], [2, 3]], [[1, 2], [3]]);\nfalse\ngap> digraph2 := DigraphReverse(digra\ ph1);\n<immutable digraph with 4 vertices, 3 edges>\ngap> IsIsomorphicDigraph(\ digraph1, digraph2,\n> [1, 1, 1, 1], [1, 1, 1, 1]);\ntrue\ngap> IsIsomorphicD\ igraph(digraph1, digraph2,\n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsom\ orphicDigraph(digraph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse]]></\ Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"\ IsomorphismDigraphs\" Label=\"for digraphs\" Arg=\"digraph1, digraph2\"/>\n <\ Returns> A permutation, or a pair of permutations, or <K>fail</K>.</Returns>\n\ <Description>\n This operation returns an isomorphism between the digraph\ s <A>digraph1</A>\n and <A>digraph2</A> if one exists, else this operation \ returns <K>fail</K>.\n <P/>\n\n An <E>isomorphism</E> from a digraph <A>\ digraph1</A> to a digraph\n <A>digraph2</A> is a bijection <C>p</C> from th\ e vertices of\n <A>digraph1</A> to the vertices of <A>digraph2</A> with the\ following\n property: for all vertices <C>i</C> and <C>j</C> of <A>digraph\ 1</A>,\n <C>[i, j]</C> is an edge of <A>digraph1</A> if and only if <C>[i ^\ p, j\n ^ p]</C> is an edge of <A>digraph2</A>. <P/>\n\n If there exis\ ts such an isomorphism, then this operation returns one.\n The form of this\ isomorphism is a permutation <C>p</C> of the vertices of\n <A>digraph1</A>\ such that <P/>\n\n <C>OnDigraphs(<A>digraph1</A>, p) = digraph2</C>.\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from &bliss; by Tommi Junttila\n and Petteri Kaski. If\ &NautyTracesInterface; is available, then &nauty; by\n Brendan Mckay and A\ dolfo Piperno can be used instead; see\n <Ref Func=\"DigraphsUseBliss\"/>, \ and <Ref Func=\"DigraphsUseNauty\"/>.\n\n <Example><![CDATA[\ngap> digraph1\ := CycleDigraph(4);\n<immutable cycle digraph with 4 vertices>\ngap> digraph2\ := CycleDigraph(5);\n<immutable cycle digraph with 5 vertices>\ngap> Isomorph\ ismDigraphs(digraph1, digraph2);\nfail\ngap> digraph1 := CompleteBipartiteDigr\ aph(10, 5);\n<immutable complete bipartite digraph with bicomponent sizes 10 a\ nd 5>\ngap> digraph2 := CompleteBipartiteDigraph(5, 10);\n<immutable complete \ bipartite digraph with bicomponent sizes 5 and 10>\ngap> p := IsomorphismDigra\ phs(digraph1, digraph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)\ngap> OnDi\ graphs(digraph1, p) = digraph2;\ntrue\n]]></Example>\n </Description>\n</ManS\ ection>\n\n <ManSection>\n <Oper Name=\"IsomorphismDigraphs\" Label=\"for \ digraphs and homogeneous lists\"\n Arg=\"digraph1, digraph2, colours1, colo\ urs2\"/>\n <Returns> A permutation, or <K>fail</K>.</Returns>\n <Description\ >\n This operation searches for an isomorphism between coloured digraphs. \ A\n coloured digraph can be specified by its underlying digraph <A>digraph1\ </A>\n and its colouring <A>colours1</A>. Let <C>n</C> be the number of ve\ rtices\n of <A>digraph1</A>. The colouring <A>colours1</A> may have one of \ the\n following two forms:\n\n <List>\n <Item>\n a list of <\ C>n</C> integers, where <A>colours</A><C>[i]</C> is the\n colour of ver\ tex <C>i</C>, using the colours <C>[1 .. m]</C> for some\n <C>m <= n\ </C>; or\n </Item>\n <Item>\n a list of non-empty disjoint li\ sts whose union is\n <C>DigraphVertices(<A>digraph</A>)</C>, such that\ \n <A>colours</A><C>[i]</C> is the list of all vertices with colour\n \ <C>i</C>.\n </Item>\n </List>\n\n An <E>isomorphism</E> betwe\ en coloured digraphs is an isomorphism between\n the underlying digraphs th\ at preserves the colourings. See <Ref\n Oper=\"IsomorphismDigraphs\" Lab\ el=\"for digraphs\"/> for more information\n about isomorphisms of digraphs\ . More precisely, let <C>f</C> be an\n isomorphism of digraphs from the dig\ raph <A>digraph1</A> (with colouring\n <A>colours1</A>) to the digraph <A>d\ igraph2</A> (with colouring\n <A>colours2</A>), and let <C>p</C> be the per\ mutation of the vertices of\n <A>digraph1</A> that corresponds to <C>f</C>.\ Then <C>f</C> preserves the\n colourings of <A>digraph1</A> and <A>digrap\ h2</A> – and hence is an\n isomorphism of coloured digraphs – if <C><A>\ colours1</A>[i] =\n <A>colours2</A>[i ^ p]</C> for all vertices <C>i</C> \ in <A>digraph1</A>.\n <P/>\n\n This operation returns such an isomorphis\ m if one exists, else it returns\n <K>fail</K>. <P/>\n\n By default, an \ isomorphism is found using the canonical labellings of the\n digraphs obtai\ ned from &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInte\ rface; is available, then &nauty; by\n Brendan Mckay and Adolfo Piperno can\ be used instead; see\n <Ref Func=\"DigraphsUseBliss\"/>, and <Ref Func=\"D\ igraphsUseNauty\"/>.\n\n <Example><![CDATA[\ngap> digraph1 := ChainDigraph(\ 4);\n<immutable chain digraph with 4 vertices>\ngap> digraph2 := ChainDigraph(\ 3);\n<immutable chain digraph with 3 vertices>\ngap> IsomorphismDigraphs(digra\ ph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], [3]]);\nfail\ngap> digraph2 := D\ igraphReverse(digraph1);\n<immutable digraph with 4 vertices, 3 edges>\ngap> c\ olours1 := [1, 1, 1, 1];;\ngap> colours2 := [1, 1, 1, 1];;\ngap> p := Isomorph\ ismDigraphs(digraph1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigra\ phs(digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> \ colours1[i ^ p]) = colours2;\ntrue\ngap> colours1 := [1, 1, 2, 2];;\ngap> colo\ urs2 := [2, 2, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colo\ urs1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\n\ gap> List(DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\n\ gap> IsomorphismDigraphs(digraph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\ \nfail]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <A\ ttr Name=\"RepresentativeOutNeighbours\" Arg=\"digraph\"/>\n <Returns>An immu\ table list of lists.</Returns>\n <Description>\n This function returns the\ list <C>out</C> of <E>out-neighbours</E> of each\n representative of the o\ rbits of the action of <Ref Attr=\"DigraphGroup\"/> on\n the vertex set of \ the digraph <A>digraph</A>. <P/>\n\n More specifically, if <C>reps</C> is t\ he list of orbit representatives,\n then a vertex <C>j</C> appears in <C>ou\ t[i]</C> each time\n there exists an edge with source <C>reps[i]</C> and ra\ nge <C>j</C> in\n <A>digraph</A>.\n <P/>\n\n If <Ref Attr=\"DigraphGr\ oup\"/> is trivial, then <Ref Attr=\"OutNeighbours\"/>\n is returned.\n\n \ <Example><![CDATA[\ngap> D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1\ , 2, 3, 5], [1, 2, 3, 4]]);\n<immutable digraph with 5 vertices, 16 edges>\nga\ p> DigraphGroup(D);\nGroup(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1\ , 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := Digr\ aph(IsMutableDigraph, [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2,\ 3, 4]]);\n<mutable digraph with 5 vertices, 16 edges>\ngap> DigraphGroup(D);\ \nGroup(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5\ ], [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := DigraphFromDigraph6Stri\ ng(\"&GYHPQgWTIIPW\");\n<immutable digraph with 8 vertices, 24 edges>\ngap> G \ := DigraphGroup(D);;\ngap> GeneratorsOfGroup(G);\n[ (1,2)(3,4)(5,6)(7,8), (1,3\ ,2,4)(5,7,6,8), (1,5)(2,6)(3,8)(4,7) ]\ngap> Set(RepresentativeOutNeighbours(D\ ), Set);\n[ [ 2, 3, 5 ] ]]]></Example>\n </Description>\n</ManSection>\n\n \ <ManSection>\n <Oper Name=\"IsDigraphIsomorphism\" Arg=\"src, ran, x\" \n \ Label=\"for digraphs and transformation or permutation\"/>\n <Oper Name=\ \"IsDigraphIsomorphism\" Arg=\"src, ran, x, col1, col2\"/>\n <Oper Name=\"IsD\ igraphAutomorphism\" Arg=\"digraph, x\" \n Label=\"for a digraph and a \ transformation or permutation\"/>\n <Oper Name=\"IsDigraphAutomorphism\" Arg=\ \"digraph, x, col\"/>\n <Returns><K>true</K> or <K>false</K>.</Returns>\n <D\ escription>\n <C>IsDigraphIsomorphism</C> returns <K>true</K> if the permut\ ation or\n transformation <A>x</A> is an isomorphism from the digraph <A>sr\ c</A> to\n the digraph <A>ran</A>.\n <P/>\n\n <C>IsDigraphAutomorphis\ m</C> returns <K>true</K> if the permutation or\n transformation <A>x</A> i\ s an automorphism of the digraph <A>digraph</A>.\n <P/>\n\n A permutatio\ n or transformation <A>x</A> is an <E>isomorphism</E> from a\n digraph <A>s\ rc</A> to a digraph <A>ran</A> if the following hold:\n <List>\n <Item\ >\n <A>x</A> is a bijection from the vertices of <A>src</A> to those of\ \n <A>ran</A>;\n </Item>\n <Item>\n <C>[u ^ <A>x</A>, \ v ^ <A>x</A>]</C> is an edge of\n <A>ran</A> if and only if <C>[u, v]</\ C> is an\n edge of <A>src</A>; and\n </Item>\n <Item>\n \ <A>x</A> fixes every <C>i</C> which is not a vertex of <A>src</A>.\n </\ Item>\n </List>\n See also <Ref Attr=\"AutomorphismGroup\" Label=\"for a\ digraph\"\n />.<P/>\n \n If <A>col1</A> and <A>col2</A>, or <A>col\ </A>, are given, then they must\n represent vertex colourings; see \n <R\ ef Oper=\"AutomorphismGroup\" Label=\"for a digraph and a homogeneous list\"/>\ \n for details of the permissible values for\n these arguments. The hom\ omorphism must then also have the property:\n \n <List>\n <Item>\n \ <C>col1[i] = col2[i ^ x]</C> for all vertices <C>i</C> of <A>src</A>,\n \ for <C>IsDigraphIsomorphism</C>. </Item>\n <Item>\n <C>col[i\ ] = col[i ^ x]</C> for all vertices <C>i</C> of <A>digraph</A>,\n for <\ C>IsDigraphAutomorphism</C>. </Item>\n </List>\n\n For some digraphs, it\ can be faster to use <C>IsDigraphAutomorphism</C>\n than to test membershi\ p in the automorphism group of <A>digraph</A>.\n\n <Example><![CDATA[\ngap>\ src := Digraph([[1], [1, 2], [1, 3]]);\n<immutable digraph with 3 vertices, 5\ edges>\ngap> IsDigraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigraphAut\ omorphism(src, (2, 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 1, \ 1]);\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ngap> I\ sDigraphAutomorphism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphism(sr\ c, (1, 4));\nfalse\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran := Di\ graph([[2, 1], [2], [2, 3]]);\n<immutable digraph with 3 vertices, 5 edges>\ng\ ap> IsDigraphIsomorphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(r\ an, src, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntrue\ng\ ap> IsDigraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nfalse\ \ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\nga\ p> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n]]></\ Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Nam\ e=\"IsDigraphColouring\" Arg=\"digraph, list\"/>\n <Oper Name=\"IsDigraphCo\ louring\" Arg=\"digraph, t\" \n Label=\"for a transformation\"/>\n <Re\ turns> <K>true</K> or <K>false</K>. </Returns>\n <Description>\n The o\ peration <C>IsDigraphColouring</C> verifies whether or not \n the list <A\ >list</A> describes a proper colouring of the digraph \n <A>digraph</A>. \ \n <P/>\n\n A list <A>list</A> describes a <E>proper colouring</E> o\ f a digraph\n <A>digraph</A> if <A>list</A> consists of positive integers\ , the length of\n <A>list</A> equals the number of vertices in <A>digraph\ </A>, and for any\n vertices <C>u, v</C> of <A>digraph</A> if <C>u</C> an\ d <C>v</C> are adjacent,\n then <C><A>list</A>[u] >< <A>list</A>[v]\ </C>.\n <P/>\n\n A transformation <A>t</A> describes a proper colour\ ing of a digraph\n <A>digraph</A>, if <C>ImageListOfTransformation(<A>t</\ A>,\n DigraphNrVertices(<A>digraph</A>))</C> is a proper colouring of \n \ <A>digraph</A>. <P/>\n\n See also <Ref Oper=\"IsDigraphHomomorphism\ \"/>.\n \n <Example><![CDATA[\ngap> D := JohnsonDigraph(5, 3);\n<imm\ utable symmetric digraph with 10 vertices, 60 edges>\ngap> IsDigraphColouring(\ D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);\ntrue\ngap> IsDigraphColouring(D, [1, 2, 3\ , 3, 2, 1, 2, 5, 6, 7]);\nfalse\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1,\ 2, 5, 6, -1]);\nfalse\ngap> IsDigraphColouring(D, [1, 2, 3]);\nfalse\ngap> Is\ DigraphColouring(D, IdentityTransformation);\ntrue\n]]></Example>\n </Descr\ iption>\n </ManSection>\n\n </Section>\n\n <Section><Heading>Homomorphisms \ of digraphs</Heading>\n\n The following methods exist to find homomorphisms\ between digraphs.\n If an argument to one of these methods is a digraph wi\ th multiple edges,\n then the multiplicity of edges will be ignored in orde\ r to perform the\n calculation; the digraph will be treated as if it has no\ multiple edges.\n\n <ManSection>\n <Func Name=\"HomomorphismDigraphsFinde\ r\" Arg=\"D1, D2, hook, user_param, max_results, hint, injective, image, parti\ al_map, colors1, colors2[, order, aut_grp]\"/>\n <Returns>The argument <A>use\ r_param</A>.</Returns>\n <Description>\n This function finds homomorphisms\ from the digraph <A>D1</A> to the digraph\n <A>D2</A> subject to the condi\ tions imposed by the other arguments as\n described below.<P/>\n\n If <C\ >f</C> and <C>g</C> are homomorphisms found by\n <C>HomomorphismDigraphsFin\ der</C>, then <C>f</C> cannot be obtained from\n <C>g</C> by right multiply\ ing by an automorphism of <A>D2</A> in\n <A>aut_grp</A>.\n\n <List>\n \ <Mark><A>hook</A></Mark>\n <Item>\n This argument should be a f\ unction or <K>fail</K>.<P/>\n\n If <A>hook</A> is a function, then it m\ ust have two arguments\n <A>user_param</A> (see below) and a transforma\ tion <C>t</C>. The\n function <C><A>hook</A>(<A>user_param</A>, t)</C> \ is called every time\n a new homomorphism <C>t</C> is found by\n \ <C>HomomorphismDigraphsFinder</C>. If the function returns <K>true</K>,\n \ then <C>HomomorphismDigraphsFinder</C> stops and does not find any\n \ further homomorphisms. This feature might be useful if you are\n sear\ ching for a homomorphism that satisfies some condition that you\n canno\ t specify via the other arguments to\n <C>HomomorphismDigraphsFinder</C\ >.\n <P/>\n\n If <A>hook</A> is <K>fail</K>, then a default func\ tion is used which\n simply adds every new homomorphism found by\n \ <C>HomomorphismDigraphsFinder</C> to <A>user_param</A>, which must be a\n \ mutable list in this case.\n </Item>\n\n <Mark><A>user_param</\ A></Mark>\n <Item>\n If <A>hook</A> is a function, then <A>user_pa\ ram</A> can be any &GAP;\n object. The object <A>user_param</A> is used\ as the first argument of\n the function <A>hook</A>. For example, <A>u\ ser_param</A> might be a\n transformation semigroup, and <C><A>hook</A>\ (<A>user_param</A>, t)</C>\n might set <A>user_param</A> to be the clos\ ure of <A>user_param</A> and\n <C>t</C>. <P/>\n\n If the value o\ f <A>hook</A> is <K>fail</K>, then the value of\n <A>user_param</A> mus\ t be a mutable list.\n </Item>\n\n <Mark><A>max_results</A></Mark>\n\ <Item>\n This argument should be a positive integer or <K>infinit\ y</K>.\n <C>HomomorphismDigraphsFinder</C> will return after it has fou\ nd\n <A>max_results</A> homomorphisms or the search is complete, whiche\ ver\n happens first.\n </Item>\n\n <Mark><A>hint</A></Mark>\n\ <Item>\n This argument should be a positive integer or <K>fail</K\ >. <P/>\n\n If <A>hint</A> is a positive integer, then only homorphisms\ of rank\n <A>hint</A> are found.<P/>\n\n If <A>hint</A> is <K>f\ ail</K>, then no restriction is put on the rank\n of homomorphisms foun\ d.\n </Item>\n\n <Mark><A>injective</A></Mark>\n <Item>\n \ This argument should be <C>0</C>, <C>1</C>, or <C>2</C>. If it is\n \ <C>2</C>, then only embeddings are found, if it is <C>1</C>, then only\n \ injective homomorphisms are found, and if it is <C>0</C> there are no\n \ restrictions imposed by this argument.\n <P/>\n\n For backwar\ ds compatibility, <A>injective</A> can also be <K>false</K>\n or <K>tru\ e</K> which correspond to the values <C>0</C> and <C>1</C>\n described \ in the previous paragraph, respectively. \n </Item>\n\n <Mark><A>ima\ ge</A></Mark>\n <Item>\n This argument should be a subset of the v\ ertices of the graph <A>D2</A>.\n <C>HomomorphismDigraphsFinder</C> onl\ y finds homomorphisms from\n <A>D1</A> to the subgraph of <A>D2</A> ind\ uced by the vertices\n <A>image</A>.\n </Item>\n\n <Mark><A>p\ artial_map</A></Mark>\n <Item>\n This argument should be a partial\ map from <A>D1</A> to <A>D2</A>,\n that is, a (not necessarily dense) \ list of vertices of the digraph\n <A>D2</A> of length no greater than t\ he number vertices in the digraph\n <A>D1</A>. <C>HomomorphismDigraphsF\ inder</C> only finds homomorphisms\n extending <A>partial_map</A> (if a\ ny).\n </Item>\n \n <Mark><A>colors1</A></Mark>\n <Item>\n\ This should be a list representing possible colours of vertices in the\ \n digraph <A>D1</A>; see \n <Ref Oper=\"AutomorphismGroup\" \n \ Label=\"for a digraph and a homogeneous list\"/> \n for detail\ s of the permissible values for this argument.\n </Item>\n \n <\ Mark><A>colors2</A></Mark>\n <Item>\n This should be a list repres\ enting possible colours of vertices in the\n digraph <A>D2</A>; see \n \ <Ref Oper=\"AutomorphismGroup\" \n Label=\"for a digraph and a\ homogeneous list\"/> \n for details of the permissible values for this\ argument.\n </Item>\n <Mark><A>order</A></Mark>\n <Item>\n \ The optional argument <A>order</A> specifies the order the\n vertic\ es in <A>D1</A> appear in the search for homomorphisms. \n The value of\ this parameter can have a large impact\n on the runtime of the functio\ n. It seems in many cases to be a good\n idea for this to be the <Ref A\ ttr=\"DigraphWelshPowellOrder\"/>, i.e.\n vertices ordered from highest\ to lowest degree.\n </Item>\n <Item>\n The optional argument\ <A>aut_grp</A> should be a subgroup of the\n automorphism group of <A>\ D2</A>. This function returns unique\n representatives of the homomorph\ isms found up to right multiplication\n by <A>aut_grp</A>. If this argu\ ment is not specific, it defaults to the\n full automorphism group of <\ A>D2</A>, which may be costly to calculate.\n </Item>\n </List>\n\n \ <Example><![CDATA[\ngap> D := ChainDigraph(10);\n<immutable chain digraph wit\ h 10 vertices>\ngap> D := DigraphSymmetricClosure(D);\n<immutable symmetric di\ graph with 10 vertices, 18 edges>\ngap> HomomorphismDigraphsFinder(D, D, fail,\ [], infinity, 2, 0,\n> [3, 4], [], fail, fail);\n[ Transformation( [ 3, 4, 3,\ 4, 3, 4, 3, 4, 3, 4 ] ), \n Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ]\ ) ]\ngap> D2 := CompleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, \ fail, [], 1, fail, 0,\n> [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformation( \ [ 1, 2, 1, 3, 4, 5, 6, 1, 2, 1 ] ) ]\ngap> func := function(user_param, t)\n> \ Add(user_param, t * user_param[1]);\n> end;;\ngap> HomomorphismDigraphsFinder(\ D, D2, func, [Transformation([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fai\ l, fail);\n[ Transformation( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, \ 5, 6, 2, 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n \ Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigrap\ hsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2\ , 3], fail, fail, fail, fail,\n> Group(()));\n[ IdentityTransformation, Transf\ ormation( [ 1, 3, 3 ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, \ 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\n\ gap> HomomorphismDigraphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], i\ nfinity, fail, 1, [1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ Ide\ ntityTransformation, Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1\ , 3 ] ) ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n \ <Oper Name=\"DigraphHomomorphism\" Arg=\"digraph1, digraph2\"/>\n <Returns> \ A transformation, or <K>fail</K>.</Returns>\n <Description>\n A homomorphi\ sm from <A>digraph1</A> to <A>digraph2</A> is a mapping from\n the vertex s\ et of <A>digraph1</A> to a subset of the vertices of\n <A>digraph2</A>, suc\ h that every pair of vertices <C>[i,j]</C> which has\n an edge <C>i->j</C> \ is mapped to a pair of vertices <C>[a,b]</C> which\n has an edge <C>a->b</C\ >. Note that non-adjacent vertices can still be\n mapped to adjacent verti\ ces. <P/>\n\n <C>DigraphHomomorphism</C> returns a single homomorphism betw\ een\n <A>digraph1</A> and <A>digraph2</A> if it exists, otherwise it return\ s\n <K>fail</K>.\n<Example><![CDATA[\ngap> gr1 := ChainDigraph(3);;\ngap> g\ r2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);\n<immutable digraph with 5 vert\ ices, 6 edges>\ngap> DigraphHomomorphism(gr1, gr1);\nIdentityTransformation\ng\ ap> map := DigraphHomomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\ \ngap> IsDigraphHomomorphism(gr1, gr2, map);\ntrue\n]]></Example>\n </Descrip\ tion>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"HomomorphismsDigraphs\ \" Arg=\"digraph1, digraph2\"/>\n <Oper Name=\"HomomorphismsDigraphsRepresent\ atives\" Arg=\"digraph1, digraph2\"/>\n <Returns> A list of transformations.<\ /Returns>\n <Description>\n <C>HomomorphismsDigraphsRepresentatives</C> fi\ nds every\n <Ref Oper=\"DigraphHomomorphism\"/> between <A>digraph1</A> and\ \n <A>digraph2</A>, up to right multiplication by an element of the\n <R\ ef Attr=\"AutomorphismGroup\" Label=\"for a digraph\"/> of <A>digraph2</A>.\n \ In other words, every homomorphism <C>f</C> between <A>digraph1</A> and\n \ <A>digraph2</A> can be written as the composition <C>f = g * x</C>,\n whe\ re <C>g</C> is one of the <C>HomomorphismsDigraphsRepresentatives</C>\n and\ <C>x</C> is an automorphism of <A>digraph2</A>.\n <P/>\n\n <C>Homomorph\ ismsDigraphs</C> returns all homomorphisms between\n <A>digraph1</A> and <A\ >digraph2</A>.\n <Example><![CDATA[\ngap> gr1 := ChainDigraph(3);;\ngap> gr\ 2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);\n<immutable digraph with 5 verti\ ces, 6 edges>\ngap> HomomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 3\ , 1 ] ), Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 1, 5, 4, 4, 5 ] \ ), Transformation( [ 2, 2, 2 ] ), \n Transformation( [ 3, 1, 3 ] ), Transform\ ation( [ 3, 1, 5, 4, 5 ] ), \n Transformation( [ 3, 3, 1 ] ), Transformation(\ [ 3, 3, 3 ] ) ]\ngap> HomomorphismsDigraphsRepresentatives(gr1, CompleteDigra\ ph(3));\n[ Transformation( [ 2, 1 ] ), Transformation( [ 2, 1, 2 ] ) ]\n]]></E\ xample>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"D\ igraphMonomorphism\" Arg=\"digraph1, digraph2\"/>\n <Returns>A transformation\ , or <K>fail</K>.</Returns>\n <Description>\n <C>DigraphMonomorphism</C> r\ eturns a single <E>injective</E> <Ref\n Oper=\"DigraphHomomorphism\"/> be\ tween <A>digraph1</A> and <A>digraph2</A>\n if one exists, otherwise it ret\ urns <K>fail</K>.\n <Example><![CDATA[\ngap> gr1 := ChainDigraph(3);;\ngap>\ gr2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);\n<immutable digraph with 5 ve\ rtices, 6 edges>\ngap> DigraphMonomorphism(gr1, gr1);\nIdentityTransformation\ \ngap> DigraphMonomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\n]]\ ></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name\ =\"MonomorphismsDigraphs\" Arg=\"digraph1, digraph2\"/>\n <Oper Name=\"Monomo\ rphismsDigraphsRepresentatives\" Arg=\"digraph1, digraph2\"/>\n <Returns>A li\ st of transformations.</Returns>\n <Description>\n These operations behave\ the same as <Ref Oper=\"HomomorphismsDigraphs\"/> and\n <Ref Oper=\"Homomo\ rphismsDigraphsRepresentatives\"/>, except they only return\n <E>injective<\ /E> homomorphisms.\n <Example><![CDATA[\ngap> gr1 := ChainDigraph(3);;\ngap\ > gr2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);\n<immutable digraph with 5 v\ ertices, 6 edges>\ngap> MonomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ \ 1, 5, 4, 4, 5 ] ), \n Transformation( [ 3, 1, 5, 4, 5 ] ) ]\ngap> Monomorphis\ msDigraphsRepresentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 \ ] ) ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <\ Oper Name=\"DigraphEpimorphism\" Arg=\"digraph1, digraph2\"/>\n <Returns>A tr\ ansformation, or <K>fail</K>.</Returns>\n <Description>\n <C>DigraphEpimor\ phism</C> returns a single <E>surjective</E> <Ref\n Oper=\"DigraphHomomor\ phism\"/> between <A>digraph1</A> and <A>digraph2</A>\n if one exists, othe\ rwise it returns <K>fail</K>.\n <Example><![CDATA[\ngap> gr1 := DigraphReve\ rse(ChainDigraph(4));\n<immutable digraph with 4 vertices, 3 edges>\ngap> gr2 \ := DigraphRemoveEdge(CompleteDigraph(3), [1, 2]);\n<immutable digraph with 3 v\ ertices, 5 edges>\ngap> DigraphEpimorphism(gr2, gr1);\nfail\ngap> DigraphEpimo\ rphism(gr1, gr2);\nTransformation( [ 3, 1, 2, 3 ] )\n]]></Example>\n </Descri\ ption>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"EpimorphismsDigraphs\ \" Arg=\"digraph1, digraph2\"/>\n <Oper Name=\"EpimorphismsDigraphsRepresenta\ tives\" Arg=\"digraph1, digraph2\"/>\n <Returns>A list of transformations.</R\ eturns>\n <Description>\n These operations behave the same as <Ref Oper=\"\ HomomorphismsDigraphs\"/> and\n <Ref Oper=\"HomomorphismsDigraphsRepresenta\ tives\"/>, except they only return\n <E>surjective</E> homomorphisms.\n \ <Example><![CDATA[\ngap> gr1 := DigraphReverse(ChainDigraph(4));\n<immutable d\ igraph with 4 vertices, 3 edges>\ngap> gr2 := DigraphSymmetricClosure(CycleDig\ raph(3));\n<immutable symmetric digraph with 3 vertices, 6 edges>\ngap> Epimor\ phismsDigraphsRepresentatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), \ Transformation( [ 3, 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap>\ EpimorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 2, 1, 3 ] ), Transfor\ mation( [ 1, 2, 3, 1 ] ), \n Transformation( [ 1, 2, 3, 2 ] ), Transformation\ ( [ 1, 3, 1, 2 ] ), \n Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1,\ 3, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3,\ 1 ] ), \n Transformation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] )\ , \n Transformation( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), \n \ Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), \n Transf\ ormation( [ 3, 1, 3, 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), \n Transformati\ on( [ 3, 2, 1, 3 ] ), Transformation( [ 3, 2, 3, 1 ] ) ]\n]]></Example>\n </D\ escription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphEmbeddin\ g\" Arg=\"digraph1, digraph2\"/>\n <Returns> A transformation, or <K>fail</K>\ .</Returns>\n <Description>\n An embedding of a digraph <A>digraph1</A> in\ to another digraph\n <A>digraph2</A> is a <Ref Oper=\"DigraphMonomorphism\"\ /> from <A>digraph1</A>\n to <A>digraph2</A> which has the additional prope\ rty that a pair of\n vertices <C>[i, j]</C> which have no edge <C>i -> j</C\ > in <A>digraph1</A> are\n\n mapped to a pair of vertices <C>[a, b]</C> whi\ ch have no edge <C>a->b</C> in\n <A>digraph2</A>.<P/>\n\n In other words\ , an embedding <C>t</C> is an isomorphism from\n <A>digraph1</A> to the <Re\ f Oper=\"InducedSubdigraph\"/> of <A>digraph2</A>\n on the image of <C>t</C\ >. <P/>\n\n <C>DigraphEmbedding</C> returns a single embedding if one exist\ s, otherwise\n it returns <K>fail</K>.\n <Example><![CDATA[\ngap> gr := \ ChainDigraph(3);\n<immutable chain digraph with 3 vertices>\ngap> DigraphEmbed\ ding(gr, CompleteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Digraph([[3], [\ 1, 4], [1], [3]]));\nTransformation( [ 2, 4, 3, 4 ] )\n]]></Example>\n </Desc\ ription>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"EmbeddingsDigraphs\ \" Arg=\"D1, D2\"/>\n <Oper Name=\"EmbeddingsDigraphsRepresentatives\" Arg=\"\ D1, D2\"/>\n <Returns>A list of transformations.</Returns>\n <Description>\n\ These operations behave the same as <Ref Oper=\"HomomorphismsDigraphs\"/> \ and\n <Ref Oper=\"HomomorphismsDigraphsRepresentatives\"/>, except they onl\ y return\n embeddings of <A>D1</A> into <A>D2</A>.<P/>\n\n See also <Ref\ Oper=\"IsDigraphEmbedding\"/>.\n <Example><![CDATA[\ngap> D1 := NullDigrap\ h(2);\n<immutable empty digraph with 2 vertices>\ngap> D2 := CycleDigraph(5);\ \n<immutable cycle digraph with 5 vertices>\ngap> EmbeddingsDigraphsRepresenta\ tives(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 \ ] ) ]\ngap> EmbeddingsDigraphs(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Tran\ sformation( [ 1, 4, 3, 4 ] ), \n Transformation( [ 2, 4, 4, 5, 1 ] ), \n Tra\ nsformation( [ 2, 5, 4, 5, 1 ] ), \n Transformation( [ 3, 1, 5, 1, 2 ] ), \n \ Transformation( [ 3, 5, 5, 1, 2 ] ), \n Transformation( [ 4, 1, 1, 2, 3 ] ),\ \n Transformation( [ 4, 2, 1, 2, 3 ] ), \n Transformation( [ 5, 2, 2, 3, 4 \ ] ), \n Transformation( [ 5, 3, 2, 3, 4 ] ) ]\n]]></Example>\n </Description\ >\n</ManSection>\n\n <ManSection>\n <Oper Name=\"IsDigraphHomomorphism\" A\ rg=\"src, ran, x\" \n Label=\"for digraphs and a permutation or transfo\ rmation\"/>\n <Oper Name=\"IsDigraphHomomorphism\" Arg=\"src, ran, x, col1, c\ ol2\"/>\n <Oper Name=\"IsDigraphEpimorphism\" Arg=\"src, ran, x\"\n La\ bel=\"for digraphs and a permutation or transformation\"/>\n <Oper Name=\"IsD\ igraphEpimorphism\" Arg=\"src, ran, x, col1, col2\"/>\n <Oper Name=\"IsDigrap\ hMonomorphism\" Arg=\"src, ran, x\"\n Label=\"for digraphs and a permut\ ation or transformation\"/>\n <Oper Name=\"IsDigraphMonomorphism\" Arg=\"src,\ ran, x, col1, col2\"/>\n <Oper Name=\"IsDigraphEndomorphism\" Arg=\"digraph,\ x\"\n Label=\"for digraphs and a permutation or transformation\"/>\n \ <Oper Name=\"IsDigraphEndomorphism\" Arg=\"digraph, x, col\"/>\n <Returns><K>\ true</K> or <K>false</K>.</Returns>\n <Description>\n <C>IsDigraphHomomorp\ hism</C> returns <K>true</K> if the permutation\n or transformation <A>x</A\ > is a homomorphism from the digraph\n <A>src</A> to the digraph <A>ran</A>\ .\n <P/>\n\n <C>IsDigraphEpimorphism</C> returns <K>true</K> if the perm\ utation\n or transformation <A>x</A> is a surjective homomorphism from the \ digraph\n <A>src</A> to the digraph <A>ran</A>.\n <P/>\n\n <C>IsDigra\ phMonomorphism</C> returns <K>true</K> if the permutation\n or transformati\ on <A>x</A> is an injective homomorphism from the digraph\n <A>src</A> to t\ he digraph <A>ran</A>.\n <P/>\n\n <C>IsDigraphEndomorphism</C> returns <\ K>true</K> if the permutation\n or transformation <A>x</A> is an endomorphi\ sm of the digraph\n <A>digraph</A>.\n <P/>\n\n A permutation or trans\ formation <A>x</A> is a <E>homomorphism</E> from a\n digraph <A>src</A> to \ a digraph <A>ran</A> if the following hold:\n <List>\n <Item>\n \ <C>[u ^ <A>x</A>, v ^ <A>x</A>]</C> is an edge of\n <A>ran</A> wheneve\ r <C>[u, v]</C> is an\n edge of <A>src</A>; and </Item>\n <Item>\n\ <A>x</A> fixes every <C>i</C> which is not a vertex of <A>src</A>.\n \ </Item>\n </List>\n See also <Ref Func=\"GeneratorsOfEndomorphismMon\ oid\"/>.<P/>\n\n\n If <A>col1</A> and <A>col2</A>, or <A>col</A>, are given\ , then they must\n represent vertex colourings; see \n <Ref Oper=\"Autom\ orphismGroup\" Label=\"for a digraph and a homogeneous list\"/> \n for deta\ ils of the permissible values for these argument. The\n homomorphism must t\ hen also have the property:\n \n <List>\n <Item>\n <C>col[i] =\ col[i ^ x]</C> for all vertices <C>i</C> of <A>digraph</A>,\n in the c\ ase of <C>IsDigraphEndomorphism</C>.</Item>\n <Item>\n <C>col1[i] \ = col2[i ^ x]</C> for all vertices <C>i</C> of <A>src</A>,\n in the cas\ es of the other operations.</Item>\n </List>\n\n See also <Ref Oper=\"Di\ graphsRespectsColouring\"/>.\n \n <Example><![CDATA[\ngap> src := Digrap\ h([[1], [1, 2], [1, 3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> \ ran := Digraph([[1], [1, 2]]);\n<immutable digraph with 2 vertices, 3 edges>\n\ gap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> I\ sDigraphHomomorphism(src, ran, Transformation([2, 1, 2]));\nfalse\ngap> IsDigr\ aphHomomorphism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphHo\ momorphism(src, src, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphHomomorp\ hism(src, ran, Transformation([1, 2, 2]),\n> [1, 2, 2\ ], [1, 2]);\ntrue\ngap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, \ 2]),\n> [2, 1, 1], [1, 2]);\nfalse\ngap> IsDigraphEnd\ omorphism(src, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphEndomorphism(s\ rc, Transformation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsDigraphEndomorphism(s\ rc, Transformation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> IsDigraphEpimorphism(s\ rc, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphMonomorphism(src, r\ an, Transformation([1, 2, 2]));\nfalse\ngap> IsDigraphEpimorphism(src, ran, Tr\ ansformation([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphism(ran, src, ());\ntr\ ue]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper \ Name=\"IsDigraphEmbedding\" Arg=\"src, ran, x\"\n Label=\"for digraphs \ and a permutation or transformation\"/>\n <Oper Name=\"IsDigraphEmbedding\" A\ rg=\"src, ran, x, col1, col2\"/>\n <Returns><K>true</K> or <K>false</K>.</Ret\ urns>\n <Description>\n <C>IsDigraphEmbedding</C> returns <K>true</K> if t\ he permutation\n or transformation <A>x</A> is a embedding of the digraph\n\ <A>src</A> into the digraph <A>ran</A>, while respecting the colourings\n \ <A>col1</A> and <A>col2</A> if given.\n <P/>\n\n A permutation or tra\ nsformation <A>x</A> is a <E>embedding</E> of a digraph\n <A>src</A> into a\ digraph <A>ran</A> if <A>x</A> is a monomorphism from\n <A>src</A> to <A>r\ an</A> and the inverse of <A>x</A> is a monomorphism from\n the subdigraph \ of <A>ran</A> induced by the image of <A>x</A> to\n <A>src</A>.\n\n See \ also <Ref Oper=\"IsDigraphHomomorphism\"/>.<P/>\n\n <Example><![CDATA[\ngap\ > src := Digraph([[1], [1, 2]]);\n<immutable digraph with 2 vertices, 3 edges>\ \ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n<immutable digraph with 3 verti\ ces, 5 edges>\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDigraph\ Embedding(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], \ [2, 1, 1]);\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]);\n\ false\ngap> ran := Digraph([[1, 2], [1, 2], [1, 3]]);\n<immutable digraph with\ 3 vertices, 6 edges>\ngap> IsDigraphMonomorphism(src, ran, IdentityTransforma\ tion);\ntrue\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\nfals\ e]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Op\ er Name=\"DigraphsRespectsColouring\" Arg=\"src, ran, x, col1, col2\"/>\n <\ Returns> <K>true</K> or <K>false</K>. </Returns>\n <Description>\n The\ operation <C>DigraphsRespectsColouring</C> verifies whether or not \n th\ e permutation or transformation <A>x</A> respects the vertex colourings\n \ <A>col1</A> and <A>col2</A> of the digraphs <A>src</A> and <A>range</A>.\n \ That is, <C>DigraphsRespectsColouring</C> returns <K>true</K> if and only i\ f for\n all vertices <C>i</C> of <A>src</A>, <C>col1[i] = col2[i ^ x]</C>\ .\n <P/>\n\n <Example><![CDATA[\ngap> src := Digraph([[1], [1, 2]]);\ \n<immutable digraph with 2 vertices, 3 edges>\ngap> ran := Digraph([[1], [1, \ 2], [1, 3]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> DigraphsResp\ ectsColouring(src, ran, (1, 2), [2, 1], [1, 2, 1]);\ntrue\ngap> DigraphsRespec\ tsColouring(src, ran, (1, 2), [2, 1], [2, 1, 1]);\nfalse\n]]></Example>\n <\ /Description>\n </ManSection>\n\n <ManSection>\n <Func Name=\"GeneratorsO\ fEndomorphismMonoid\" Arg=\"digraph[, colors][, limit]\"/>\n <Attr Name=\"Gen\ eratorsOfEndomorphismMonoidAttr\" Arg=\"digraph\"/>\n <Returns> A list of tra\ nsformations.</Returns>\n <Description>\n An endomorphism of <A>digraph</A\ > is a homomorphism\n <Ref Oper=\"DigraphHomomorphism\"/> from <A>digraph</\ A> back to itself.\n\n <C>GeneratorsOfEndomorphismMonoid</C>, called with a\ single argument,\n returns a generating set for the monoid of all endomorp\ hisms of\n <A>digraph</A>. If <A>digraph</A> belongs to <Ref\n Filt=\ \"IsImmutableDigraph\"/>, then the value of\n <C>GeneratorsOfEndomorphismMo\ noid</C> will not be recomputed on future\n calls.<P/>\n\n If the <A>col\ ors</A> argument is specified, then\n <C>GeneratorsOfEndomorphismMonoid</C>\ will return a generating\n set for the monoid of endomorphisms which respe\ ct the given colouring.\n The colouring <A>colors</A> can be in one of two \ forms: <P/>\n\n <List>\n <Item>\n A list of positive integers o\ f size the number of vertices of\n <A>digraph</A>, where <A>colors</A><\ C>[i]</C> is the colour of vertex\n <C>i</C>.\n </Item>\n <It\ em>\n A list of lists, such that <A>colors</A><C>[i]</C> is a list of a\ ll\n vertices with colour <C>i</C>.\n </Item>\n </List>\n\n \ If the <A>limit</A> argument is specified, then it will return only the\n f\ irst <A>limit</A> homomorphisms, where <A>limit</A> must be a positive\n in\ teger or <C>infinity</C>. <P/>\n\n <Example><![CDATA[\ngap> gr := Digraph(L\ ist([1 .. 3], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ T\ ransformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransfor\ mation, Transformation( [ 1, 2, 1 ] ), \n Transformation( [ 1, 2, 2 ] ), Tran\ sformation( [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 ] ) ]\ngap> Generator\ sOfEndomorphismMonoid(gr, 3);\n[ Transformation( [ 1, 3, 2 ] ), Transformation\ ( [ 2, 1 ] ), \n IdentityTransformation ]\ngap> gr := CompleteDigraph(3);;\ng\ ap> GeneratorsOfEndomorphismMonoid(gr);\n[ Transformation( [ 2, 3, 1 ] ), Tran\ sformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap> GeneratorsOfEndomor\ phismMonoid(gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransfor\ mation ]\ngap> GeneratorsOfEndomorphismMonoid(gr, [[1], [2, 3]]);\n[ Transform\ ation( [ 1, 3, 2 ] ), IdentityTransformation ]\n]]></Example>\n </Description\ >\n</ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphColouring\" Arg=\"\ digraph, n\"\n Label=\"for a digraph and a number of colours\"/>\n <Return\ s> A transformation, or <K>fail</K>.</Returns>\n <Description>\n A <E>prop\ er colouring</E> of a digraph is a labelling of its vertices in\n such a wa\ y that adjacent vertices have different labels. A <E>proper\n <C>n</C>-co\ louring</E> is a proper colouring that uses exactly <C>n</C>\n colours. Eq\ uivalently, a proper (<C>n</C>-)colouring of a digraph can be\n defined to \ be a <Ref Oper=\"DigraphEpimorphism\"/> from a digraph onto the\n complete \ digraph (with <C>n</C> vertices); see <Ref\n Oper=\"CompleteDigraph\"/>. \ Note that a digraph with loops (<Ref\n Prop=\"DigraphHasLoops\"/>) does \ not have a proper <C>n</C>-colouring for\n any value <C>n</C>.\n <P/>\n\ \n If <A>digraph</A> is a digraph and <A>n</A> is a non-negative integer, t\ hen\n <C>DigraphColouring(<A>digraph</A>, <A>n</A>)</C> returns an epimorph\ ism\n from <A>digraph</A> onto the complete digraph with <A>n</A> vertices \ if one\n exists, else it returns <K>fail</K>. <P/>\n\n See also <Ref Att\ r=\"DigraphGreedyColouring\" Label=\"for a digraph\"/> and \n <P/>\n\n N\ ote that a digraph with at least two vertices has a 2-colouring if and only\n \ if it is bipartite, see <Ref Prop=\"IsBipartiteDigraph\"/>.\n\n <Example\ ><![CDATA[\ngap> DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphC\ olouring(ChainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := \ DigraphColouring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ng\ ap> IsDigraphColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransfo\ rmation( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n]]></Example>\n </Description>\n\ </ManSection>\n\n <ManSection>\n <Oper Name=\"DigraphGreedyColouring\" \n \ Arg=\"digraph, order\" \n Label=\"for a digraph and vertex order\ \"/>\n <Oper Name=\"DigraphGreedyColouring\" \n Arg=\"digraph, func\" \ \n Label=\"for a digraph and vertex order function\"/>\n <Attr Name=\"\ DigraphGreedyColouring\" \n Arg=\"digraph\" \n Label=\"for a dig\ raph\"/>\n <Returns> A transformation, or <K>fail</K>.</Returns>\n <Descript\ ion>\n A <E>proper colouring</E> of a digraph is a labelling of its vertice\ s in\n such a way that adjacent vertices have different labels. Note that a\ digraph \n with loops (<Ref Prop=\"DigraphHasLoops\"/>) does not have any \ proper \n colouring.\n <P/>\n \n If <A>digraph</A> is a digraph an\ d <A>order</A> is a dense list consisting \n of all of the vertices of <A>d\ igraph</A> (in any order), then \n <C>DigraphGreedyColouring</C>\n uses\ a greedy algorithm with the specified order to obtain some proper\n colour\ ing of <A>digraph</A>, which may not use the minimal number of\n colours. <\ P/>\n \n If <A>digraph</A> is a digraph and <A>func</A> is a function wh\ ose argument \n is a digraph, and that returns a dense list <A>order</A>, t\ hen \n <C>DigraphGreedyColouring(<A>digraph</A>, <A>func</A>)</C> returns\n\ <C>DigraphGreedyColouring(<A>digraph</A>, <A>func</A>(<A>digraph</A>))</C>\ .\n <P/>\n\n If the optional second argument (either a list or a functio\ n), is not \n specified, then <Ref Attr=\"DigraphWelshPowellOrder\"/> is us\ ed by default. \n <P/>\n \n See also\n <Ref Oper=\"DigraphColourin\ g\" \n Label=\"for a digraph and a number of colours\"/>.\n <P/>\n\ \n <Example><![CDATA[\ngap> DigraphGreedyColouring(ChainDigraph(10));\nTran\ sformation( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\ngap> DigraphGreedyColouring(Ch\ ainDigraph(10), [1 .. 10]);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] \ )\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr\ Name=\"DigraphWelshPowellOrder\" Arg=\"digraph\"/>\n <Returns> A list of the\ vertices.</Returns>\n <Description>\n <C>DigraphWelshPowellOrder</C> retu\ rns a list of all of the vertices of \n the digraph <A>digraph</A> ordered \ according to the sum of the number of \n out- and in-neighbours, from highe\ st to lowest. \n <P/>\n\n <Example><![CDATA[\ngap> DigraphWelshPowellOrd\ er(Digraph([[4], [9], [9], [], \n> [4, 6, \ 9], [1], [], [], \n> [4, 5], [4, 5]]));\n[\ 5, 9, 4, 1, 6, 10, 2, 3, 7, 8 ]\n]]></Example>\n </Description>\n</ManSectio\ n>\n\n <ManSection>\n <Attr Name=\"ChromaticNumber\" Arg=\"digraph\"/>\n \ <Returns> A non-negative integer.</Returns>\n <Description>\n A <E>proper \ colouring</E> of a digraph is a labelling of its\n vertices in such a way t\ hat adjacent vertices have different labels.\n Equivalently, a proper digra\ ph colouring can be defined to be a <Ref\n Oper=\"DigraphEpimorphism\"/> \ from a digraph onto a complete digraph. <P/>\n\n If <A>digraph</A> is a dig\ raph without loops (see <Ref\n Prop=\"DigraphHasLoops\"/>, then <C>Chroma\ ticNumber</C> returns the least\n non-negative integer <C>n</C> such that t\ here is a proper colouring of\n <A>digraph</A> with <C>n</C> colours. In o\ ther words, for a digraph with at\n least one vertex, <C>ChromaticNumber</C\ > returns the least number <C>n</C>\n such that <C>DigraphColouring(<A>digr\ aph</A>, n)</C> does not return\n <K>fail</K>. See <Ref Oper=\"DigraphColou\ ring\"\n Label=\"for a digraph and a number of colours\"/>.\n\n <Examp\ le><![CDATA[\ngap> ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(\ CompleteDigraph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5)\ );\n2\ngap> ChromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> Ch\ romaticNumber(NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\ \n<mutable digraph with 10 vertices, 30 edges>\ngap> ChromaticNumber(D);\n3\n]\ ]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr\ Name = \"DigraphCore\" Arg = \"D\"/>\n <Returns>A list of positive integer\ s.</Returns>\n <Description>\n If <A>D</A> is a digraph, then <C>Digra\ phCore</C> returns a list of vertices\n corresponding to the <C>core</C> \ of <A>D</A>. In particular, the subdigraph\n of <A>D</A> induced by this \ list is isomorphic to the core of <A>D</A>.<P/>\n\n The <E>core</E> of a di\ graph <C>D</C> is the minimal subdigraph <A>C</A> of\n <C>D</C> which is a \ homomorphic image of <C>D</C>. The core of a digraph \n is unique up to iso\ morphism.\n <Example><![CDATA[\ngap> D := DigraphSymmetricClosure(CycleDi\ graph(8));\n<immutable symmetric digraph with 8 vertices, 16 edges>\ngap> Digr\ aphCore(D);\n[ 1, 2 ]\ngap> D := PetersenGraph();\n<immutable digraph with 10 \ vertices, 30 edges>\ngap> DigraphCore(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMu\ tableDigraph, [[3], [3], [4], [5], [2]]);\n<mutable digraph with 5 vertices, 5\ edges>\ngap> DigraphCore(D);\n[ 2, 3, 4, 5 ]\n]]></Example>\n </Descriptio\ n>\n </ManSection>\n\n </Section>\n\n</Chapter>\n <!-- homomorphisms -->\n \ <Chapter Label=\"Finding cliques and independent sets\">\n <Heading>Finding c\ liques and independent sets</Heading>\n\n In &Digraphs;, a <E>clique</E> of\ a digraph is a set of mutually adjacent\n vertices of the digraph, and an \ <E>independent set</E> is a set of mutually\n non-adjacent vertices of the \ digraph. A <E>maximal clique</E> is a clique\n which is not properly contai\ ned in another clique, and a <E>maximal\n independent set</E> is an indep\ endent set which is not properly contained\n in another independent set. U\ sing this definition in &Digraphs;, cliques\n and independent sets are both\ permitted, but not required, to contain vertices\n at which there is a loo\ p. Another name for a clique is a <E>complete\n subgraph</E>.\n <P/>\ \n\n &Digraphs; provides extensive functionality for computing cliques and\ \n independent sets of a digraph, whether maximal or not. The fundamental\ \n algorithm used in most of the methods in &Digraphs; to calculate cliques\ \n and independent sets is a version of the Bron-Kerbosch algorithm.\n C\ alculating the cliques and independent sets of a digraph is a well-known\n \ and hard problem, and searching for cliques or independent sets in a\n digr\ aph can be very lengthy, even for a digraph with a small number of\n vertic\ es. &Digraphs; uses several strategies to increase the performance\n of th\ ese calculations.\n <P/>\n\n From the definition of cliques and independ\ ent sets, it follows that the\n presence of loops and multiple edges in a d\ igraph is irrelevant to the\n existence of cliques and independent sets in \ the digraph. See <Ref\n Prop=\"DigraphHasLoops\"/> and <Ref Prop=\"IsMul\ tiDigraph\"/> for more\n information about these properties. Therefore giv\ en a digraph\n <A>digraph</A>, the cliques and independent sets of <A>digra\ ph</A> are\n equal to the cliques and independent sets of the digraph:\n\n \ <List>\n <Item>\n <C>DigraphRemoveLoops(DigraphRemoveAllMultipl\ eEdges(</C><A>digraph</A><C>))</C>.\n </Item>\n </List>\n\n See <Re\ f Oper=\"DigraphRemoveLoops\"/> and <Ref\n Oper=\"DigraphRemoveAllMultipl\ eEdges\"/> for more information about these\n attributes. Furthermore, the\ cliques of this digraph are equal to the\n cliques of the digraph formed b\ y removing any edge <C>[u,v]</C> for which\n the corresponding reverse edge\ <C>[v,u]</C> is not present. Therefore,\n overall, the cliques of <A>digr\ aph</A> are equal to the cliques of the\n symmetric digraph:\n\n <List>\ \n <Item>\n <C>MaximalSymmetricSubdigraphWithoutLoops(</C><A>digra\ ph</A><C>)</C>.\n </Item>\n </List>\n\n See <Ref Oper=\"MaximalSymm\ etricSubdigraphWithoutLoops\"/> for more\n information about this. The <Ref\ Attr=\"AutomorphismGroup\"\n Label=\"for a digraph\" /> of this symmetri\ c digraph contains the\n automorphism group of <A>digraph</A> as a subgroup\ . By performing the\n search for maximal cliques with the help of this lar\ ger automorphism group\n to reduce the search space, the computation time m\ ay be reduced. The\n functions and attributes which return representatives\ of cliques of\n <A>digraph</A> will return orbit representatives of clique\ s under the\n action of the automorphism group of the <E>maximal symmetric \ subdigraph\n without loops</E> on sets of vertices.<P/>\n\n The indepe\ ndent sets of a digraph are equal to the independent sets of the\n <Ref Ope\ r=\"DigraphSymmetricClosure\"/>. Therefore, overall, the independent\n set\ s of <A>digraph</A> are equal to the independent sets of the symmetric\n di\ graph:\n\n <List>\n <Item>\n <C>DigraphSymmetricClosure(Digraph\ RemoveLoops(DigraphRemoveAllMultipleEdges(\n </C><A>digraph</A><C>)))</\ C>.\n </Item>\n </List>\n\n Again, the automorphism group of this s\ ymmetric digraph contains the\n automorphism group of <A>digraph</A>. Since\ a search for independent sets\n is equal to a search for cliques in the <R\ ef Oper=\"DigraphDual\"/>, the\n methods used in &Digraphs; usually transfo\ rm a search for independent sets\n into a search for cliques, as described \ above. The functions and\n attributes which return representatives of inde\ pendent sets of\n <A>digraph</A> will return orbit representatives of indep\ endent sets under\n the action of the automorphism group of the <E>symmetri\ c closure</E> of the\n digraph formed by removing loops and multiple edges.\ <P/>\n\n Please note that in &Digraphs;, cliques and independent sets are n\ ot\n required to be maximal. Some authors use the word clique to mean\n \ <E>maximal</E> clique, and some authors use the phrase independent set to\n \ mean <E>maximal</E> independent set, but please note that &Digraphs; does\n \ not use this definition.\n <P/>\n\n <Section><Heading>Finding cliques</H\ eading>\n <ManSection>\n<Oper Name=\"IsClique\" Arg=\"digraph, l\"/>\n<Oper\ Name=\"IsMaximalClique\" Arg=\"digraph, l\"/>\n<Returns><K>true</K> or <K>fal\ se</K>.</Returns>\n<Description>\n If <A>digraph</A> is a digraph and <A>l</A\ > is a duplicate-free list of\n vertices of <A>digraph</A>, then\n <C>IsCliq\ ue(</C><A>digraph</A><C>,</C><A>l</A><C>)</C> returns <K>true</K>\n if <A>l\ </A> is a <E>clique</E> of <A>digraph</A> and <K>false</K> if it is\n not. \ Similarly,\n <C>IsMaximalClique(</C><A>digraph</A><C>,</C><A>l</A><C>)</C>\ returns\n <K>true</K> if <A>l</A> is a <E>maximal clique</E> of <A>digraph\ </A> and\n <K>false</K> if it is not. <P/>\n\n A <E>clique</E> of a dig\ raph is a set of mutually adjacent vertices of the\n digraph. A <E>maximal \ clique</E> is a clique that is not properly\n contained in another clique. \ A clique is permitted, but not required, to\n contain vertices at which the\ re is a loop.\n <Example><![CDATA[\ngap> D := CompleteDigraph(4);;\ngap> Is\ Clique(D, [1, 3, 2]);\ntrue\ngap> IsMaximalClique(D, [1, 3, 2]);\nfalse\ngap> \ IsMaximalClique(D, DigraphVertices(D));\ntrue\ngap> D := Digraph([[1, 2, 4, 4]\ , [1, 3, 4], [2, 1], [1, 2]]);\n<immutable multidigraph with 4 vertices, 11 ed\ ges>\ngap> IsClique(D, [2, 3, 4]);\nfalse\ngap> IsMaximalClique(D, [1, 2, 4]);\ \ntrue\ngap> D := CompleteDigraph(IsMutableDigraph, 4);;\ngap> IsClique(D, [1,\ 3, 2]);\ntrue]]></Example>\n </Description>\n</ManSection>\n\n <ManSectio\ n>\n <Func Name=\"CliquesFinder\" Arg=\"digraph, hook, user_param, limit, inc\ lude,\n exclude, max, size, reps\"/>\n <Returns>The argument <A>user_param\ </A>.</Returns>\n <Description>\n This function finds cliques of the digra\ ph <A>digraph</A> subject to the\n conditions imposed by the other argument\ s as described below. Note\n that a clique is represented by the immutable \ list of the vertices that\n it contains.\n <P/>\n\n Let <C>G</C> deno\ te the automorphism group of the maximal symmetric\n subdigraph of <A>digra\ ph</A> without loops (see <Ref\n Attr=\"AutomorphismGroup\" Label=\"for a\ digraph\"/> and <Ref\n Oper=\"MaximalSymmetricSubdigraphWithoutLoops\"/>\ ).\n\n <List>\n <Mark><A>hook</A></Mark>\n <Item>\n This a\ rgument should be a function or <K>fail</K>.<P/>\n\n If <A>hook</A> is \ a function, then it should have two arguments\n <A>user_param</A> (see \ below) and a clique <C>c</C>. The function\n <C><A>hook</A>(<A>user_par\ am</A>, c)</C> is called every time a new\n clique <C>c</C> is found by\ <C>CliquesFinder</C>.<P/>\n\n If <A>hook</A> is <K>fail</K>, then a de\ fault function is used that\n simply adds every new clique found by <C>\ CliquesFinder</C> to\n <A>user_param</A>, which must be a list in this \ case.\n </Item>\n\n <Mark><A>user_param</A></Mark>\n <Item>\n \ If <A>hook</A> is a function, then <A>user_param</A> can be any &GAP;\n \ object. The object <A>user_param</A> is used as the first argument for\ \n the function <A>hook</A>. For example, <A>user_param</A> might be a\ \n list, and <C><A>hook</A>(<A>user_param</A>, c)</C> might add the siz\ e\n of the clique <C>c</C> to the list <A>user_param</A>. <P/>\n\n \ If the value of <A>hook</A> is <K>fail</K>, then the value of\n <A>u\ ser_param</A> must be a list.\n </Item>\n\n <Mark><A>limit</A></Mark\ >\n <Item>\n This argument should be a positive integer or <K>infi\ nity</K>.\n <C>CliquesFinder</C> will return after it has found\n \ <A>limit</A> cliques or the search is complete.\n </Item>\n\n <Mar\ k><A>include</A> and <A>exclude</A></Mark>\n <Item>\n These argume\ nts should each be a (possibly empty) duplicate-free list\n of vertices\ of <A>digraph</A> (i.e. positive integers less than the\n number of ve\ rtices of <A>digraph</A>). <P/>\n\n <C>CliquesFinder</C> will only look\ for cliques containing all of the\n vertices in <A>include</A> and con\ taining none of the vertices in\n <A>exclude</A>. <P/>\n\n Note \ that the search may be much more efficient if each of these lists\n is \ invariant under the action of <C>G</C> on sets of vertices.\n </Item>\n\n\ <Mark><A>max</A></Mark>\n <Item>\n This argument should be <\ K>true</K> or <K>false</K>. If <A>max</A> is\n true then <C>CliquesFin\ der</C> will only search for <E>maximal</E>\n cliques. If <K>max</K> is\ <K>false</K> then non-maximal cliques may be\n found.\n </Item>\n\ \n <Mark><A>size</A></Mark>\n <Item>\n This argument should b\ e <K>fail</K> or a positive integer.\n If <A>size</A> is a positive int\ eger then <C>CliquesFinder</C> will\n only search for cliques that cont\ ain precisely <A>size</A> vertices.\n If <A>size</A> is <K>fail</K> the\ n cliques of any size may be found.\n </Item>\n\n <Mark><A>reps</A><\ /Mark>\n <Item>\n This argument should be <K>true</K> or <K>false<\ /K>.<P/>\n\n If <A>reps</A> is <K>true</K> then the arguments <A>includ\ e</A> and\n <A>exclude</A> are each required to be invariant under the \ action of\n <C>G</C> on sets of vertices. In this case, <C>CliquesFind\ er</C> will\n find representatives of the orbits of the desired cliques\ under the\n action of <C>G</C>, <E>although representatives may be ret\ urned that\n are in the same orbit</E>.\n If <A>reps</A> is fa\ lse then <C>CliquesFinder</C> will not take this into\n consideration.<\ P/>\n\n For a digraph such that <C>G</C> is non-trivial, the search for\ \n clique representatives can be much more efficient than the search fo\ r\n all cliques.\n </Item>\n </List>\n\n This function uses \ a version of the Bron-Kerbosch algorithm.\n <Example><![CDATA[\ngap> D := C\ ompleteDigraph(5);\n<immutable complete digraph with 5 vertices>\ngap> user_pa\ ram := [];;\ngap> f := function(a, b) # Calculate size of clique\n> AddSet(\ user_param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity,\ [], [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> Cliq\ uesFinder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, \ 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, \ 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D\ , fail, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> Cl\ iquesFinder(D, fail, [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 \ ], [ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3\ ], [], true, 3, false);\n[ ]\ngap> D := CompleteDigraph(IsMutableDigraph, 5);\ \n<mutable digraph with 5 vertices, 20 edges>\ngap> user_param := [];;\ngap> f\ := function(a, b) # Calculate size of clique\n> AddSet(user_param, Size(b)\ );\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], [], false, fai\ l,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFinder(D, fail, \ [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ]\ , [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [3], false, fai\ l, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fail, [], infinit\ y, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> CliquesFinder(D, fail\ , [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1\ , 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], [], true, 3, fal\ se);\n[ ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n\ <Func Name=\"DigraphClique\" Arg=\"digraph[, include[, exclude[, size]]]\"/>\ \n <Func Name=\"DigraphMaximalClique\" Arg=\"digraph[, include[, exclude[,\n \ size]]]\"/>\n <Returns>An immutable list of positive integers, or <K>fail<\ /K>.</Returns>\n <Description>\n If <A>digraph</A> is a digraph, then thes\ e functions returns a clique\n of <A>digraph</A> if one exists that satisfi\ es the arguments, else it\n returns <K>fail</K>. A clique is defined by th\ e set of vertices that it\n contains; see <Ref Oper=\"IsClique\"/> and <Ref\ Oper=\"IsMaximalClique\"/>.<P/>\n\n The optional arguments <A>include</A> \ and <A>exclude</A> must each be a\n (possibly empty) duplicate-free list of\ vertices of <A>digraph</A>, and\n the optional argument <A>size</A> must b\ e a positive integer. By default,\n <A>include</A> and <A>exclude</A> are \ empty. These functions will search for\n a clique of <A>digraph</A> that in\ cludes the vertices of <A>include</A>\n but does not include any vertices i\ n <A>exclude</A>; if the argument\n <A>size</A> is supplied, then additiona\ lly the clique will be required to\n contain precisely <A>size</A> vertices\ .<P/>\n\n If <A>include</A> is not a clique, then these functions return <K\ >fail</K>.\n Otherwise, the functions behave in the following way, dependin\ g on the\n number of arguments:\n\n <List>\n <Mark>One or two argum\ ents</Mark>\n <Item>\n If one or two arguments are supplied, then \ <C>DigraphClique</C> and\n <C>DigraphMaximalClique</C> greedily enlarge\ the clique <A>include</A>\n until it cannot continue. The result is g\ uaranteed to be a maximal\n clique. This may or may not return an answe\ r more quickly than using\n <Ref Func=\"DigraphMaximalCliques\"/> with \ a limit of 1.\n </Item>\n <Mark>Three arguments</Mark>\n <Item>\ \n If three arguments are supplied, then <C>DigraphClique</C> greedily\ \n enlarges the clique <A>include</A> until it cannot continue, althoug\ h\n this clique may not be maximal.<P/>\n\n Given three argument\ s, <C>DigraphMaximalClique</C> returns the maximal\n clique returned by\ <C>DigraphMaximalCliques(</C><A>digraph</A><C>,\n </C><A>include</A><C\ >, </C><A>exclude</A><C>, 1)</C> if one exists,\n else <K>fail</K>.\n \ </Item>\n <Mark>Four arguments</Mark>\n <Item>\n If four \ arguments are supplied, then <C>DigraphClique</C> returns the\n clique \ returned by <C>DigraphCliques(</C><A>digraph</A><C>,\n </C><A>include\ </A><C>, </C><A>exclude</A><C>, 1,\n </C><A>size</A><C>)</C> if one exi\ sts, else <K>fail</K>. This clique may\n not be maximal.<P/> Given four\ arguments, <C>DigraphMaximalClique</C>\n returns the maximal clique re\ turned by\n <C>DigraphMaximalCliques(</C><A>digraph</A><C>, </C><A>incl\ ude</A><C>,\n </C><A>exclude</A><C>, 1, </C><A>size</A><C>)</C> if one \ exists, else\n <K>fail</K>.\n </Item>\n </List>\n\n <Example\ ><![CDATA[\ngap> D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n<immu\ table digraph with 5 vertices, 9 edges>\ngap> IsSymmetricDigraph(D);\nfalse\ng\ ap> DigraphClique(D);\n[ 5 ]\ngap> DigraphMaximalClique(D);\n[ 5 ]\ngap> Digra\ phClique(D, [1, 2]);\n[ 1, 2, 3 ]\ngap> DigraphMaximalClique(D, [1, 3]);\n[ 1,\ 3, 2 ]\ngap> DigraphClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalClique\ (D, [1], [3, 4]);\nfail\ngap> DigraphClique(D, [1, 5]);\nfail\ngap> DigraphCli\ que(D, [], [], 2);\n[ 1, 2 ]\ngap> D := Digraph(IsMutableDigraph,\n> \ [[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n<mutable digraph with 5 vert\ ices, 9 edges>\ngap> IsSymmetricDigraph(D);\nfalse\ngap> DigraphClique(D);\n[ \ 5 ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Func\ Name=\"DigraphMaximalCliques\" Arg=\"digraph[, include[, exclude[, limit[, si\ ze]]]]\"/>\n <Func Name=\"DigraphMaximalCliquesReps\" Arg=\"digraph[, include\ [, exclude[,\n limit[, size]]]]\"/>\n <Func Name=\"DigraphCliques\" Arg=\"\ digraph[, include[, exclude[,\n limit[, size]]]]\"/>\n <Func Name=\"Digrap\ hCliquesReps\" Arg=\"digraph[, include[,\n exclude[, limit[, size]]]]\"/>\n\ <Attr Name=\"DigraphMaximalCliquesAttr\" Arg=\"digraph\"/>\n <Attr Name=\"D\ igraphMaximalCliquesRepsAttr\" Arg=\"digraph\"/>\n <Returns>An immutable list\ of lists of positive integers.</Returns>\n <Description>\n If <A>digraph<\ /A> is digraph, then these functions and attributes use <Ref\n Func=\"Cli\ quesFinder\"/> to return cliques of <A>digraph</A>. A\n clique is defined b\ y the set of vertices that it contains; see <Ref\n Oper=\"IsClique\"/> an\ d <Ref Oper=\"IsMaximalClique\"/>.<P/>\n\n The optional arguments <A>includ\ e</A> and <A>exclude</A> must each be a\n (possibly empty) list of vertices\ of <A>digraph</A>, the optional argument\n <A>limit</A> must be either a p\ ositive integer or <C>infinity</C>, and the\n optional argument <A>size</A>\ must be a positive integer. If not\n specified, then <A>include</A> and <\ A>exclude</A> are chosen to be empty\n lists, and <A>limit</A> is set to <C\ >infinity</C>. <P/>\n\n The functions will return as many suitable cliques \ as possible, up to the\n number <A>limit</A>. These functions will find cl\ iques that contain all\n of the vertices of <A>include</A> but do not conta\ in any of the\n vertices of <A>exclude</A>. The argument <A>size</A> restr\ icts the search\n to those cliques that contain precisely <A>size</A> verti\ ces.\n If the function or attribute has <C>Maximal</C> in its name, then on\ ly\n maximal cliques will be returned; otherwise non-maximal cliques may be\ \n returned. <P/>\n\n Let <C>G</C> denote the automorphism group of maxi\ mal symmetric subdigraph\n of <A>digraph</A> without loops (see <Ref Attr=\ \"AutomorphismGroup\"\n Label=\"for a digraph\"/> and <Ref\n Oper=\"\ MaximalSymmetricSubdigraphWithoutLoops\"/>).\n\n <List>\n <Mark>Distin\ ct cliques</Mark>\n <Item>\n <C>DigraphMaximalCliques</C> and <C>D\ igraphCliques</C> each return a\n duplicate-free list of at most <A>lim\ it</A> cliques of <A>digraph</A>\n that satisfy the arguments.<P/>\n\n \ The computation may be significantly faster if <A>include</A> and\n \ <A>exclude</A> are invariant under the action of <C>G</C>\n on sets\ of vertices.\n </Item>\n\n <Mark>Orbit representatives of cliques</\ Mark>\n <Item>\n To use <C>DigraphMaximalCliquesReps</C> or <C>Dig\ raphCliquesReps</C>,\n the arguments <A>include</A> and <A>exclude</A> \ must each be invariant\n under the action of <C>G</C> on sets of vertic\ es.<P/>\n\n If this is the case, then <C>DigraphMaximalCliquesReps</C> \ and\n <C>DigraphCliquesReps</C> each return a duplicate-free list of at\ most\n <A>limit</A> orbits representatives (under the action of <C>G</\ C> on\n sets vertices) of cliques of <A>digraph</A> that satisfy the\n \ arguments.<P/>\n\n The representatives are not guaranteed to be \ in distinct orbits.\n However, if fewer than <A>lim</A> results are ret\ urned, then there\n will be at least one representative from each orbit\ of maximal cliques.\n </Item>\n </List>\n\n <Example><![CDATA[\nga\ p> D := Digraph([\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]);\n\ <immutable digraph with 6 vertices, 14 edges>\ngap> IsSymmetricDigraph(D);\ntr\ ue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ \ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ngap>\ DigraphMaximalCliquesReps(D);\n[ [ 1, 2, 3 ], [ 3, 4 ] ]\ngap> Orbit(G, [1, 2\ , 3], OnSets);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]\ngap> Orbit(G, [3, 4], OnSets);\n\ [ [ 3, 4 ] ]\ngap> DigraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ] ]\ \ngap> DigraphMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCliq\ ues(D, [1], [5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph(I\ sMutableDigraph, [\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]);\ \n<mutable digraph with 6 vertices, 14 edges>\ngap> IsSymmetricDigraph(D);\ntr\ ue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ \ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]]]></E\ xample>\n </Description>\n</ManSection>\n\n <ManSection>\n <Attr Name=\"C\ liqueNumber\" Arg=\"digraph\"/>\n <Returns>A non-negative integer.</Returns>\ \n <Description>\n If <A>digraph</A> is a digraph, then <C>CliqueNumber(<A\ >digraph</A>)</C>\n returns the largest integer <C>n</C> such that <A>digra\ ph</A> contains a\n clique with <C>n</C> vertices as an induced subdigraph.\ \n <P/>\n\n A <E>clique</E> of a digraph is a set of mutually adjacent v\ ertices of the\n digraph. Loops and multiple edges are ignored for the purp\ ose of\n determining the clique number of a digraph. \n <Example><![CDAT\ A[\ngap> D := CompleteDigraph(4);;\ngap> CliqueNumber(D);\n4\ngap> D := Digrap\ h([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n<immutable multidigraph with 4 \ vertices, 11 edges>\ngap> CliqueNumber(D);\n3\ngap> D := CompleteDigraph(IsMut\ ableDigraph, 4);;\ngap> CliqueNumber(D);\n4]]></Example>\n</Description>\n</Ma\ nSection>\n\n </Section>\n\n <Section><Heading>Finding independent sets</Hea\ ding>\n <ManSection>\n <Oper Name=\"IsIndependentSet\" Arg=\"digraph, l\"/\ >\n <Oper Name=\"IsMaximalIndependentSet\" Arg=\"digraph, l\"/>\n <Returns><\ K>true</K> or <K>false</K>.</Returns>\n <Description>\n If <A>digraph</A> \ is a digraph and <A>l</A> is a duplicate-free list of\n vertices of <A>digr\ aph</A>, then\n <C>IsIndependentSet(</C><A>digraph</A><C>,</C><A>l</A><C>)<\ /C> returns\n <K>true</K> if <A>l</A> is an <E>independent set</E> of <A>di\ graph</A> and\n <K>false</K> if it is not. Similarly,\n <C>IsMaximalInd\ ependentSet(</C><A>digraph</A><C>,</C><A>l</A><C>)</C>\n returns <K>true</K\ > if <A>l</A> is a <E>maximal independent set</E> of\n <A>digraph</A> and <\ K>false</K> if it is not. <P/>\n\n An <E>independent set</E> of a digraph \ is a set of mutually non-adjacent\n vertices of the digraph. A <E>maximal i\ ndependent set</E> is an independent\n set that is not properly contained i\ n another independent set. An\n independent set is permitted, but not requi\ red, to contain vertices at\n which there is a loop.\n <Example><![CDATA\ [\ngap> D := CycleDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsM\ aximalIndependentSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfa\ lse\ngap> IsIndependentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, \ [2, 4]);\ntrue\ngap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndepend\ entSet(D, [1]);\ntrue]]></Example>\n </Description>\n</ManSection>\n\n <Ma\ nSection>\n <Func Name=\"DigraphIndependentSet\"\n Arg=\"digraph[, include\ [, exclude[, size]]]\"/>\n <Func Name=\"DigraphMaximalIndependentSet\"\n A\ rg=\"digraph[, include[, exclude[,\n size]]]\"/>\n <Returns>An immutable l\ ist of positive integers, or <K>fail</K>.</Returns>\n <Description>\n If <\ A>digraph</A> is a digraph, then these functions returns an independent\n s\ et of <A>digraph</A> if one exists that satisfies the arguments, else it\n \ returns <K>fail</K>. An independent set is defined by the set of vertices\n \ that it contains; see <Ref Oper=\"IsIndependentSet\"/> and <Ref\n Oper=\ \"IsMaximalIndependentSet\"/>.<P/>\n\n The optional arguments <A>include</A\ > and <A>exclude</A> must each be a\n (possibly empty) duplicate-free list \ of vertices of <A>digraph</A>, and the\n optional argument <A>size</A> must\ be a positive integer. By default,\n <A>include</A> and <A>exclude</A> ar\ e empty. These functions will search\n for an independent set of <A>digraph\ </A> that includes the vertices of\n <A>include</A> but does not include an\ y vertices in <A>exclude</A>;\n if the argument <A>size</A> is supplied, th\ en additionally the independent\n set will be required to contain precisely\ <A>size</A> vertices.<P/>\n\n If <A>include</A> is not an independent set,\ then these functions return\n <K>fail</K>. Otherwise, the functions behav\ e in the following way,\n depending on the number of arguments:\n\n <Lis\ t>\n <Mark>One or two arguments</Mark>\n <Item>\n If one or t\ wo arguments are supplied, then <C>DigraphIndependentSet</C>\n and <C>D\ igraphMaximalIndependentSet</C> greedily enlarge the\n independent set \ <A>include</A> until it cannot continue. The result\n is guaranteed to\ be a maximal independent set. This may or may not\n return an answer m\ ore quickly than using <Ref\n Func=\"DigraphMaximalIndependentSets\"/\ > with a limit of 1.\n </Item>\n <Mark>Three arguments</Mark>\n \ <Item>\n If three arguments are supplied, then <C>DigraphIndependentSe\ t</C>\n greedily enlarges the independent set <A>include</A> until it c\ annot\n continue, although this independent set may not be maximal.<P/>\ \n\n Given three arguments, <C>DigraphMaximalIndependentSet</C> returns\ the\n maximal independent set returned by\n <C>DigraphMaximalIn\ dependentSets(</C><A>digraph</A><C>,\n </C><A>include</A><C>, </C><A>ex\ clude</A><C>, 1)</C> if one exists,\n else <K>fail</K>.\n </Item>\ \n <Mark>Four arguments</Mark>\n <Item>\n If four arguments a\ re supplied, then <C>DigraphIndependentSet</C>\n returns the independen\ t set returned by\n <C>DigraphIndependentSets(</C><A>digraph</A><C>, </\ C><A>include</A><C>,\n </C><A>exclude</A><C>, 1, </C><A>size</A><C>)</C\ > if one exists, else\n <K>fail</K>. This independent set may not be ma\ ximal.<P/> Given four\n arguments, <C>DigraphMaximalIndependentSet</C> \ returns the maximal\n independent set returned by\n <C>DigraphMa\ ximalIndependentSets(</C><A>digraph</A><C>,\n </C><A>include</A><C>, \ </C><A>exclude</A><C>, 1,\n </C><A>size</A><C>)</C> if one exists, else\ <K>fail</K>.\n </Item>\n </List>\n <Example><![CDATA[\ngap> D := C\ hainDigraph(6);\n<immutable chain digraph with 6 vertices>\ngap> DigraphIndepe\ ndentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ \ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalInde\ pendentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, [2, 4], [6]\ );\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap> D\ igraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndependent\ Set(D, [1], [], 2);\nfail\ngap> DigraphMaximalIndependentSet(D, [1], [], 3);\n\ [ 1, 3, 5 ]\ngap> D := ChainDigraph(IsMutableDigraph, 6);\n<mutable digraph wi\ th 6 vertices, 5 edges>\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> Dig\ raphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, \ 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ \ngap> DigraphIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIn\ dependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [1], [], 2)\ ;\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ngap> Dig\ raphMaximalIndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]]]></Example>\n </Descr\ iption>\n</ManSection>\n\n <ManSection>\n <Func Name=\"DigraphMaximalIndep\ endentSets\" Arg=\"digraph[, include[, exclude[,\n limit[, size]]]]\"/>\n \ <Func Name=\"DigraphMaximalIndependentSetsReps\" Arg=\"digraph[, include[,\n \ exclude[, limit[, size]]]]\"/>\n <Func Name=\"DigraphIndependentSets\" Arg=\ \"digraph[, include[,\n exclude[, limit[, size]]]]\"/>\n <Func Name=\"Digr\ aphIndependentSetsReps\" Arg=\"digraph[, include[,\n exclude[, limit[, size\ ]]]]\"/>\n <Attr Name=\"DigraphMaximalIndependentSetsAttr\" Arg=\"digraph\"/>\ \n <Attr Name=\"DigraphMaximalIndependentSetsRepsAttr\" Arg=\"digraph\"/>\n \ <Returns>An immutable list of lists of positive integers.</Returns>\n <Descri\ ption>\n If <A>digraph</A> is digraph, then these functions and attributes \ use <Ref\n Func=\"CliquesFinder\"/> to return independent sets of <A>digr\ aph</A>. An\n independent set is defined by the set of vertices that it con\ tains; see\n <Ref Oper=\"IsMaximalIndependentSet\"/> and <Ref\n Oper=\ \"IsIndependentSet\"/>.<P/>\n\n The optional arguments <A>include</A> and <\ A>exclude</A> must each be a\n (possibly empty) list of vertices of <A>digr\ aph</A>, the optional argument\n <A>limit</A> must be either a positive int\ eger or <C>infinity</C>, and the\n optional argument <A>size</A> must be a \ positive integer. If not\n specified, then <A>include</A> and <A>exclude</\ A> are chosen to be empty\n lists, and <A>limit</A> is set to <C>infinity</\ C>. <P/>\n\n The functions will return as many suitable independent sets as\ possible, up\n to the number <A>limit</A>. These functions will find inde\ pendent sets\n that contain all of the vertices of <A>include</A> but do no\ t\n contain any of the vertices of <A>exclude</A> The argument <A>size</A>\ \n restricts the search to those cliques that contain precisely <A>size</A>\ \n vertices. If the function or attribute has <C>Maximal</C> in its name,\ \n then only maximal independent sets will be returned; otherwise non-maxim\ al\n independent sets may be returned.\n <P/>\n\n Let <C>G</C> denote\ the <Ref Attr=\"AutomorphismGroup\" Label=\"for a digraph\"/>\n of the <Re\ f Oper=\"DigraphSymmetricClosure\"/> of the digraph formed from\n <A>digrap\ h</A> by removing loops and ignoring the multiplicity of edges.\n\n <List>\ \n <Mark>Distinct independent sets</Mark>\n <Item>\n <C>Digra\ phMaximalIndependentSets</C> and <C>DigraphIndependentSets</C>\n each r\ eturn a duplicate-free list of at most <A>limit</A> independent\n sets \ of <A>digraph</A> that satisfy the arguments.<P/>\n\n The computation m\ ay be significantly faster if <A>include</A> and\n <A>exclude</A> are i\ nvariant under the action of <C>G</C> on sets of\n vertices.\n </I\ tem>\n\n <Mark>Representatives of distinct orbits of independent sets</Ma\ rk>\n <Item>\n To use <C>DigraphMaximalIndependentSetsReps</C> or\ \n <C>DigraphIndependentSetsReps</C>, the arguments <A>include</A> and\ \n <A>exclude</A> must each be invariant under the action of <C>G</C> o\ n\n sets of vertices.<P/>\n\n If this is the case, then <C>Digra\ phMaximalIndependentSetsReps</C> and\n <C>DigraphIndependentSetsReps</C\ > each return a list of\n at most <A>limit</A> orbits representatives (\ under the action of\n <C>G</C> on sets of vertices) of independent sets\ of <A>digraph</A>\n that satisfy the arguments. <P/>\n\n The re\ presentatives are not guaranteed to be in distinct orbits.\n However, i\ f <A>lim</A> is not specified, or fewer than <A>lim</A>\n results are r\ eturned, then there will be at least one representative\n from each orb\ it of maximal independent sets.\n </Item>\n </List>\n\n <Example><!\ [CDATA[\ngap> D := CycleDigraph(5);\n<immutable cycle digraph with 5 vertices>\ \ngap> DigraphMaximalIndependentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndepe\ ndentSetsReps(D);\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets\ (D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaxim\ alIndependentSets(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSet\ s(D, [], [4, 5]);\n[ [ 1 ], [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentS\ ets(D, [], [4, 5], 1, 2);\n[ [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigra\ ph, 5);\n<mutable digraph with 5 vertices, 5 edges>\ngap> DigraphMaximalIndepe\ ndentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ],\ [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ]\ , [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\ \n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]);\n[ [ 1 ]\ , [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5], 1, 2);\ \n[ [ 1, 3 ] ]]]></Example>\n </Description>\n</ManSection>\n\n </Section>\n\ </Chapter>\n <!-- cliques -->\n <Chapter Label=\"Visualising and IO\"><Headin\ g>Visualising and IO</Heading>\n\n <Section><Heading>Visualising a digraph</H\ eading>\n <ManSection>\n <Func Name=\"Splash\" Arg=\"str, [opts]\"/>\n\ <Returns>Nothing.</Returns>\n <Description>\n This function attemp\ ts to convert the string <A>str</A> into a pdf\n document and open this d\ ocument, i.e. to splash it all over your monitor.<P/>\n\n The string <A>s\ tr</A> must correspond to a valid <C>dot</C> or\n <C>LaTeX</C> text file \ and you must have have <C>GraphViz</C> and\n <C>pdflatex</C> installed on\ your computer. For details about these file\n formats, see <URL>https:/\ /www.latex-project.org</URL> and\n <URL>https://www.graphviz.org</URL>.<P\ />\n\n This function is provided to allow convenient, immediate viewing o\ f the\n pictures produced by the function <Ref Attr=\"DotDigraph\"/>.<P/>\ \n\n The optional second argument <A>opts</A> should be a record with\n \ components corresponding to various options, given below.\n\n <List>\ \n <Mark>path</Mark>\n <Item>\n this should be a string\ representing the path to the directory where\n you want <C>Splash</C\ > to do its work. The default value of this\n option is <C>\"~/\"</C>\ .\n </Item>\n\n <Mark>directory</Mark>\n <Item>\n \ this should be a string representing the name of the directory in\n \ <C>path</C> where you want <C>Splash</C> to do its work. This function\n \ will create this directory if does not already exist. <P/>\n\n T\ he default value of this option is <C>\"tmp.viz\"</C> if the option\n \ <C>path</C> is present, and the result of\n <Ref Func=\"DirectoryTem\ porary\" BookName=\"ref\"/> is used otherwise.\n </Item>\n\n <Ma\ rk>filename</Mark>\n <Item>\n this should be a string represen\ ting the name of the file where\n <A>str</A> will be written. The de\ fault value of this option is\n <C>\"vizpicture\"</C>.\n </Ite\ m>\n\n <Mark>viewer</Mark>\n <Item>\n this should be a \ string representing the name of the program which\n should open the f\ iles produced by <C>GraphViz</C> or <C>pdflatex</C>.\n </Item>\n\n \ <Mark>type</Mark>\n <Item>\n this option can be used to spe\ cify that the string <A>str</A> contains\n a &LaTeX; or <C>dot</C> do\ cument. You can specify this option in\n <A>str</A> directly by makin\ g the first line <C>\"%latex\"</C> or\n <C>\"//dot\"</C>. There is n\ o default value for this option, this\n option must be specified in <\ A>str</A> or in <A>opt.type</A>.\n </Item>\n\n <Mark>engine</Mar\ k>\n <Item>\n this option can be used to specify the GraphViz \ engine to use\n to render a <C>dot</C> document. The valid choices ar\ e <C>\"dot\"</C>,\n <C>\"neato\"</C>, <C>\"circo\"</C>, <C>\"twopi\"<\ /C>, <C>\"fdp\"</C>,\n <C>\"sfdp\"</C>, and <C>\"patchwork\"</C>. Ple\ ase refer to the\n GraphViz documentation for details on these engine\ s.\n The default value for this option is <C>\"dot\"</C>, and it\n \ must be specified in <A>opt.engine</A>.\n </Item>\n\n <Ma\ rk>filetype</Mark>\n <Item>\n this should be a string represen\ ting the type of file which\n <C>Splash</C> should produce. For &LaTe\ X; files, this option is\n ignored and the default value <C>\"pdf\"</\ C> is used.\n </Item>\n </List>\n\n This function was written\ by Attila Egri-Nagy and Manuel Delgado with some\n minor changes by J. D\ . Mitchell.<P/>\n <Log><![CDATA[\ngap> Splash(DotDigraph(RandomDigraph(4)\ ));\n]]></Log>\n </Description>\n </ManSection>\n\n <ManSection>\n \ <Attr Name=\"DotDigraph\" Arg=\"digraph\"/>\n <Oper Name=\"DotColoredDigra\ ph\" Arg=\"digraph, vert, edge\"/>\n <Oper Name=\"DotVertexLabelledDigraph\ \" Arg=\"digraph\"/>\n <Oper Name=\"DotVertexColoredDigraph\" Arg=\"digraph\ , vert\"/>\n <Oper Name=\"DotEdgeColoredDigraph\" Arg=\"digraph, edge\"/>\n\ <Returns>A string.</Returns>\n <Description>\n <C>DotDigraph</C> p\ roduces a graphical representation of the digraph\n <A>digraph</A>. Verti\ ces are displayed as circles, numbered consistently\n with <A>digraph</A>\ . Edges are displayed as arrowed lines between\n vertices, with the arrow\ head of each line pointing towards the range\n of the edge.<P/>\n\n \ <C>DotColoredDigraph</C> differs from <C>DotDigraph</C> only in\n that th\ e values in given in the two lists are used to color the vertices and\n e\ dges of the graph when displayed. The list for vertex colours should be\n \ a list of length equal to the number of vertices, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. The list for edge colours should be a list of lists\n \ with the same shape of the outneighbours of the digraph that contains strings\ \n that correspond to colours accepted by the graphviz software. If the \ lists \n are not the appropriate size, or have holes then the function wi\ ll return\n an error.<P/>\n\n <C>DotVertexColoredDigraph</C> differs\ from <C>DotDigraph</C> only in\n that the values in given in the list ar\ e used to color the vertices\n of the graph when displayed. The list for \ vertex colours should be\n a list of length equal to the number of vertic\ es, containing strings that\n are accepted by the graphviz software, whic\ h is the one used for graph\n representation. If the list is not the appr\ opriate size, or has holes \n then the function will return an error.<P/>\ \n\n <C>DotEdgeColoredDigraph</C> differs from <C>DotDigraph</C> only in\ \n that the values in given in the list are used to color the vertices an\ d\n edges of the graph when displayed. The list for edge colours should b\ e a list\n of lists with the same shape of the outneighbours of the digra\ ph that contains\n strings that correspond to colours accepted by the gra\ phviz software. If the \n list is not the appropriate size, or has holes \ then the function will return\n an error.<P/>\n\n <C>DotVertexLabell\ edDigraph</C> differs from <C>DotDigraph</C> only in\n that the values in\ <Ref Oper=\"DigraphVertexLabels\"/> are used to label\n the vertices in \ the produced picture rather than the numbers <C>1</C> to\n the number of \ vertices of the digraph. <P/>\n\n The output is in <C>dot</C> format (als\ o known as <C>GraphViz</C>)\n format. For details about this file format,\ and information about how to\n display or edit this format see <URL>http\ s://www.graphviz.org</URL>. <P/>\n\n The string returned by <C>DotDigraph\ </C> or\n <C>DotVertexLabelledDigraph</C> can be written to a file using\ \n the command <Ref Func=\"FileString\" BookName=\"GAPDoc\"/>.<P/>\n \ <Log><![CDATA[\ngap> adj := List([1 .. 4], x -> [1, 1, 1, 1]);\n[ [ 1, 1, 1, \ 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] ]\ngap> adj[1][3] := 0;\n0\ \ngap> gr := DigraphByAdjacencyMatrix(adj);\n<immutable digraph with 4 vertice\ s, 15 edges>\ngap> D := CompleteDigraph(4);\n<immutable complete digraph with \ 4 vertices>\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcol\ ors[2] := \"red\";; \ngap> vertcolors[3] := \"green\";; vertcolors[4] := \"yel\ low\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := \ [];;\ngap> edgecolors[3] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2]\ := \"lightblue\";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4\ ] := \"purple\";;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2\ ][3] := \"pink\";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3\ ][1] := \"lightblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolor\ s[3][4] := \"purple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgec\ olors[4][2] := \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(D\ otColoredDigraph(D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shap\ e=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[color=gr\ een, style=filled]\n4[color=yellow, style=filled]\n1 -> 2[color=lightblue]\n1 \ -> 3[color=pink]\n1 -> 4[color=purple]\n2 -> 1[color=lightblue]\n2 -> 3[color=\ pink]\n2 -> 4[color=purple]\n3 -> 1[color=lightblue]\n3 -> 2[color=pink]\n3 ->\ 4[color=purple]\n4 -> 1[color=lightblue]\n4 -> 2[color=pink]\n4 -> 3[color=pu\ rple]\n}\ngap> D := EmptyDigraph(3);\n<immutable empty digraph with 3 vertices\ >\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \ \"red\";;\ngap> vertcolors[3] := \"green\";;\ngap> edgecolors := [];;\ngap> ed\ gecolors[1] := [];; edgecolors[2] := [];; \ngap> edgecolors[3] := [];;\ngap> P\ rint(DotColoredDigraph(D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode\ [shape=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[co\ lor=green, style=filled]\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n<immutabl\ e digraph with 3 vertices, 4 edges>\ngap> vertcolors := [];;\ngap> vertcolors[\ 1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> vertcolors[3] := \"pu\ rple\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] :=\ [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgeco\ lors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2]\ := \"red\";;\ngap> Print(DotSymmetricColoredDigraph(D, vertcolors, edgecolors\ ));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[color=blue, style=filled]\n2[\ color=pink, style=filled]\n3[color=purple, style=filled]\n1 -- 2[color=green]\ \n2 -- 3[color=red]\n}\ngap> D := Digraph([[2, 3], [1, 3], [1]]);\n<immutable \ digraph with 3 vertices, 5 edges>\ngap> vertcolors := [];;\ngap> vertcolors[1]\ := \"blue\";; vertcolors[2] := \"red\";;\ngap> vertcolors[3] := \"green\";;\n\ gap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap\ > edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"orange\";; edgecolors[1][3\ ] := \"yellow\";;\ngap> edgecolors[2][1] := \"orange\";; edgecolors[2][3] := \ \"pink\";;\ngap> edgecolors[3][1] := \"yellow\";;\ngap> Print(DotColoredDigrap\ h(D, vertcolors, edgecolors));;\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[c\ olor=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, style=fil\ led]\n1 -> 2[color=orange]\n1 -> 3[color=yellow]\n2 -> 1[color=orange]\n2 -> 3\ [color=pink]\n3 -> 1[color=yellow]\n}\ngap> D := Digraph(IsMutableDigraph, [[2\ , 3], [1, 3], [1]]);\n<mutable digraph with 3 vertices, 5 edges>\ngap> vertcol\ ors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap>\ vertcolors[3] := \"green\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := \ [];; edgecolors[2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] \ := \"orange\";; edgecolors[1][3] := \"yellow\";;\ngap> edgecolors[2][1] := \"o\ range\";; edgecolors[2][3] := \"pink\";;\ngap> edgecolors[3][1] := \"yellow\";\ ;\ngap> Print(DotColoredDigraph(D, vertcolors, edgecolors));;\n//dot\ndigraph \ hgn{\nnode [shape=circle]\n1[color=blue, style=filled]\n2[color=red, style=fil\ led]\n3[color=green, style=filled]\n1 -> 2[color=orange]\n1 -> 3[color=yellow]\ \n2 -> 1[color=orange]\n2 -> 3[color=pink]\n3 -> 1[color=yellow]\n}\ngap> D;\n\ <mutable digraph with 3 vertices, 5 edges>\ngap> DotSymmetricDigraph(gr2){[12 \ .. 70]};\n\" hgn{\\nnode [shape=circle]\\n\\n1\\n2\\n3\\n4\\n1 -- 2\\n2 -- 3\\\ n3 -- 3\\n3 -\"\ngap> DotSymmetricDigraph(gr1);\nError, the argument <D> must \ be a symmetric digraph,\ngap> D := CompleteDigraph(4);\n<immutable complete di\ graph with 4 vertices>\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\ \";; vertcolors[2] := \"red\";; \ngap> vertcolors[3] := \"green\";; vertcolors\ [4] := \"yellow\";;\ngap> Print(DotVertexColoredDigraph(D, vertcolors));\n//do\ t\ndigraph hgn{\nnode [shape=circle]\n1[color=blue, style=filled]\n2[color=red\ , style=filled]\n3[color=green, style=filled]\n4[color=yellow, style=filled]\n\ 1 -> 2\n1 -> 3\n1 -> 4\n2 -> 1\n2 -> 3\n2 -> 4\n3 -> 1\n3 -> 2\n3 -> 4\n4 -> 1\ \n4 -> 2\n4 -> 3\n}\ngap> D := CompleteDigraph(4);\n<immutable complete digrap\ h with 4 vertices>\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgeco\ lors[2] := [];;\ngap> edgecolors[3] := [];; edgecolors[4] := [];; \ngap> edgec\ olors[1][2] := \"lightblue\";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edge\ colors[1][4] := \"purple\";;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> e\ dgecolors[2][3] := \"pink\";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> e\ dgecolors[3][1] := \"lightblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap\ > edgecolors[3][4] := \"purple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ \ngap> edgecolors[4][2] := \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\n\ gap> Print(DotEdgeColoredDigraph(D, edgecolors));\n//dot\ndigraph hgn{\nnode [\ shape=circle]\n1\n2\n3\n4\n1 -> 2[color=lightblue]\n1 -> 3[color=pink]\n1 -> 4\ [color=purple]\n2 -> 1[color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[color=purp\ le]\n3 -> 1[color=lightblue]\n3 -> 2[color=pink]\n3 -> 4[color=purple]\n4 -> 1\ [color=lightblue]\n4 -> 2[color=pink]\n4 -> 3[color=purple]\n}\ngap> FileStrin\ g(\"dot/k4.dot\", DotDigraph(gr));\n154]]></Log>\n </Description>\n </ManS\ ection>\n\n <ManSection>\n <Attr Name=\"DotSymmetricDigraph\" Arg=\"di\ graph\"/>\n <Oper Name=\"DotSymmetricColoredDigraph\" Arg=\"digraph, vert, \ edge\"/>\n <Oper Name=\"DotSymmetricVertexColoredDigraph\" Arg=\"digraph, v\ ert\"/>\n <Oper Name=\"DotSymmetricEdgeColoredDigraph\" Arg=\"digraph, edge\ \"/>\n <Returns>A string.</Returns>\n <Description>\n This function\ produces a graphical representation of the symmetric\n digraph <A>digrap\ h</A>. <C>DotSymmetricDigraph</C> will return an\n error if <A>digraph</A\ > is not a symmetric digraph. See\n <Ref Prop=\"IsSymmetricDigraph\"/>.<P\ />\n\n The function <C>DotSymmetricColoredDigraph</C> differs from <C>Dot\ Digraph</C>\n only in that the values given in the two lists are used to \ color the vertices\n and edges of the graph when displayed. The list for \ vertex colours should be\n a list of length equal to the number of vertic\ es, containing strings that\n are accepted by the graphviz software, whic\ h is the one used for graph\n representation. The list for edge colours \ should be a list of lists\n with the same shape of the outneighbours of t\ he digraph that contains strings \n that correspond to colours accepted b\ y the graphviz software.\n If the list is not the appropriate size, or ha\ s holes then the function\n will return an error. <P/>\n\n The func\ tion <C>DotSymmetricVertexColoredDigraph</C> differs from <C>DotDigraph</C>\n \ only in that the values in given in the list is used to color the vertice\ s\n of the graph when displayed. The list for vertex colours should be\n \ a list of length equal to the number of vertices, containing strings that\ \n are accepted by the graphviz software, which is the one used for graph\ \n representation. If the list is not the appropriate size, or has holes \ \n then the function will return an error. <P/>\n\n The function <C\ >DotSymmetricEdgeColoredDigraph</C> differs from <C>DotDigraph</C>\n only\ in that the values given in the list are used to color the edges\n of th\ e graph when displayed. The list for edge colours should be\n a list of l\ ists with the same shape of the outneighbours, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error. <P/>\n\n Vertices are displayed \ as circles, numbered consistently with\n <A>digraph</A>. Since <A>digraph\ </A> is symmetric, for every non-loop\n edge there is a complementary edg\ e with opposite source and range.\n <C>DotSymmetricDigraph</C> displays e\ ach pair of complementary edges\n as a single line between the relevant v\ ertices, with no arrowhead.<P/>\n\n The output is in <C>dot</C> format (a\ lso known as <C>GraphViz</C>)\n format. For details about this file forma\ t, and information about how\n to display or edit this format see\n \ <URL>https://www.graphviz.org</URL>. <P/>\n\n The string returned by <C>D\ otSymmetricDigraph</C> can be written to a\n file using the command <Ref \ Func=\"FileString\" BookName=\"GAPDoc\"/>.<P/>\n\n <Log><![CDATA[\ngap> s\ tar := Digraph([[2, 2, 3, 4], [1, 1], [1], [1, 4]]);\n<immutable multidigraph \ with 4 vertices, 9 edges>\ngap> IsSymmetricDigraph(star);\ntrue\ngap> FileStri\ ng(\"dot/star.dot\", DotSymmetricDigraph(gr));\ngap> D := Digraph([[2], [1, 3]\ , [2]]);\n<immutable digraph with 3 vertices, 4 edges>\ngap> vertcolors := [];\ ;\ngap> vertcolors[1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> ve\ rtcolors[3] := \"purple\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := []\ ;; edgecolors[2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] :=\ \"green\";; edgecolors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \"red\"\ ;; edgecolors[3][2] := \"red\";;\ngap> Print(DotSymmetricColoredDigraph(D, ver\ tcolors, edgecolors));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[color=blue\ , style=filled]\n2[color=pink, style=filled]\n3[color=purple, style=filled]\n1\ -- 2[color=green]\n2 -- 3[color=red]\n}\ngap> D := Digraph([[2], [1, 3], [2]]\ );\n<immutable digraph with 3 vertices, 4 edges>\ngap> vertcolors := [];;\ngap\ > vertcolors[1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> vertcolo\ rs[3] := \"purple\";;\ngap> Print(DotSymmetricVertexColoredDigraph(D, vertcolo\ rs));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[color=blue, style=filled]\n\ 2[color=pink, style=filled]\n3[color=purple, style=filled]\n1 -- 2\n2 -- 3\n}\ \ngap> D := Digraph([[2], [1, 3], [2]]);\n<immutable digraph with 3 vertices, \ 4 edges>\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] :=\ [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgeco\ lors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2]\ := \"red\";;\ngap> Print(DotSymmetricEdgeColoredDigraph(D, edgecolors));\n//d\ ot\ngraph hgn{\nnode [shape=circle]\n\n1\n2\n3\n1 -- 2[color=green]\n2 -- 3[co\ lor=red]\n}\n83]]></Log>\n </Description>\n </ManSection>\n\n <ManSec\ tion>\n <Attr Name=\"DotPartialOrderDigraph\" Arg=\"digraph\"/>\n <Retur\ ns>A string.</Returns>\n <Description>\n This function produces a grap\ hical representation of a partial order\n digraph <A>digraph</A>. <C>DotP\ artialOrderDigraph</C> will return an error\n if <A>digraph</A> is not a \ partial order digraph. See <Ref\n Prop=\"IsPartialOrderDigraph\"/>.<P/>\ \n\n Since <A>digraph</A> is a partial order, it is both reflexive and\n \ transitive. The output of <C>DotPartialOrderDigraph</C> is the\n <Re\ f Attr=\"DotDigraph\"/> of the\n <Ref Oper=\"DigraphReflexiveTransitiveRe\ duction\"/> of <A>digraph</A>.<P/>\n\n The output is in <C>dot</C> format\ (also known as <C>GraphViz</C>)\n format. For details about this file fo\ rmat, and information about how\n to display or edit this format see\n \ <URL>https://www.graphviz.org</URL>. <P/>\n\n The string returned by <\ C>DotPartialOrderDigraph</C> can be written to a\n file using the command\ <Ref Func=\"FileString\" BookName=\"GAPDoc\"/>.<P/>\n\n <Log><![CDATA[\n\ gap> poset := Digraph([[1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigra\ ph(gr);\ntrue\ngap> FileString(\"dot/poset.dot\", DotPartialOrderDigraph(gr));\ \n83]]></Log>\n </Description>\n </ManSection>\n\n <ManSection>\n \ <Attr Name=\"DotPreorderDigraph\" Arg=\"digraph\"/>\n <Attr Name=\"DotQuasi\ orderDigraph\" Arg=\"digraph\"/>\n <Returns>A string.</Returns>\n <Descr\ iption>\n This function produces a graphical representation of a preorder\ \n digraph <A>digraph</A>. <C>DotPreorderDigraph</C> will return an error\ \n if <A>digraph</A> is not a preorder digraph. See <Ref\n Prop=\"\ IsPreorderDigraph\"/>.<P/>\n\n A preorder digraph is reflexive and transi\ tive but in general it is\n not anti-symmetric and may have strongly conn\ ected components\n containing more than one vertex. The <Ref Oper=\"Quoti\ entDigraph\"/>\n <A>Q</A> obtained by forming the quotient of <A>digraph<\ /A> by the\n partition of its vertices into the strongly connected compon\ ents\n satisfies <Ref Prop=\"IsPartialOrderDigraph\"/>. Thus every vertex\ of\n <A>Q</A> corresponds to a strongly connected component of <A>digrap\ h</A>.\n The output of <C>DotPreorderDigraph</C> displays the \n <Re\ f Oper=\"DigraphReflexiveTransitiveReduction\"/> of <A>Q</A> with\n verti\ ces displayed as rounded rectangles labelled by all of the vertices\n of \ <A>digraph</A> in the corresponding strongly connected component. <P/>\n\n \ The output is in <C>dot</C> format (also known as <C>GraphViz</C>)\n fo\ rmat. For details about this file format, and information about how\n to \ display or edit this format see\n <URL>https://www.graphviz.org</URL>. <P\ />\n\n The string returned by <C>DotPreorderDigraph</C> can be written to\ a\n file using the command <Ref Func=\"FileString\" BookName=\"GAPDoc\"/\ >.<P/>\n\n <Log><![CDATA[\ngap> preset := Digraph([[1, 2, 4, 5], [1, 2, 4\ , 5], [3, 4], [4], [1, 2, 4, 5]);\ngap> IsPreorderDigraph(gr);\ntrue\ngap> Fil\ eString(\"dot/preset.dot\", DotProrderDigraph(gr));\n83]]></Log>\n </Descri\ ption>\n </ManSection>\n\n <ManSection>\n <Oper Name=\"DotHighlighted\ Digraph\" Arg=\"digraph, verts [, colour1, colour2]\"/>\n <Returns>A string\ .</Returns>\n <Description>\n <C>DotHighlightedDigraph</C> produces a \ graphical represenation of the\n digraph <A>digraph</A>, where the vertic\ es in the list <A>verts</A>, and\n edges between them, are drawn with col\ our <A>colour1</A> and all other\n vertices and edges in <A>digraph</A> a\ re drawn with colour <A>colour2</A>.\n If <A>colour1</A> and <A>colour2</\ A> are not given then\n <C>DotHighlightedDigraph</C> uses black and grey \ respectively. <P/>\n\n Note that <C>DotHighlightedDigraph</C> does not va\ lidate the colours\n <A>colour1</A> and <A>colour2</A> - consult the Grap\ hViz documentation to\n see what is available.\n\n See <Ref Attr=\"D\ otDigraph\"/> for more details on the output.<P/>\n\n <Log><![CDATA[\ngap\ > digraph := Digraph([[2, 3], [2], [1, 3]]);\n<digraph with 3 vertices, 5 edge\ s>\ngap> FileString(\"dot/my_digraph.dot\",\n> DotHighlightedDigraph(digraph, \ [1, 2], \"red\", \"black\"));\n264]]></Log>\n </Description>\n </ManSectio\ n>\n\n </Section>\n\n <Section><Heading>Reading and writing digraphs to a fi\ le</Heading>\n This section describes different ways to store and read grap\ hs\n from a file in the &Digraphs; package.\n <List>\n <Mark> Graph\ 6 </Mark>\n <Item>\n\t<E> Graph6 </E> is a graph data format for storing \ undirected graphs\n\twith no multiple edges nor loops of size up to <M> 2^{36}\ - 1 </M> in\n\tprintable chracters. The format consists of two parts. The fir\ st part\n\tuses one to eight bytes to store the number of vertices. And the se\ cond\n\tpart is the upper half of the adjacency matrix converted into ASCII\n\ \tcharacters. For a more detail description see <URL Text='Graph6'>\n\thttp://\ cs.anu.edu.au/~bdm/data/formats.txt</URL>.\n </Item>\n\n <Mark> Spar\ se6 </Mark>\n <Item>\n\t<E> Sparse6 </E> is a graph data format for stori\ ng undirected graphs\n\twith possibly multiple edges or loops. The maximal num\ ber of vertices\n\tallowed is <M> 2^{36} - 1 </M>. The format consists of two \ parts. The\n\tfirst part uses one to eight bytes to store the number of vertic\ es.\n\tAnd the second part only stores information about the edges.\n\tTherefo\ re, the <E> Sparse6 </E> format return a more compact encoding\n\tthen <E> Gra\ ph6 </E> for sparse graph, i.e. graphs where the number\n\tof edges is much l\ ess than the number of vertices squared. For a more\n\tdetail description see\ <URL Text='Sparse6'>\n http://cs.anu.edu.au/~bdm/data/formats.txt</URL\ >.\n </Item>\n\n <Mark> Digraph6 </Mark>\n <Item>\n\t<E>Digraph\ 6</E> is a new format based on <E> Graph6 </E>, but designed\n\tfor digraphs. \ The entire adjacency matrix is stored, and\n\ttherefore there is support for d\ irected edges and single-vertex loops.\n\tHowever, multiple edges are not supp\ orted.\n </Item>\n\n <Mark> DiSparse6 </Mark>\n <Item>\n\t<E>Di\ Sparse6</E> is a new format based on <E> Sparse6 </E>, but\n\tdesigned for dig\ raphs. In this format the list of edges is\n\tpartitioned into inceasing and d\ ecreasing edges, depending whether the\n\tedge has its source bigger than the \ range. Then both sets of edges are\n\twritten separetly in <E> Sparse6 </E> fo\ rmat with a separation symbol\n\tin between.\n </Item>\n </List>\n \ <ManSection>\n <Attr Name=\"String\" Arg=\"digraph\"/>\n <Oper Name=\"PrintS\ tring\" Arg=\"digraph\"/>\n <Returns>A string.</Returns>\n <Description>\n \ Returns a string <C>string</C> such that <C>EvalString(string)</C>\n is e\ qual to <A>digraph</A>, and has the same mutability.\n See <Ref Func=\"Eval\ String\" BookName=\"ref\" />.\n <P/>\n\n The methods installed for <C>St\ ring</C> make some attempts to\n ensure that <C>string</C> has as short a l\ ength as possible, but\n there may exist shorter strings that also evaluate\ to <A>digraph</A>.\n <P/>\n\n It is possible that <C>string</C> may con\ tain escaped special\n characters. To obtain a representation of <A>digraph\ </A> that\n can be entered as GAP input, please use\n <Ref Func=\"Print\ \" BookName=\"ref\" />.\n Note that <C>Print</C> for a digraph delegates t\ o\n <C>PrintString</C>, which delegates to <C>String</C>.\n <Log><![CDAT\ A[\ngap> D := CycleDigraph(3);\n<immutable cycle digraph with 3 vertices>\ngap\ > Print(D);\nCycleDigraph(3);\ngap> G := PetersenGraph(IsMutableDigraph);\n<mu\ table digraph with 10 vertices, 30 edges>\ngap> String(G);\n\"DigraphFromGraph\ 6String(IsMutableDigraph, \\\"IheA@GUAo\\\");\"\ngap> Print(last);\nDigraphFro\ mGraph6String(IsMutableDigraph, \"IheA@GUAo\");\ngap> DigraphFromGraph6String(\ IsMutableDigraph, \"IheA@GUAo\");\n<mutable digraph with 10 vertices, 30 edges\ >\n]]></Log>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Nam\ e=\"DigraphFromGraph6String\" Arg=\"[filt, ]str\"/>\n <Oper Name=\"Digraph\ FromDigraph6String\" Arg=\"[filt, ]str\"/>\n <Oper Name=\"DigraphFromSparse6\ String\" Arg=\"[filt, ]str\"/>\n <Oper Name=\"DigraphFromDiSparse6String\" \ Arg=\"[filt, ]str\"/>\n <Returns>A digraph.</Returns>\n <Description>\n I\ f <A>str</A> is a string encoding a graph in Graph6, Digraph6, Sparse6 or\n \ DiSparse6 format, then the corresponding function returns a digraph. In the\n\ case of either Graph6 or Sparse6, formats which do not support directed\n \ edges, this will be a digraph such that for every edge, the edge going in\n\ the opposite direction is also present.<P/>\n\n Each of these functions\ takes an optional first argument <A>filt</A>,\n which should be either <Re\ f Filt=\"IsMutableDigraph\"/>\n or <Ref Filt=\"IsImmutableDigraph\"/>,\n \ and which specifies whether the output digraph shall\n be mutable or immut\ able.\n If no first argument is provided, then an immutable\n digraph is\ returned by default.\n <Example><![CDATA[\ngap> DigraphFromGraph6String(\"\ ?\");\n<immutable empty digraph with 0 vertices>\ngap> DigraphFromGraph6String\ (\"C]\");\n<immutable symmetric digraph with 4 vertices, 8 edges>\ngap> Digrap\ hFromGraph6String(\"H?AAEM{\");\n<immutable symmetric digraph with 9 vertices,\ 22 edges>\ngap> DigraphFromDigraph6String(\"&?\");\n<immutable empty digraph \ with 0 vertices>\ngap> DigraphFromDigraph6String(IsMutableDigraph, \"&DOOOW?\"\ );\n<mutable digraph with 5 vertices, 5 edges>\ngap> DigraphFromDigraph6String\ (\"&CQFG\");\n<immutable digraph with 4 vertices, 6 edges>\ngap> DigraphFromDi\ graph6String(\"&IM[SrKLc~lhesbU[F_\");\n<immutable digraph with 10 vertices, 5\ 1 edges>\ngap> DigraphFromDiSparse6String(\".CaWBGA?b\");\n<immutable multidig\ raph with 4 vertices, 9 edges>\n]]></Example>\n </Description>\n</ManSectio\ n>\n\n <ManSection>\n <Oper Name=\"Graph6String\" Arg=\"digraph\"/>\n <Op\ er Name=\"Digraph6String\" Arg=\"digraph\"/>\n <Oper Name=\"Sparse6String\" A\ rg=\"digraph\"/>\n <Oper Name=\"DiSparse6String\" Arg=\"digraph\"/>\n <Retur\ ns>A string.</Returns>\n <Description>\n These four functions return a hig\ hly compressed string fully describing the\n digraph <A>digraph</A>. <P/>\n\ \n Graph6 and Digraph6 are formats best used on small, dense graphs, if\n \ applicable. For larger, sparse graphs use <E>Sparse6</E> and\n <E>Dispars\ e6</E> (this latter also preserves multiple edges). <P/>\n\n See <Ref Func=\ \"WriteDigraphs\"/>.\n <Example><![CDATA[\ngap> gr := Digraph([[2, 3], [1],\ [1]]);\n<immutable digraph with 3 vertices, 4 edges>\ngap> Sparse6String(gr);\ \n\":Bc\"\ngap> DiSparse6String(gr);\n\".Bc{f\"\n]]></Example>\n </Descriptio\ n>\n</ManSection>\n\n <ManSection>\n <Func Name=\"DigraphFile\" Arg=\"file\ name [, coder][, mode]\"/>\n <Returns>An IO file object.</Returns>\n <Descri\ ption>\n If <A>filename</A> is a string representing the name of a file, th\ en\n <C>DigraphFile</C> returns an &IO; package file object for that file.\ \n <P/>\n\n If the optional argument <A>coder</A> is specified\n and \ is a function which either encodes a digraph as a string, or decodes a\n st\ ring into a digraph, then this function will be used when reading or\n writ\ ing to the returned file object. If the optional argument <A>coder</A>\n is\ not specified, then the encoding of the digraphs in the returned file\n ob\ ject must be specified in the the file extension. The file extension must\n \ be one of: <C>.g6</C>, <C>.s6</C>, <C>.d6</C>, <C>.ds6</C>, <C>.txt</C>,\n \ <C>.p</C>, or <C>.pickle</C>; more details of these file formats is given\n \ below. <P/>\n\n If the optional argument <A>mode</A> is specified, then i\ t must be one of:\n <C>\"w\"</C> (for write), <C>\"a\"</C> (for append), or\ <C>\"r\"</C> (for read).\n If <A>mode</A> is not specified, then <C>\"r\"<\ /C> is used by default. <P/>\n\n If <A>filename</A> ends in one of: <C>.gz<\ /C>, <C>.bz2</C>, or\n <C>.xz</C>, then the digraphs which are read from, o\ r written to, the\n returned file object are decompressed, or compressed, a\ ppropriately.\n <P/>\n\n The file object returned by <C>DigraphFile</C> \ can be given as the first\n argument for either of the functions <Ref Func=\ \"ReadDigraphs\"/> or <Ref\n Func=\"WriteDigraphs\"/>. The purpose of thi\ s is to reduce the overhead of\n recreating the file object inside the func\ tions <Ref Func=\"ReadDigraphs\"/>\n or <Ref Func=\"WriteDigraphs\"/> when,\ for example, reading or writing many\n digraphs in a loop.\n <P/>\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n <List>\n <Mark>graph6 (.g6)</Mark>\n <Item>\n\tA standa\ rd and widely-used format for undirected graphs, with no support\n\tfor loops \ or multiple edges. Only symmetric graphs are allowed -- each\n\tedge is combi\ ned with its converse edge to produce a single undirected\n\tedge. This forma\ t is best used for \"dense\" graphs -- those with many\n\tedges per vertex.\n \ </Item>\n <Mark>sparse6 (.s6)</Mark>\n <Item>\n\tUnlike graph6,\ sparse6 has support for loops and multiple edges.\n\tHowever, its use is stil\ l limited to symmetric graphs. This format is\n\tbetter-suited to \"sparse\" \ graphs -- those with few edges per vertex.\n </Item>\n <Mark>digraph\ 6 (.d6)</Mark>\n <Item>\n\tThis format is based on graph6, but stores dir\ ection information -\n\ttherefore is not limited to symmetric graphs. Loops a\ re allowed, but\n\tmultiple edges are not. Best compression with \"dense\" gr\ aphs.\n </Item>\n <Mark>disparse6 (.ds6)</Mark>\n <Item>\n\tAny\ type of digraph can be encoded in disparse6: directions, loops, and\n\tmultip\ le edges are all allowed. Similar to sparse6, this has the best\n\tcompressio\ n rate with \"sparse\" graphs.\n </Item>\n <Mark>plain text (.txt)</\ Mark>\n <Item>\n\tThis is a human-readable format which stores graphs in \ the form\n <C>0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8</C> i.e. pairs of \ vertices\n describing edges in a graph. More specifically, the vertices\ making up\n one edge must be separated by a single space, and pairs of\ vertices must\n be separated by two spaces. <P/>\n\n See <Ref O\ per=\"ReadPlainTextDigraph\"/> for a more flexible way to store\n digra\ phs in a plain text file. <P/>\n </Item>\n\n <Mark>pickled (<F>.p</F\ > or <F>.pickle</F>)</Mark>\n <Item>\n Digraphs are pickled using \ the &IO; package. This is particularly good\n when the <Ref Attr=\"Digr\ aphGroup\"/> is non-trivial.\n </Item>\n </List>\n\n <Example><![CD\ ATA[\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ \ngap> file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> W\ riteDigraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file\ );;\ngap> file := DigraphFile(filename, \"r\");;\ngap> ReadDigraphs(file, 9);\ \n<immutable digraph with 3 vertices, 5 edges>\ngap> IO_Close(file);;\n]]></Ex\ ample>\n </Description>\n</ManSection>\n\n <ManSection>\n <Func Name=\"\ ReadDigraphs\" Arg=\"filename[, decoder][, n]\"/>\n <Returns>A digraph, or a \ list of digraphs.</Returns>\n <Description>\n If <A>filename</A> is a stri\ ng containing the name of a file containing\n encoded digraphs or an &IO; f\ ile object created using <Ref\n Func=\"DigraphFile\"/>, then <C>ReadDigra\ phs</C> returns the digraphs\n encoded in the file as a list. Note that if\ <A>filename</A> is a\n compressed file, which has been compressed appropri\ ately to give a filename\n extension of <C>.gz</C>, <C>.bz2</C>, or <C>.xz<\ /C>, then\n <C>ReadDigraphs</C> can read <A>filename</A> without it first n\ eeding to be\n decompressed.\n <P/>\n\n If the optional argument <A>d\ ecoder</A> is specified\n and is a function which decodes a string into a d\ igraph,\n then <C>ReadDigraphs</C> will use <A>decoder</A> to decode the di\ graphs\n contained in <A>filename</A>.<P/>\n\n If the optional argument \ <A>n</A> is specified, then <C>ReadDigraphs</C>\n returns the <A>n</A>th di\ graph encoded in the file <A>filename</A>.\n <P/>\n\n If the optional ar\ gument <A>decoder</A> is not specified, then\n <C>ReadDigraphs</C> will ded\ uce which decoder to use based on the filename\n extension of <A>filename</\ A> (after removing the compression-related\n filename extensions <C>.gz</C>\ , <C>.bz2</C>, and <C>.xz</C>). For example,\n if the filename extension i\ s <C>.g6</C>, then <C>ReadDigraphs</C> will use\n the graph6 decoder <Ref O\ per=\"DigraphFromGraph6String\"/>.\n <P/>\n\n The currently supported fi\ le formats, and associated filename extensions,\n are:\n <List>\n <\ Mark>graph6 (.g6)</Mark>\n <Item>\n\tA standard and widely-used format fo\ r undirected graphs, with no support\n\tfor loops or multiple edges. Only sym\ metric graphs are allowed -- each\n\tedge is combined with its converse edge t\ o produce a single undirected\n\tedge. This format is best used for \"dense\"\ graphs -- those with many\n\tedges per vertex.\n </Item>\n <Mark>sp\ arse6 (.s6)</Mark>\n <Item>\n\tUnlike graph6, sparse6 has support for loo\ ps and multiple edges.\n\tHowever, its use is still limited to symmetric graph\ s. This format is\n\tbetter-suited to \"sparse\" graphs -- those with few edg\ es per vertex.\n </Item>\n <Mark>digraph6 (.d6)</Mark>\n <Item>\ \n\tThis format is based on graph6, but stores direction information -\n\tther\ efore is not limited to symmetric graphs. Loops are allowed, but\n\tmultiple \ edges are not. Best compression with \"dense\" graphs.\n </Item>\n \ <Mark>disparse6 (.ds6)</Mark>\n <Item>\n\tAny type of digraph can be enco\ ded in disparse6: directions, loops, and\n\tmultiple edges are all allowed. S\ imilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graph\ s.\n </Item>\n <Mark>plain text (.txt)</Mark>\n <Item>\n\tThis \ is a human-readable format which stores graphs in the form\n <C>0 7 0 \ 8 1 7 2 8 3 8 4 8 5 8 6 8</C> i.e. pairs of vertices\n describing\ edges in a graph. More specifically, the vertices making up\n one edge\ must be separated by a single space, and pairs of vertices must\n be s\ eparated by two spaces. <P/>\n\n See <Ref Oper=\"ReadPlainTextDigraph\"\ /> for a more flexible way to store\n digraphs in a plain text file. <P\ />\n<!--\n Just so I remember:\n <C>ReadPlainTextDigraph(\"data-\ local/soc-Epinions1.txt.gz\", \"\\t\", 1,\n '#');</C>\n-->\n </Ite\ m>\n\n <Mark>pickled (<F>.p</F> or <F>.pickle</F>)</Mark>\n <Item>\n\ Digraphs are pickled using the &IO; package. This is particularly good\ \n when the <Ref Attr=\"DigraphGroup\"/> is non-trivial.\n </Item>\ \n </List>\n\n <Example><![CDATA[\ngap> ReadDigraphs(\n> Concatenation(D\ IGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 10);\n<immutable symmetric digraph wit\ h 5 vertices, 8 edges>\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"\ /data/graph5.g6.gz\"), 17);\n<immutable symmetric digraph with 5 vertices, 12 \ edges>\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/tree9.4.tx\ t\"));\n[ <immutable digraph with 9 vertices, 8 edges>, \n <immutable digraph\ with 9 vertices, 8 edges>, \n <immutable digraph with 9 vertices, 8 edges>, \ \n <immutable digraph with 9 vertices, 8 edges>, \n <immutable digraph with \ 9 vertices, 8 edges>, \n <immutable digraph with 9 vertices, 8 edges>, \n <i\ mmutable digraph with 9 vertices, 8 edges>, \n <immutable digraph with 9 vert\ ices, 8 edges>, \n <immutable digraph with 9 vertices, 8 edges>, \n <immutab\ le digraph with 9 vertices, 8 edges>, \n <immutable digraph with 9 vertices, \ 8 edges>, \n <immutable digraph with 9 vertices, 8 edges>, \n <immutable dig\ raph with 9 vertices, 8 edges>, \n <immutable digraph with 9 vertices, 8 edge\ s> ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <F\ unc Name=\"WriteDigraphs\" Arg=\"filename, digraphs[, encoder] [, mode]\"/>\n \ <Description>\n If <A>digraphs</A> is a list of digraphs or a digraph and \ <A>filename</A>\n is a string or an &IO; file object created using <Ref Fun\ c=\"DigraphFile\"/>,\n then <C>WriteDigraphs</C> writes the digraphs to the\ file represented by\n <A>filename</A>. If the supplied filename ends in o\ ne of the extensions\n <C>.gz</C>, <C>.bz2</C>, or <C>.xz</C>, then the fil\ e will be compressed\n appropriately. Excluding these extensions, if the f\ ile ends with an\n extension in the list below, the corresponding graph for\ mat will be used to\n encode it. If such an extension is not included, an \ appropriate format\n will be chosen intelligently, and an extension appende\ d, to minimise file\n size.\n <P/>\n\n <!-- TODO: encode and mode. --\ >\n\n For more verbose information on the progress of the function, set the\ info\n level of <A>InfoDigraphs</A> to 1 or higher, using <C>SetInfoLevel<\ /C>.<P/>\n\n The currently supported file formats are:\n <List>\n <\ Mark>graph6 (.g6)</Mark>\n <Item>\n\tA standard and widely-used format fo\ r undirected graphs, with no support\n\tfor loops or multiple edges. Only sym\ metric graphs are allowed -- each\n\tedge is combined with its converse edge t\ o produce a single undirected\n\tedge. This format is best used for \"dense\"\ graphs -- those with many\n\tedges per vertex.\n </Item>\n <Mark>sp\ arse6 (.s6)</Mark>\n <Item>\n\tUnlike graph6, sparse6 has support for loo\ ps and multiple edges.\n\tHowever, its use is still limited to symmetric graph\ s. This format is\n\tbetter-suited to \"sparse\" graphs -- those with few edg\ es per vertex.\n </Item>\n <Mark>digraph6 (.d6)</Mark>\n <Item>\ \n\tThis format is based on graph6, but stores direction information -\n\tther\ efore is not limited to symmetric graphs. Loops are allowed, but\n\tmultiple \ edges are not. Best compression with \"dense\" graphs.\n </Item>\n \ <Mark>disparse6 (.ds6)</Mark>\n <Item>\n\tAny type of digraph can be enco\ ded in disparse6: directions, loops, and\n\tmultiple edges are all allowed. S\ imilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graph\ s.\n </Item>\n\n <Mark>plain text (.txt)</Mark>\n <Item>\n\tThi\ s is a human-readable format which stores graphs in the form\n <C>0 7 \ 0 8 1 7 2 8 3 8 4 8 5 8 6 8</C> i.e. pairs of vertices\n describi\ ng edges in a graph. More specifically, the vertices making up\n one ed\ ge must be separated by a single space, and pairs of vertices must\n be\ separated by two spaces. <P/>\n\n See <Ref Oper=\"ReadPlainTextDigraph\ \"/> for a more flexible way to store\n digraphs in a plain text file. \ <P/>\n<!--\n Just so I remember:\n <C>ReadPlainTextDigraph(\"dat\ a-local/soc-Epinions1.txt.gz\", \"\\t\", 1, '#');</C>\n-->\n </Item>\n \ <Mark>pickled (<F>.p</F> or <F>.pickle</F>)</Mark>\n <Item>\n D\ igraphs are pickled using the &IO; package. This is particularly good\n \ when the <Ref Attr=\"DigraphGroup\"/> is non-trivial.\n </Item>\n </L\ ist>\n\n <Example><![CDATA[\ngap> grs := [];;\ngap> grs[1] := Digraph([]);\ \n<immutable empty digraph with 0 vertices>\ngap> grs[2] := Digraph([[1, 3], [\ 2], [1, 2]]);\n<immutable digraph with 3 vertices, 5 edges>\ngap> grs[3] := Di\ graph([\n> [6, 7], [6, 9], [1, 3, 4, 5, 8, 9],\n> [1, 2, 3, 4, 5, 6, 7, 10], [\ 1, 5, 6, 7, 10], [2, 4, 5, 9, 10],\n> [3, 4, 5, 6, 7, 8, 9, 10], [1, 3, 5, 7, \ 8, 9], [1, 2, 5],\n> [1, 2, 4, 6, 7, 8]]);\n<immutable digraph with 10 vertice\ s, 51 edges>\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6\ .gz\");;\ngap> WriteDigraphs(filename, grs, \"w\");\nIO_OK\ngap> ReadDigraphs(\ filename);\n[ <immutable empty digraph with 0 vertices>, \n <immutable digrap\ h with 3 vertices, 5 edges>, \n <immutable digraph with 10 vertices, 51 edges\ > ]]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Fu\ nc Name=\"IteratorFromDigraphFile\" Arg=\"filename [, decoder]\"/>\n <Returns\ >An iterator.</Returns>\n <Description>\n If <A>filename</A> is a string r\ epresenting the name of a file containing\n encoded digraphs, then <C>Itera\ torFromDigraphFile</C> returns an iterator\n for which the value of <Ref Op\ er=\"NextIterator\" BookName=\"ref\"/> is the\n next digraph encoded in the\ file.\n <P/>\n\n If the optional argument <A>decoder</A> is specified a\ nd is a function\n which decodes a string into a digraph, then <C>IteratorF\ romDigraphFile</C>\n will use <A>decoder</A> to decode the digraphs contain\ ed in <A>filename</A>.\n <P/>\n\n The purpose of this function is to eas\ ily allow looping over digraphs\n encoded in a file when loading all of the\ encoded digraphs would require\n too much memory. <P/>\n\n To see what \ file types are available, see <Ref Func=\"WriteDigraphs\"/>.\n\n <Example><\ ![CDATA[\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\ \");;\ngap> file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\ \n> WriteDigraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Clo\ se(file);;\ngap> iter := IteratorFromDigraphFile(filename);\n<iterator>\ngap> \ for x in iter do od;\n]]></Example>\n </Description>\n</ManSection>\n\n <M\ anSection>\n <Func Name=\"DigraphPlainTextLineEncoder\" Arg=\"delimiter1, [de\ limiter2,] offset\"/>\n <Oper Name=\"DigraphPlainTextLineDecoder\" Arg=\"deli\ miter1, [delimiter2,] offset\"/>\n <Returns>A string.</Returns>\n <Descripti\ on>\n These two functions return a function which encodes or decodes a digr\ aph in\n a plain text format.<P/>\n\n <A>DigraphPlainTextLineEncoder</A>\ returns a function which takes a single\n digraph as an argument. The fun\ ction returns a string describing the edges\n of that digraph; each edge is\ written as a pair of integers separated by the\n string <A>delimiter2</A>,\ and the edges themselves are separated by the\n string <A>delimiter1</A>. \ <A>DigraphPlainTextLineDecoder</A> returns the\n corresponding decoder fun\ ction, which takes a string argument in this format\n and returns a digraph\ .<P/>\n\n If only one delimiter is passed as an argument to\n <A>Digraph\ PlainTextLineDecoder</A>, it will return a function which decodes\n a singl\ e edge, returning its contents as a list of integers.<P/>\n\n The argument \ <A>offset</A> should be an integer, which will describe a\n number to be ad\ ded to each vertex before it is encoded, or after it is\n decoded. This ma\ y be used, for example, to label vertices starting at 0\n instead of 1.<P/>\ \n\n Note that the number of vertices of a digraph is not stored, and so ve\ rtices\n which are not connected to any edge may be lost.\n\n <Example><\ ![CDATA[\ngap> gr := Digraph([[2, 3], [1], [1]]);\n<immutable digraph with 3 v\ ertices, 4 edges>\ngap> enc := DigraphPlainTextLineEncoder(\" \", \" \", -1);\ ;\ngap> dec := DigraphPlainTextLineDecoder(\" \", \" \", 1);;\ngap> enc(gr);\ \n\"0 1 0 2 1 0 2 0\"\ngap> dec(last);\n<immutable digraph with 3 vertices,\ 4 edges>\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\ \n <Oper Name=\"TournamentLineDecoder\" Arg=\"str\"/>\n <Returns>A digraph.<\ /Returns>\n <Description>\n This function takes a string <A>str</A>, decod\ es it, and then returns the\n tournament [see <Ref Prop=\"IsTournament\"/>]\ which it defines, according to\n the following rules. <P/>\n\n The char\ acters of the string <A>str</A> represent the entries in the upper\n triang\ le of a tournament's adjacency matrix. The number of vertices\n <C>n</C> wi\ ll be detected from the length of the string and will be as\n large as poss\ ible. <P/>\n\n The first character represents the possible edge <C>1 -> 2<\ /C>, the second\n represents <C>1 -> 3</C> and so on until <C>1 -> n</C>; t\ hen the following\n character represents <C>2 -> 3</C>, and so on up to the\ character which\n represents the edge <C>n-1 -> n</C>. <P/>\n\n If a ch\ aracter of the string with corresponding edge <C>i -> j</C> is equal\n to <\ C>1</C>, then the edge <C>i -> j</C> is present in the tournament.\n Otherw\ ise, the edge <C>i -> j</C> is present instead. In this way, all the\n pos\ sible edges are encoded one-by-one.\n\n <Example><![CDATA[\ngap> gr := Tour\ namentLineDecoder(\"100001\");\n<immutable digraph with 4 vertices, 6 edges>\n\ gap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1, 2 ] ]\n]]></Example>\ \n </Description>\n</ManSection>\n\n <ManSection>\n <Oper Name=\"Adjacenc\ yMatrixUpperTriangleLineDecoder\" Arg=\"str\"/>\n <Returns>A digraph.</Return\ s>\n <Description>\n This function takes a string <A>str</A>, decodes it, \ and then returns the\n topologically sorted digraph [see <Ref Attr=\"Digrap\ hTopologicalSort\"/>]\n which it defines, according to the following rules.\ <P/>\n\n The characters of the string <A>str</A> represent the entries in \ the upper\n triangle of a digraph's adjacency matrix. The number of vertice\ s <C>n</C>\n will be detected from the length of the string and will be as \ large as\n possible. <P/>\n\n The first character represents the possib\ le edge <C>1 -> 2</C>, the second\n represents <C>1 -> 3</C> and so on unti\ l <C>1 -> n</C>; then the following\n character represents <C>2 -> 3</C>, a\ nd so on up to the character which\n represents the edge <C>n-1 -> n</C>. \ If a character of the string with\n corresponding edge <C>i -> j</C> is equ\ al to <C>1</C>, then this edge is\n present in the digraph. Otherwise, it i\ s not present. In this way, all the\n possible edges are encoded one-by-on\ e. <P/>\n\n In particular, note that there exists no edge <C>[i, j]</C> if \ <M>j \\leq\n i</M>. In order words, the digraph will be topologically so\ rted.\n\n <Example><![CDATA[\ngap> gr := AdjacencyMatrixUpperTriangleLineDe\ coder(\"100001\");\n<immutable digraph with 4 vertices, 2 edges>\ngap> OutNeig\ hbours(gr);\n[ [ 2 ], [ ], [ 4 ], [ ] ]\ngap> gr := AdjacencyMatrixUpperTria\ ngleLineDecoder(\"111111x111\");\n<immutable digraph with 5 vertices, 9 edges>\ \ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 3, 4 ], [ 4, 5 ], [ 5 ], [ ] ]\ \n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <Oper \ Name=\"TCodeDecoder\" Arg=\"str\"/>\n <Returns>A digraph.</Returns>\n <Descr\ iption>\n If <A>str</A> is a string consisting of at least two non-negative\ integers\n separated by spaces, then this function will attempt to return \ the digraph\n which it defines as a TCode string. <P/>\n\n The first int\ eger of the string defines the number of vertices <C>v</C> in\n the digraph\ , and the second defines the number of edges <C>e</C>. The\n following <C>\ 2e</C> integers should be vertex numbers in the range <C>[0 ..\n v-1]</C>\ . These integers are read in pairs and define the digraph's\n edges. This \ function will return an error if <A>str</A> has fewer than\n <C>2e+2</C> en\ tries. <P/>\n\n Note that the vertex numbers will be incremented by 1 in th\ e digraph\n returned. Hence the string fragment <C>0 6</C> will describe t\ he edge\n <C>[1,7]</C>.\n <Example><![CDATA[\ngap> gr := TCodeDecoder(\"\ 3 2 0 2 2 1\");\n<immutable digraph with 3 vertices, 2 edges>\ngap> OutNeighbo\ urs(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeDecoder(\"12 3 0 10 5 2 8 8\ \");\n<immutable digraph with 12 vertices, 3 edges>\ngap> OutNeighbours(gr);\n\ [ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ], [ 9 ], [ ], \n [ ], [ \ ] ]\n]]></Example>\n </Description>\n</ManSection>\n\n <ManSection>\n <O\ per Name=\"PlainTextString\" Arg=\"digraph\"/>\n <Oper Name=\"DigraphFromPlai\ nTextString\" Arg=\"s\"/>\n <Returns>A string.</Returns>\n <Description>\n \ <A>PlainTextString</A> takes a single digraph, and returns a string\n des\ cribing the edges of that digraph. <A>DigraphFromPlainTextString</A>\n tak\ es such a string and returns the digraph which it describes. Each edge\n i\ s written as a pair of integers separated by a single space. The edges\n t\ hemselves are separated by a double space. Vertex numbers are reduced by\n \ 1 when they are encoded, so that vertices in the string are labelled\n sta\ rting at 0.<P/>\n\n Note that the number of vertices of a digraph is not st\ ored, and so vertices\n which are not connected to any edge may be lost.<P/\ >\n\n \n The operation <C>DigraphFromPlainTextString</C>\n takes an o\ ptional first argument <Ref Filt=\"IsMutableDigraph\"/>\n or <Ref Filt=\"Is\ ImmutableDigraph\"/>, which specifies whether the output digraph shall\n be\ mutable or immutable. If no first argument is provided, then an immutable\n \ digraph is returned by default.\n\n <Example><![CDATA[\ngap> gr := Digrap\ h([[2, 3], [1], [1]]);\n<immutable digraph with 3 vertices, 4 edges>\ngap> Pla\ inTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFromPlainTextString(las\ t);\n<immutable digraph with 3 vertices, 4 edges>\n]]></Example>\n </Descript\ ion>\n</ManSection>\n\n <ManSection>\n <Func Name=\"WritePlainTextDigraph\ \" Arg=\"filename, digraph, delimiter, offset\"/>\n <Oper Name=\"ReadPlainTex\ tDigraph\" Arg=\"filename, delimiter, offset, ignore\"/>\n <Description>\n \ These functions write and read a single digraph in a human-readable plain\n \ text format as follows: each line contains a single edge, and each edge is\n\ written as a pair of integers separated by the string <A>delimiter</A>.<P/\ >\n\n <A>filename</A> should be the name of a file which will be written to\ or\n read from, and <A>offset</A> should be an integer which is added to e\ ach\n vertex number as it is written or read. For example, if\n <C>Writ\ ePlainTextDigraph</C> is called with <A>offset</A> <C>-1</C>, then\n the ve\ rtices will be numbered in the file starting from 0 instead of 1 -\n <C>Rea\ dPlainTextDigraph</C> would then need to be called with <A>offset</A>\n <C>\ 1</C> to convert back to the original graph.<P/>\n\n <A>ignore</A> should b\ e a list of characters which will be ignored when\n reading the graph.\n \ <Example><![CDATA[\ngap> gr := Digraph([[1, 2, 3], [1, 1], [2]]);\n<immutable\ multidigraph with 3 vertices, 6 edges>\ngap> filename := Concatenation(DIGRAP\ HS_Dir(), \"/tst/out/plain.txt\");;\ngap> WritePlainTextDigraph(filename, gr, \ \",\", -1);\ngap> ReadPlainTextDigraph(filename, \",\", 1, ['/', '%']);\n<immu\ table multidigraph with 3 vertices, 6 edges>\n]]></Example>\n </Description>\ \n</ManSection>\n\n <ManSection>\n <Oper Name=\"WriteDIMACSDigraph\" Arg=\ \"filename, digraph\"/>\n <Oper Name=\"ReadDIMACSDigraph\" Arg=\"filename\"/>\ \n <Description>\n These operations write or read the single symmetric dig\ raph <A>digraph</A>\n to or from a file in DIMACS format, as appropriate. T\ he operation\n <C>WriteDIMACSDigraph</C> records the vertices and edges of \ <A>digraph</A>.\n The vertex labels of <A>digraph</A> will be recorded only\ if they are\n integers. See <Ref Prop=\"IsSymmetricDigraph\"/> and <Ref\n \ Oper=\"DigraphVertexLabels\"/>.<P/>\n\n The first argument <A>filename\ </A> should be the name of the file which will\n be written to or read from\ . A file can contain one symmetric digraph in\n DIMACS format. If <A>filen\ ame</A> ends in one of <C>.gz</C>, <C>.bz2</C>,\n or <C>.xz</C>, then the f\ ile is compressed, or decompressed, appropriately.\n <P/>\n\n The DIMACS\ format is described as follows. Each line in the DIMACS file has\n one of\ four types:\n <List>\n <Item>\n A line beginning with <C>c</C>\ and followed by any number of characters\n is a comment line, and is i\ gnored.\n </Item>\n <Item>\n A line beginning with <C>p</C> d\ efines the numbers of vertices and edges\n the digraph. This line has \ the format <C>p edge <nr_vertices>\n <nr_edges></C>, wher\ e <C><nr_vertices></C> and\n <C><nr_edges></C> are replaced\ by the relevant integers. There\n must be exactly one such line in the\ file, and it must occur before any\n of the following kinds of line.<P\ />\n\n Although it is required to be present, the value of\n <C>\ <nr_edges></C> will be ignored. The correct number of edges\n wil\ l be deduced from the rest of the information in the file.\n </Item>\n \ <Item>\n A line of the form <C>e <v> <w></C>, where <C>&\ lt;v></C>\n and <C><w></C> are integers in the range <C>[1 ..\ \n <nr_vertices>]</C>, specifies that there is a (symmetric) ed\ ge\n in the digraph between the vertices <C><v></C> and\n \ <C><w></C>. A symmetric edge only needs to be defined once; an\n \ additional line <C>e <v> <w></C>, or <C>e <w>\n <\ ;v></C>, will be interpreted as an additional, multiple, edge.\n Loo\ ps are permitted.\n </Item>\n <Item>\n A line of the form <C>\ n <v> <label></C>, where\n <C><v></C> is an integer i\ n the range <C>[1 ..\n <nr_vertices>]</C> and <C><label><\ /C> is an integer,\n signifies that the vertex <C><v></C> has the\ label\n <C><label></C> in the digraph. If a label is not specifi\ ed for a\n vertex, then <C>ReadDIMACSDigraph</C> will assign the label\ \n <C>1</C>, according to the DIMACS specification.\n </Item>\n \ </List>\n\n A detailed definition of the DIMACS format can be found at\n \ <URL>http://mat.gsia.cmu.edu/COLOR/general/ccformat.ps</URL>, in Section\n \ 2.1. Note that optional descriptor lines, as described in Section 2.1, will\ \n be ignored.\n <P/>\n\n\n <Example><![CDATA[\ngap> gr := Digraph([[\ 2], [1, 3, 4], [2, 4], [2, 3]]);\n<immutable digraph with 4 vertices, 8 edges>\ \ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigraph(filename, gr);;\ngap\ > ReadDIMACSDigraph(filename);\n<immutable digraph with 4 vertices, 8 edges>]]\ ></Example>\n </Description>\n</ManSection>\n\n </Section>\n</Chapter>\n <!-\ - visualising, io -->\n</Body>\n\n<Appendix>\n <Heading>\n Grape to Digrap\ hs Command Map\n </Heading>\n Below is a table of &Grape; commands with the \ &Digraphs; counterparts.\n The sections in this chapter correspond to the cha\ pters in the &Grape;\n manual.\n\n <Section>\n <Heading>\n Functions\ to construct and modify graphs\n </Heading>\n <Alt Not=\"HTML\">\n \ <E>\n The table in this section contains more information when viewed \ in\n html format.\n </E>\n </Alt>\n\n <Table Align=\"|l|l|l|\ \">\n <HorLine/>\n <Row>\n <Item>\n &Grape; command\ \n </Item>\n <Item>\n &Digraphs; command\n </Ite\ m>\n <Item>\n <Alt Only=\"HTML\">\n Comments\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>Graph</C>\n </Item>\n <Item>\n \ <Ref Oper=\"Digraph\"/>\n </Item>\n <Item>\n <Alt O\ nly=\"HTML\">\n Used with the arguments of <C>Graph</C> in &Grape;, <\ Ref\n Oper=\"Digraph\"/> will return an equivalent digraph. <P/>\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row\ >\n <Item>\n <C>EdgeOrbitsGraph</C>\n </Item>\n \ <Item>\n <Ref Oper=\"EdgeOrbitsDigraph\"/>\n </Item>\n \ <Item>\n <Alt Only=\"HTML\">\n The operation <Ref Oper=\"Edg\ eOrbitsDigraph\"/> behaves the same as\n <C>EdgeOrbitsDigraph</C> in \ &Grape;.\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\ \n <Row>\n <Item>\n <C>NullGraph</C>\n </Item>\n \ <Item>\n <Ref Oper=\"NullDigraph\"/>\n <Alt Only=\"HTM\ L\">\n or\n <Ref Oper=\"EmptyDigraph\"/>\n </Al\ t>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ Note that, unlike in &Grape;, the arguments of <Ref\n Oper=\"N\ ullDigraph\"/> and <Ref Oper=\"EmptyDigraph\"/> are just\n non-nega\ tive integers. Since digraphs in &Digraphs; are immutable,\n there \ is no reason (although it is possible) to create a null digraph\n a\ nd then add edges, as you might be used to doing in &Grape;.\n </Alt>\ \n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Ite\ m>\n <C>CompleteGraph</C>\n </Item>\n <Item>\n \ <Ref Oper=\"CompleteDigraph\"/>\n </Item>\n <Item>\n <\ Alt Only=\"HTML\">\n Note that, unlike in &Grape;, the argument to \ <Ref\n Oper=\"CompleteDigraph\"/> is simply a non-negative intege\ r\n indicating the number of vertices. By default a complete digrap\ h in\n &Digraphs; does not have any loops. If you want a complete d\ igraph\n with loops at every vertex, then use <Ref\n O\ per=\"DigraphAddAllLoops\"/>.\n </Alt>\n </Item>\n </Row>\ \n <HorLine/>\n\n <Row>\n <Item>\n <C>JohnsonGraph</\ C>\n </Item>\n <Item>\n <Ref Oper=\"JohnsonDigraph\"/>\ \n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ The &Grape; function <C>JohnsonGraph</C> is equivalent to\n <Ref Op\ er=\"JohnsonDigraph\"/>.\n </Alt>\n </Item>\n </Row>\n \ <HorLine/>\n\n <Row>\n <Item>\n <C>CayleyGraph</C>\n \ </Item>\n <Item>\n <Ref Oper=\"CayleyDigraph\"/>\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n The oper\ ation <Ref Oper=\"CayleyDigraph\"/> behaves the same as\n <C>Cayley\ Graph</C> in &Grape;.\n </Alt>\n </Item>\n </Row>\n \ <HorLine/>\n\n <Row>\n <Item>\n <C>AddEdgeOrbit</C>\n \ </Item>\n <Item>\n <Ref Oper=\"DigraphAddEdgeOrbit\"/>\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n Note\ that, unlike in &Grape;, digraphs in &Digraphs; are immutable,\n a\ nd this operation returns a new digraph, i.e. it does not modify the\n \ digraph given as an argument.\n </Alt>\n </Item>\n </\ Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>RemoveEdge\ Orbit</C>\n </Item>\n <Item>\n <Ref Oper=\"DigraphRemov\ eEdgeOrbit\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\ \n Note that, unlike in &Grape;, digraphs in &Digraphs; are immutab\ le,\n and this operation returns a new digraph, i.e. it does not mo\ dify the\n digraph given as an argument.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n \ <C>AssignVertexNames</C>\n </Item>\n <Item>\n <Ref\ Oper=\"SetDigraphVertexLabels\"/>\n <Alt Only=\"HTML\">\n \ and\n <Ref Oper=\"SetDigraphVertexLabel\"/>\n </Alt>\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n Use \ <Ref Oper=\"SetDigraphVertexLabel\"/> to set an individual\n vertex\ label and <Ref Oper=\"SetDigraphVertexLabels\"/> to set all of\n t\ he vertex labels.\n </Alt>\n </Item>\n </Row>\n <Hor\ Line/>\n </Table>\n </Section>\n\n <Section>\n <Heading>\n Functi\ ons to inspect graphs, vertices and edges\n </Heading>\n <Alt Not=\"HTML\ \">\n <E>\n The table in this section contains more information wh\ en viewed in\n html format.\n </E>\n </Alt>\n\n <Table Align\ =\"|l|l|l|\">\n <HorLine/>\n\n <Row>\n <Item>\n &Gra\ pe; command\n </Item>\n <Item>\n &Digraphs; command\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n Comm\ ents\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n \ <Row>\n <Item>\n <C>IsGraph</C>\n </Item>\n <\ Item>\n <Ref Filt=\"IsDigraph\"/>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n <Ref Filt=\"IsDigraph\"/> is a categ\ ory and not a function like\n <C>IsGraph</C>.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n\ <C>OrderGraph</C>\n </Item>\n <Item>\n <Ref A\ ttr=\"DigraphNrVertices\"/>\n </Item>\n <Item>\n <Alt O\ nly=\"HTML\">\n The operation <Ref Attr=\"DigraphNrVertices\"/> beh\ aves the same as\n <C>OrderGraph</C> in &Grape;.\n </Alt>\ \n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Ite\ m>\n <C>IsVertex(<A>graph</A>, <A>v</A>)</C>\n </Item>\n \ <Item>\n <C><A>v</A> in DigraphVertices(<A>digraph</A>)</C>\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n The attr\ ibute <Ref Attr=\"DigraphVertices\"/> can be used to return\n the l\ ist of vertices of a digraph, or if <A>v</A> is known to be\n a pos\ itive integer, then you can simply do\n <C><A>v</A> <= DigraphNr\ Vertices(<A>digraph</A>)</C>\n </Alt>\n </Item>\n </Row>\ \n <HorLine/>\n\n <Row>\n <Item>\n <C>VertexName</C>\ \n </Item>\n <Item>\n <Ref Oper=\"DigraphVertexLabel\"/\ >\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ Unlike <C>VertexName</C> in &Grape;, the return value of\n <Ref Op\ er=\"DigraphVertexLabel\"/> is mutable.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>Vert\ exNames</C>\n </Item>\n <Item>\n <Ref Oper=\"DigraphVer\ texLabels\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\ \n Unlike <C>VertexNames</C> in &Grape;, the return value of\n \ <Ref Oper=\"DigraphVertexLabels\"/> is mutable.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n \ <C>Vertices</C>\n </Item>\n <Item>\n <Ref Attr=\ \"DigraphVertices\"/>\n </Item>\n <Item>\n <Alt Only=\"\ HTML\">\n The vertices of a digraph in &Digraphs; is always the ran\ ge <C>[1\n .. n]</C> where <C>n</C> is the number of vertices of \ the\n digraph. The operation <Ref Attr=\"DigraphVertices\"/> behav\ es the\n same as <C>Vertices</C> in &Grape;.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n \ <C>VertexDegree</C>\n </Item>\n <Item>\n <Ref \ Oper=\"OutDegreeOfVertex\"/>\n <Alt Only=\"HTML\">\n (see \ also: <Ref Oper=\"InDegreeOfVertex\"/>)\n </Alt>\n </Item>\n \ <Item>\n <Alt Only=\"HTML\">\n The operation <Ref Op\ er=\"OutDegreeOfVertex\"/> behaves the same as\n <C>VertexDegree</C\ > in &Grape;.\n </Alt>\n </Item>\n </Row>\n <HorLine\ />\n\n <Row>\n <Item>\n <C>VertexDegrees</C>\n </I\ tem>\n <Item>\n <Ref Attr=\"OutDegreeSet\"/>\n <Alt O\ nly=\"HTML\">\n (see also:\n <Ref Attr=\"OutDegreeSequen\ ce\"/>,\n <Ref Attr=\"OutDegrees\"/>,\n <Ref Attr=\"InDe\ grees\"/>,\n <Ref Attr=\"InDegreeSequence\"/>,\n <Ref At\ tr=\"InDegreeSet\"/>)\n </Alt>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n The operation <Ref Attr=\"OutDegreeSet\ \"/> behaves the same as\n <C>VertexDegrees</C> in &Grape;.\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>IsLoopy</C>\n </Item>\n <Item>\n \ <Ref Prop=\"DigraphHasLoops\"/>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n The operation <Ref Prop=\"DigraphHasLoops\ \"/> behaves the same as\n <C>IsLoopy</C> in &Grape;.\n </\ Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>IsSimpleGraph</C>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n <Ref Prop=\"IsSymmetricDigraph\"/>\n \ and not <Ref Prop=\"DigraphHasLoops\"/>\n </Alt>\n \ <Alt Not=\"HTML\">\n <Ref Prop=\"IsSymmetricDigraph\"/>\n \ </Alt>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ The &Grape; function <C>IsSimpleGraph</C> is equivalent to\n \ <Ref Prop=\"IsSymmetricDigraph\"/>\n and not <Ref Prop=\"DigraphHa\ sLoops\"/>.\n </Alt>\n </Item>\n </Row>\n <HorLine/>\ \n\n <Row>\n <Item>\n <C>Adjacency</C>\n </Item>\n\ <Item>\n <Ref Oper=\"OutNeighboursOfVertex\"/>\n <Al\ t Only=\"HTML\">\n (see also <Ref Oper=\"InNeighboursOfVertex\"/>)\ \n </Alt>\n </Item>\n <Item>\n <Alt Only=\"HTM\ L\">\n The operation <Ref Oper=\"OutNeighboursOfVertex\"/> behaves \ the same\n as <C>Adjacency</C> in &Grape;.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n \ <C>IsEdge</C>\n </Item>\n <Item>\n <Ref Oper=\"I\ sDigraphEdge\" Label=\"for digraph and list\"/>\n </Item>\n <Ite\ m>\n <Alt Only=\"HTML\">\n The arguments of\n <\ Ref Oper=\"IsDigraphEdge\" Label=\"for digraph and list\"/>\n can b\ e a digraph and an edge, or a digraph and two vertices. In the\n fi\ rst form the operation\n <Ref Oper=\"IsDigraphEdge\" Label=\"for di\ graph and list\"/>\n behaves the same as <C>IsEdge</C> in &Grape;.\ \n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <\ Row>\n <Item>\n <C>DirectedEdges</C>\n </Item>\n \ <Item>\n <Ref Attr=\"DigraphEdges\"/>\n </Item>\n <Ite\ m>\n <Alt Only=\"HTML\">\n The attribute <Ref Attr=\"Digra\ phEdges\"/> behaves the same\n as <C>DirectedEdges</C> in &Grape;.\ \n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <\ Row>\n <Item>\n <C>UndirectedEdges</C>\n </Item>\n \ <Item>\n None\n </Item>\n <Item>\n <Alt Onl\ y=\"HTML\">\n There is no analogous function in &Digraphs;, and no \ plans to\n include one.\n </Alt>\n </Item>\n <\ /Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>Distance<\ /C>\n </Item>\n <Item>\n <Ref Oper=\"DigraphShortestDis\ tance\"\n Label=\"for a digraph and two vertices\"/>\n </\ Item>\n <Item>\n <Alt Only=\"HTML\">\n The operatio\ n <Ref Oper=\"DigraphShortestDistance\"\n Label=\"for a digraph and\ two vertices\"/> behaves the same as\n <C>Distance</C> in &Grape;.\ \n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <\ Row>\n <Item>\n <C>Diameter</C>\n </Item>\n <Ite\ m>\n <Ref Attr=\"DigraphDiameter\"/>\n </Item>\n <Item>\ \n <Alt Only=\"HTML\">\n The attribute <Ref Attr=\"Digraph\ Diameter\"/> behaves the same\n as <C>Diameter</C> in &Grape;.\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\ \n <Item>\n <C>Girth</C>\n </Item>\n <Item>\n \ <Ref Attr=\"DigraphUndirectedGirth\"/>\n </Item>\n <Item>\ \n <Alt Only=\"HTML\">\n The attribute <Ref Attr=\"Digraph\ UndirectedGirth\"/> behaves the same\n as <C>Girth</C> in &Grape; e\ xcept that in the case that the\n argument is a forest, <K>infinity\ </K> is returned by &Digraphs;\n whereas <C>-1</C> is returned by &\ Grape;.\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n\ <Row>\n <Item>\n <C>IsConnectedGraph</C>\n </Item\ >\n <Item>\n <Ref Prop=\"IsStronglyConnectedDigraph\"/>\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n The att\ ribute <Ref Prop=\"IsStronglyConnectedDigraph\"/> behaves the\n sam\ e as <C>IsConnectedGraph</C> in &Grape;.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>IsB\ ipartite</C>\n </Item>\n <Item>\n <Ref Prop=\"IsBiparti\ teDigraph\"/>\n <Alt Only=\"HTML\">\n and <Ref Prop=\"IsSy\ mmetricDigraph\"/>\n </Alt>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n The &Grape; function <C>IsBipartite</C> is\ equivalent to\n <Ref Prop=\"IsBipartiteDigraph\"/> and <Ref\n \ Prop=\"IsSymmetricDigraph\"/>.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>IsNu\ llGraph</C>\n </Item>\n <Item>\n <Ref Prop=\"IsNullDigr\ aph\"/>\n <Alt Only=\"HTML\">\n or <Ref Prop=\"IsEmptyDigr\ aph\"/>\n </Alt>\n </Item>\n <Item>\n <Alt Onl\ y=\"HTML\">\n The &Grape; function <C>IsNullGraph</C> is equivalent\ to\n <Ref Prop=\"IsEmptyDigraph\"/> and its synonym <Ref\n \ Prop=\"IsNullDigraph\"/>.\n </Alt>\n </Item>\n </Ro\ w>\n <HorLine/>\n\n <Row>\n <Item>\n <C>IsCompleteGr\ aph</C>\n </Item>\n <Item>\n <Ref Prop=\"IsCompleteDigr\ aph\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ The &Grape; function <C>IsCompleteGraph</C> is roughly\n equ\ ivalent to\n <Ref Prop=\"IsCompleteDigraph\"/> except that in &Digr\ aphs;\n a complete digraph has <E>no loops</E>.\n </Alt>\n\ </Item>\n </Row>\n <HorLine/>\n\n\n </Table>\n </Section>\ \n\n <Section>\n <Heading>\n Functions to determine regularity proper\ ties of graphs\n </Heading>\n <Alt Not=\"HTML\">\n <E>\n The\ table in this section contains more information when viewed in\n html \ format.\n </E>\n </Alt>\n\n <Table Align=\"|l|l|l|\">\n <HorLi\ ne/>\n\n <Row>\n <Item>\n &Grape; command\n </Item\ >\n <Item>\n &Digraphs; command\n </Item>\n <Ite\ m>\n <Alt Only=\"HTML\">\n Comments\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n \ <C>IsRegularGraph</C>\n </Item>\n <Item>\n <Re\ f Prop=\"IsOutRegularDigraph\"/>\n <Alt Only=\"HTML\">\n (\ see also\n <Ref Prop=\"IsInRegularDigraph\"/>\n and\n \ <Ref Prop=\"IsRegularDigraph\"/>)\n </Alt>\n </Item>\ \n <Item>\n <Alt Only=\"HTML\">\n The &Grape; funct\ ion <C>IsRegularGraph</C> is\n equivalent to <Ref Prop=\"IsOutRegul\ arDigraph\"/>.\n </Alt>\n </Item>\n </Row>\n <HorLin\ e/>\n\n <Row>\n <Item>\n <C>LocalParameters</C>\n \ </Item>\n <Item>\n None\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n The &Grape; function <C>LocalParameters<\ /C> has no equivalent\n function in &Digraphs;, although this might\ change in the future.\n </Alt>\n </Item>\n </Row>\n \ <HorLine/>\n\n <Row>\n <Item>\n <C>GlobalParameters</C>\ \n </Item>\n <Item>\n None\n </Item>\n <I\ tem>\n <Alt Only=\"HTML\">\n The &Grape; function <C>Global\ Parameters</C> has no equivalent\n function in &Digraphs;, although\ this might change in the future.\n </Alt>\n </Item>\n </\ Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>IsDistance\ Regular</C>\n </Item>\n <Item>\n <Ref Prop=\"IsDistance\ RegularDigraph\"/>\n </Item>\n <Item>\n <Alt Only=\"HTM\ L\">\n The &Grape; function <C>IsDistanceRegular</C> is\n \ equivalent to <Ref Prop=\"IsDistanceRegularDigraph\"/>.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\ \n <C>CollapsedAdjacencyMat</C>\n </Item>\n <Item>\n \ None\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ The &Grape; function <C>CollapsedAdjacencyMat</C> has no equivalent\ \n function in &Digraphs;, although this might change in the future\ .\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n \ <Row>\n <Item>\n <C>OrbitalGraphColadjMats</C>\n </Item\ >\n <Item>\n None\n </Item>\n <Item>\n <\ Alt Only=\"HTML\">\n The &Grape; function <C>OrbitalGraphColadjMats\ </C> has no equivalent\n function in &Digraphs;, although this migh\ t change in the future.\n </Alt>\n </Item>\n </Row>\n \ <HorLine/>\n\n <Row>\n <Item>\n <C>VertexTransitiveDRGs\ </C>\n </Item>\n <Item>\n None\n </Item>\n \ <Item>\n <Alt Only=\"HTML\">\n The &Grape; function <C>Ve\ rtexTransitiveDRGs</C> has no equivalent\n function in &Digraphs;, \ although this might change in the future.\n </Alt>\n </Item>\n\ </Row>\n <HorLine/>\n\n </Table>\n\n </Section>\n\n <Section>\ \n <Heading>\n Some special vertex subsets of a graph\n </Heading>\ \n <Alt Not=\"HTML\">\n <E>\n The table in this section contain\ s more information when viewed in\n html format.\n </E>\n </Alt\ >\n\n <Table Align=\"|l|l|l|\">\n <HorLine/>\n\n <Row>\n <\ Item>\n &Grape; command\n </Item>\n <Item>\n &\ Digraphs; command\n </Item>\n <Item>\n <Alt Only=\"HTML\ \">\n Comments\n </Alt>\n </Item>\n </Row>\n \ <HorLine/>\n\n <Row>\n <Item>\n <C>ConnectedComponent\ </C>\n </Item>\n <Item>\n <Ref Oper=\"DigraphConnectedC\ omponent\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n\ When applied to a simple di/graph and a vertex,\n <C>Co\ nnectedComponent</C> in &Grape; is equivalent to\n <Ref Oper=\"Digr\ aphConnectedComponent\"/>.\n <Ref Oper=\"DigraphConnectedComponent\ \"/> can also be applied to\n non-simple graphs (in the language of\ &Grape;).\n </Alt>\n </Item>\n </Row>\n <HorLine/>\ \n\n <Row>\n <Item>\n <C>ConnectedComponents</C>\n \ </Item>\n <Item>\n <Ref Attr=\"DigraphConnectedComponents\"/>\ \n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ When applied to a simple di/graph,\n <C>ConnectedComponents</C> in \ &Grape; is equivalent to\n <Ref Attr=\"DigraphConnectedComponents\"\ />.\n <Ref Attr=\"DigraphConnectedComponents\"/> can also be applie\ d to\n non-simple graphs (in the language of &Grape;).\n <\ /Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>Bicomponents</C>\n </Item>\n <Item>\n \ <Ref Attr=\"DigraphBicomponents\"/>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n When applied to a simple bipartite di/\ graph,\n <C>Bicomponents</C> in &Grape; is equivalent to\n \ <Ref Attr=\"DigraphBicomponents\"/>.\n The definition of <Ref Pr\ op=\"IsBipartiteDigraph\"/> does not require\n its argument to be s\ imple (in the language of &Grape;), i.e. the\n bipartite graphs in \ &Grape; are a proper subset of the bipartite\n digraphs in &Digraph\ s;.\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n \ <Row>\n <Item>\n <C>DistanceSet</C>\n </Item>\n \ <Item>\n <Ref Oper=\"DigraphDistanceSet\"\n Label=\"\ for a digraph, a pos int, and an int\"/>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n <C>DistanceSet</C> in &Grape; is equiv\ alent to\n <Ref Oper=\"DigraphDistanceSet\"\n Label=\"\ for a digraph, a pos int, and an int\"/> in &Digraphs;.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n \ <C>Layers</C>\n </Item>\n <Item>\n <Ref Oper=\ \"DigraphLayers\"/>\n </Item>\n <Item>\n <Alt Only=\"HT\ ML\">\n <C>Layers</C> in &Grape; is equivalent to\n <Ref\ Oper=\"DigraphLayers\"/> in &Digraphs;.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>Ind\ ependentSet</C>\n </Item>\n <Item>\n <Ref Func=\"Digrap\ hIndependentSet\"/>\n </Item>\n <Item>\n <Alt Only=\"HT\ ML\">\n <C>IndependentSet</C> in &Grape; is equivalent to\n \ <Ref Func=\"DigraphIndependentSet\"/> in &Digraphs;.\n </Alt>\n \ </Item>\n </Row>\n <HorLine/>\n\n </Table>\n\n </Section>\ \n\n <Section>\n <Heading>\n Functions to construct new graphs from o\ ld\n </Heading>\n <Alt Not=\"HTML\">\n <E>\n The table in th\ is section contains more information when viewed in\n html format.\n \ </E>\n </Alt>\n\n <Table Align=\"|l|l|l|\">\n <HorLine/>\n\n \ <Row>\n <Item>\n &Grape; command\n </Item>\n <\ Item>\n &Digraphs; command\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n Comments\n </Alt>\n </Item>\ \n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>\ InducedSubgraph</C>\n </Item>\n <Item>\n <Ref Oper=\"In\ ducedSubdigraph\"/>\n </Item>\n <Item>\n <Alt Only=\"HT\ ML\">\n <Ref Oper=\"InducedSubdigraph\"/> in &Digraphs; does not al\ low a third\n argument <C>G</C> specifying a group of automorphisms\ on the\n resultant graph. This may be included in a future release\ .\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n \ <Row>\n <Item>\n <C>DistanceSetInduced</C>\n </Item>\n \ <Item>\n None\n </Item>\n <Item>\n <Alt\ Only=\"HTML\">\n There is no analogous function in &Digraphs;, and\ no plans to\n include one.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>Dista\ nceGraph</C>\n </Item>\n <Item>\n <Ref Oper=\"DistanceD\ igraph\" Label=\"for digraph and list\"/>\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n <C>DistanceGraph</C> in &Grape; is eq\ uivalent to\n <Ref Oper=\"DistanceDigraph\" Label=\"for digraph and\ list\"/> in\n &Digraphs;.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>Comple\ mentGraph</C>\n </Item>\n <Item>\n <Ref Oper=\"DigraphD\ ual\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ <Ref Oper=\"DigraphDual\"/> in &Digraphs; is equivalent to\n \ <C>ComplementGraph</C> in &Grape; with the optional argument\n <A>\ comploops</A> set to <K>true</K> (loops/nonloops are\n complemented\ ).\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n \ <Row>\n <Item>\n <C>PointGraph</C>\n </Item>\n \ <Item>\n None\n </Item>\n <Item>\n <Alt Only=\ \"HTML\">\n There is no analogous function in &Digraphs;, and no pl\ ans to\n include one.\n </Alt>\n </Item>\n </R\ ow>\n <HorLine/>\n\n <Row>\n <Item>\n <C>EdgeGraph</\ C>\n </Item>\n <Item>\n <Ref Oper=\"EdgeUndirectedDigra\ ph\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ <C>EdgeGraph</C> in &Grape; is equivalent to\n <Ref Oper=\"Ed\ geUndirectedDigraph\"/> in &Digraphs;.\n </Alt>\n </Item>\n \ </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>Switc\ hedGraph</C>\n </Item>\n <Item>\n None\n </Item>\ \n <Item>\n <Alt Only=\"HTML\">\n There is no analo\ gous function in &Digraphs;, and no plans to\n include one.\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>UnderlyingGraph</C>\n </Item>\n <Item\ >\n <Ref Oper=\"DigraphSymmetricClosure\"/>\n </Item>\n \ <Item>\n <Alt Only=\"HTML\">\n <C>UnderlyingGraph</C> in \ &Grape; is equivalent to\n <Ref Oper=\"DigraphSymmetricClosure\"/> \ in &Digraphs;.\n </Alt>\n </Item>\n </Row>\n <HorLin\ e/>\n\n <Row>\n <Item>\n <C>QuotientGraph</C>\n </\ Item>\n <Item>\n <Ref Oper=\"QuotientDigraph\"/>\n </It\ em>\n <Item>\n <Alt Only=\"HTML\">\n <Ref Oper=\"Qu\ otientDigraph\"/> in &Digraphs; takes a partition\n <C>p</C> of the\ vertices of the digraph, rather than a set of\n generating pairs. \ Hence, the equivalence might not be\n group-invariant. Also note \ that in &Digraphs; multiple edges may be\n created, whereas in &Gra\ pe; they are effectively combined into a\n single edge.\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>BipartiteDouble</C>\n </Item>\n <Item>\n \ <Ref Oper=\"BipartiteDoubleDigraph\"/>\n </Item>\n <Ite\ m>\n <Alt Only=\"HTML\">\n <C>BipartiteDouble</C> in &Grap\ e; is equivalent to\n <Ref Oper=\"BipartiteDoubleDigraph\"/> in &Di\ graphs;.\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\ \n <Row>\n <Item>\n <C>GeodesicsGraph</C>\n </Item\ >\n <Item>\n None\n </Item>\n <Item>\n \ <Alt Only=\"HTML\">\n There is no analogous function in &Digraphs;,\ and no plans to\n include one.\n </Alt>\n </Item>\ \n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>\ CollapsedIndependentOrbitsGraph</C>\n </Item>\n <Item>\n \ None\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ There is no analogous function in &Digraphs;, although this feature\n \ might be implemented in the future.\n </Alt>\n </Item\ >\n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C\ >CollapsedCompleteOrbitsGraph</C>\n </Item>\n <Item>\n \ None\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ There is no analogous function in &Digraphs;, although this feature\n \ might be implemented in the future.\n </Alt>\n </Item>\ \n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>\ NewGroupGraph</C>\n </Item>\n <Item>\n None\n </\ Item>\n <Item>\n <Alt Only=\"HTML\">\n There is no \ analogous function in &Digraphs;. However, the same\n results woul\ d be possible by calling <Ref Oper=\"DigraphCopy\"/> on\n the digra\ ph to produce a copy <C>gr</C>, and then applying\n <C>SetDigraphGr\ oup(gr, G)</C>.\n </Alt>\n </Item>\n </Row>\n <HorLi\ ne/>\n\n </Table>\n\n </Section>\n\n <Section>\n <Heading>\n Vert\ ex-Colouring and Complete Subgraphs\n </Heading>\n <Alt Not=\"HTML\">\n \ <E>\n The table in this section contains more information when vie\ wed in\n html format.\n </E>\n </Alt>\n\n <Table Align=\"|l|\ l|l|\">\n <HorLine/>\n\n <Row>\n <Item>\n &Grape; co\ mmand\n </Item>\n <Item>\n &Digraphs; command\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n Comments\n\ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Ro\ w>\n <Item>\n <C>VertexColouring</C>\n </Item>\n \ <Item>\n <Ref Attr=\"DigraphGreedyColouring\" Label=\"for a digraph\ \"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ <C>VertexColouring</C> in &Grape; is equivalent to <Ref\n Attr=\ \"DigraphGreedyColouring\" Label=\"for a digraph\"/> in &Digraphs;,\n \ except it returns a transformation rather than a list of vertex\n \ colors.\n </Alt>\n </Item>\n </Row>\n <HorLine/>\n\ \n <Row>\n <Item>\n <C>CompleteSubgraphs</C>\n </I\ tem>\n <Item>\n <Ref Func=\"DigraphCliques\"/>\n </Item\ >\n <Item>\n <Alt Only=\"HTML\">\n The functionalit\ y of <C>CompleteSubgraphs</C> in &Grape; is provided\n by <Ref Func\ =\"DigraphCliques\"/> and <Ref Func=\"DigraphMaximalCliques\"/>\n i\ n &Digraphs;.\n </Alt>\n </Item>\n </Row>\n <HorLine\ />\n\n <Row>\n <Item>\n <C>CompleteSubgraphsOfGivenSize</\ C>\n </Item>\n <Item>\n <Ref Func=\"DigraphCliques\"/>\ \n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ Much of the functionality of <C>CompleteSubgraphsOfGivenSize</C> in\n \ &Grape; is provided by <Ref Func=\"DigraphCliques\"/> and <Ref\n \ Func=\"DigraphMaximalCliques\"/> in &Digraphs;, except that the\n \ functionality relating to vertex weights and partial colourings is\n \ not implemented.\n </Alt>\n </Item>\n </Row>\n <\ HorLine/>\n\n </Table>\n </Section>\n\n <Section>\n <Heading>\n A\ utomorphism groups and isomorphism testing for graphs\n </Heading>\n <Al\ t Not=\"HTML\">\n <E>\n The table in this section contains more in\ formation when viewed in\n html format.\n </E>\n </Alt>\n\n \ <Table Align=\"|l|l|l|\">\n <HorLine/>\n\n <Row>\n <Item>\n \ &Grape; command\n </Item>\n <Item>\n &Digraphs;\ command\n </Item>\n <Item>\n <Alt Only=\"HTML\">\n \ Comments\n </Alt>\n </Item>\n </Row>\n <HorL\ ine/>\n\n <Row>\n <Item>\n <C>AutGroupGraph</C>\n \ </Item>\n <Item>\n <Ref Attr=\"AutomorphismGroup\" Label=\"for\ a digraph\"/>\n </Item>\n <Item>\n <Alt Only=\"HTML\">\ \n <C>AutGroupGraph</C> in &Grape; is equivalent to\n <R\ ef Attr=\"AutomorphismGroup\" Label=\"for a digraph\"/> in &Digraphs;.\n \ </Alt>\n </Item>\n </Row>\n <HorLine/>\n\n <Row>\n \ <Item>\n <C>GraphIsomorphism</C>\n </Item>\n <Ite\ m>\n <Ref Oper=\"IsomorphismDigraphs\" Label=\"for digraphs\"/>\n \ </Item>\n <Item>\n <Alt Only=\"HTML\">\n <C>Gra\ phIsomorphism</C> in &Grape; is equivalent to <Ref\n Oper=\"Isomorp\ hismDigraphs\" Label=\"for digraphs\"/> in &Digraphs;.\n Note howev\ er that <C>GraphIsomorphism</C>'s optional argument\n <A>firstunbin\ dcanon</A> is not applicable, since the canonical\n labellings of a\ digraph are not portable between sessions.\n </Alt>\n </Item>\ \n </Row>\n <HorLine/>\n\n <Row>\n <Item>\n <C>\ IsIsomorphicGraph</C>\n </Item>\n <Item>\n <Ref Oper=\"\ IsIsomorphicDigraph\" Label=\"for digraphs\"/>\n </Item>\n <Item\ >\n <Alt Only=\"HTML\">\n <C>IsIsomorphicGraph</C> in &Gra\ pe; is equivalent to <Ref\n Oper=\"IsIsomorphicDigraph\" Label=\"fo\ r digraphs\"/> in &Digraphs;.\n Note however that <C>IsIsomorphicGr\ aph</C>'s optional argument\n <A>firstunbindcanon</A> is not applic\ able, since the canonical\n labellings of a digraph are not portabl\ e between sessions.\n </Alt>\n </Item>\n </Row>\n <H\ orLine/>\n\n <Row>\n <Item>\n <C>GraphIsomorphismClassRep\ resentatives</C>\n </Item>\n <Item>\n None\n </I\ tem>\n <Item>\n <Alt Only=\"HTML\">\n There is no a\ nalogous function in &Digraphs;, although this feature\n might be i\ mplemented in the future.\n </Alt>\n </Item>\n </Row>\n \ <HorLine/>\n\n </Table>\n </Section>\n</Appendix>\n <!-- Grape to Dig\ raphs command map -->\n\n<Bibliography Databases=\"digraphs\" />\n<TheIndex/>\ \n\n</Book>\n", inputorigins := [ [ 1, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main.\ xml", 1 ], [ 36, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main\ .xml", 2 ], [ 90, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main\ .xml", 3 ], [ 132, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main\ .xml", 4 ], [ 201, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main\ .xml", 5 ], [ 249, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main\ .xml", 6 ], [ 291, "/builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/main\ .xml", 7 ], [ 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class=\"Heading\">Creating digraphs</spa\ n>", Definitions := "<span class=\"Heading\">Definitions</span>", ("Finding cliques and independent sets") := "<span class=\"Heading\">Fin\ ding cliques and independent sets</span>", ("For those in a hurry") := "<span class=\"Heading\">For those in a hurr\ y</span>", Homomorphisms := "<span class=\"Heading\">Homomorphisms</span>", ("Installing Digraphs") := "<span class=\"Heading\">Installing <strong c\ lass=\"pkg\">Digraphs</strong></span>", ("Isomorphisms and canonical labellings") := "<span class=\"Heading\">Is\ omorphisms and canonical labellings</span>", Operators := "<span class=\"Heading\">Operators</span>", ("Optional package dependencies") := "<span class=\"Heading\">Optional p\ ackage dependencies</span>", Planarity := "<span class=\"Heading\">Planarity</span>", ("Properties of digraphs") := "<span class=\"Heading\">Properties of dig\ raphs</span>", ("Rebuilding the documentation") := "<span class=\"Heading\">Rebuilding \ the documentation</span>", ("Testing your installation") := "<span class=\"Heading\">Testing your i\ nstallation</span>", ("The Digraphs package") := "<span class=\"Heading\">\n The <strong c\ lass=\"pkg\">Digraphs</strong> package\n </span>", ("The Grape package") := "<span class=\"Heading\">The Grape package</spa\ n>", ("Visualising and IO") := "<span class=\"Heading\">Visualising and IO</sp\ an>" ), linelength := 76, mathmode := "MathJax", name := "WHOLEDOCUMENT", next := 490033, root := ~, six := [ [ "Title page", ".", [ 0, 0, 0 ], 1, 1, "title page", "X7D2C85EC87DD46E5" ], [ "Abstract", ".-2", [ 0, 0, 2 ], 61, 2, "abstract", "X7AA6C5737B711C89" ], [ "Copyright", ".-1", [ 0, 0, 1 ], 67, 2, "copyright", "X81488B807F2A1CF1" ], [ "Acknowledgements", ".-3", [ 0, 0, 3 ], 79, 2, "acknowledgements", "X82A988D47DFAFCFA" ], [ "Table of Contents", ".-4", [ 0, 0, 4 ], 86, 3, "table of contents", "X8537FEB07AF2BEC8" ], [ "\033[1X\033[33X\033[0;-2YThe \033[5XDigraphs\033[105X\033[101X\027\\ 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369, 370, 371, 372, 373, 374, 41, 375, 376, 377, 378, 379, 380, 381, 382, 384, 383, 385, 387, 386, 391, 395, 394, 393, 392, 390, 389, 388, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 42, 43, 408, 409, 410, 411, 412, 418, 417, 416, 413, 414, 415, 419, 44, 421, 420, 422, 423, 429, 428, 426, 425, 424, 427, 45, 46, 430, 431, 432, 433, 434, 435, 439, 438, 437, 436, 440, 441, 442, 443, 47, 445, 444, 448, 449, 447, 446, 453, 452, 451, 450, 454, 455, 456, 457, 459, 458, 460, 461, 462, 464, 463, 466, 465, 467, 468, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58 ], start := 0, stop := 490032, toctext := "<div class=\"ContChap\"><a href=\"chap1_mj.html#X7F202ABA780E595\ D\">1 <span class=\"Heading\">\n The <strong class=\"pkg\">Digraphs</strong\ > package\n </span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><spa\ n class=\"nocss\"> </span><a href=\"chap1_mj.html#X7DFB63A97E67C0A1\">1.1\ <span class=\"Heading\">Introduction</span></a>\n</span>\n<div class=\"ContSS\ Block\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap1_mj.html#X84541F61810C741D\">1.1-1 <span class=\"Heading\">De\ finitions</span></a>\n</span>\n</div></div>\n</div>\n<div class=\"ContChap\"><\ a href=\"chap2_mj.html#X817088B27F90D596\">2 <span class=\"Heading\">Installin\ g <strong class=\"pkg\">Digraphs</strong></span></a>\n<div class=\"ContSect\">\ <span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap2_mj.\ html#X7DA3059C79842BF3\">2.1 <span class=\"Heading\">For those in a hurry</spa\ n></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap2_mj.html#X7C1F2E6D860DBDE\ C\">2.1-1 <span class=\"Heading\">Configuration options</span></a>\n</span>\n<\ /div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"noc\ ss\"> </span><a href=\"chap2_mj.html#X780AA9D97EBCA95D\">2.2 <span class=\ \"Heading\">Optional package dependencies</span></a>\n</span>\n<div class=\"Co\ ntSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap2_mj.html#X8493C7587FCF6D8B\">2.2-1 <span class=\"Heading\ \">The Grape package</span></a>\n</span>\n</div></div>\n<div class=\"ContSect\ \"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap2_\ mj.html#X849F6196875A6DF5\">2.3 <span class=\"Heading\">Compiling the kernel m\ odule</span></a>\n</span>\n</div>\n<div class=\"ContSect\"><span class=\"tocli\ ne\"><span class=\"nocss\"> </span><a href=\"chap2_mj.html#X857CBE5484CF7\ 03A\">2.4 <span class=\"Heading\">Rebuilding the documentation</span></a>\n</s\ pan>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"\ nocss\">  </span><a href=\"chap2_mj.html#X870631C38610AC25\">2.4-1 D\ igraphsMakeDoc</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\ \"tocline\"><span class=\"nocss\"> </span><a href=\"chap2_mj.html#X7862D3\ F37C5BBDEF\">2.5 <span class=\"Heading\">Testing your installation</span></a>\ \n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span cla\ ss=\"nocss\">  </span><a href=\"chap2_mj.html#X86AF4DAE80B978DA\">2.\ 5-1 DigraphsTestInstall</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap2_mj.html#X7A20088B7C406C4A\">2.5-2\ DigraphsTestStandard</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap2_mj.html#X7885EF3E785D4298\">2.5-3 Di\ graphsTestExtreme</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\"><a\ href=\"chap3_mj.html#X7D34861E863A5D93\">3 <span class=\"Heading\">Creating d\ igraphs</span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span clas\ s=\"nocss\"> </span><a href=\"chap3_mj.html#X7D34861E863A5D93\">3.1 <span\ class=\"Heading\">Creating digraphs</span></a>\n</span>\n<div class=\"ContSSB\ lock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap3_mj.html#X7877ADC77F85E630\">3.1-1 IsDigraph</a></span>\n<span\ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"cha\ p3_mj.html#X7D7EDF83820ED6F5\">3.1-2 IsMutableDigraph</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.h\ tml#X7CAFAA89804F80BD\">3.1-3 IsImmutableDigraph</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X\ 7E749324800B38A5\">3.1-4 IsCayleyDigraph</a></span>\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X80F1B6D2\ 8478D8B9\">3.1-5 IsDigraphWithAdjacencyFunction</a></span>\n<span class=\"Cont\ SS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X8\ 6E798B779515678\">3.1-6 DigraphByOutNeighboursType</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html\ #X834843057CE86655\">3.1-7 Digraph</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap3_mj.html#X8023FE387A3AB6\ 09\">3.1-8 DigraphByAdjacencyMatrix</a></span>\n<span class=\"ContSS\"><br /><\ span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7F37B6768349E\ 269\">3.1-9 DigraphByEdges</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap3_mj.html#X7B75C1D680757D6F\">3.1\ -10 EdgeOrbitsDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap3_mj.html#X81BC49B57EAADEFB\">3.1-11 D\ igraphByInNeighbours</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap3_mj.html#X7FCADADC7EC28478\">3.1-12 Ca\ yleyDigraph</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"to\ cline\"><span class=\"nocss\"> </span><a href=\"chap3_mj.html#X83608C407C\ C8836D\">3.2 <span class=\"Heading\">Changing representations</span></a>\n</sp\ an>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap3_mj.html#X7FBC4BDB82FBEDD2\">3.2-1 As\ BinaryRelation</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\   </span><a href=\"chap3_mj.html#X83CA27E67F685EB3\">3.2-2 AsDigraph\ </a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </\ span><a href=\"chap3_mj.html#X7B335342839E5146\">3.2-3 Graph</a></span>\n<span\ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"cha\ p3_mj.html#X7C4F13E080EC16B0\">3.2-4 AsGraph</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7C5\ 360B2799943F3\">3.2-5 AsTransformation</a></span>\n</div></div>\n<div class=\"\ ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\ \"chap3_mj.html#X85126078848B420A\">3.3 <span class=\"Heading\">New digraphs f\ rom old</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X83D\ 93A8A8251E6F9\">3.3-1 DigraphImmutableCopy</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X8399F7\ 427B227228\">3.3-2 DigraphImmutableCopyIfImmutable</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html\ #X83C51DA182CCEA2F\">3.3-3 InducedSubdigraph</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7CA\ F093B85A93D2F\">3.3-4 ReducedDigraph</a></span>\n<span class=\"ContSS\"><br />\ <span class=\"nocss\">  </span><a href=\"chap3_mj.html#X829E3EAC7C4B\ 3B1E\">3.3-5 MaximalSymmetricSubdigraph</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X79BA6A668\ 46D5A95\">3.3-6 MaximalAntiSymmetricSubdigraph</a></span>\n<span class=\"ContS\ S\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7D\ D9766C86D3ED20\">3.3-7 UndirectedSpanningForest</a></span>\n<span class=\"Cont\ SS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7\ D1D26D27F5B56C2\">3.3-8 QuotientDigraph</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X78DECD268\ 11EFD7C\">3.3-9 DigraphReverse</a></span>\n<span class=\"ContSS\"><br /><span \ class=\"nocss\">  </span><a href=\"chap3_mj.html#X7F71D99D852B130F\"\ >3.3-10 DigraphDual</a></span>\n<span class=\"ContSS\"><br /><span class=\"noc\ ss\">  </span><a href=\"chap3_mj.html#X874883DD7DD450C4\">3.3-11 Dig\ raphSymmetricClosure</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap3_mj.html#X7A6C419080AD41DE\">3.3-12 Di\ graphTransitiveClosure</a></span>\n<span class=\"ContSS\"><br /><span class=\"\ nocss\">  </span><a href=\"chap3_mj.html#X82AD17517E273600\">3.3-13 \ DigraphTransitiveReduction</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap3_mj.html#X83B2506D79453208\">3.3\ -14 DigraphAddVertex</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap3_mj.html#X8134BEE7786BD3A7\">3.3-15 Di\ graphAddVertices</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap3_mj.html#X7E58CC4880627658\">3.3-16 Digra\ phAddEdge</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap3_mj.html#X7BE5C7028760B053\">3.3-17 DigraphAddEdg\ eOrbit</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &n\ bsp;</span><a href=\"chap3_mj.html#X8693A61B7F752C76\">3.3-18 DigraphAddEdges<\ /a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </s\ pan><a href=\"chap3_mj.html#X7B634A2B83C08B16\">3.3-19 DigraphRemoveVertex</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap3_mj.html#X7E290E847A5A299A\">3.3-20 DigraphRemoveVertices</a><\ /span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap3_mj.html#X8433E3BC7E5EA6BF\">3.3-21 DigraphRemoveEdge</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap3_mj.html#X85981D9187F49018\">3.3-22 DigraphRemoveEdgeOrbit</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap3_mj.html#X87093FDA7F88E732\">3.3-23 DigraphRemoveEdges</a></span>\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap3_mj.html#X79324AF7818C0C02\">3.3-24 DigraphRemoveLoops</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap3_mj.html#X7DCCD0247897A3DE\">3.3-25 DigraphRemoveAllMultipleEdges</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap3_mj.html#X7821753D85402A8C\">3.3-26 DigraphReverseEdges</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap3_mj.html#X814F1DFC83DB273F\">3.3-27 DigraphDisjointUnion</a></span>\n<s\ pan class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"\ chap3_mj.html#X7DA997697D310E44\">3.3-28 DigraphEdgeUnion</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_\ mj.html#X7DDFC759860E3390\">3.3-29 DigraphJoin</a></span>\n<span class=\"ContS\ S\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7D\ 625CC87DBFFDED\">3.3-30 DigraphCartesianProduct</a></span>\n<span class=\"Cont\ SS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X8\ 4D24DC9833B54A5\">3.3-31 DigraphDirectProduct</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X879\ 5087A78FE7D54\">3.3-32 DigraphCartesianProductProjections</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_\ mj.html#X84FB64F185B804C2\">3.3-33 DigraphDirectProductProjections</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X8595BF937B749F22\">3.3-34 LineDigraph</a></span>\n<span cla\ ss=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_m\ j.html#X8364C6F17A1680CB\">3.3-35 LineUndirectedDigraph</a></span>\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj\ .html#X7FB8B48C87C0ED16\">3.3-36 DoubleDigraph</a></span>\n<span class=\"ContS\ S\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7C\ 6E6CB284982C7A\">3.3-37 BipartiteDoubleDigraph</a></span>\n<span class=\"ContS\ S\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X81\ 67A50A83256ED1\">3.3-38 DigraphAddAllLoops</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X865436\ 437DF95FEF\">3.3-39 DistanceDigraph</a></span>\n<span class=\"ContSS\"><br /><\ span class=\"nocss\">  </span><a href=\"chap3_mj.html#X86F9CCEA839AB\ C48\">3.3-40 DigraphClosure</a></span>\n<span class=\"ContSS\"><br /><span cla\ ss=\"nocss\">  </span><a href=\"chap3_mj.html#X7B5AC5FE859F4D80\">3.\ 3-41 DigraphMycielskian</a></span>\n</div></div>\n<div class=\"ContSect\"><spa\ n class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap3_mj.html\ #X85B5D9B97F5187B7\">3.4 <span class=\"Heading\">Random digraphs</span></a>\n<\ /span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap3_mj.html#X86CF9F66788B2A24\">3.4-1\ RandomDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\   </span><a href=\"chap3_mj.html#X78FE275E7E77D56F\">3.4-2 RandomMul\ tiDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap3_mj.html#X7D36B5E57F055051\">3.4-3 RandomTourname\ nt</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ </span><a href=\"chap3_mj.html#X7A023E4787682475\">3.4-4 RandomLattice</a></sp\ an>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class\ =\"nocss\"> </span><a href=\"chap3_mj.html#X7C76D1DC7DAF03D3\">3.5 <span \ class=\"Heading\">Standard examples</span></a>\n</span>\n<div class=\"ContSSBl\ ock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap3_mj.html#X870594FC866AC88E\">3.5-1 ChainDigraph</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap3_mj.html#X812417E278198D9C\">3.5-2 CompleteDigraph</a></span>\n<span class\ =\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.\ html#X8795B0AD856014FA\">3.5-3 CompleteBipartiteDigraph</a></span>\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj\ .html#X873F29CC863241F8\">3.5-4 CompleteMultipartiteDigraph</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 3_mj.html#X80C29DDE876FFBEB\">3.5-5 CycleDigraph</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X\ 80DAE31A79FEFD40\">3.5-6 EmptyDigraph</a></span>\n<span class=\"ContSS\"><br /\ ><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X80ED9CE7858\ 19607\">3.5-7 JohnsonDigraph</a></span>\n<span class=\"ContSS\"><br /><span cl\ ass=\"nocss\">  </span><a href=\"chap3_mj.html#X823F43217A6C375D\">3\ .5-8 PetersenGraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap3_mj.html#X7B5441F386BD105E\">3.5-9 Gener\ alisedPetersenGraph</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\">\ <a href=\"chap4_mj.html#X7AD6F77E7D95C996\">4 <span class=\"Heading\">Operator\ s</span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"no\ css\"> </span><a href=\"chap4_mj.html#X84E0B5B88358C96B\">4.1 <span class\ =\"Heading\">Operators for digraphs</span></a>\n</span>\n<div class=\"ContSSBl\ ock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap4_mj.html#X829B911D7EFD2D85\">4.1-1 IsSubdigraph</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap4_mj.html#X833C3299787E2309\">4.1-2 IsUndirectedSpanningTree</a></span>\n</\ div></div>\n</div>\n<div class=\"ContChap\"><a href=\"chap5_mj.html#X8739F6CD7\ 8C90B14\">5 <span class=\"Heading\">Attributes and operations</span></a>\n<div\ class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span\ ><a href=\"chap5_mj.html#X7E814B6478F7D015\">5.1 <span class=\"Heading\">Verti\ ces and edges</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.htm\ l#X7C45F7D878D896AC\">5.1-1 DigraphVertices</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7C6F1\ 9B57CB2E882\">5.1-2 DigraphNrVertices</a></span>\n<span class=\"ContSS\"><br /\ ><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7D1C6A4D7EC\ EC317\">5.1-3 DigraphEdges</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap5_mj.html#X85E1CFDD7E164AD0\">5.1\ -4 DigraphNrEdges</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap5_mj.html#X7BDABAF07917462B\">5.1-5 Digrap\ hNrLoops</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \  </span><a href=\"chap5_mj.html#X85D5E08280914EE4\">5.1-6 DigraphSinks</a\ ></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap5_mj.html#X7F5C6268839BE98C\">5.1-7 DigraphSources</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap5_mj.html#X785C30378064CF47\">5.1-8 DigraphTopologicalSort</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap5_mj.html#X7CA91E4B7904F793\">5.1-9 DigraphVertexLabel</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap5_mj.html#X7E51F2FE87140B32\">5.1-10 DigraphVertexLabels</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 5_mj.html#X79FAEACC7F438C2F\">5.1-11 DigraphEdgeLabel</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.h\ tml#X7C24851087D4A8FB\">5.1-12 DigraphEdgeLabels</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X\ 7EFAF01B7A155157\">5.1-13 DigraphInEdges</a></span>\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7BECFE66\ 87ECD028\">5.1-14 DigraphOutEdges</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap5_mj.html#X7BB8ED88835F07B\ 4\">5.1-15 IsDigraphEdge</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap5_mj.html#X848FED0B7B4ACD1F\">5.1-1\ 6 IsMatching</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X7B7A67277B1C9A02\">5.1-17 DigraphMax\ imalMatching</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X78E9847A858788D1\">5.1-18 DigraphMax\ imumMatching</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"t\ ocline\"><span class=\"nocss\"> </span><a href=\"chap5_mj.html#X7D7CE8328\ 187D0DF\">5.2 <span class=\"Heading\">Neighbours and degree</span></a>\n</span\ >\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"noc\ ss\">  </span><a href=\"chap5_mj.html#X7DC2CD70830BEE60\">5.2-1 Adja\ cencyMatrix</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nb\ sp; </span><a href=\"chap5_mj.html#X87FA0A727CDB060B\">5.2-2 Characterist\ icPolynomial</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X8507DC4F794780C1\">5.2-3 BooleanAdja\ cencyMatrix</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nb\ sp; </span><a href=\"chap5_mj.html#X7AFCE34A7A04D5C1\">5.2-4 DigraphAdjac\ encyFunction</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X7FDEBF3279759961\">5.2-5 DigraphRang\ e</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap5_mj.html#X7E9880767AE68E00\">5.2-6 OutNeighbours</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap5_mj.html#X85C7AA5A81DA6E11\">5.2-7 InNeighbours</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5\ _mj.html#X7F5ACE807D1BC2E2\">5.2-8 OutDegrees</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7AD\ DFBFD7A365775\">5.2-9 InDegrees</a></span>\n<span class=\"ContSS\"><br /><span\ class=\"nocss\">  </span><a href=\"chap5_mj.html#X7A09EB648070276D\ \">5.2-10 OutDegreeOfVertex</a></span>\n<span class=\"ContSS\"><br /><span cla\ ss=\"nocss\">  </span><a href=\"chap5_mj.html#X83315B0186850806\">5.\ 2-11 OutNeighboursOfVertex</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap5_mj.html#X7C9CD0527CB9E6EF\">5.2\ -12 InDegreeOfVertex</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap5_mj.html#X7C0DA18B8291F302\">5.2-13 In\ NeighboursOfVertex</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap5_mj.html#X83271F607BD809CF\">5.2-14 Digr\ aphLoops</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \  </span><a href=\"chap5_mj.html#X7DDB33B686B3A2C6\">5.2-15 PartialOrderDi\ graphMeetOfVertices</a></span>\n<span class=\"ContSS\"><br /><span class=\"noc\ ss\">  </span><a href=\"chap5_mj.html#X7BEAE1C78267F54D\">5.2-16 Deg\ reeMatrix</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap5_mj.html#X865390B08331936B\">5.2-17 LaplacianMatr\ ix</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><\ span class=\"nocss\"> </span><a href=\"chap5_mj.html#X8537F4088400DC48\">\ 5.3 <span class=\"Heading\">Reachability and connectivity</span></a>\n</span>\ \n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap5_mj.html#X7F16B9EB8398459C\">5.3-1 Digra\ phDiameter</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbs\ p; </span><a href=\"chap5_mj.html#X8104A9D37BCD8A05\">5.3-2 DigraphShorte\ stDistance</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbs\ p; </span><a href=\"chap5_mj.html#X81F99BC67E9D050F\">5.3-3 DigraphShorte\ stDistances</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nb\ sp; </span><a href=\"chap5_mj.html#X8223718079D98A82\">5.3-4 DigraphLonge\ stDistanceFromVertex</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap5_mj.html#X7CB7DDCD84621D38\">5.3-5 Dig\ raphDistanceSet</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"\ >  </span><a href=\"chap5_mj.html#X79A3DA4078CF3C90\">5.3-6 DigraphG\ irth</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nbs\ p;</span><a href=\"chap5_mj.html#X8374B7357EC189C1\">5.3-7 DigraphOddGirth</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap5_mj.html#X84688B337BDDBB09\">5.3-8 DigraphUndirectedGirth</a><\ /span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap5_mj.html#X842FAD6A7B835977\">5.3-9 DigraphConnectedComponents</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap5_mj.html#X8484EC557810CD31\">5.3-10 DigraphConnectedComponen\ t</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap5_mj.html#X833ECD6B7A84944C\">5.3-11 DigraphStronglyConnec\ tedComponents</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\ nbsp; </span><a href=\"chap5_mj.html#X7EFCB5017D662254\">5.3-12 DigraphSt\ ronglyConnectedComponent</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap5_mj.html#X7F1B5A2782F598B1\">5.3-1\ 3 DigraphBicomponents</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap5_mj.html#X7DDE06E47E605DD7\">5.3-14 A\ rticulationPoints</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap5_mj.html#X84D5A125848BD800\">5.3-15 Bridg\ es</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ </span><a href=\"chap5_mj.html#X865590147BD1C507\">5.3-16 StrongOrientation</a\ ></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap5_mj.html#X853D0B0981A33433\">5.3-17 DigraphPeriod</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap5_mj.html#X864A31A8809F61C2\">5.3-18 DigraphFloydWarshall</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap5_mj.html#X7FBAB09E7C0BE5CF\">5.3-19 IsReachable</a></span>\n<span class\ =\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.\ html#X7CAE81D97B43DDF3\">5.3-20 VerticesReachableFrom</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.h\ tml#X8039170B82A32257\">5.3-21 DigraphPath</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X80E9D6\ 45843973A6\">5.3-22 DigraphShortestPath</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7C0416FE7\ A69CA2C\">5.3-23 IteratorOfPaths</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap5_mj.html#X7ECD16838704FAAA\ \">5.3-24 DigraphAllSimpleCircuits</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap5_mj.html#X7C735C4E86BDD5\ F6\">5.3-25 DigraphLongestSimpleCircuit</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X870E04307\ C5F213F\">5.3-26 DigraphLayers</a></span>\n<span class=\"ContSS\"><br /><span \ class=\"nocss\">  </span><a href=\"chap5_mj.html#X7B2E42327DA118E0\"\ >5.3-27 DigraphDegeneracy</a></span>\n<span class=\"ContSS\"><br /><span class\ =\"nocss\">  </span><a href=\"chap5_mj.html#X827C2BD17A4547E3\">5.3-\ 28 DigraphDegeneracyOrdering</a></span>\n<span class=\"ContSS\"><br /><span cl\ ass=\"nocss\">  </span><a href=\"chap5_mj.html#X863FDFC4839A3B82\">5\ .3-29 HamiltonianPath</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap5_mj.html#X82F30D5681466BC6\">5.3-30 N\ rSpanningTrees</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\   </span><a href=\"chap5_mj.html#X79352A8286D1D8F6\">5.3-31 DigraphD\ ijkstra</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"toclin\ e\"><span class=\"nocss\"> </span><a href=\"chap5_mj.html#X82F900777D677F\ 55\">5.4 <span class=\"Heading\">Cayley graphs of groups</span></a>\n</span>\n\ <div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap5_mj.html#X7A000B1D7CCF7093\">5.4-1 GroupO\ fCayleyDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\   </span><a href=\"chap5_mj.html#X8528455987D7D2BF\">5.4-2 Generator\ sOfCayleyDigraph</a></span>\n</div></div>\n<div class=\"ContSect\"><span class\ =\"tocline\"><span class=\"nocss\"> </span><a href=\"chap5_mj.html#X790FD\ 6647ECCAE3C\">5.5 <span class=\"Heading\">Associated semigroups</span></a>\n</\ span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap5_mj.html#X87D5C60D7B0C1309\">5.5-1\ AsSemigroup</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X7C6D5EC27C51066B\">5.5-2 AsSemigroup\ </a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><sp\ an class=\"nocss\"> </span><a href=\"chap5_mj.html#X7E2305528492DDC0\">5.\ 6 <span class=\"Heading\">Planarity</span></a>\n</span>\n<div class=\"ContSSBl\ ock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap5_mj.html#X7DC478637E8C190D\">5.6-1 KuratowskiPlanarSubdigraph</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap5_mj.html#X78E8F09A8286501B\">5.6-2 KuratowskiOuterPlanarSubd\ igraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &n\ bsp;</span><a href=\"chap5_mj.html#X84E3947E7D39BA64\">5.6-3 PlanarEmbedding</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap5_mj.html#X85DFB8C18088711F\">5.6-4 OuterPlanarEmbedding</a><\ /span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap5_mj.html#X806D2D6B85E0B269\">5.6-5 SubdigraphHomeomorphicToK23<\ /a></span>\n</div></div>\n</div>\n<div class=\"ContChap\"><a href=\"chap6_mj.h\ tml#X7ADDEFD478D470D5\">6 <span class=\"Heading\">Properties of digraphs</span\ ></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&\ nbsp;</span><a href=\"chap6_mj.html#X7F49388F8245F0E9\">6.1 <span class=\"Head\ ing\">Edge properties</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 6_mj.html#X7D92935C7D535187\">6.1-1 DigraphHasLoops</a></span>\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.htm\ l#X7DB1BC2286FC08E2\">6.1-2 IsAntiSymmetricDigraph</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html\ #X860CFB0C8665F356\">6.1-3 IsBipartiteDigraph</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X790\ A4DBF8533516E\">6.1-4 IsCompleteBipartiteDigraph</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X\ 81F28D4D879FE3B2\">6.1-5 IsCompleteDigraph</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7FD1A1\ 5779FEC341\">6.1-6 IsCompleteMultipartiteDigraph</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X\ 7E894CCF7B1C27AE\">6.1-7 IsEmptyDigraph</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X83762C257\ DED2751\">6.1-8 IsEquivalenceDigraph</a></span>\n<span class=\"ContSS\"><br />\ <span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7E8F37E585DA\ ED52\">6.1-9 IsFunctionalDigraph</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap6_mj.html#X7BB84CFC7E8B2B26\ \">6.1-10 IsMultiDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap6_mj.html#X8462D8E2792B23F6\">6.1-1\ 1 IsReflexiveDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap6_mj.html#X81B3EA7887219860\">6.1-12 Is\ SymmetricDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap6_mj.html#X7DD8D1A185EBE865\">6.1-13 IsTou\ rnament</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap6_mj.html#X7F0835667F29F0C0\">6.1-14 IsTransitiveDig\ raph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nbs\ p;</span><a href=\"chap6_mj.html#X8617726C7829F796\">6.1-15 IsPreorderDigraph<\ /a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </s\ pan><a href=\"chap6_mj.html#X82BAE6D37D49A145\">6.1-16 IsPartialOrderDigraph</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap6_mj.html#X78D3E17B7F737516\">6.1-17 IsMeetSemilatticeDigraph\ </a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><sp\ an class=\"nocss\"> </span><a href=\"chap6_mj.html#X7AAF896982EC22FA\">6.\ 2 <span class=\"Heading\">Regularity</span></a>\n</span>\n<div class=\"ContSSB\ lock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap6_mj.html#X7A3F78B37E31D9E1\">6.2-1 IsInRegularDigraph</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap6_mj.html#X7F0A806E7BCB413C\">6.2-2 IsOutRegularDigraph</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap6_mj.html#X7AF9DB1E7DB12306\">6.2-3 IsRegularDigraph</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6\ _mj.html#X7E2082AC7CAE59CD\">6.2-4 IsDistanceRegularDigraph</a></span>\n</div>\ </div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">\  </span><a href=\"chap6_mj.html#X780DAEB37C5E07FD\">6.3 <span class=\"Hea\ ding\">Connectivity and cycles</span></a>\n</span>\n<div class=\"ContSSBlock\"\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap6_mj.html#X7BD08D4478218F77\">6.3-1 IsAcyclicDigraph</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap6_mj.html#X79563B297C220FB5\">6.3-2 IsChainDigraph</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.h\ tml#X83C08C0B7EC1A91F\">6.3-3 IsConnectedDigraph</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X\ 838FAF2D825977BE\">6.3-4 IsBiconnectedDigraph</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7D2\ A6572820B7F24\">6.3-5 IsBridgelessDigraph</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7B37B94\ 67C68C208\">6.3-6 IsStronglyConnectedDigraph</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X80E\ 883967EBE839E\">6.3-7 IsAperiodicDigraph</a></span>\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7B46EA6C\ 7B2DF2FB\">6.3-8 IsDirectedTree</a></span>\n<span class=\"ContSS\"><br /><span\ class=\"nocss\">  </span><a href=\"chap6_mj.html#X80FC20FA7AC4BC2A\ \">6.3-9 IsUndirectedTree</a></span>\n<span class=\"ContSS\"><br /><span class\ =\"nocss\">  </span><a href=\"chap6_mj.html#X79DFBB198525544E\">6.3-\ 10 IsEulerianDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap6_mj.html#X79EFEBC37C2D262D\">6.3-11 Is\ HamiltonianDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap6_mj.html#X7E91320B8028F5D8\">6.3-12 IsCy\ cleDigraph</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"toc\ line\"><span class=\"nocss\"> </span><a href=\"chap6_mj.html#X7E230552849\ 2DDC0\">6.4 <span class=\"Heading\">Planarity</span></a>\n</span>\n<div class=\ \"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nb\ sp;</span><a href=\"chap6_mj.html#X8606D415858C40AA\">6.4-1 IsPlanarDigraph</a\ ></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap6_mj.html#X8251E8B187E7F059\">6.4-2 IsOuterPlanarDigraph</a></\ span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span cla\ ss=\"nocss\"> </span><a href=\"chap6_mj.html#X83DC8AD283C41326\">6.5 <spa\ n class=\"Heading\">Homomorphisms and transformations</span></a>\n</span>\n<di\ v class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\ nbsp; </span><a href=\"chap6_mj.html#X78C0B2637985648B\">6.5-1 IsDigraphC\ ore</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap6_mj.html#X8361CA6E8401FF26\">6.5-2 IsEdgeTransitive</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap6_mj.html#X7B26E4447A1611E9\">6.5-3 IsVertexTransitive</a></spa\ n>\n</div></div>\n</div>\n<div class=\"ContChap\"><a href=\"chap7_mj.html#X849\ 75388859F203D\">7 <span class=\"Heading\">Homomorphisms</span></a>\n<div class\ =\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a hr\ ef=\"chap7_mj.html#X8200192B80AD2071\">7.1 <span class=\"Heading\">Acting on d\ igraphs</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X808\ 972017C486F1F\">7.1-1 OnDigraphs</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap7_mj.html#X7AFBA7608498F9CE\ \">7.1-2 OnMultiDigraphs</a></span>\n</div></div>\n<div class=\"ContSect\"><sp\ an class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap7_mj.htm\ l#X7E48B9F87A0F22D4\">7.2 <span class=\"Heading\">Isomorphisms and canonical l\ abellings</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"Cont\ SS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X8\ 3E593F3855B122E\">7.2-1 DigraphsUseNauty</a></span>\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X858C3212\ 7A190175\">7.2-2 AutomorphismGroup</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap7_mj.html#X7E7B0D88865A89\ F6\">7.2-3 BlissAutomorphismGroup</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap7_mj.html#X857758B18144C0C\ D\">7.2-4 NautyAutomorphismGroup</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap7_mj.html#X877732B1783C391B\ \">7.2-5 AutomorphismGroup</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap7_mj.html#X7DA2C4FE837FFE01\">7.2\ -6 AutomorphismGroup</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap7_mj.html#X7D676FE67A6684FF\">7.2-7 Bli\ ssCanonicalLabelling</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap7_mj.html#X87DA265D803DB337\">7.2-8 Bli\ ssCanonicalLabelling</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap7_mj.html#X877B1D377EC197D7\">7.2-9 Bli\ ssCanonicalDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap7_mj.html#X803ACEDA7BBAC5B3\">7.2-10 Digr\ aphGroup</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \  </span><a href=\"chap7_mj.html#X8096C5287E459279\">7.2-11 DigraphOrbits<\ /a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </s\ pan><a href=\"chap7_mj.html#X8386028782F2D3FF\">7.2-12 DigraphOrbitReps</a></s\ pan>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a\ href=\"chap7_mj.html#X8657604E87A25E5F\">7.2-13 DigraphSchreierVector</a></sp\ an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a \ href=\"chap7_mj.html#X78913684795FB256\">7.2-14 DigraphStabilizer</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap7_mj.html#X7B4F24B283C9EE28\">7.2-15 IsIsomorphicDigraph</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap7_mj.html#X7D4A1FA8868DA930\">7.2-16 IsIsomorphicDigraph</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 7_mj.html#X8219FE6C839D9457\">7.2-17 IsomorphismDigraphs</a></span>\n<span cla\ ss=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_m\ j.html#X7ED93C0F86D9D34F\">7.2-18 IsomorphismDigraphs</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.h\ tml#X7E1C806D81DFE15E\">7.2-19 RepresentativeOutNeighbours</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7\ _mj.html#X80183D4A7C51365A\">7.2-20 IsDigraphIsomorphism</a></span>\n<span cla\ ss=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_m\ j.html#X7F16B8B3825A627A\">7.2-21 IsDigraphColouring</a></span>\n</div></div>\ \n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> \ </span><a href=\"chap7_mj.html#X7E1A02FE8384C03C\">7.3 <span class=\"Heading\"\ >Homomorphisms of digraphs</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap7_mj.html#X79ABF0E783FD67C7\">7.3-1 HomomorphismDigraphsFinder</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap7_mj.html#X85E9B019877AD7FE\">7.3-2 DigraphHomomorphism</a></span>\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap7_mj.html#X8653EDA183E06D05\">7.3-3 HomomorphismsDigraphs</a></span>\n<s\ pan class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"\ chap7_mj.html#X82D0FCD87D47ACA8\">7.3-4 DigraphMonomorphism</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 7_mj.html#X84B4BDB984C2A9A8\">7.3-5 MonomorphismsDigraphs</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_\ mj.html#X7CB5AD9F861684FD\">7.3-6 DigraphEpimorphism</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.h\ tml#X7DFB1F5D873937B3\">7.3-7 EpimorphismsDigraphs</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html\ #X7AD04CC186E86CCA\">7.3-8 DigraphEmbedding</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X82EBD\ F137EEDC5FD\">7.3-9 EmbeddingsDigraphs</a></span>\n<span class=\"ContSS\"><br \ /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X80FE9FCB79\ 718A61\">7.3-10 IsDigraphHomomorphism</a></span>\n<span class=\"ContSS\"><br /\ ><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X7D6B5A08879\ 03810\">7.3-11 IsDigraphEmbedding</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap7_mj.html#X7F80709A80CBFA9\ 8\">7.3-12 DigraphsRespectsColouring</a></span>\n<span class=\"ContSS\"><br />\ <span class=\"nocss\">  </span><a href=\"chap7_mj.html#X7E93B268823F\ 6478\">7.3-13 GeneratorsOfEndomorphismMonoid</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X86D\ 0A74B83D6B9A7\">7.3-14 DigraphColouring</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X7AB7200D8\ 31013C1\">7.3-15 DigraphGreedyColouring</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X7F9CB3B27\ B9590DB\">7.3-16 DigraphWelshPowellOrder</a></span>\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X807D4935\ 8529A37C\">7.3-17 ChromaticNumber</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap7_mj.html#X863487EF7D64EF5\ 6\">7.3-18 DigraphCore</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\ \"><a href=\"chap8_mj.html#X7A7E0B957932DF44\">8 <span class=\"Heading\">Findi\ ng cliques and independent sets</span></a>\n<div class=\"ContSect\"><span clas\ s=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap8_mj.html#X7884\ 74E38446039B\">8.1 <span class=\"Heading\">Finding cliques</span></a>\n</span>\ \n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap8_mj.html#X80670E3E7C36183F\">8.1-1 IsCli\ que</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap8_mj.html#X7A3BC9FF8169ABCD\">8.1-2 CliquesFinder</a></s\ pan>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a\ href=\"chap8_mj.html#X789854AC7B466402\">8.1-3 DigraphClique</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap8_mj.html#X84EA2F9482B8D4AF\">8.1-4 DigraphMaximalCliques</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 8_mj.html#X78427A8B81FEB457\">8.1-5 CliqueNumber</a></span>\n</div></div>\n<di\ v class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </spa\ n><a href=\"chap8_mj.html#X7E31253287708348\">8.2 <span class=\"Heading\">Find\ ing independent sets</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap8\ _mj.html#X790733B67B0BC894\">8.2-1 IsIndependentSet</a></span>\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap8_mj.htm\ l#X84350678782BFAB7\">8.2-2 DigraphIndependentSet</a></span>\n<span class=\"Co\ ntSS\"><br /><span class=\"nocss\">  </span><a href=\"chap8_mj.html#\ X7D2C78EB81A798A9\">8.2-3 DigraphMaximalIndependentSets</a></span>\n</div></di\ v>\n</div>\n<div class=\"ContChap\"><a href=\"chap9_mj.html#X81714DBC80CC4BCB\ \">9 <span class=\"Heading\">Visualising and IO</span></a>\n<div class=\"ContS\ ect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"cha\ p9_mj.html#X7CF67F8278D9469C\">9.1 <span class=\"Heading\">Visualising a digra\ ph</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X83B33187\ 84E78415\">9.1-1 Splash</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap9_mj.html#X7F9B99C478EE093A\">9.1-2\ DotDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nb\ sp; </span><a href=\"chap9_mj.html#X78DA31D1869D2F94\">9.1-3 DotSymmetric\ Digraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap9_mj.html#X84120D447E2CAC23\">9.1-4 DotPartialOrderD\ igraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &n\ bsp;</span><a href=\"chap9_mj.html#X7F5653C8854FAEA6\">9.1-5 DotPreorderDigrap\ h</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap9_mj.html#X808D12CF81E2C19F\">9.1-6 DotHighlightedDigraph<\ /a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><spa\ n class=\"nocss\"> </span><a href=\"chap9_mj.html#X81C9B1DC791C0BBD\">9.2\ <span class=\"Heading\">Reading and writing digraphs to a file</span></a>\n</\ span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap9_mj.html#X81FB5BE27903EC32\">9.2-1\ String</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap9_mj.html#X873D6EB8836A3C61\">9.2-2 DigraphFromGraph\ 6String</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap9_mj.html#X782A052979EA2376\">9.2-3 Graph6String</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap9_mj.html#X7845ACDA7D4D333D\">9.2-4 DigraphFile</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap9_mj.html#X7E8142007ED0CB9F\">9.2-5 ReadDigraphs</a></span>\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.htm\ l#X82A4F0767C9EFC87\">9.2-6 WriteDigraphs</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X86D531B\ 287733561\">9.2-7 IteratorFromDigraphFile</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X7AAA358\ 67B5D8BF4\">9.2-8 DigraphPlainTextLineEncoder</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X780\ 8D99079295C22\">9.2-9 TournamentLineDecoder</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X7DF22\ D1687D096D6\">9.2-10 AdjacencyMatrixUpperTriangleLineDecoder</a></span>\n<span\ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"cha\ p9_mj.html#X875393338486D77D\">9.2-11 TCodeDecoder</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html\ #X8540FC21796A793F\">9.2-12 PlainTextString</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X81BD2\ 3697857B244\">9.2-13 WritePlainTextDigraph</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X804FDF\ 6A7ED64049\">9.2-14 WriteDIMACSDigraph</a></span>\n</div></div>\n</div>\n<div \ class=\"ContChap\"><a href=\"chapA_mj.html#X800DAF137BA8E25B\">A <span class=\ \"Heading\">\n Grape to Digraphs Command Map\n </span></a>\n<div class=\"C\ ontSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\ \"chapA_mj.html#X7CC8A9DB86F9F0F4\">A.1 <span class=\"Heading\">\n Functi\ ons to construct and modify graphs\n </span></a>\n</span>\n</div>\n<div cla\ ss=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a \ href=\"chapA_mj.html#X7C38A6397D2770FB\">A.2 <span class=\"Heading\">\n F\ unctions to inspect graphs, vertices and edges\n </span></a>\n</span>\n</di\ v>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&nbs\ p;</span><a href=\"chapA_mj.html#X78A5A34783C70CD7\">A.3 <span class=\"Heading\ \">\n Functions to determine regularity properties of graphs\n </span>\ </a>\n</span>\n</div>\n<div class=\"ContSect\"><span class=\"tocline\"><span c\ lass=\"nocss\"> </span><a href=\"chapA_mj.html#X7C74978D85F7AF5C\">A.4 <s\ pan class=\"Heading\">\n Some special vertex subsets of a graph\n </sp\ an></a>\n</span>\n</div>\n<div class=\"ContSect\"><span class=\"tocline\"><spa\ n class=\"nocss\"> </span><a href=\"chapA_mj.html#X7E51B92F7E89E266\">A.5\ <span class=\"Heading\">\n Functions to construct new graphs from old\n \ </span></a>\n</span>\n</div>\n<div class=\"ContSect\"><span class=\"tocline\ \"><span class=\"nocss\"> </span><a href=\"chapA_mj.html#X7A61E3FC7A2BB50\ E\">A.6 <span class=\"Heading\">\n Vertex-Colouring and Complete Subgraph\ s\n </span></a>\n</span>\n</div>\n<div class=\"ContSect\"><span class=\"toc\ line\"><span class=\"nocss\"> </span><a href=\"chapA_mj.html#X7DFC61EC7CB\ DBC7E\">A.7 <span class=\"Heading\">\n Automorphism groups and isomorphis\ m testing for graphs\n </span></a>\n</span>\n</div>\n</div>\n<div class=\"C\ ontChap\"><a href=\"chapBib_mj.html\"><span class=\"Heading\">References</span\ ></a></div>\n<div class=\"ContChap\"><a href=\"chapInd_mj.html\"><span class=\ \"Heading\">Index</span></a></div>\n<div class=\"ContChap\"><a href=\"chap1_mj\ .html#X7F202ABA780E595D\">1 <span class=\"Heading\">\n The <strong class=\"\ pkg\">Digraphs</strong> package\n </span></a>\n<div class=\"ContSect\"><span \ class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap1_mj.html#X\ 7DFB63A97E67C0A1\">1.1 <span class=\"Heading\">Introduction</span></a>\n</span\ >\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"noc\ ss\">  </span><a href=\"chap1_mj.html#X84541F61810C741D\">1.1-1 <spa\ n class=\"Heading\">Definitions</span></a>\n</span>\n</div></div>\n</div>\n<di\ v class=\"ContChap\"><a href=\"chap2_mj.html#X817088B27F90D596\">2 <span class\ =\"Heading\">Installing <strong class=\"pkg\">Digraphs</strong></span></a>\n<d\ iv class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </sp\ an><a href=\"chap2_mj.html#X7DA3059C79842BF3\">2.1 <span class=\"Heading\">For\ those in a hurry</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap2_mj\ .html#X7C1F2E6D860DBDEC\">2.1-1 <span class=\"Heading\">Configuration options<\ /span></a>\n</span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocli\ ne\"><span class=\"nocss\"> </span><a href=\"chap2_mj.html#X780AA9D97EBCA\ 95D\">2.2 <span class=\"Heading\">Optional package dependencies</span></a>\n</\ span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap2_mj.html#X8493C7587FCF6D8B\">2.2-1\ <span class=\"Heading\">The Grape package</span></a>\n</span>\n</div></div>\n\ <div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </\ span><a href=\"chap2_mj.html#X849F6196875A6DF5\">2.3 <span class=\"Heading\">C\ ompiling the kernel module</span></a>\n</span>\n</div>\n<div class=\"ContSect\ \"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap2_\ mj.html#X857CBE5484CF703A\">2.4 <span class=\"Heading\">Rebuilding the documen\ tation</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap2_mj.html#X870\ 631C38610AC25\">2.4-1 DigraphsMakeDoc</a></span>\n</div></div>\n<div class=\"C\ ontSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\ \"chap2_mj.html#X7862D3F37C5BBDEF\">2.5 <span class=\"Heading\">Testing your i\ nstallation</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"Co\ ntSS\"><br /><span class=\"nocss\">  </span><a href=\"chap2_mj.html#\ X86AF4DAE80B978DA\">2.5-1 DigraphsTestInstall</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap2_mj.html#X7A2\ 0088B7C406C4A\">2.5-2 DigraphsTestStandard</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap2_mj.html#X7885EF\ 3E785D4298\">2.5-3 DigraphsTestExtreme</a></span>\n</div></div>\n</div>\n<div \ class=\"ContChap\"><a href=\"chap3_mj.html#X7D34861E863A5D93\">3 <span class=\ \"Heading\">Creating digraphs</span></a>\n<div class=\"ContSect\"><span class=\ \"tocline\"><span class=\"nocss\"> </span><a href=\"chap3_mj.html#X7D3486\ 1E863A5D93\">3.1 <span class=\"Heading\">Creating digraphs</span></a>\n</span>\ \n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap3_mj.html#X7877ADC77F85E630\">3.1-1 IsDig\ raph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nbs\ p;</span><a href=\"chap3_mj.html#X7D7EDF83820ED6F5\">3.1-2 IsMutableDigraph</a\ ></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap3_mj.html#X7CAFAA89804F80BD\">3.1-3 IsImmutableDigraph</a></sp\ an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a \ href=\"chap3_mj.html#X7E749324800B38A5\">3.1-4 IsCayleyDigraph</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap3_mj.html#X80F1B6D28478D8B9\">3.1-5 IsDigraphWithAdjacencyFunction</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap3_mj.html#X86E798B779515678\">3.1-6 DigraphByOutNeighboursType</a></\ span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><\ a href=\"chap3_mj.html#X834843057CE86655\">3.1-7 Digraph</a></span>\n<span cla\ ss=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_m\ j.html#X8023FE387A3AB609\">3.1-8 DigraphByAdjacencyMatrix</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_\ mj.html#X7F37B6768349E269\">3.1-9 DigraphByEdges</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X\ 7B75C1D680757D6F\">3.1-10 EdgeOrbitsDigraph</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X81BC4\ 9B57EAADEFB\">3.1-11 DigraphByInNeighbours</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7FCADA\ DC7EC28478\">3.1-12 CayleyDigraph</a></span>\n</div></div>\n<div class=\"ContS\ ect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"cha\ p3_mj.html#X83608C407CC8836D\">3.2 <span class=\"Heading\">Changing representa\ tions</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7FBC4\ BDB82FBEDD2\">3.2-1 AsBinaryRelation</a></span>\n<span class=\"ContSS\"><br />\ <span class=\"nocss\">  </span><a href=\"chap3_mj.html#X83CA27E67F68\ 5EB3\">3.2-2 AsDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"\ nocss\">  </span><a href=\"chap3_mj.html#X7B335342839E5146\">3.2-3 G\ raph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nbs\ p;</span><a href=\"chap3_mj.html#X7C4F13E080EC16B0\">3.2-4 AsGraph</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X7C5360B2799943F3\">3.2-5 AsTransformation</a></span>\n</div\ ></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"\ > </span><a href=\"chap3_mj.html#X85126078848B420A\">3.3 <span class=\"He\ ading\">New digraphs from old</span></a>\n</span>\n<div class=\"ContSSBlock\">\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X83D93A8A8251E6F9\">3.3-1 DigraphImmutableCopy</a></span>\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap3_mj.html#X8399F7427B227228\">3.3-2 DigraphImmutableCopyIfImmutable</a><\ /span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap3_mj.html#X83C51DA182CCEA2F\">3.3-3 InducedSubdigraph</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X7CAF093B85A93D2F\">3.3-4 ReducedDigraph</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3\ _mj.html#X829E3EAC7C4B3B1E\">3.3-5 MaximalSymmetricSubdigraph</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap3_mj.html#X79BA6A66846D5A95\">3.3-6 MaximalAntiSymmetricSubdigraph</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap3_mj.html#X7DD9766C86D3ED20\">3.3-7 UndirectedSpanningForest</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap3_mj.html#X7D1D26D27F5B56C2\">3.3-8 QuotientDigraph</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap3_mj.html#X78DECD26811EFD7C\">3.3-9 DigraphReverse</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.h\ tml#X7F71D99D852B130F\">3.3-10 DigraphDual</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X874883\ DD7DD450C4\">3.3-11 DigraphSymmetricClosure</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7A6C4\ 19080AD41DE\">3.3-12 DigraphTransitiveClosure</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X82A\ D17517E273600\">3.3-13 DigraphTransitiveReduction</a></span>\n<span class=\"Co\ ntSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#\ X83B2506D79453208\">3.3-14 DigraphAddVertex</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X8134B\ EE7786BD3A7\">3.3-15 DigraphAddVertices</a></span>\n<span class=\"ContSS\"><br\ /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7E58CC488\ 0627658\">3.3-16 DigraphAddEdge</a></span>\n<span class=\"ContSS\"><br /><span\ class=\"nocss\">  </span><a href=\"chap3_mj.html#X7BE5C7028760B053\ \">3.3-17 DigraphAddEdgeOrbit</a></span>\n<span class=\"ContSS\"><br /><span c\ lass=\"nocss\">  </span><a href=\"chap3_mj.html#X8693A61B7F752C76\">\ 3.3-18 DigraphAddEdges</a></span>\n<span class=\"ContSS\"><br /><span class=\"\ nocss\">  </span><a href=\"chap3_mj.html#X7B634A2B83C08B16\">3.3-19 \ DigraphRemoveVertex</a></span>\n<span class=\"ContSS\"><br /><span class=\"noc\ ss\">  </span><a href=\"chap3_mj.html#X7E290E847A5A299A\">3.3-20 Dig\ raphRemoveVertices</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap3_mj.html#X8433E3BC7E5EA6BF\">3.3-21 Digr\ aphRemoveEdge</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\ nbsp; </span><a href=\"chap3_mj.html#X85981D9187F49018\">3.3-22 DigraphRe\ moveEdgeOrbit</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\ nbsp; </span><a href=\"chap3_mj.html#X87093FDA7F88E732\">3.3-23 DigraphRe\ moveEdges</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap3_mj.html#X79324AF7818C0C02\">3.3-24 DigraphRemove\ Loops</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nb\ sp;</span><a href=\"chap3_mj.html#X7DCCD0247897A3DE\">3.3-25 DigraphRemoveAllM\ ultipleEdges</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap3_mj.html#X7821753D85402A8C\">3.3-26 DigraphRev\ erseEdges</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap3_mj.html#X814F1DFC83DB273F\">3.3-27 DigraphDisjoi\ ntUnion</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap3_mj.html#X7DA997697D310E44\">3.3-28 DigraphEdgeUnio\ n</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap3_mj.html#X7DDFC759860E3390\">3.3-29 DigraphJoin</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap3_mj.html#X7D625CC87DBFFDED\">3.3-30 DigraphCartesianProduct</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap3_mj.html#X84D24DC9833B54A5\">3.3-31 DigraphDirectProduct</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X8795087A78FE7D54\">3.3-32 DigraphCartesianProductProjection\ s</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap3_mj.html#X84FB64F185B804C2\">3.3-33 DigraphDirectProductP\ rojections</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbs\ p; </span><a href=\"chap3_mj.html#X8595BF937B749F22\">3.3-34 LineDigraph<\ /a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </s\ pan><a href=\"chap3_mj.html#X8364C6F17A1680CB\">3.3-35 LineUndirectedDigraph</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap3_mj.html#X7FB8B48C87C0ED16\">3.3-36 DoubleDigraph</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X7C6E6CB284982C7A\">3.3-37 BipartiteDoubleDigraph</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap3_mj.html#X8167A50A83256ED1\">3.3-38 DigraphAddAllLoops</a></span>\n<s\ pan class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"\ chap3_mj.html#X865436437DF95FEF\">3.3-39 DistanceDigraph</a></span>\n<span cla\ ss=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_m\ j.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X\ 7B5AC5FE859F4D80\">3.3-41 DigraphMycielskian</a></span>\n</div></div>\n<div cl\ ass=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a\ href=\"chap3_mj.html#X85B5D9B97F5187B7\">3.4 <span class=\"Heading\">Random d\ igraphs</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X86C\ F9F66788B2A24\">3.4-1 RandomDigraph</a></span>\n<span class=\"ContSS\"><br /><\ span class=\"nocss\">  </span><a href=\"chap3_mj.html#X78FE275E7E77D\ 56F\">3.4-2 RandomMultiDigraph</a></span>\n<span class=\"ContSS\"><br /><span \ class=\"nocss\">  </span><a href=\"chap3_mj.html#X7D36B5E57F055051\"\ >3.4-3 RandomTournament</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap3_mj.html#X7A023E4787682475\">3.4-4\ RandomLattice</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\ \"tocline\"><span class=\"nocss\"> </span><a href=\"chap3_mj.html#X7C76D1\ DC7DAF03D3\">3.5 <span class=\"Heading\">Standard examples</span></a>\n</span>\ \n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap3_mj.html#X870594FC866AC88E\">3.5-1 Chain\ Digraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap3_mj.html#X812417E278198D9C\">3.5-2 CompleteDigraph<\ /a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </s\ pan><a href=\"chap3_mj.html#X8795B0AD856014FA\">3.5-3 CompleteBipartiteDigraph\ </a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </\ span><a href=\"chap3_mj.html#X873F29CC863241F8\">3.5-4 CompleteMultipartiteDig\ raph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nbs\ p;</span><a href=\"chap3_mj.html#X80C29DDE876FFBEB\">3.5-5 CycleDigraph</a></s\ pan>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a\ href=\"chap3_mj.html#X80DAE31A79FEFD40\">3.5-6 EmptyDigraph</a></span>\n<span\ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"cha\ p3_mj.html#X80ED9CE785819607\">3.5-7 JohnsonDigraph</a></span>\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap3_mj.htm\ l#X823F43217A6C375D\">3.5-8 PetersenGraph</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap3_mj.html#X7B5441F\ 386BD105E\">3.5-9 GeneralisedPetersenGraph</a></span>\n</div></div>\n</div>\n<\ div class=\"ContChap\"><a href=\"chap4_mj.html#X7AD6F77E7D95C996\">4 <span cla\ ss=\"Heading\">Operators</span></a>\n<div class=\"ContSect\"><span class=\"toc\ line\"><span class=\"nocss\"> </span><a href=\"chap4_mj.html#X84E0B5B8835\ 8C96B\">4.1 <span class=\"Heading\">Operators for digraphs</span></a>\n</span>\ \n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap4_mj.html#X829B911D7EFD2D85\">4.1-1 IsSub\ digraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap4_mj.html#X833C3299787E2309\">4.1-2 IsUndirectedSpan\ ningTree</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\"><a href=\"c\ hap5_mj.html#X8739F6CD78C90B14\">5 <span class=\"Heading\">Attributes and oper\ ations</span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class\ =\"nocss\"> </span><a href=\"chap5_mj.html#X7E814B6478F7D015\">5.1 <span \ class=\"Heading\">Vertices and edges</span></a>\n</span>\n<div class=\"ContSSB\ lock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap5_mj.html#X7C45F7D878D896AC\">5.1-1 DigraphVertices</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap5_mj.html#X7C6F19B57CB2E882\">5.1-2 DigraphNrVertices</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap5_mj.html#X7D1C6A4D7ECEC317\">5.1-3 DigraphEdges</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html\ #X85E1CFDD7E164AD0\">5.1-4 DigraphNrEdges</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7BDABAF\ 07917462B\">5.1-5 DigraphNrLoops</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap5_mj.html#X85D5E08280914EE4\ \">5.1-6 DigraphSinks</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap5_mj.html#X7F5C6268839BE98C\">5.1-7 Di\ graphSources</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X785C30378064CF47\">5.1-8 DigraphTopo\ logicalSort</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nb\ sp; </span><a href=\"chap5_mj.html#X7CA91E4B7904F793\">5.1-9 DigraphVerte\ xLabel</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &n\ bsp;</span><a href=\"chap5_mj.html#X7E51F2FE87140B32\">5.1-10 DigraphVertexLab\ els</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap5_mj.html#X79FAEACC7F438C2F\">5.1-11 DigraphEdgeLabel</a\ ></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap5_mj.html#X7C24851087D4A8FB\">5.1-12 DigraphEdgeLabels</a></sp\ an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a \ href=\"chap5_mj.html#X7EFAF01B7A155157\">5.1-13 DigraphInEdges</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap5_mj.html#X7BECFE6687ECD028\">5.1-14 DigraphOutEdges</a></span>\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj\ .html#X7BB8ED88835F07B4\">5.1-15 IsDigraphEdge</a></span>\n<span class=\"ContS\ S\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X84\ 8FED0B7B4ACD1F\">5.1-16 IsMatching</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap5_mj.html#X7B7A67277B1C9A\ 02\">5.1-17 DigraphMaximalMatching</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap5_mj.html#X78E9847A858788\ D1\">5.1-18 DigraphMaximumMatching</a></span>\n</div></div>\n<div class=\"Cont\ Sect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"ch\ ap5_mj.html#X7D7CE8328187D0DF\">5.2 <span class=\"Heading\">Neighbours and deg\ ree</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7DC2CD7\ 0830BEE60\">5.2-1 AdjacencyMatrix</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap5_mj.html#X87FA0A727CDB060\ B\">5.2-2 CharacteristicPolynomial</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap5_mj.html#X8507DC4F794780\ C1\">5.2-3 BooleanAdjacencyMatrix</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap5_mj.html#X7AFCE34A7A04D5C\ 1\">5.2-4 DigraphAdjacencyFunction</a></span>\n<span class=\"ContSS\"><br /><s\ pan class=\"nocss\">  </span><a href=\"chap5_mj.html#X7FDEBF32797599\ 61\">5.2-5 DigraphRange</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap5_mj.html#X7E9880767AE68E00\">5.2-6\ OutNeighbours</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\   </span><a href=\"chap5_mj.html#X85C7AA5A81DA6E11\">5.2-7 InNeighbo\ urs</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap5_mj.html#X7F5ACE807D1BC2E2\">5.2-8 OutDegrees</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap5_mj.html#X7ADDFBFD7A365775\">5.2-9 InDegrees</a></span>\n<span class\ =\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.\ html#X7A09EB648070276D\">5.2-10 OutDegreeOfVertex</a></span>\n<span class=\"Co\ ntSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#\ X83315B0186850806\">5.2-11 OutNeighboursOfVertex</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X\ 7C9CD0527CB9E6EF\">5.2-12 InDegreeOfVertex</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7C0DA1\ 8B8291F302\">5.2-13 InNeighboursOfVertex</a></span>\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X83271F60\ 7BD809CF\">5.2-14 DigraphLoops</a></span>\n<span class=\"ContSS\"><br /><span \ class=\"nocss\">  </span><a href=\"chap5_mj.html#X7DDB33B686B3A2C6\"\ >5.2-15 PartialOrderDigraphMeetOfVertices</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7BEAE1C\ 78267F54D\">5.2-16 DegreeMatrix</a></span>\n<span class=\"ContSS\"><br /><span\ class=\"nocss\">  </span><a href=\"chap5_mj.html#X865390B08331936B\ \">5.2-17 LaplacianMatrix</a></span>\n</div></div>\n<div class=\"ContSect\"><s\ pan class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap5_mj.ht\ ml#X8537F4088400DC48\">5.3 <span class=\"Heading\">Reachability and connectivi\ ty</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7F16B9EB\ 8398459C\">5.3-1 DigraphDiameter</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap5_mj.html#X8104A9D37BCD8A05\ \">5.3-2 DigraphShortestDistance</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap5_mj.html#X81F99BC67E9D050F\ \">5.3-3 DigraphShortestDistances</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap5_mj.html#X8223718079D98A8\ 2\">5.3-4 DigraphLongestDistanceFromVertex</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7CB7DD\ CD84621D38\">5.3-5 DigraphDistanceSet</a></span>\n<span class=\"ContSS\"><br /\ ><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X79A3DA4078C\ F3C90\">5.3-6 DigraphGirth</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap5_mj.html#X8374B7357EC189C1\">5.3\ -7 DigraphOddGirth</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap5_mj.html#X84688B337BDDBB09\">5.3-8 Digra\ phUndirectedGirth</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap5_mj.html#X842FAD6A7B835977\">5.3-9 Digrap\ hConnectedComponents</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\ css\">  </span><a href=\"chap5_mj.html#X8484EC557810CD31\">5.3-10 Di\ graphConnectedComponent</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap5_mj.html#X833ECD6B7A84944C\">5.3-1\ 1 DigraphStronglyConnectedComponents</a></span>\n<span class=\"ContSS\"><br />\ <span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7EFCB5017D66\ 2254\">5.3-12 DigraphStronglyConnectedComponent</a></span>\n<span class=\"Cont\ SS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7\ F1B5A2782F598B1\">5.3-13 DigraphBicomponents</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7DD\ E06E47E605DD7\">5.3-14 ArticulationPoints</a></span>\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X84D5A12\ 5848BD800\">5.3-15 Bridges</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap5_mj.html#X865590147BD1C507\">5.3\ -16 StrongOrientation</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap5_mj.html#X853D0B0981A33433\">5.3-17 D\ igraphPeriod</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap5_mj.html#X864A31A8809F61C2\">5.3-18 DigraphFlo\ ydWarshall</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbs\ p; </span><a href=\"chap5_mj.html#X7FBAB09E7C0BE5CF\">5.3-19 IsReachable<\ /a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </s\ pan><a href=\"chap5_mj.html#X7CAE81D97B43DDF3\">5.3-20 VerticesReachableFrom</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap5_mj.html#X8039170B82A32257\">5.3-21 DigraphPath</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap5_mj.html#X80E9D645843973A6\">5.3-22 DigraphShortestPath</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap5_mj.html#X7C0416FE7A69CA2C\">5.3-23 IteratorOfPaths</a></span>\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj\ .html#X7ECD16838704FAAA\">5.3-24 DigraphAllSimpleCircuits</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_\ mj.html#X7C735C4E86BDD5F6\">5.3-25 DigraphLongestSimpleCircuit</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap5_mj.html#X870E04307C5F213F\">5.3-26 DigraphLayers</a></span>\n<span class=\ \"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.h\ tml#X7B2E42327DA118E0\">5.3-27 DigraphDegeneracy</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X\ 827C2BD17A4547E3\">5.3-28 DigraphDegeneracyOrdering</a></span>\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.htm\ l#X863FDFC4839A3B82\">5.3-29 HamiltonianPath</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X82F\ 30D5681466BC6\">5.3-30 NrSpanningTrees</a></span>\n<span class=\"ContSS\"><br \ /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X79352A8286\ D1D8F6\">5.3-31 DigraphDijkstra</a></span>\n</div></div>\n<div class=\"ContSec\ t\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap5\ _mj.html#X82F900777D677F55\">5.4 <span class=\"Heading\">Cayley graphs of grou\ ps</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><b\ r /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7A000B1D\ 7CCF7093\">5.4-1 GroupOfCayleyDigraph</a></span>\n<span class=\"ContSS\"><br /\ ><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X8528455987D\ 7D2BF\">5.4-2 GeneratorsOfCayleyDigraph</a></span>\n</div></div>\n<div class=\ \"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a hre\ f=\"chap5_mj.html#X790FD6647ECCAE3C\">5.5 <span class=\"Heading\">Associated s\ emigroups</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"Cont\ SS\"><br /><span class=\"nocss\">  </span><a href=\"chap5_mj.html#X8\ 7D5C60D7B0C1309\">5.5-1 AsSemigroup</a></span>\n<span class=\"ContSS\"><br /><\ span class=\"nocss\">  </span><a href=\"chap5_mj.html#X7C6D5EC27C510\ 66B\">5.5-2 AsSemigroup</a></span>\n</div></div>\n<div class=\"ContSect\"><spa\ n class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap5_mj.html\ #X7E2305528492DDC0\">5.6 <span class=\"Heading\">Planarity</span></a>\n</span>\ \n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap5_mj.html#X7DC478637E8C190D\">5.6-1 Kurat\ owskiPlanarSubdigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap5_mj.html#X78E8F09A8286501B\">5.6-2 Ku\ ratowskiOuterPlanarSubdigraph</a></span>\n<span class=\"ContSS\"><br /><span c\ lass=\"nocss\">  </span><a href=\"chap5_mj.html#X84E3947E7D39BA64\">\ 5.6-3 PlanarEmbedding</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap5_mj.html#X85DFB8C18088711F\">5.6-4 Ou\ terPlanarEmbedding</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap5_mj.html#X806D2D6B85E0B269\">5.6-5 Subdi\ graphHomeomorphicToK23</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\ \"><a href=\"chap6_mj.html#X7ADDEFD478D470D5\">6 <span class=\"Heading\">Prope\ rties of digraphs</span></a>\n<div class=\"ContSect\"><span class=\"tocline\">\ <span class=\"nocss\"> </span><a href=\"chap6_mj.html#X7F49388F8245F0E9\"\ >6.1 <span class=\"Heading\">Edge properties</span></a>\n</span>\n<div class=\ \"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nb\ sp;</span><a href=\"chap6_mj.html#X7D92935C7D535187\">6.1-1 DigraphHasLoops</a\ ></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </spa\ n><a href=\"chap6_mj.html#X7DB1BC2286FC08E2\">6.1-2 IsAntiSymmetricDigraph</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap6_mj.html#X860CFB0C8665F356\">6.1-3 IsBipartiteDigraph</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap6_mj.html#X790A4DBF8533516E\">6.1-4 IsCompleteBipartiteDigraph</a></\ span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><\ a href=\"chap6_mj.html#X81F28D4D879FE3B2\">6.1-5 IsCompleteDigraph</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap6_mj.html#X7FD1A15779FEC341\">6.1-6 IsCompleteMultipartiteDigraph</a><\ /span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span>\ <a href=\"chap6_mj.html#X7E894CCF7B1C27AE\">6.1-7 IsEmptyDigraph</a></span>\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap6_mj.html#X83762C257DED2751\">6.1-8 IsEquivalenceDigraph</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap6_mj.html#X7E8F37E585DAED52\">6.1-9 IsFunctionalDigraph</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6\ _mj.html#X7BB84CFC7E8B2B26\">6.1-10 IsMultiDigraph</a></span>\n<span class=\"C\ ontSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html\ #X8462D8E2792B23F6\">6.1-11 IsReflexiveDigraph</a></span>\n<span class=\"ContS\ S\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X81\ B3EA7887219860\">6.1-12 IsSymmetricDigraph</a></span>\n<span class=\"ContSS\">\ <br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7DD8D1\ A185EBE865\">6.1-13 IsTournament</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap6_mj.html#X7F0835667F29F0C0\ \">6.1-14 IsTransitiveDigraph</a></span>\n<span class=\"ContSS\"><br /><span c\ lass=\"nocss\">  </span><a href=\"chap6_mj.html#X8617726C7829F796\">\ 6.1-15 IsPreorderDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap6_mj.html#X82BAE6D37D49A145\">6.1-1\ 6 IsPartialOrderDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap6_mj.html#X78D3E17B7F737516\">6.1-1\ 7 IsMeetSemilatticeDigraph</a></span>\n</div></div>\n<div class=\"ContSect\"><\ span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap6_mj.h\ tml#X7AAF896982EC22FA\">6.2 <span class=\"Heading\">Regularity</span></a>\n</s\ pan>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"\ nocss\">  </span><a href=\"chap6_mj.html#X7A3F78B37E31D9E1\">6.2-1 I\ sInRegularDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap6_mj.html#X7F0A806E7BCB413C\">6.2-2 IsOutR\ egularDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\ nbsp; </span><a href=\"chap6_mj.html#X7AF9DB1E7DB12306\">6.2-3 IsRegularD\ igraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &n\ bsp;</span><a href=\"chap6_mj.html#X7E2082AC7CAE59CD\">6.2-4 IsDistanceRegular\ Digraph</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"toclin\ e\"><span class=\"nocss\"> </span><a href=\"chap6_mj.html#X780DAEB37C5E07\ FD\">6.3 <span class=\"Heading\">Connectivity and cycles</span></a>\n</span>\n\ <div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap6_mj.html#X7BD08D4478218F77\">6.3-1 IsAcyc\ licDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbs\ p; </span><a href=\"chap6_mj.html#X79563B297C220FB5\">6.3-2 IsChainDigrap\ h</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap6_mj.html#X83C08C0B7EC1A91F\">6.3-3 IsConnectedDigraph</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap6_mj.html#X838FAF2D825977BE\">6.3-4 IsBiconnectedDigraph</a></s\ pan>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a\ href=\"chap6_mj.html#X7D2A6572820B7F24\">6.3-5 IsBridgelessDigraph</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap6_mj.html#X7B37B9467C68C208\">6.3-6 IsStronglyConnectedDigraph</a></sp\ an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a \ href=\"chap6_mj.html#X80E883967EBE839E\">6.3-7 IsAperiodicDigraph</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap6_mj.html#X7B46EA6C7B2DF2FB\">6.3-8 IsDirectedTree</a></span>\n<span cla\ ss=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_m\ j.html#X80FC20FA7AC4BC2A\">6.3-9 IsUndirectedTree</a></span>\n<span class=\"Co\ ntSS\"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#\ X79DFBB198525544E\">6.3-10 IsEulerianDigraph</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X79E\ FEBC37C2D262D\">6.3-11 IsHamiltonianDigraph</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X7E913\ 20B8028F5D8\">6.3-12 IsCycleDigraph</a></span>\n</div></div>\n<div class=\"Con\ tSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"c\ hap6_mj.html#X7E2305528492DDC0\">6.4 <span class=\"Heading\">Planarity</span><\ /a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span \ class=\"nocss\">  </span><a href=\"chap6_mj.html#X8606D415858C40AA\"\ >6.4-1 IsPlanarDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"\ nocss\">  </span><a href=\"chap6_mj.html#X8251E8B187E7F059\">6.4-2 I\ sOuterPlanarDigraph</a></span>\n</div></div>\n<div class=\"ContSect\"><span cl\ ass=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap6_mj.html#X83\ DC8AD283C41326\">6.5 <span class=\"Heading\">Homomorphisms and transformations\ </span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br \ /><span class=\"nocss\">  </span><a href=\"chap6_mj.html#X78C0B26379\ 85648B\">6.5-1 IsDigraphCore</a></span>\n<span class=\"ContSS\"><br /><span cl\ ass=\"nocss\">  </span><a href=\"chap6_mj.html#X8361CA6E8401FF26\">6\ .5-2 IsEdgeTransitive</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap6_mj.html#X7B26E4447A1611E9\">6.5-3 Is\ VertexTransitive</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\"><a \ href=\"chap7_mj.html#X84975388859F203D\">7 <span class=\"Heading\">Homomorphis\ ms</span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"n\ ocss\"> </span><a href=\"chap7_mj.html#X8200192B80AD2071\">7.1 <span clas\ s=\"Heading\">Acting on digraphs</span></a>\n</span>\n<div class=\"ContSSBlock\ \">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a \ href=\"chap7_mj.html#X808972017C486F1F\">7.1-1 OnDigraphs</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_\ mj.html#X7AFBA7608498F9CE\">7.1-2 OnMultiDigraphs</a></span>\n</div></div>\n<d\ iv class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </sp\ an><a href=\"chap7_mj.html#X7E48B9F87A0F22D4\">7.2 <span class=\"Heading\">Iso\ morphisms and canonical labellings</span></a>\n</span>\n<div class=\"ContSSBlo\ ck\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><\ a href=\"chap7_mj.html#X83E593F3855B122E\">7.2-1 DigraphsUseNauty</a></span>\n\ <span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap7_mj.html#X858C32127A190175\">7.2-2 AutomorphismGroup</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 7_mj.html#X7E7B0D88865A89F6\">7.2-3 BlissAutomorphismGroup</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7\ _mj.html#X857758B18144C0CD\">7.2-4 NautyAutomorphismGroup</a></span>\n<span cl\ ass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_\ mj.html#X877732B1783C391B\">7.2-5 AutomorphismGroup</a></span>\n<span class=\"\ ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.htm\ l#X7DA2C4FE837FFE01\">7.2-6 AutomorphismGroup</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X7D6\ 76FE67A6684FF\">7.2-7 BlissCanonicalLabelling</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X87D\ A265D803DB337\">7.2-8 BlissCanonicalLabelling</a></span>\n<span class=\"ContSS\ \"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X877\ B1D377EC197D7\">7.2-9 BlissCanonicalDigraph</a></span>\n<span class=\"ContSS\"\ ><br /><span class=\"nocss\">  </span><a href=\"chap7_mj.html#X803AC\ EDA7BBAC5B3\">7.2-10 DigraphGroup</a></span>\n<span class=\"ContSS\"><br /><sp\ an class=\"nocss\">  </span><a href=\"chap7_mj.html#X8096C5287E45927\ 9\">7.2-11 DigraphOrbits</a></span>\n<span class=\"ContSS\"><br /><span class=\ \"nocss\">  </span><a href=\"chap7_mj.html#X8386028782F2D3FF\">7.2-1\ 2 DigraphOrbitReps</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\ s\">  </span><a href=\"chap7_mj.html#X8657604E87A25E5F\">7.2-13 Digr\ aphSchreierVector</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap7_mj.html#X78913684795FB256\">7.2-14 Digra\ phStabilizer</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\ bsp; </span><a href=\"chap7_mj.html#X7B4F24B283C9EE28\">7.2-15 IsIsomorph\ icDigraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap7_mj.html#X7D4A1FA8868DA930\">7.2-16 IsIsomorphicD\ igraph</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &n\ bsp;</span><a href=\"chap7_mj.html#X8219FE6C839D9457\">7.2-17 IsomorphismDigra\ phs</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap7_mj.html#X7ED93C0F86D9D34F\">7.2-18 IsomorphismDigraphs\ </a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </\ span><a href=\"chap7_mj.html#X7E1C806D81DFE15E\">7.2-19 RepresentativeOutNeigh\ bours</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nb\ sp;</span><a href=\"chap7_mj.html#X80183D4A7C51365A\">7.2-20 IsDigraphIsomorph\ ism</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap7_mj.html#X7F16B8B3825A627A\">7.2-21 IsDigraphColouring<\ /a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><spa\ n class=\"nocss\"> </span><a href=\"chap7_mj.html#X7E1A02FE8384C03C\">7.3\ <span class=\"Heading\">Homomorphisms of digraphs</span></a>\n</span>\n<div c\ lass=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbs\ p; </span><a href=\"chap7_mj.html#X79ABF0E783FD67C7\">7.3-1 HomomorphismD\ igraphsFinder</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\ nbsp; </span><a href=\"chap7_mj.html#X85E9B019877AD7FE\">7.3-2 DigraphHom\ omorphism</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap7_mj.html#X8653EDA183E06D05\">7.3-3 HomomorphismsD\ igraphs</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap7_mj.html#X82D0FCD87D47ACA8\">7.3-4 DigraphMonomorph\ ism</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  \ ;</span><a href=\"chap7_mj.html#X84B4BDB984C2A9A8\">7.3-5 MonomorphismsDigraph\ s</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  <\ /span><a href=\"chap7_mj.html#X7CB5AD9F861684FD\">7.3-6 DigraphEpimorphism</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap7_mj.html#X7DFB1F5D873937B3\">7.3-7 EpimorphismsDigraphs</a></s\ pan>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a\ href=\"chap7_mj.html#X7AD04CC186E86CCA\">7.3-8 DigraphEmbedding</a></span>\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap7_mj.html#X82EBDF137EEDC5FD\">7.3-9 EmbeddingsDigraphs</a></span>\n<span\ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"cha\ p7_mj.html#X80FE9FCB79718A61\">7.3-10 IsDigraphHomomorphism</a></span>\n<span \ class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap\ 7_mj.html#X7D6B5A0887903810\">7.3-11 IsDigraphEmbedding</a></span>\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj\ .html#X7F80709A80CBFA98\">7.3-12 DigraphsRespectsColouring</a></span>\n<span c\ lass=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7\ _mj.html#X7E93B268823F6478\">7.3-13 GeneratorsOfEndomorphismMonoid</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap7_mj.html#X86D0A74B83D6B9A7\">7.3-14 DigraphColouring</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap7_mj.html#X7AB7200D831013C1\">7.3-15 DigraphGreedyColouring</a></span>\n<spa\ n class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"ch\ ap7_mj.html#X7F9CB3B27B9590DB\">7.3-16 DigraphWelshPowellOrder</a></span>\n<sp\ an class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"c\ hap7_mj.html#X807D49358529A37C\">7.3-17 ChromaticNumber</a></span>\n<span clas\ s=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\"chap7_mj\ .html#X863487EF7D64EF56\">7.3-18 DigraphCore</a></span>\n</div></div>\n</div>\ \n<div class=\"ContChap\"><a href=\"chap8_mj.html#X7A7E0B957932DF44\">8 <span \ class=\"Heading\">Finding cliques and independent sets</span></a>\n<div class=\ \"ContSect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a hre\ f=\"chap8_mj.html#X788474E38446039B\">8.1 <span class=\"Heading\">Finding cliq\ ues</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><\ br /><span class=\"nocss\">  </span><a href=\"chap8_mj.html#X80670E3\ E7C36183F\">8.1-1 IsClique</a></span>\n<span class=\"ContSS\"><br /><span clas\ s=\"nocss\">  </span><a href=\"chap8_mj.html#X7A3BC9FF8169ABCD\">8.1\ -2 CliquesFinder</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\ \">  </span><a href=\"chap8_mj.html#X789854AC7B466402\">8.1-3 Digrap\ hClique</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &\ nbsp;</span><a href=\"chap8_mj.html#X84EA2F9482B8D4AF\">8.1-4 DigraphMaximalCl\ iques</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nb\ sp;</span><a href=\"chap8_mj.html#X78427A8B81FEB457\">8.1-5 CliqueNumber</a></\ span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span cla\ ss=\"nocss\"> </span><a href=\"chap8_mj.html#X7E31253287708348\">8.2 <spa\ n class=\"Heading\">Finding independent sets</span></a>\n</span>\n<div class=\ \"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\"> &nb\ sp;</span><a href=\"chap8_mj.html#X790733B67B0BC894\">8.2-1 IsIndependentSet</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap8_mj.html#X84350678782BFAB7\">8.2-2 DigraphIndependentSet</a>\ </span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span\ ><a href=\"chap8_mj.html#X7D2C78EB81A798A9\">8.2-3 DigraphMaximalIndependentSe\ ts</a></span>\n</div></div>\n</div>\n<div class=\"ContChap\"><a href=\"chap9_m\ j.html#X81714DBC80CC4BCB\">9 <span class=\"Heading\">Visualising and IO</span>\ </a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&n\ bsp;</span><a href=\"chap9_mj.html#X7CF67F8278D9469C\">9.1 <span class=\"Headi\ ng\">Visualising a digraph</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<\ span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a href=\ \"chap9_mj.html#X83B3318784E78415\">9.1-1 Splash</a></span>\n<span class=\"Con\ tSS\"><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#X\ 7F9B99C478EE093A\">9.1-2 DotDigraph</a></span>\n<span class=\"ContSS\"><br /><\ span class=\"nocss\">  </span><a href=\"chap9_mj.html#X78DA31D1869D2\ F94\">9.1-3 DotSymmetricDigraph</a></span>\n<span class=\"ContSS\"><br /><span\ class=\"nocss\">  </span><a href=\"chap9_mj.html#X84120D447E2CAC23\ \">9.1-4 DotPartialOrderDigraph</a></span>\n<span class=\"ContSS\"><br /><span\ class=\"nocss\">  </span><a href=\"chap9_mj.html#X7F5653C8854FAEA6\ \">9.1-5 DotPreorderDigraph</a></span>\n<span class=\"ContSS\"><br /><span cla\ ss=\"nocss\">  </span><a href=\"chap9_mj.html#X808D12CF81E2C19F\">9.\ 1-6 DotHighlightedDigraph</a></span>\n</div></div>\n<div class=\"ContSect\"><s\ pan class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chap9_mj.ht\ ml#X81C9B1DC791C0BBD\">9.2 <span class=\"Heading\">Reading and writing digraph\ s to a file</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"Co\ ntSS\"><br /><span class=\"nocss\">  </span><a href=\"chap9_mj.html#\ X81FB5BE27903EC32\">9.2-1 String</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap9_mj.html#X873D6EB8836A3C61\ \">9.2-2 DigraphFromGraph6String</a></span>\n<span class=\"ContSS\"><br /><spa\ n class=\"nocss\">  </span><a href=\"chap9_mj.html#X782A052979EA2376\ \">9.2-3 Graph6String</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\ ocss\">  </span><a href=\"chap9_mj.html#X7845ACDA7D4D333D\">9.2-4 Di\ graphFile</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \ ; </span><a href=\"chap9_mj.html#X7E8142007ED0CB9F\">9.2-5 ReadDigraphs</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap9_mj.html#X82A4F0767C9EFC87\">9.2-6 WriteDigraphs</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap9_mj.html#X86D531B287733561\">9.2-7 IteratorFromDigraphFile</a></span>\ \n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hre\ f=\"chap9_mj.html#X7AAA35867B5D8BF4\">9.2-8 DigraphPlainTextLineEncoder</a></s\ pan>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a\ href=\"chap9_mj.html#X7808D99079295C22\">9.2-9 TournamentLineDecoder</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap9_mj.html#X7DF22D1687D096D6\">9.2-10 AdjacencyMatrixUpperTriangleLin\ eDecoder</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"> \  </span><a href=\"chap9_mj.html#X875393338486D77D\">9.2-11 TCodeDecoder</\ a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </sp\ an><a href=\"chap9_mj.html#X8540FC21796A793F\">9.2-12 PlainTextString</a></spa\ n>\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a h\ ref=\"chap9_mj.html#X81BD23697857B244\">9.2-13 WritePlainTextDigraph</a></span\ >\n<span class=\"ContSS\"><br /><span class=\"nocss\">  </span><a hr\ ef=\"chap9_mj.html#X804FDF6A7ED64049\">9.2-14 WriteDIMACSDigraph</a></span>\n<\ /div></div>\n</div>\n<div class=\"ContChap\"><a href=\"chapA_mj.html#X800DAF13\ 7BA8E25B\">A <span class=\"Heading\">\n Grape to Digraphs Command Map\n </\ span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\ \"> </span><a href=\"chapA_mj.html#X7CC8A9DB86F9F0F4\">A.1 <span class=\"\ Heading\">\n Functions to construct and modify graphs\n </span></a>\n<\ /span>\n</div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"\ nocss\"> </span><a href=\"chapA_mj.html#X7C38A6397D2770FB\">A.2 <span cla\ ss=\"Heading\">\n Functions to inspect graphs, vertices and edges\n </\ span></a>\n</span>\n</div>\n<div class=\"ContSect\"><span class=\"tocline\"><s\ pan class=\"nocss\"> </span><a href=\"chapA_mj.html#X78A5A34783C70CD7\">A\ .3 <span class=\"Heading\">\n Functions to determine regularity propertie\ s of graphs\n </span></a>\n</span>\n</div>\n<div class=\"ContSect\"><span c\ lass=\"tocline\"><span class=\"nocss\"> </span><a href=\"chapA_mj.html#X7\ C74978D85F7AF5C\">A.4 <span class=\"Heading\">\n Some special vertex subs\ ets of a graph\n </span></a>\n</span>\n</div>\n<div class=\"ContSect\"><spa\ n class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chapA_mj.html\ #X7E51B92F7E89E266\">A.5 <span class=\"Heading\">\n Functions to construc\ t new graphs from old\n </span></a>\n</span>\n</div>\n<div class=\"ContSect\ \"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"chapA_\ mj.html#X7A61E3FC7A2BB50E\">A.6 <span class=\"Heading\">\n Vertex-Colouri\ ng and Complete Subgraphs\n </span></a>\n</span>\n</div>\n<div class=\"Cont\ Sect\"><span class=\"tocline\"><span class=\"nocss\"> </span><a href=\"ch\ apA_mj.html#X7DFC61EC7CBDBC7E\">A.7 <span class=\"Heading\">\n Automorphi\ sm groups and isomorphism testing for graphs\n </span></a>\n</span>\n</div>\ \n</div>\n<div class=\"ContChap\"><a href=\"chapBib_mj.html\"><span class=\"He\ ading\">References</span></a></div>\n<div class=\"ContChap\"><a href=\"chapInd\ _mj.html\"><span class=\"Heading\">Index</span></a></div>\n" ) gap> gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.6 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io). maintained by: The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/NautyTracesInterface/ Report issues at https://github.com/gap-packages/NautyTracesInterface/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.5 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.4.1 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf A. Wilson (https://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (https://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── #I File: /builddir/build/BUILD/digraphs-1.4.1/../pkg/digraphs-1.4.1/doc/manual.lab written. gap> gap> + rm -fr ../pkg + RPM_EC=0 ++ jobs -p + exit 0 Executing(%install): /bin/sh -e /var/tmp/rpm-tmp.c1aINC + umask 022 + cd /builddir/build/BUILD + '[' /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64 '!=' / ']' + rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64 ++ dirname /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64 + mkdir -p /builddir/build/BUILDROOT + mkdir /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64 + cd digraphs-1.4.1 + mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap/pkg/digraphs-1.4.1/doc + cp -a bin data gap notebooks tst VERSIONS PackageInfo.g init.g makedoc.g read.g /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap/pkg/digraphs-1.4.1 + cp -p doc/digraphs.bib doc/lefttoc.css doc/manual.css doc/nocolorprompt.css doc/ragged.css doc/times.css doc/toggless.css doc/chap0.html doc/chap0_mj.html doc/chap1.html doc/chap1_mj.html doc/chap2.html doc/chap2_mj.html doc/chap3.html doc/chap3_mj.html doc/chap4.html doc/chap4_mj.html doc/chap5.html doc/chap5_mj.html doc/chap6.html doc/chap6_mj.html doc/chap7.html doc/chap7_mj.html doc/chap8.html doc/chap8_mj.html doc/chap9.html doc/chap9_mj.html doc/chapA.html doc/chapA_mj.html doc/chapBib.html doc/chapBib_mj.html doc/chapInd.html doc/chapInd_mj.html doc/chooser.html doc/manual.js doc/rainbow.js doc/toggless.js doc/manual.lab doc/manual.pdf doc/manual.six doc/chap0.txt doc/chap1.txt doc/chap2.txt doc/chap3.txt doc/chap4.txt doc/chap5.txt doc/chap6.txt doc/chap7.txt doc/chap8.txt doc/chap9.txt doc/chapA.txt doc/chapBib.txt doc/chapInd.txt doc/attr.xml doc/cliques.xml doc/constructors.xml doc/digraph.xml doc/display.xml doc/examples.xml doc/grahom.xml doc/grape.xml doc/io.xml doc/isomorph.xml doc/labels.xml doc/main.xml doc/oper.xml doc/orbits.xml doc/planar.xml doc/prop.xml doc/utils.xml doc/z-appA.xml doc/z-chap0.xml doc/z-chap1.xml doc/z-chap2.xml doc/z-chap3.xml doc/z-chap4.xml doc/z-chap5.xml doc/z-chap6.xml doc/z-chap8.xml doc/z-chap9.xml doc/z-title.xml /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap/pkg/digraphs-1.4.1/doc + /usr/lib/rpm/find-debuginfo.sh -j5 --strict-build-id -m -i --build-id-seed 1.4.1-1.fc35 --unique-debug-suffix -1.4.1-1.fc35.aarch64 --unique-debug-src-base gap-pkg-digraphs-1.4.1-1.fc35.aarch64 --run-dwz --dwz-low-mem-die-limit 10000000 --dwz-max-die-limit 50000000 -S debugsourcefiles.list /builddir/build/BUILD/digraphs-1.4.1 explicitly decompress any DWARF compressed ELF sections in /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap/pkg/digraphs-1.4.1/bin/aarch64-redhat-linux-gnu-default64-kv7/digraphs.so extracting debug info from /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap/pkg/digraphs-1.4.1/bin/aarch64-redhat-linux-gnu-default64-kv7/digraphs.so original debug info size: 2636kB, size after compression: 2480kB /usr/lib/rpm/sepdebugcrcfix: Updated 1 CRC32s, 0 CRC32s did match. 1036 blocks + /usr/lib/rpm/check-buildroot + /usr/lib/rpm/redhat/brp-ldconfig + /usr/lib/rpm/brp-compress + /usr/lib/rpm/redhat/brp-strip-lto /usr/bin/strip + /usr/lib/rpm/brp-strip-static-archive /usr/bin/strip + /usr/lib/rpm/redhat/brp-python-bytecompile '' 1 0 + /usr/lib/rpm/brp-python-hardlink + /usr/lib/rpm/redhat/brp-mangle-shebangs *** WARNING: ./usr/src/debug/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/extern/bliss-0.73/graph.cc is executable but has no shebang, removing executable bit Executing(%check): /bin/sh -e /var/tmp/rpm-tmp.iOSYhL + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.1 + export LC_ALL=C.UTF-8 + LC_ALL=C.UTF-8 + cd tst + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.11.1 of 2021-03-02 │ GAP │ https://www.gap-system.org └───────┘ Architecture: aarch64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.1, PrimGrp 3.4.1, SmallGrp 1.4.2, TransGrp 3.1 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.6 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io). maintained by: The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/NautyTracesInterface/ Report issues at https://github.com/gap-packages/NautyTracesInterface/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.5 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.4.1 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf A. Wilson (https://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (https://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: testinstall.tst msecs: 444 true gap> Digraphs package: standard/attr.tst msecs: 3150 true gap> Digraphs package: standard/cliques.tst msecs: 869 true gap> Digraphs package: standard/constructors.tst msecs: 75 true gap> Digraphs package: standard/display.tst msecs: 65 true gap> Digraphs package: standard/examples.tst msecs: 32 true gap> Digraphs package: standard/grahom.tst msecs: 922 true gap> Digraphs package: standard/grape.tst msecs: 33 true gap> Digraphs package: standard/io.tst msecs: 4357 true gap> Digraphs package: standard/isomorph.tst msecs: 796 true gap> Digraphs package: standard/labels.tst msecs: 20 true gap> Digraphs package: standard/oper.tst msecs: 1128 true gap> Digraphs package: standard/orbits.tst msecs: 8 true gap> Digraphs package: standard/planar.tst msecs: 77 true gap> Digraphs package: standard/prop.tst msecs: 5208 true gap> gap> + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.11.1 of 2021-03-02 │ GAP │ https://www.gap-system.org └───────┘ Architecture: aarch64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.1, PrimGrp 3.4.1, SmallGrp 1.4.2, TransGrp 3.1 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.6 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io). maintained by: The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/NautyTracesInterface/ Report issues at https://github.com/gap-packages/NautyTracesInterface/issues ──────────────────────────────────────────────────────────────────���────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.5 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.4.1 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf A. Wilson (https://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (https://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: standard/digraph.tst msecs: 7437 gap> + find /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/lib/gap/pkg/digraphs-1.4.1 -size 0 -delete + RPM_EC=0 ++ jobs -p + exit 0 Processing files: gap-pkg-digraphs-1.4.1-1.fc35.aarch64 Executing(%doc): /bin/sh -e /var/tmp/rpm-tmp.UJeVUH + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.1 + DOCDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/doc/gap-pkg-digraphs + export LC_ALL=C + LC_ALL=C + export DOCDIR + /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/doc/gap-pkg-digraphs + cp -pr CHANGELOG.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/doc/gap-pkg-digraphs + cp -pr README.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/doc/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.6vs739 + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.1 + LICENSEDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/licenses/gap-pkg-digraphs + export LC_ALL=C + LC_ALL=C + export LICENSEDIR + /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/licenses/gap-pkg-digraphs + cp -pr GPL /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/licenses/gap-pkg-digraphs + cp -pr LICENSE /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64/usr/share/licenses/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Provides: bundled(bliss) = 0.73 gap-pkg-digraphs = 1.4.1-1.fc35 gap-pkg-digraphs(aarch-64) = 1.4.1-1.fc35 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Requires: ld-linux-aarch64.so.1()(64bit) ld-linux-aarch64.so.1(GLIBC_2.17)(64bit) libc.so.6()(64bit) libc.so.6(GLIBC_2.17)(64bit) libgcc_s.so.1()(64bit) libgcc_s.so.1(GCC_3.0)(64bit) libgcc_s.so.1(GCC_3.3.1)(64bit) libgcc_s.so.1(GCC_4.2.0)(64bit) libm.so.6()(64bit) libplanarity.so.0()(64bit) libstdc++.so.6()(64bit) libstdc++.so.6(CXXABI_1.3)(64bit) libstdc++.so.6(CXXABI_1.3.9)(64bit) libstdc++.so.6(GLIBCXX_3.4)(64bit) libstdc++.so.6(GLIBCXX_3.4.29)(64bit) rtld(GNU_HASH) Recommends: gap-pkg-grape(aarch-64) gap-pkg-nautytracesinterface(aarch-64) Processing files: gap-pkg-digraphs-doc-1.4.1-1.fc35.noarch Provides: gap-pkg-digraphs-doc = 1.4.1-1.fc35 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Processing files: gap-pkg-digraphs-debugsource-1.4.1-1.fc35.aarch64 Provides: gap-pkg-digraphs-debugsource = 1.4.1-1.fc35 gap-pkg-digraphs-debugsource(aarch-64) = 1.4.1-1.fc35 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Processing files: gap-pkg-digraphs-debuginfo-1.4.1-1.fc35.aarch64 Provides: debuginfo(build-id) = d4015ada29d3954eb52ecd1f2e686931dc141ab1 gap-pkg-digraphs-debuginfo = 1.4.1-1.fc35 gap-pkg-digraphs-debuginfo(aarch-64) = 1.4.1-1.fc35 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Recommends: gap-pkg-digraphs-debugsource(aarch-64) = 1.4.1-1.fc35 Checking for unpackaged file(s): /usr/lib/rpm/check-files /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64 Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debugsource-1.4.1-1.fc35.aarch64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-1.4.1-1.fc35.aarch64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debuginfo-1.4.1-1.fc35.aarch64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-doc-1.4.1-1.fc35.noarch.rpm Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.S4m8RF + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.1 + /usr/bin/rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.1-1.fc35.aarch64 + RPM_EC=0 ++ jobs -p + exit 0 Child return code was: 0