&Digraphs;

\n[BM06] Boyer, J. M.\ and Myrvold, W. J.,\n On the Cutting Edge: Simplified\ O(n) Planarity by Edge\n Addition,\n in Gr\ aph Algorithms and Applications 5,\n {WORLD} {S\ CIENTIFIC}\n (2006),\n 241--273.\n

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

\n[Gab00] Gabow, H. N.,\n Path-based depth-first search for strong and biconnected\n \ components,\n Information Processing Letters,\n 74 (34)\n (\ 2000),\n 107 - 114.\ \n

\n[JK07] Junttila, T. and\ Kaski, P. (Applegate, D., Brodal, G. S., Panario,\ D. and Sedgewick, R., Eds.),\n Engineering an effi\ cient canonical labeling tool for large and\n sparse graphs,\n \ in Proceedings of the Ninth Workshop on Algorithm Eng\ ineering and\n Experiments and the Fourth Workshop on Analytic Algorithms and\ \n Combinatorics,\n SIAM\n\ (2007),\n 135--149\ .\n

\n[MP14] McKay, B. D. an\ d Piperno, A.,\n Practical graph isomorphism, II,\ \n Journal of Symbolic Computation,\n 60 (0)\n (2014),\n 94 - 112.\n

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

1 \n The Digraphs\ package\n \n

1.1 \ Introduction\n\n

\n
1.1-1 Def\ initions\n\n

2 Installing Digraphs\n
2.1 For those in a hurry\n\n
\n
2.2 Optional package\ dependencies\n\n
\n
  2.2-1 The Grape package\n\n
\n
2.\ 3 Compiling the kernel module\n\n\n
&nb\ sp;2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
  \ 2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\ \n
  2.5-3 DigraphsTestExtreme\n
\n
\n", 3 := "
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
<\ span class=\"nocss\">  3.1-1 IsDigraph\n
  3.1-2 Is\ MutableDigraph\n
\   3.1-3 IsImmutab\ leDigraph\n
\ ; 3.1-4 IsCayleyDigrap\ h\n
  <\ /span>3.1-5 IsDigraphWithAdjacency\ Function\n
\ 3.1-6 DigraphByOutNei\ ghboursType\n
&nb\ sp; 3.1-7 Digraph<\ /span>\n
  \ 3.1-8 DigraphByAdjacencyMatrix\ \n
  3.1-9 DigraphByEdges\n\
  3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n<\ div class=\"ContSect\"> 3.2 Ch\ anging representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2-2 AsDigraph\n  3.2-3 Graph\n
  3.2-4 As\ Graph\n
&nb\ sp;3.2-5 AsTransformation\n
\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
3.4 Random digraphs\n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 RandomMultiDigraph\n
  3.4-3 RandomTournament\n<\ br />  3.4-4 RandomLattice\n
\n
3.5 Standard examples\n\n
\n
  3.5-1 ChainDigraph\n
  3.5-2\ CompleteDigraph\n
  3.5-3 Comple\ teBipartiteDigraph\n
  3.5-4 Compl\ eteMultipartiteDigraph\n
  3.5-5 C\ ycleDigraph\n
&nb\ sp; 3.5-6 EmptyDigraph\ \n
  3.5-7 JohnsonDigraph\n
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n", 4 := "
4 Operators\n
4.1 Operators for digraphs\n\n
\n
  4.1-1 IsSubdigraph\n
  4.1-2 Is\ UndirectedSpanningTree\n
\n
\n", 5 := "
5 Attributes and operations\n
5.1 Vertices and\ edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 Di\ graphNrEdges\n
&n\ bsp; 5.1-5 DigraphSink\ s\n
  <\ /span>5.1-6 DigraphSources\n
  5.1-7 DigraphTopologicalSort\n
  5.1-8 DigraphVertexLabel\n<\ span class=\"ContSS\">
  5.1-9 DigraphVertexLabels\n
  5.1-10 DigraphEdgeLabel\n
  5.1-11 DigraphEdgeLabels\n
  5.1-12 DigraphInEdges\n\
  5.1-13 DigraphOutEdges\n
<\ span class=\"nocss\">  5.1-14 IsDigraphEdge\n
  5.1\ -15 IsMatching\n
\n
5.2 Neighbours and degree\n\n
\n
  5.2-1 A\ djacencyMatrix\n
\   5.2-2 Character\ isticPolynomial\n
  5.2-3 BooleanA\ djacencyMatrix\n
\   5.2-4 DigraphAd\ jacencyFunction\n
  5.2-5 DigraphR\ ange\n
&nbs\ p;5.2-6 OutNeighbours\n
  <\ a href=\"chap5_mj.html#X85C7AA5A81DA6E11\">5.2-7 InNeighbours\n
  5.2-8 OutDegrees\n
  5.2-9 InDegrees\n
  5.2-10 OutDegreeOfVertex\n
  \ 5.2-11 OutNeighboursOfVertex\n
  5\ .2-12 InDegreeOfVertex\n
  5.2-13 \ InNeighboursOfVertex\n
  5.2-14 Di\ graphLoops\n
&nbs\ p; 5.2-15 PartialOrder\ DigraphMeetOfVertices\n
  5.2-16 D\ egreeMatrix\n
&nb\ sp; 5.2-17 LaplacianMa\ trix\n
\n
5.3 Reachability and connectivity\n\n
\n
  5.3-1 Dig\ raphDiameter\n
&n\ bsp; 5.3-2 DigraphShor\ testDistance\n
&n\ bsp; 5.3-3 DigraphShor\ testDistances\n
&\ nbsp; 5.3-4 DigraphLon\ gestDistanceFromVertex\n
  5.3-5 D\ igraphDistanceSet\n
  5.3-6 Digrap\ hGirth\n
&n\ bsp;5.3-7 DigraphOddGirth\n
  5.3-8 DigraphUndirectedGirth\n
  5.3-9 DigraphConnectedComponents\ \n
  5.3-10 DigraphConnectedCompon\ ent\n
  \ ;5.3-11 DigraphStronglyConn\ ectedComponents\n
  5.3-12 Digraph\ StronglyConnectedComponent\n
  5.3\ -13 DigraphBicomponents\n
  5.3-1\ 4 ArticulationPoints\n
  5.3-15 Di\ graphPeriod\n
&nb\ sp; 5.3-16 DigraphFloy\ dWarshall\n
\ ; 5.3-17 IsReachable\n
  5.3-18 DigraphPath\n\
  5.3-19 DigraphShortestPath\n
  5.3-20 IteratorOfPaths\n
  5.3-21 DigraphAllSimpleCircuits\n
  5.3-22 DigraphLongestSimpleCircuit\n
  5.3-23 DigraphLayers\n
  5.3-24 DigraphDegeneracy\n
  5.3-25 DigraphDegeneracyOrdering\n
  5.3-26 HamiltonianPath\n
  5.3-27 NrSpanningTrees\n
\n
5.4 Cayley graphs \ of groups\n\n
\n
  5.4-1 GroupOfCayleyDigraph\n
  5.4-2 GeneratorsOfCayleyDigraph\n
\n 5.5 Assoc\ iated semigroups\n\n
\n
  5.5-1 AsSemigroup\n\
  5.5-2 AsSemigroup\n
\n
5.6 Planarity\n\ \n
\n
  5.6-\ 1 KuratowskiPlanarSubdigraph\n
  5\ .6-2 KuratowskiOuterPlanarSubdigraph\n
\   5.6-3 PlanarEmbedding\n
  5\ .6-4 OuterPlanarEmbedding\n
  5.6-\ 5 SubdigraphHomeomorphicToK23\n
\n
\n", 6 := "
6 Properties of digraphs\n
6.1 Edge propertie\ s\n\n
\n  6.1-1 DigraphHasLoops\n
  6.1-2 IsAntiSymmetricDigraph\n
  6.1-3 IsBipartiteDigraph\n
  6.\ 1-4 IsCompleteBipartiteDigraph\n
  6.1-5 IsCompleteDigraph\n
  6.1-6\ IsCompleteMultipartiteDigraph\n
  6.1-7 IsEmptyDigraph\n
  6.1-8 Is\ EquivalenceDigraph\n
  6.1-9 IsFun\ ctionalDigraph\n
\   6.1-10 IsMultiD\ igraph\n
&n\ bsp;6.1-11 IsReflexiveDigra\ ph\n
  \ 6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\ \n
  6.1-14 IsTransitiveDigraph\n<\ span class=\"ContSS\">
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n\
  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectiv\ ity and cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  6.3-3 IsConnectedDigraph\n
  6.3-4 IsBiconnectedDigraph\n
  \ 6.3-5 IsStronglyConnectedDigraph\n
  6.3-6 IsAperiodicDigraph\n
  6.\ 3-7 IsDirectedTree\n
  6.3-8 IsUnd\ irectedTree\n
&nb\ sp; 6.3-9 IsEulerianDi\ graph\n
&nb\ sp;6.3-10 IsHamiltonianDigr\ aph\n
  \ ;6.3-11 IsCycleDigraph<\ /span>\n
\n
6.4 Planarity\n\n
\n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n
&n\ bsp;6.5 Homomorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\ \n
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\n
\n", 7 := "
7 Homomorphisms\n
7.1 Acting on digraphs\n\n
\n
  7.1-1 OnDigraphs\n
  7.1-2 OnMu\ ltiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n
\n
  7.2-1 DigraphsUseNauty\n
  7.\ 2-2 AutomorphismGroup\n
  7.2-3 Bl\ issAutomorphismGroup\n
  7.2-4 Nau\ tyAutomorphismGroup\n
  7.2-5 Auto\ morphismGroup\n
&\ nbsp; 7.2-6 Automorphi\ smGroup\n
&\ nbsp;7.2-7 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-8 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-9 BlissCanonicalDi\ graph\n
&nb\ sp;7.2-10 DigraphGroup<\ /span>\n
  \ 7.2-11 DigraphOrbits\n<\ span class=\"ContSS\">
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  7.2-17 IsomorphismDigraphs\n
  7.2-18 IsomorphismDigraphs\n\
  7.2-19 RepresentativeOutNeighbours\n
  7.2-20 IsDigraphIsomorphism\n
  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphis\ ms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n  7.3-7 EpimorphismsDigraphs\n
  7.3-8 DigraphEmbedding\n
  7.3-9 EmbeddingsDigraphs\n
  7.3-\ 10 IsDigraphHomomorphism\n
  7.3-1\ 1 IsDigraphEmbedding\n
  7.3-12 Di\ graphsRespectsColouring\n
  7.3-1\ 3 GeneratorsOfEndomorphismMonoid\n
  7.3-14 DigraphColouring\n
  7.3\ -15 DigraphGreedyColouring\n
  7.3\ -16 DigraphWelshPowellOrder\n
  7.\ 3-17 ChromaticNumber\n
  7.3-18 Di\ graphCore\n
\n
\n", 8 := "
8 Finding cliques and independent sets\n\
8.1 F\ inding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8\ .1-3 DigraphClique\n
  8.1-4 Digra\ phMaximalCliques\n
  8.1-5 Clique\ Number\n
\n
8.2 Finding independent sets\n\n\
\n
  8.2-1 IsInde\ pendentSet\n
&nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n
\ 8.2-3 DigraphMaximalI\ ndependentSets\n
\n
\n", 9 := "
9 Visualising and IO\n
9.1 Visualising a digra\ ph\n\n
\n  9.1-1 Splash\n
  9.1-2\ DotDigraph\n
&nb\ sp; 9.1-3 DotSymmetric\ Digraph\n
&\ nbsp;9.1-4 DotPartialOrderD\ igraph\n
&n\ bsp;9.1-5 DotPreorderDigrap\ h\n
  <\ /span>9.1-6 DotHighlightedDigraph<\ /a>\n
\n
9.2\ Reading and writing digraphs to a file\n\n
\n
  9.2-1\ DigraphFromGraph6String\n
  9.2-2\ Graph6String\n
&\ nbsp; 9.2-3 DigraphFil\ e\n
  <\ /span>9.2-4 ReadDigraphs\n
  9.2-5 WriteDigraphs\n
  9.2-6 IteratorFromDigraphFile\n
  9.2-7 DigraphPlainTextLineEncoder\n
  9.2-8 TournamentLineDecoder\n
  9.2-9 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-10 TCodeDecoder\ \n
  9.2-11 PlainTextString\n
  9.2-12 WritePlainTextDigraph\n
  9.2-13 WriteDIMACSDigraph\n
\n
\n", A := "
A \n Grape to Digraphs Command Map\n \n
\ ;A.1 \n Functions to construct and modify graphs\n \n\ \n
\n
A.2 \n Functions to inspect graphs, vertices and edges\n \ \n\n
\n
A.3 \n Functions to determine regularity properties of \ graphs\n \n\n
\n
A.4 \n Some special vertex subsets o\ f a graph\n \n\n
\n
A.5 \n Functions to construct new\ graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring an\ d Complete Subgraphs\n \n\n
\n
A.7 \n Automorphism g\ roups and isomorphism testing for graphs\n \n\n
\n\n" ), chapsectlinkstext := rec( 1 := "
1 \n The Digraphs\ package\n \n
1.1 \ Introduction\n\n
\n
  1.1-1 Def\ initions\n\n
\n
\n", 2 := "
2 Installing Digraphs\n
2.1 For those in a hurry\n\n
\n
2.2 Optional package\ dependencies\n\n
\n
  2.2-1 The Grape package\n\n
\n
2.\ 3 Compiling the kernel module\n\n\n
&nb\ sp;2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
  \ 2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\ \n
  2.5-3 DigraphsTestExtreme\n
\n
\n", 3 := "
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
<\ span class=\"nocss\">  3.1-1 IsDigraph\n
  3.1-2 Is\ MutableDigraph\n
\   3.1-3 IsImmutab\ leDigraph\n
\ ; 3.1-4 IsCayleyDigrap\ h\n
  <\ /span>3.1-5 IsDigraphWithAdjacency\ Function\n
\ 3.1-6 DigraphByOutNei\ ghboursType\n
&nb\ sp; 3.1-7 Digraph<\ /span>\n
  \ 3.1-8 DigraphByAdjacencyMatrix\ \n
  3.1-9 DigraphByEdges\n\
  3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n<\ div class=\"ContSect\"> 3.2 Ch\ anging representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2-2 AsDigraph\n  3.2-3 Graph\n
  3.2-4 As\ Graph\n
&nb\ sp;3.2-5 AsTransformation\n
\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
3.4 Random digraphs\n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 RandomMultiDigraph\n
  3.4-3 RandomTournament\n<\ br />  3.4-4 RandomLattice\n
\n
3.5 Standard examples\n\n
\n
  3.5-1 ChainDigraph\n
  3.5-2\ CompleteDigraph\n
  3.5-3 Comple\ teBipartiteDigraph\n
  3.5-4 Compl\ eteMultipartiteDigraph\n
  3.5-5 C\ ycleDigraph\n
&nb\ sp; 3.5-6 EmptyDigraph\ \n
  3.5-7 JohnsonDigraph\n
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n", 4 := "
4 Operators\n
4.1 Operators for digraphs\n\n
\n
  4.1-1 IsSubdigraph\n
  4.1-2 Is\ UndirectedSpanningTree\n
\n
\n", 5 := "
5 Attributes and operations\n
5.1 Vertices and\ edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 Di\ graphNrEdges\n
&n\ bsp; 5.1-5 DigraphSink\ s\n
  <\ /span>5.1-6 DigraphSources\n
  5.1-7 DigraphTopologicalSort\n
  5.1-8 DigraphVertexLabel\n<\ span class=\"ContSS\">
  5.1-9 DigraphVertexLabels\n
  5.1-10 DigraphEdgeLabel\n
  5.1-11 DigraphEdgeLabels\n
  5.1-12 DigraphInEdges\n\
  5.1-13 DigraphOutEdges\n
<\ span class=\"nocss\">  5.1-14 IsDigraphEdge\n
  5.1\ -15 IsMatching\n
\n
5.2 Neighbours and degree\n\n
\n
  5.2-1 A\ djacencyMatrix\n
\   5.2-2 Character\ isticPolynomial\n
  5.2-3 BooleanA\ djacencyMatrix\n
\   5.2-4 DigraphAd\ jacencyFunction\n
  5.2-5 DigraphR\ ange\n
&nbs\ p;5.2-6 OutNeighbours\n
  <\ a href=\"chap5_mj.html#X85C7AA5A81DA6E11\">5.2-7 InNeighbours\n
  5.2-8 OutDegrees\n
  5.2-9 InDegrees\n
  5.2-10 OutDegreeOfVertex\n
  \ 5.2-11 OutNeighboursOfVertex\n
  5\ .2-12 InDegreeOfVertex\n
  5.2-13 \ InNeighboursOfVertex\n
  5.2-14 Di\ graphLoops\n
&nbs\ p; 5.2-15 PartialOrder\ DigraphMeetOfVertices\n
  5.2-16 D\ egreeMatrix\n
&nb\ sp; 5.2-17 LaplacianMa\ trix\n
\n
5.3 Reachability and connectivity\n\n
\n
  5.3-1 Dig\ raphDiameter\n
&n\ bsp; 5.3-2 DigraphShor\ testDistance\n
&n\ bsp; 5.3-3 DigraphShor\ testDistances\n
&\ nbsp; 5.3-4 DigraphLon\ gestDistanceFromVertex\n
  5.3-5 D\ igraphDistanceSet\n
  5.3-6 Digrap\ hGirth\n
&n\ bsp;5.3-7 DigraphOddGirth\n
  5.3-8 DigraphUndirectedGirth\n
  5.3-9 DigraphConnectedComponents\ \n
  5.3-10 DigraphConnectedCompon\ ent\n
  \ ;5.3-11 DigraphStronglyConn\ ectedComponents\n
  5.3-12 Digraph\ StronglyConnectedComponent\n
  5.3\ -13 DigraphBicomponents\n
  5.3-1\ 4 ArticulationPoints\n
  5.3-15 Di\ graphPeriod\n
&nb\ sp; 5.3-16 DigraphFloy\ dWarshall\n
\ ; 5.3-17 IsReachable\n
  5.3-18 DigraphPath\n\
  5.3-19 DigraphShortestPath\n
  5.3-20 IteratorOfPaths\n
  5.3-21 DigraphAllSimpleCircuits\n
  5.3-22 DigraphLongestSimpleCircuit\n
  5.3-23 DigraphLayers\n
  5.3-24 DigraphDegeneracy\n
  5.3-25 DigraphDegeneracyOrdering\n
  5.3-26 HamiltonianPath\n
  5.3-27 NrSpanningTrees\n
\n
5.4 Cayley graphs \ of groups\n\n
\n
  5.4-1 GroupOfCayleyDigraph\n
  5.4-2 GeneratorsOfCayleyDigraph\n
\n 5.5 Assoc\ iated semigroups\n\n
\n
  5.5-1 AsSemigroup\n\
  5.5-2 AsSemigroup\n
\n
5.6 Planarity\n\ \n
\n
  5.6-\ 1 KuratowskiPlanarSubdigraph\n
  5\ .6-2 KuratowskiOuterPlanarSubdigraph\n
\   5.6-3 PlanarEmbedding\n
  5\ .6-4 OuterPlanarEmbedding\n
  5.6-\ 5 SubdigraphHomeomorphicToK23\n
\n
\n", 6 := "
6 Properties of digraphs\n
6.1 Edge propertie\ s\n\n
\n  6.1-1 DigraphHasLoops\n
  6.1-2 IsAntiSymmetricDigraph\n
  6.1-3 IsBipartiteDigraph\n
  6.\ 1-4 IsCompleteBipartiteDigraph\n
  6.1-5 IsCompleteDigraph\n
  6.1-6\ IsCompleteMultipartiteDigraph\n
  6.1-7 IsEmptyDigraph\n
  6.1-8 Is\ EquivalenceDigraph\n
  6.1-9 IsFun\ ctionalDigraph\n
\   6.1-10 IsMultiD\ igraph\n
&n\ bsp;6.1-11 IsReflexiveDigra\ ph\n
  \ 6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\ \n
  6.1-14 IsTransitiveDigraph\n<\ span class=\"ContSS\">
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n\
  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectiv\ ity and cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  6.3-3 IsConnectedDigraph\n
  6.3-4 IsBiconnectedDigraph\n
  \ 6.3-5 IsStronglyConnectedDigraph\n
  6.3-6 IsAperiodicDigraph\n
  6.\ 3-7 IsDirectedTree\n
  6.3-8 IsUnd\ irectedTree\n
&nb\ sp; 6.3-9 IsEulerianDi\ graph\n
&nb\ sp;6.3-10 IsHamiltonianDigr\ aph\n
  \ ;6.3-11 IsCycleDigraph<\ /span>\n
\n
6.4 Planarity\n\n
\n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n
&n\ bsp;6.5 Homomorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\ \n
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\n
\n", 7 := "
7 Homomorphisms\n
7.1 Acting on digraphs\n\n
\n
  7.1-1 OnDigraphs\n
  7.1-2 OnMu\ ltiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n
\n
  7.2-1 DigraphsUseNauty\n
  7.\ 2-2 AutomorphismGroup\n
  7.2-3 Bl\ issAutomorphismGroup\n
  7.2-4 Nau\ tyAutomorphismGroup\n
  7.2-5 Auto\ morphismGroup\n
&\ nbsp; 7.2-6 Automorphi\ smGroup\n
&\ nbsp;7.2-7 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-8 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-9 BlissCanonicalDi\ graph\n
&nb\ sp;7.2-10 DigraphGroup<\ /span>\n
  \ 7.2-11 DigraphOrbits\n<\ span class=\"ContSS\">
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  7.2-17 IsomorphismDigraphs\n
  7.2-18 IsomorphismDigraphs\n\
  7.2-19 RepresentativeOutNeighbours\n
  7.2-20 IsDigraphIsomorphism\n
  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphis\ ms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n  7.3-7 EpimorphismsDigraphs\n
  7.3-8 DigraphEmbedding\n
  7.3-9 EmbeddingsDigraphs\n
  7.3-\ 10 IsDigraphHomomorphism\n
  7.3-1\ 1 IsDigraphEmbedding\n
  7.3-12 Di\ graphsRespectsColouring\n
  7.3-1\ 3 GeneratorsOfEndomorphismMonoid\n
  7.3-14 DigraphColouring\n
  7.3\ -15 DigraphGreedyColouring\n
  7.3\ -16 DigraphWelshPowellOrder\n
  7.\ 3-17 ChromaticNumber\n
  7.3-18 Di\ graphCore\n
\n
\n", 8 := "
8 Finding cliques and independent sets\n\
8.1 F\ inding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8\ .1-3 DigraphClique\n
  8.1-4 Digra\ phMaximalCliques\n
  8.1-5 Clique\ Number\n
\n
8.2 Finding independent sets\n\n\
\n
  8.2-1 IsInde\ pendentSet\n
&nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n
\ 8.2-3 DigraphMaximalI\ ndependentSets\n
\n
\n", 9 := "
9 Visualising and IO\n
9.1 Visualising a digra\ ph\n\n
\n  9.1-1 Splash\n
  9.1-2\ DotDigraph\n
&nb\ sp; 9.1-3 DotSymmetric\ Digraph\n
&\ nbsp;9.1-4 DotPartialOrderD\ igraph\n
&n\ bsp;9.1-5 DotPreorderDigrap\ h\n
  <\ /span>9.1-6 DotHighlightedDigraph<\ /a>\n
\n
9.2\ Reading and writing digraphs to a file\n\n
\n
  9.2-1\ DigraphFromGraph6String\n
  9.2-2\ Graph6String\n
&\ nbsp; 9.2-3 DigraphFil\ e\n
  <\ /span>9.2-4 ReadDigraphs\n
  9.2-5 WriteDigraphs\n
  9.2-6 IteratorFromDigraphFile\n
  9.2-7 DigraphPlainTextLineEncoder\n
  9.2-8 TournamentLineDecoder\n
  9.2-9 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-10 TCodeDecoder\ \n
  9.2-11 PlainTextString\n
  9.2-12 WritePlainTextDigraph\n
  9.2-13 WriteDIMACSDigraph\n
\n
\n", A := "
A \n Grape to Digraphs Command Map\n \n
\ ;A.1 \n Functions to construct and modify graphs\n \n\ \n
\n
A.2 \n Functions to inspect graphs, vertices and edges\n \ \n\n
\n
A.3 \n Functions to determine regularity properties of \ graphs\n \n\n
\n
A.4 \n Some special vertex subsets o\ f a graph\n \n\n
\n
A.5 \n Functions to construct new\ graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring an\ d Complete Subgraphs\n \n\n
\n
A.7 \n Automorphism g\ roups and isomorphism testing for graphs\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 http://www.maths.qmul.ac.uk/~leonard/grape/Grape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suiteedge-addition-planarity-suite\">\n ht\ tp://pallini.di.uniroma1.it/nauty\">\n https://github.com/sebasguts/NautyTracesInterfac\ eNautyTracesInterface\">\n \n http://gap-pack\ ages.github.io/io/\nIO\"\ >\n \n http://http://ga\ p-packages.github.io/orb/\n Orb\">\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 := 1686, root := ~, start := 1661, stop := 1685 ), rec( attributes := rec( ), content := [ rec( content := "Version 1.1.1", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Version", next := 1725, root := ~, start := 1689, stop := 1724 ), 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 := 1791, root := ~, start := 1754, stop := 1790 ), 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 := 1852, root := ~, start := 1796, stop := 1851 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 1864, root := ~, start := 1729, stop := 1863 ), 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 := 1942, root := ~, start := 1900, stop := 1941 ), 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 := 1994, root := ~, start := 1947, stop := 1993 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2006, root := ~, start := 1868, stop := 2005 ), 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 := 2072, root := ~, start := 2040, stop := 2071 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://www-groups.mcs.st-\ andrews.ac.uk/~jamesm", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2144, root := ~, start := 2077, stop := 2143 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2156, root := ~, start := 2010, stop := 2155 ), rec( attributes := rec( ), content := [ rec( content := "Michael 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 := 2224, root := ~, start := 2187, stop := 2223 ), 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 := 2275, root := ~, start := 2229, stop := 2274 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2287, root := ~, start := 2160, stop := 2286 ), 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 := 2352, root := ~, start := 2318, stop := 2351 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://wilf.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2392, root := ~, start := 2357, stop := 2391 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2404, root := ~, start := 2291, stop := 2403 ), 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 := 2439, root := ~, start := 2408, stop := 2438 ), 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 := 2474, root := ~, start := 2443, stop := 2473 ), 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 := 2507, root := ~, start := 2478, stop := 2506 ), 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 := 2536, root := ~, start := 2511, stop := 2535 ), 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 := 2578, root := ~, start := 2540, stop := 2577 ), 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 := 2614, root := ~, start := 2582, stop := 2613 ), rec( attributes := rec( ), content := [ rec( content := "Chris Russell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2648, root := ~, start := 2618, stop := 2647 ), 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 := 2679, root := ~, start := 2652, stop := 2678 ), 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 := 2712, root := ~, start := 2683, stop := 2711 ), rec( attributes := rec( ), content := [ rec( content := "© 2014-20 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 := 2862, root := ~, start := 2858, stop := 2861 ), 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 http://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 := 3051, root := ~, start := 2965, stop := 3050 ), 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 := 3192, root := ~, start := 2716, stop := 3191 ), 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 := 3330, root := ~, start := 3196, stop := 3329 ), 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 := 3625, root := ~, start := 3334, stop := 3624 ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "TitlePage", next := 3639, root := ~, start := 1647, stop := 3638 ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "TableOfContents", next := 3660, root := ~, start := 3642, stop := 3659 ), 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 := 3761, root := ~, start := 3712, stop := 3760 ), 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 := 3821, root := ~, start := 3764, stop := 3820 ), 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 := 3863, root := ~, start := 3832, stop := 3862 ), rec( content := "\n\n This is the manual\ for version 1.1.1 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 := 4039, root := ~, start := 4013, stop := 4038 ), 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 := 4077, root := ~, start := 4044, stop := 4076 ), 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 := 4113, root := ~, start := 4082, stop := 4112 ), rec( attributes := rec( ), content := [ rec( content := "Michael C\ . Torpey, and", count := [ 1, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 11 ], name := "Item", next := 4153, root := ~, start := 4118, stop := 4152 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. W\ ilson.", count := [ 1, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 13 ], name := "Item", next := 4186, root := ~, start := 4158, stop := 4185 ) ], count := [ 1, 1, 0, 4 ], mathmode := "Text", name := "List", next := 4196, root := ~, start := 4002, stop := 4195 ), rec( content := "\n Additional contribut\ ions were made by:\n ", count := [ 1, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Stuart Bu\ rrell,", count := [ 1, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 17 ], name := "Item", next := 4279, root := ~, start := 4251, stop := 4278 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Ci\ rpons,", count := [ 1, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 19 ], name := "Item", next := 4312, root := ~, start := 4284, stop := 4311 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elli\ ott,", count := [ 1, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 21 ], name := "Item", next := 4343, root := ~, start := 4317, stop := 4342 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn,\ ", count := [ 1, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 23 ], name := "Item", next := 4370, root := ~, start := 4348, stop := 4369 ), rec( attributes := rec( ), content := [ rec( content := "Christoph\ er Jefferson,", count := [ 1, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 25 ], name := "Item", next := 4410, root := ~, start := 4375, stop := 4409 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pf\ eiffer,", count := [ 1, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 27 ], name := "Item", next := 4444, root := ~, start := 4415, stop := 4443 ), rec( attributes := rec( ), content := [ rec( content := "Chris Rus\ sell,", count := [ 1, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 29 ], name := "Item", next := 4476, root := ~, start := 4449, stop := 4475 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smit\ h, and", count := [ 1, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 31 ], name := "Item", next := 4509, root := ~, start := 4481, stop := 4508 ), rec( attributes := rec( ), content := [ rec( content := "Murray Wh\ yte.", count := [ 1, 1, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 33 ], name := "Item", next := 4540, root := ~, start := 4514, stop := 4539 ) ], count := [ 1, 1, 0, 16 ], mathmode := "Text", name := "List", next := 4550, root := ~, start := 4240, stop := 4549 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 35 ], 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, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "P", next := 4812, root := ~, start := 4808, stop := 4811 ), rec( content := "\n\n If the ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 36 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 36 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package is available, i\ t will be loaded automatically.\n Digraphs created with the ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package can be converte\ d to ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 36 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 36 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\n graphs with ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ) , content := 0, count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Ref", next := 4994, root := ~, start := 4975, stop := 4993 ), rec( content := ", and conversely ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 36 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 36 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " graphs can be\n conver\ ted to ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " objects with ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph" ), content := 0, count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Ref", next := 5093, root := ~, start := 5072, stop := 5092 ), rec( content := ". ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 36 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 36 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not\n required for \ ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to run. ", count := [ 1, 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "P", next := 5149, root := ~, start := 5145, stop := 5148 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 37 ], 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, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 37 ], 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, 37 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " tool ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Cite", next := 5188, root := ~, start := 5170, stop := 5187 ), 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, 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 rely on ", count := [ 1, 1, 0, 37 ], 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, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 37 ], 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, 37 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ". If the ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/sebasguts/NautyTrac\ esInterface", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 1, 1, 0, 37 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 1, 1, 0, 37 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], name := "Package", next := 173, root := ~, start := 134, stop := 172 ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), rec( content := "\n package for GAP is a\ vailable then it is also possible to use ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 1, 1, 0, 37 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 1, 1, 0, 37 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Cite", next := 5550, root := ~, start := 5528, stop := 5549 ), rec( content := " for computing automorph\ ism groups and canonical forms\n in ", 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 := ". See Section \n ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Isomorphisms and canoni\ cal labellings" ), content := 0, count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Ref", next := 5688, root := ~, start := 5637, stop := 5687 ), rec( content := " for more details. \n \ ", 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 := 5715, root := ~, start := 5711, stop := 5714 ), 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, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "MakeImmutable" ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 6196, root := ~, start := 6150, stop := 6195 ), 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, 38 ], 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 := 6827, root := ~, start := 6797, stop := 6826 ), 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 := 6930, root := ~, start := 6926, stop := 6929 ), 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 := 6952, root := ~, start := 6938, stop := 6951 ), 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 := 6975, root := ~, start := 6953, stop := 6974 ), 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 := 7020, root := ~, start := 6993, stop := 7019 ), 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 := 7062, root := ~, start := 7052, stop := 7061 ), 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 := 7096, root := ~, start := 7086, stop := 7095 ), 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 := 7145, root := ~, start := 7121, stop := 7144 ), 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 := 7175, root := ~, start := 7162, stop := 7174 ), 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 := 7192, root := ~, start := 7180, stop := 7191 ), 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 := 7286, root := ~, start := 7275, stop := 7285 ), 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 := 7341, root := ~, start := 7330, stop := 7340 ), 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 := 7426, root := ~, start := 7407, stop := 7425 ), 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 := 7509, root := ~, start := 7505, stop := 7508 ), 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 := 7537, root := ~, start := 7517, stop := 7536 ), 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 := 7649, root := ~, start := 7603, stop := 7648 ), 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 := 7684, root := ~, start := 7674, stop := 7683 ), 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 := 7706, root := ~, start := 7696, stop := 7705 ), 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 := 7731, root := ~, start := 7717, stop := 7730 ), 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 := 7761, root := ~, start := 7735, stop := 7760 ), 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 := 7860, root := ~, start := 7837, stop := 7859 ), 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 := 7908, root := ~, start := 7886, stop := 7907 ), 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 := 7937, root := ~, start := 7916, stop := 7936 ), 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 := 8026, root := ~, start := 8010, stop := 8025 ), 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 := 8033, root := ~, start := 8029, stop := 8032 ), 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 := 8056, root := ~, start := 8043, stop := 8055 ), 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 := 8128, root := ~, start := 8076, stop := 8127 ), 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 := 8151, root := ~, start := 8141, stop := 8150 ), 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 := 8263, root := ~, start := 8217, stop := 8262 ), 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 := 8339, root := ~, start := 8329, stop := 8338 ), 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 := 8366, root := ~, start := 8356, stop := 8365 ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8370, root := ~, start := 8314, stop := 8369 ), 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 := 8435, root := ~, start := 8425, stop := 8434 ), 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 := 8460, root := ~, start := 8452, stop := 8459 ), 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 := 8486, root := ~, start := 8472, stop := 8485 ), 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 := 8607, root := ~, start := 8598, stop := 8606 ), 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 := 8632, root := ~, start := 8611, stop := 8631 ), 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 := 8704, root := ~, start := 8700, stop := 8703 ), rec( content := "\n\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 1 ], mathmode := "Text", name := "Subsection", next := 8721, root := ~, start := 6765, stop := 8720 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 8734, root := ~, start := 3823, stop := 8733 ), rec( content := "\n", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 8745, root := ~, start := 3671, stop := 8744 ), 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 := 8844, root := ~, start := 8804, stop := 8843 ), 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 := 8930, root := ~, start := 8891, stop := 8929 ), 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 := 9020, root := ~, start := 9016, stop := 9019 ), 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 \ 4.9.0 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 := "http://www.gap-sy\ stem.org", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "URL", next := 9286, root := ~, start := 9250, stop := 9285 ), 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 := 9291, root := ~, start := 9287, stop := 9290 ), 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 http://gap-packages\ .github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), 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 := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), 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 http://gap-packages\ .github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), 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 := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), 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 := 9582, root := ~, start := 9421, stop := 9581 ), 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 http://http://gap-\ packages.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL\ ", next := 86, root := ~, start := 17, stop := 85 ), rec( content := "\n " , count := [ 2, 1, 0, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 98, root := ~, start := 1, stop := 97 ), 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 := 137, root := ~, start := 115 , stop := 136 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 98, stop := 142 ), 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 http://http://gap-\ packages.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL\ ", next := 86, root := ~, start := 17, stop := 85 ), rec( content := "\n " , count := [ 2, 1, 0, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 98, root := ~, start := 1, stop := 97 ), 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 := 137, root := ~, start := 115 , stop := 136 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 98, stop := 142 ), 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 := 9862, root := ~, start := 9590, stop := 9861 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "B", next := 9914, root := ~, start := 9885, stop := 9913 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 11 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ to be available; see Section\n ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Ref", next := 10054, root := ~, start := 10020, stop := 10053 ), rec( content := " for ful\ l details. To use these\n functions make sure\n that the ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 11 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ version 4.8.1 or higher is available.\n If ", count := [ 2, 1, 0, 11 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 11 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not \ available, then ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 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 listed in Subsec\ tion\n ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Ref", next := 10376, root := ~, start := 10342, stop := 10375 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Item", next := 10405, root := ~, start := 9870, stop := 10404 ), rec( attributes := rec( ), content := [ rec( content := "\n \ download the package archive ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.1.1.tar.gz", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "F", next := 10484, root := ~, start := 10457, stop := 10483 ), rec( content := " from\n \ ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "the Dig\ raph package webpage" ), content := [ rec( content := "http://gap-packages.git\ hub.io/Digraphs/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL", next := 10583, root := ~, start := 10498, stop := 10582 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Item", next := 10598, root := ~, start := 10413, stop := 10597 ), rec( attributes := rec( ), content := [ rec( content := "\n \ unzip and untar the file, this should create a directory called\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.1.1", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "F", next := 10713, root := ~, start := 10693, stop := 10712 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Item", next := 10728, root := ~, start := 10606, stop := 10727 ), rec( attributes := rec( ), content := [ rec( content := "\n \ locate the ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 10772, root := ~, start := 10762, stop := 10771 ), rec( content := " directo\ ry of your ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " directo\ ry, which contains\n the directories ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ib", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 10857, root := ~, start := 10847, stop := 10856 ), rec( content := ", ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ oc", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 10869, root := ~, start := 10859, stop := 10868 ), rec( content := " and so \ on. Move the directory\n ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.1.1", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 10928, root := ~, start := 10908, stop := 10927 ), rec( content := " into th\ e ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 10948, root := ~, start := 10938, stop := 10947 ), rec( content := " directo\ ry.\n ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], name := "Item", next := 10973, root := ~, start := 10736, stop := 10972 ), rec( attributes := rec( ), content := [ rec( content := "\n \ it is necessary to compile the ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n Inside the ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg/digraphs-1.1.1", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "F", next := 11090, root := ~, start := 11066, stop := 11089 ), rec( content := " directo\ ry, type\n ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ n./configure\nmake", count := [ 2, 1, 0, 20 ], name := "PCDATA", root := ~ ) ] , count := [ 2, 1, 0, 20 ], mathmode := "Text", name := "Listing", next := 11163, root := ~, start := 11115, stop := 11162 ), rec( content := "\n \ Further information about this step can be found in Section\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Compili\ ng the kernel module" ), content := 0, count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "Ref", next := 11281, root := ~, start := 11240, stop := 11280 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], name := "Item", next := 11296, root := ~, start := 10981, stop := 11295 ), rec( attributes := rec( ), content := [ rec( content := "\n \ start ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " in the \ usual way (i.e. type ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "C", next := 11369, root := ~, start := 11359, stop := 11368 ), rec( content := " at the \ command line).\n ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], name := "Item", next := 11405, root := ~, start := 11304, stop := 11404 ), rec( attributes := rec( ), content := [ rec( content := "\n \ type ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ oadPackage(\"digraphs\");", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "C", next := 11464, root := ~, start := 11433, stop := 11463 ), rec( content := "\n \ ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 25 ], name := "Item", next := 11478, root := ~, start := 11413, stop := 11477 ) ], count := [ 2, 1, 0, 6 ], mathmode := "Text", name := "List", next := 11490, root := ~, start := 9407, stop := 11489 ), 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, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Testing your installati\ on" ), content := 0, count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "Ref", next := 11657, root := ~, start := 11618, stop := 11656 ), rec( content := ".\n ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 11671, root := ~, start := 8848, stop := 11670 ), rec( content := "\n\n ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 1, 0, 27 ], 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 := 11936, root := ~, start := 11888, stop := 11935 ), 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\ tp://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 := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), 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 := 122, root := ~, start := 101, stop := 121 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), 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\ tp://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 := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), 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 := 122, root := ~, start := 101, stop := 121 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), 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 := 12231, root := ~, start := 12227, stop := 12230 ), 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 := 12399, root := ~, start := 12363, stop := 12398 ), 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\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113, stop := 136 ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), 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 := 12544, root := ~, start := 12525, stop := 12543 ) , 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 := 12575, root := ~, start := 12519, stop := 12574 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Attr := "AsGraph" ), content := 0, count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "Ref", next := 12611, root := ~, start := 12590, stop := 12610\ ), rec( content := " with a digraph argument", count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 7 ], name := "Item", next := 12642, root := ~, start := 12584, stop := 12641 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Oper := "Digraph" ), content := 0, count := [ 2, 2, 1, 9 ], mathmode := \ "Text", name := "Ref", next := 12678, root := ~, start := 12657, stop := 12677\ ), 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 := "http://www.math\ s.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 9 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name\ := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 9 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113, stop :=\ 136 ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\ graph argument", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 2, 2, 1, 9 ], name := "Item", next := 12715, root := ~, start := 12651, stop := 12714 ) ], count := [ 2, 2, 1, 4 ], mathmode := "Text", name := "List", next := 12729, root := ~, start := 12504, stop := 12728 ), 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\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 11 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113, stop := 136 ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), 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 := 12886, root := ~, start := 12318, stop := 12885 ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 12899, root := ~, start := 11836, stop := 12898 ), 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 := 13160, root := ~, start := 13114, stop := 13159 ), 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 := 13340, root := ~, start := 13336, stop := 13339 ), 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 := 13420, root := ~, start := 13416, stop := 13419 ), 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-1.1.\ 1", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "F", next := 13493, root := ~, start := 13469, stop := 13492 ), 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 := 13563, root := ~, start := 13514, stop := 13562 ), 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 := 13572, root := ~, start := 13568, stop := 13571 ), 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 := 13950, root := ~, start := 13946, stop := 13949 ), 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 := 14598, root := ~, start := 13064, stop := 14597 ), 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 := 14861, root := ~, start := 14814, stop := 14860 ), 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 := 15138, root := ~, start := 15109, stop := 15137 ), 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 := 15197, root := ~, start := 15160, stop := 15196 ), 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 := 15226, root := ~, start := 15200, stop := 15225 ), 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 := 15356, root := ~, start := 15229, stop := 15355 ) ], count := [ 2, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 15370, root := ~, start := 15145, stop := 15369 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 15384, root := ~, start := 14763, stop := 15383 ), 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 := 15641, root := ~, start := 15597, stop := 15640 ), 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 := 15820, root := ~, start := 15787, stop := 15819 ), 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 := 15891, root := ~, start := 15857, stop := 15890 ), 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 := 15897, root := ~, start := 15893, stop := 15896 ), 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 := 16015, root := ~, start := 15975, stop := 16014 ), 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/gap-packages/Digraphs/issues", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "URL", next := 16210, root := ~, start := 16125, stop := 16209 ), 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 := 16313, root := ~, start := 16272, stop := 16312 ), 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 := 16336, root := ~, start := 16325, stop := 16335 ), 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 := 16352, root := ~, start := 16340, stop := 16351 ), rec( content := ".", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 16363, root := ~, start := 16316, stop := 16362 ), 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 := 16674, root := ~, start := 16640, stop := 16673 ), rec( content := ".\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 16692, root := ~, start := 16366, stop := 16691 ) ], count := [ 2, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 16706, root := ~, start := 16257, stop := 16705 ), 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 := 16769, root := ~, start := 16727, stop := 16768 ), 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 := 16792, root := ~, start := 16781, stop := 16791 ), 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 := 16808, root := ~, start := 16796, stop := 16807 ), rec( content := ".", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 16819, root := ~, start := 16772, stop := 16818 ), 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 := 17005, root := ~, start := 17001, stop := 17004 ), 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 := 17107, root := ~, start := 17070, stop := 17106 ), 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 := 17177, root := ~, start := 17140, stop := 17176 ), rec( content := ".\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 17195, root := ~, start := 16822, stop := 17194 ) ], count := [ 2, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 17209, root := ~, start := 16712, stop := 17208 ), 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 := 17271, root := ~, start := 17230, stop := 17270 ), 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 := 17301, root := ~, start := 17274, stop := 17300 ), 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 := 17577, root := ~, start := 17573, stop := 17576 ), 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 := 17652, root := ~, start := 17633, stop := 17651 ), 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 := 17709, root := ~, start := 17694, stop := 17708 ), 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 := 17764, root := ~, start := 17745, stop := 17763 ), 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 := 17839, root := ~, start := 17304, stop := 17838 ) ], count := [ 2, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 17853, root := ~, start := 17215, stop := 17852 ), 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 := 17868, root := ~, start := 15549, stop := 17867 ), 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 := 18042, root := ~, start := 8764, stop := 18041 ), 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 := 18137, root := ~, start := 18101, stop := 18136 ), 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 := 18196, root := ~, start := 18192, stop := 18195 ), 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 := 18245, root := ~, start := 18209, stop := 18244 ), 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 := 18305, root := ~, start := 18265, stop := 18304 ), 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 := 18404, root := ~, start := 18388, stop := 18403 ), 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 := 18494, root := ~, start := 18465, stop := 18493 ), 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 := 18522, root := ~, start := 18496, stop := 18521 ), 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 := 18559, root := ~, start := 18532, stop := 18558 ), rec( content := ".\n " , count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 18579, root := ~, start := 18308, stop := 18578 ) ], count := [ 3, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 18593, root := ~, start := 18250, stop := 18592 ), 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 := 18661, root := ~, start := 18614, stop := 18660 ), 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 := 18705, root := ~, start := 18682, stop := 18704 ), 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 := 18747, root := ~, start := 18723, stop := 18746 ), 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 := 18795, root := ~, start := 18752, stop := 18794 ), 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 := 18974, root := ~, start := 18923, stop := 18973 ), 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 := 18984, root := ~, start := 18980, stop := 18983 ), 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 := 19126, root := ~, start := 19084, stop := 19125 ), 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 := 19165, root := ~, start := 18664, stop := 19164 ) ], count := [ 3, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 19179, root := ~, start := 18599, stop := 19178 ), 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 := 19249, root := ~, start := 19200, stop := 19248 ), 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 := 19295, root := ~, start := 19270, stop := 19294 ), 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 := 19340, root := ~, start := 19316, stop := 19339 ), 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 := 19392, root := ~, start := 19367, stop := 19391 ), 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 := 19516, root := ~, start := 19465, stop := 19515 ), 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 := 19522, root := ~, start := 19518, stop := 19521 ), 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 := 19642, root := ~, start := 19617, stop := 19641 ), 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 := 19700, root := ~, start := 19658, stop := 19699 ), 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 := 19720, root := ~, start := 19716, stop := 19719 ), 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 := 19772, root := ~, start := 19740, stop := 19771 ), 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 := 19954, root := ~, start := 19252, stop := 19953 ) ], count := [ 3, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 19968, root := ~, start := 19185, stop := 19967 ), 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 := 20035, root := ~, start := 19989, stop := 20034 ), 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 := 20078, root := ~, start := 20056, stop := 20077 ), 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 := 20115, root := ~, start := 20099, stop := 20114 ), 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 := 20239, root := ~, start := 20212, stop := 20238 ), 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 := 20322, root := ~, start := 20038, stop := 20321 ) ], count := [ 3, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 20336, root := ~, start := 19974, stop := 20335 ), 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 := 20418, root := ~, start := 20357, stop := 20417 ), 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 := 20476, root := ~, start := 20439, stop := 20475 ), 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 := 20513, root := ~, start := 20497, stop := 20512 ), 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 := 20551, root := ~, start := 20537, stop := 20550 ), 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 := 20634, root := ~, start := 20421, stop := 20633 ) ], count := [ 3, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 20648, root := ~, start := 20342, stop := 20647 ), 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 := 20709, root := ~, start := 20669, stop := 20708 ), rec( attributes := rec( Name := "DigraphFamily" ), content := 0, count := [ 3, 1, 6, 4 ], mathmode := "Text", name := "Fam", next := 20739, root := ~, start := 20712, stop := 20738 ), 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 := 20821, root := ~, start := 20788, stop := 20820 ), 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 := 20877, root := ~, start := 20857, stop := 20876 ), rec( content := ".\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "Description", next := 20895, root := ~, start := 20742, stop := 20894 ) ], count := [ 3, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 20909, root := ~, start := 20654, stop := 20908 ), 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 := 20991, root := ~, start := 20930, stop := 20990 ), 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 := 21084, root := ~, start := 20994, stop := 21083 ), 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 := 21201, root := ~, start := 21087, stop := 21200 ), 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 := 21233, root := ~, start := 21204, stop := 21232 ), 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 := 21296, root := ~, start := 21285, stop := 21295 ), 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 := 21442, root := ~, start := 21431, stop := 21441 ), 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 := 21476, root := ~, start := 21446, stop := 21475 ), 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 := 21544, root := ~, start := 21533, stop := 21543 ), 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 := 21584, root := ~, start := 21548, stop := 21583 ), 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 := 21669, root := ~, start := 21658, stop := 21668 ), 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 := 21727, root := ~, start := 21691, stop := 21726 ), 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 := 21813, root := ~, start := 21765, stop := 21812 ), 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 := 21848, root := ~, start := 21838, stop := 21847 ) , 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 := 21923, root := ~, start := 21915, stop :=\ 21922 ), 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 := 21947, root\ := ~, start := 21937, stop := 21946 ), rec( content := ")", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "C", next := 21952, root := ~, start := 21\ 927, stop := 21951 ), 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 := 22038, root := ~, start := 22023, stop := 22037 ), 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 := 22064, root := ~, start := 22054, stop := 220\ 63 ), rec( content := ")]", count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "C", next := 22070, root := ~, start := 22038, stop := 22069 ), 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 := 22132, root := ~, start := 22122, stop := 22\ 131 ), 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 := 22138, root := ~, star\ t := 22134, stop := 22137 ), 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 :=\ 22201, root := ~, start := 22193, stop := 22200 ), 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 := 22213, root := ~, start := 22205, stop :=\ 22212 ), 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 := 22245, root := ~, start := 22237, stop := 22244 ), 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 := 22265, root := ~, start := 22255, stop := 222\ 64 ), rec( content := "[i]", count := [ 3, 1, 7, 10 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 22272, root := ~, start := 22252, stop := 22271 ), 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 := 22317, root := ~, start := 22309, stop := 22316 ), 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 :=\ 22343, root := ~, start := 22335, stop := 22342 ), 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 := 22356, root := ~, start := 22348, stop :=\ 22355 ), 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 := 22388, root := \ ~, start := 22378, stop := 22387 ), rec( content := "[i]", count := [ 3, 1, 7,\ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 10 ], mathmode := "Text", name := "C", next := 22395, root := ~, start := 22\ 375, stop := 22394 ), 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 := 22422, root := ~, start := 22414, stop := 22\ 421 ), 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 := 22447, root := ~, start := 22439, stop := 22446 ), 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 :=\ 22475, root := ~, start := 22467, stop := 22474 ), 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 := 22487, root := ~, start := 22479, stop :=\ 22486 ), rec( content := ".\n ", count := [ 3, 1, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], name := "Item\ ", next := 22502, root := ~, start := 21820, stop := 22501 ), 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 := 22538, root := ~, start := 22510, stop := 22537 ), 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 := 22573, root := ~, start := 22563, stop := 2257\ 2 ), 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 := 22617, root := ~, start := 22604, stop := 22\ 616 ), 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 := 22642, root := ~, start := 22630, stop := 22641 ), 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 := 22747, root := ~, st\ art := 22737, stop := 22746 ), rec( content := ")]", count := [ 3, 1, 7, 12 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ] , math\ mode := "Text", name := "C", next := 22753, root := ~, start := 22721, stop :=\ 22752 ), 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 \ := 22837, root := ~, start := 22822, stop := 22836 ), 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 := 22863, root := ~, start := 22853, stop := 22862 ), re\ c( content := ")]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C" , next\ := 22869, root := ~, start := 22837, stop := 22868 ), 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 := 22906, root := ~, start\ := 22893, stop := 22905 ), rec( content := ")", count := [ 3, 1, 7, 12 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "C", next := 22911, root := ~, start := 22883, stop :=\ 22910 ), 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 := 22937, root := ~, start := 22929, stop := 22936 ), 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 := 2297\ 8, root := ~, start := 22965, stop := 22977 ), rec( content := ")]", count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 22984, root := ~, star\ t := 22949, stop := 22983 ), 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 :\ = 23049, root := ~, start := 23033, stop := 23048 ), 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 := 23082, root := \ ~, start := 23067, stop := 23081 ), 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 := 23123, root := ~, start := 2309\ 6, stop := 23122 ), 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 := 23154, root := ~, start := 23128, stop := 23153 ), 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 := 23168, root := ~, start :=\ 23164, stop := 23167 ), 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 := 23260, root := ~, start := 23250, stop := 23259 ), rec( conte\ nt := ".\n ", count := [ 3, 1, 7, 13 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 7, 12 ], name := "Item", next := 23275, root := ~, start := 225\ 45, stop := 23274 ), 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 := 2332\ 5, root := ~, start := 23283, stop := 23324 ), 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\ 3360, root := ~, start := 23350, stop := 23359 ), 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 := 23393, root := \ ~, start := 23380, stop := 23392 ), 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 := 23418, root := ~, start := 2340\ 6, stop := 23417 ), 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 := 23516, root := ~, start := 23506, stop := 23515 ), rec( content := "]", c\ ount := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next := 23521, root := \ ~, start := 23497, stop := 23520 ), 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 := 23605, root := ~, start := 23590, stop := 23604 ), 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 := 23624, root := ~, start := 23614, stop := 236\ 23 ), rec( content := "]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "C", next := 23629, root := ~, start := 23605, stop := 23628 ), 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 := 23666, r\ oot := ~, start := 23653, stop := 23665 ), rec( content := ")", count := [ 3, \ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "C", next := 23671, root := ~, start \ := 23643, stop := 23670 ), 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 := 23697, root := ~, start := 23689, stop :=\ 23696 ), 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 := 23738, root := ~, start := 23725, stop := 23737 ), rec( conten\ t := ")]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next :=\ 23744, root := ~, start := 23709, stop := 23743 ), 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 := 23809, root := ~, start := 23793, stop := 23808 ), 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\ := 23842, root := ~, start := 23827, stop := 23841 ), 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 := 23883, root :\ = ~, start := 23856, stop := 23882 ), 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 := 23914, root := ~, start := 23888, stop \ := 23913 ), rec( content := ".\n ", count := [ 3, 1, 7, 15 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], name := "Ite\ m", next := 23929, root := ~, start := 23332, stop := 23928 ), 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 := 23975, root := ~, start := 23937, stop := 23974 ), 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 := 24011, root := ~, start := 24000, stop := 240\ 10 ), 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 := 24037, root := ~, start := 24026, stop := 24036 ), 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 := 24111, root := ~, start := 24100 , stop\ := 24110 ), 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 := 24128, root := ~, start := 24117, stop := 24127 ), 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 :=\ 24156, root := ~, start := 24145, stop := 24155 ), 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 := 24172, root := ~, start := 2416\ 0, stop := 24171 ), 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 := 2426\ 1, root := ~, start := 24250, stop := 24260 ), rec( content := ")]", count := \ [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 24267, root := ~, star\ t := 24234, stop := 24266 ), 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 := 24308, root := ~, star\ t := 24300, stop := 24307 ), 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 := 24327, root := ~, start := 24319, stop := 24326\ ), 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 := 24365, root :=\ ~, start := 24354, stop := 24364 ), 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 := 24377, root := ~, start := 24366, stop := 24\ 376 ), 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 := 24393, root := ~, start := 24382, stop := 24392 ), rec( content := "[j\ ])", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 24401 , root\ := ~, start := 24351, stop := 24400 ), 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 := 24421, root := ~, start := 24410, stop :=\ 24420 ), rec( content := ".\n ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], name := "Ite\ m", next := 24436, root := ~, start := 23982, stop := 24435 ), 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 := 24495, root := ~, start := 24444, stop := 24494 ), 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 := 24563, root := ~, star\ t := 24539, stop := 24562 ), 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\ 4569, root := ~, start := 24565, stop := 24568 ), 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 := 24591, root := ~, star\ t := 24583, stop := 24590 ), 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 := 24639, root := ~ , star\ t := 24628, stop := 24638 ), 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 := 24673, root := ~, start := 24663, stop\ := 24672 ), 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 := 24693, root := ~, start := 24683, stop := 24692 ), 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 := 24751, root := ~, start := 24740, stop := 24\ 750 ), 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 := 24790, root := ~, start := 24779, stop := 24789 ), 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 := \ 24814, root := ~, start := 24802, stop := 24813 ), 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 := 24839, root := ~, star\ t := 24829, stop := 24838 ), 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\ := 24908, root := ~, start := 24897, stop := 24907 ), 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 := 24965, root := ~, start := 24957, stop := 24964 ), 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 := 25\ 065, root := ~, start := 25054, stop := 25064 ), 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 := 25098, root := \ ~, start := 25085, stop := 25097 ), 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 := 25126, root := ~, start := 25\ 112, stop := 25125 ), 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 := 25141, root := ~, start := 25130, stop := 25140 ), 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 := 25147, root := ~, start := 25143, stop :=\ 25146 ), 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 := 25189, root := ~, start := 25181, stop :=\ 25188 ), 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 := 25219, root := ~, start := 25208, stop := 25218 ), 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 := 25281, root := ~, start := 25255, stop := 25280 ), 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 := 25360, root := ~, start := 25331, stop \ := 25359 ), rec( content := ".\n ", count := [ 3, 1, 7, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 19 ], name := "Ite\ m", next := 25375, root := ~, start := 24502, stop := 25374 ), 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 := 25421, root := ~, start := 25383, stop := 25420 ), 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 := 25456, root := ~, start := 25446, stop := 2545\ 5 ), 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\ tp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 3, 1, 7, 23 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmod\ e := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ] , coun\ t := [ 3, 1, 7, 23 ], name := "Alt", next := 93, root := ~, start := 1, stop :\ = 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes\ := rec( ), content := [ rec( content := "Grape", count := [ 3, 1, 7, 23 ], n\ ame := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Package", n\ ext := 137, root := ~, start := 113, stop := 136 ) ], count := [ 3, 1, 7, 23 ] , math\ mode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 14\ 2 ), 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 := 25544, root := \ ~, start := 25530, stop := 25543 ), 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 := 25564, root := ~, start := 25553, stop :=\ 25563 ), 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 := 25636, root := \ ~, start := 25626, stop := 25635 ), rec( content := ".\n ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Ite\ m", next := 25651, root := ~, start := 25428, stop := 25650 ), 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 := 25693, root := ~, start := 25659, stop := 25692 ), 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 := 25728, root := ~, start := 25718, stop := 2572\ 7 ), 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 := 25779, root := ~, start := 25764, stop :=\ 25778 ), rec( content := " for\n some posititve integer ", count := [ \ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "n", count := [ 3, 1, 7, 25 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "M", next := 25823, root := ~, start := 25815, stop :=\ 25822 ), 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 := 25893, root := \ ~, start := 25883, stop := 25892 ), 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 := 25967, root := ~, start := 25959, stop := 25966 ), 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 := 26020, root := ~, start := 26012, stop := 26019 ), 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 :=\ 26047, root := ~, start := 26039, stop := 26046 ), 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 := 26075, root := \ ~, start := 26063, stop := 26074 ), 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 := 2612\ 7, root := ~, start := 26117, stop := 26126 ), rec( content := ".\n ", coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], name := "Ite\ m", next := 26142, root := ~, start := 25700, stop := 26141 ) ], count := [ 3, 1, 7, 8 ], mathmode := "Text", name := "List", next := 26154, root := ~, start := 21752, stop := 26153 ), 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 := 26896, root := ~, start := 26160, stop := 26895 ), 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 := 27163, root := ~, start := 27144, stop := 27162 ), 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 := 27281, root := ~, start := 27255, stop := 27280 ), 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));\n\ngap> ac\ t := \\*;\n\ngap> digraph := Digraph(group, List(f), act, adj\ );\n\n", count := [ 3, 1, 7, 30 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 7, 30 ], mathmode := "Text", name := "Example", next := 28276, root := ~, start := 27283, stop := 28275 ), rec( content := "\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Description", next := 28293, root := ~, start := 21236, stop := 28292 ) ], count := [ 3, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 28307, root := ~, start := 20915, stop := 28306 ), 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 := 28386, root := ~, start := 28328, stop := 28385 ), 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 := 28418, root := ~, start := 28389, stop := 28417 ), 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 := 28481, root := ~, start := 28470, stop := 28480 ), 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 := 28627, root := ~, start := 28616, stop := 28626 ), 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 := 28661, root := ~, start := 28631, stop := 28660 ), 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 := 28729, root := ~, start := 28718, stop := 28728 ), 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 := 28769, root := ~, start := 28733, stop := 28768 ), 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 := 28854, root := ~, start := 28843, stop := 28853 ), 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 := 28912, root := ~, start := 28876, stop := 28911 ), 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 := 28936, root := ~, start := 28932, stop := 28935 ), 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 := 28956, root := ~, start := 28945, stop := 28955 ), 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 := 29043, root := ~, start := 29014, stop := 29042 ), 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 := 29120, root := ~, start := 29109, stop := 29119 ), 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 := 29126, root := ~, start := 29122, stop := 29125 ), 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 := 29161, root := ~, start := 29150, stop := 29160 ), 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 := 29253, root := ~, start := 29239, stop := 29252 ), 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 := 29264, root := ~, start := 29253, stop := 29263 ), 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 := 29272, root := ~, start := 29264, stop := 29271 ), 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 := 29317, root := ~, start := 29305, stop := 29316 ), 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 := 29348, root := ~, start := 29337, stop := 29347 ), 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 := 29361, root := ~, start := 29348, stop := 29360 ), 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 := 29376, root := ~, start := 29365, stop := 29375 ), 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 := 29382, root := ~, start := 29378, stop := 29381 ), 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 := 29978, root := ~, start := 29388, stop := 29977 ), rec( content := "\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 29995, root := ~, start := 28421, stop := 29994 ) ], count := [ 3, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 30009, root := ~, start := 28313, stop := 30008 ), 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 := 30083, root := ~, start := 30030, stop := 30082 ), 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 := 30115, root := ~, start := 30086, stop := 30114 ), 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 := 30178, root := ~, start := 30167, stop := 30177 ), 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 := 30324, root := ~, start := 30313, stop := 30323 ), 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 := 30358, root := ~, start := 30328, stop := 30357 ), 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 := 30426, root := ~, start := 30415, stop := 30425 ), 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 := 30466, root := ~, start := 30430, stop := 30465 ), 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 := 30551, root := ~, start := 30540, stop := 30550 ), 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 := 30609, root := ~, start := 30573, stop := 30608 ), 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 := 30633, root := ~, start := 30629, stop := 30632 ), 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 := 30653, root := ~, start := 30642, stop := 30652 ), 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 := 30780, root := ~, start := 30772, stop := 30779 ), 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 := 30821, root := ~, start := 30810, stop := 30820 ), 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 := 30826, root := ~, start := 30822, stop := 30825 ), 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 := 30872, root := ~, start := 30864, stop := 30871 ), 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 := 30915, root := ~, start := 30907, stop := 30914 ) , rec( content := "\ >= m", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 30924, root := ~, start := 30904, stop := 30923 ), 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 := 30939, root := ~, start := 30931, stop := 30938 ), 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 := 31015, root := ~, start := 31007, stop := 31014 ), 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 := 31049, root := ~, start := 31038, stop := 31048 ), 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 := 31055, root := ~, start := 31051, stop := 31054 ), 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 := 31091, root := ~, start := 31065, stop := 31090 ), 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 := 31621, root := ~, start := 31098, stop := 31620 ), rec( content := "\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 31638, root := ~, start := 30118, stop := 31637 ) ], count := [ 3, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 31652, root := ~, start := 30015, stop := 31651 ), 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 := 31725, root := ~, start := 31673, stop := 31724 ), 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 := 31776, root := ~, start := 31728, stop := 31775 ), 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 := 31808, root := ~, start := 31800, stop := 31807 ), 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 := 31845, root := ~, start := 31833, stop := 31844 ), 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 := 31891, root := ~, start := 31883, stop := 31890 ), 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 := 31936, root := ~, start := 31928, stop := 31935 ), 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 := 31964, root := ~, start := 31956, stop := 31963 ) , rec( content := "\ ]", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "C", next := 31969, root := ~, start := 31947, stop := 31968 ), 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 := 32045, root := ~, start := 32037, stop := 32044 ), 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 := 32117, root := ~, start := 32103, stop := 32116 ), 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 := 32172, root := ~, start := 32164, stop := 32171 ), 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 := 32250, root := ~, start := 32246, stop := 32249 ), 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 := 32295, root := ~, start := 32287, stop := 32294 ), 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 := 32378, root := ~, start := 32370, stop := 32377 ), 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 := 32402, root := ~, start := 32398, stop := 32401 ), 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 := 32760, root := ~, start := 32408, stop := 32759 ), rec( content := "\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 32777, root := ~, start := 31779, stop := 32776 ) ], count := [ 3, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 32791, root := ~, start := 31658, stop := 32790 ), 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 := 32867, root := ~, start := 32812, stop := 32866 ), rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbors" ), content := 0, count := [ 3, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 32924, root := ~, start := 32870, stop := 32923 ), 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 := 32956, root := ~, start := 32927, stop := 32955 ), 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 := 33017, root := ~, start := 33006, stop := 33016 ), 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 := 33159, root := ~, start := 33148, stop := 33158 ), 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 := 33193, root := ~, start := 33163, stop := 33192 ), 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 := 33259, root := ~, start := 33248, stop := 33258 ), 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 := 33297, root := ~, start := 33263, stop := 33296 ), 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 := 33380, root := ~, start := 33369, stop := 33379 ), 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 := 33436, root := ~, start := 33402, stop := 33435 ), 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 := 33460, root := ~, start := 33456, stop := 33459 ), 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 := 33478, root := ~, start := 33467, stop := 33477 ), 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 := 33563, root := ~, start := 33552, stop := 33562 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 33569, root := ~, start := 33536, stop := 33568 ), 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 := 33637, root := ~, start := 33626, stop := 33636 ), 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 := 33664, root := ~, start := 33653, stop := 33663 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 33670, root := ~, start := 33637, stop := 33669 ), 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 := 33727, root := ~, start := 33716, stop := 33726 ), 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 := 33791, root := ~, start := 33783, stop := 33790 ), 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 := 33817, root := ~, start := 33809, stop := 33816 ), 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 := 33829, root := ~, start := 33821, stop := 33828 ), 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 := 33854, root := ~, start := 33843, stop := 33853 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 33861, root := ~, start := 33840, stop := 33860 ), 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 := 33867, root := ~, start := 33863, stop := 33866 ), 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 := 33882, root := ~, start := 33874, stop := 33881 ), 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 := 33909, root := ~, start := 33898, stop := 33908 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 33916, root := ~, start := 33895, stop := 33915 ), 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 := 33943, root := ~, start := 33935, stop := 33942 ), 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 := 33970, root := ~, start := 33962, stop := 33969 ), 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 := 33999, root := ~, start := 33991, stop := 33998 ), 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 := 34011, root := ~, start := 34003, stop := 34010 ), 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 := 34017, root := ~, start := 34013, stop := 34016 ), 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 := 34051, root := ~, start := 34025, stop := 34050 ), 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 := 34587, root := ~, start := 34058, stop := 34586 ), rec( content := "\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 34604, root := ~, start := 32959, stop := 34603 ) ], count := [ 3, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 34618, root := ~, start := 32797, stop := 34617 ), 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 := 34683, root := ~, start := 34639, stop := 34682 ), 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 := 34726, root := ~, start := 34686, stop := 34725 ), 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 := 34759, root := ~, start := 34751, stop := 34758 ), 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 := 34792, root := ~, start := 34781, stop := 34791 ), 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 := 34830, root := ~, start := 34822, stop := 34829 ), 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 := 34928, root := ~, start := 34920, stop := 34927 ), 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 := 34957, root := ~, start := 34946, stop := 34956 ), 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 := 35000, root := ~, start := 34992, stop := 34999 ), 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 := 35051, root := ~, start := 35043, stop := 35050 ), 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 := 35074, root := ~, start := 35066, stop := 35073 ), 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 := 35127, root := ~, start := 35119, stop := 35126 ), 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 := 35142, root := ~, start := 35131, stop := 35141 ), 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 := 35169, root := ~, start := 35153, stop := 35168 ), 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 := 35175, root := ~, start := 35171, stop := 35174 ), 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 := 35302, root := ~, start := 35291, stop := 35301 ), 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 := 35354, root := ~, start := 35346, stop := 35353 ), 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 := 35379, root := ~, start := 35375, stop := 35378 ), 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 := 35482, root := ~, start := 35447, stop := 35481 ), 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 := 35502, root := ~, start := 35494, stop := 35501 ), 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 := 35581, root := ~, start := 35547, stop := 35580 ), 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 := 35617, root := ~, start := 35606, stop := 35616 ), 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 := 35679, root := ~, start := 35640, stop := 35678 ), 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 := 35684, root := ~, start := 35680, stop := 35683 ), 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 := 36240, root := ~, start := 35757, stop := 36239 ), rec( content := "\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 36257, root := ~, start := 34729, stop := 36256 ) ], count := [ 3, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 36271, root := ~, start := 34624, stop := 36270 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 36285, root := ~, start := 18200, stop := 36284 ), 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 := 36341, root := ~, start := 36298, stop := 36340 ), 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 := 36406, root := ~, start := 36361, stop := 36405 ), 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 := 36446, root := ~, start := 36409, stop := 36445 ), 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 := 36484, root := ~, start := 36470, stop := 36483 ), 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 := 36545, root := ~, start := 36537, stop := 36544 ), 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 := 36647, root := ~, start := 36634, stop := 36646 ), 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 := 36674, root := ~, start := 36662, stop := 36673 ), 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 := 36732, root := ~, start := 36720, stop := 36731 ), 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 := 36761, root := ~, start := 36747, stop := 36760 ), 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 := 36771, root := ~, start := 36767, stop := 36770 ), 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 := 36953, root := ~, start := 36772, stop := 36952 ), rec( content := "\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 36970, root := ~, start := 36449, stop := 36969 ) ], count := [ 3, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 36984, root := ~, start := 36346, stop := 36983 ), 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 := 37054, root := ~, start := 37005, stop := 37053 ), 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 := 37091, root := ~, start := 37080, stop := 37090 ), rec( content := ".", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 37102, root := ~, start := 37057, stop := 37101 ), 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 := 37165, root := ~, start := 37154, stop := 37164 ), 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 := 37311, root := ~, start := 37300, stop := 37310 ), 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 := 37345, root := ~, start := 37315, stop := 37344 ), 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 := 37413, root := ~, start := 37402, stop := 37412 ), 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 := 37453, root := ~, start := 37417, stop := 37452 ), 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 := 37538, root := ~, start := 37527, stop := 37537 ), 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 := 37596, root := ~, start := 37560, stop := 37595 ), 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 := 37620, root := ~, start := 37616, stop := 37619 ), 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 := 37641, root := ~, start := 37629, stop := 37640 ), 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 := 37675, root := ~, start := 37667, stop := 37674 ), 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 := 37747, root := ~, start := 37735, stop := 37746 ), 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 := 37768, root := ~, start := 37760, stop := 37767 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 37773, root := ~, start := 37751, stop := 37772 ), 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 := 37823, root := ~, start := 37815, stop := 37822 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 37828, root := ~, start := 37806, stop := 37827 ), 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 := 37851, root := ~, start := 37835, stop := 37850 ), 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 := 37900, root := ~, start := 37892, stop := 37899 ), 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 := 37937, root := ~, start := 37925, stop := 37936 ), 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 := 37976, root := ~, start := 37943, stop := 37975 ), 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 := 37986, root := ~, start := 37982, stop := 37985 ), 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 := 38046, root := ~, start := 38030, stop := 38045 ), 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 := 38060, root := ~, start := 38052, stop := 38059 ), 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 := 38096, root := ~, start := 38088, stop := 38095 ), 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 := 38117, root := ~, start := 38109, stop := 38116 ) , rec( content := "\ ]", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 38122, root := ~, start := 38100, stop := 38121 ), 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 := 38171, root := ~, start := 38163, stop := 38170 ), 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 := 38210, root := ~, start := 38198, stop := 38209 ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 38214, root := ~, start := 38193, stop := 38213 ), 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 := 38224, root := ~, start := 38220, stop := 38223 ), 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 := 38270, root := ~, start := 38262, stop := 38269 ), 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 := 38342, root := ~, start := 38330, stop := 38341 ), 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 := 38402, root := ~, start := 38390, stop := 38401 ), 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 := 38423, root := ~, start := 38415, stop := 38422 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 38428, root := ~, start := 38406, stop := 38427 ), 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 := 38486, root := ~, start := 38478, stop := 38485 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 38491, root := ~, start := 38469, stop := 38490 ), 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 := 38523, root := ~, start := 38511, stop := 38522 ) , 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 := 38533, root := ~, start := 38525, stop := 38532 ) , rec( content := "\ )", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 38538, root := ~, start := 38498, stop := 38537 ), 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 := 38562, root := ~, start := 38551, stop := 38561 ), 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 := 39055, root := ~, start := 38569, stop := 39054 ), rec( content := "\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 39072, root := ~, start := 37105, stop := 39071 ) ], count := [ 3, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 39086, root := ~, start := 36990, stop := 39085 ), 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 := 39141, root := ~, start := 39107, stop := 39140 ), 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 := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph.", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 39187, root := ~, start := 39144, stop := 39186 ), 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 := 39225, root := ~, start := 39211, stop := 39224 ), 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 := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), 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 := 39376, root := ~, start := 39362, stop := 39375 ), 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 := 39382, root := ~, start := 39378, stop := 39381 ), 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 := 39405, root := ~, start := 39391, stop := 39404 ), 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 := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 4 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), 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 := 39673, root := ~, start := 39630, stop := 39672 ), 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 := 39687, root := ~, start := 39673, stop := 39686 ), 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 := 39696, root := ~, start := 39687, stop := 39695 ), 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 := 40349, root := ~, start := 39703, stop := 40348 ), rec( content := "\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 40366, root := ~, start := 39190, stop := 40365 ) ], count := [ 3, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 40380, root := ~, start := 39092, stop := 40379 ), 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 := 40437, root := ~, start := 40401, stop := 40436 ), 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 := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 4, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), 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 := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph.", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 40483, root := ~, start := 40440, stop := 40482 ), 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 := 40521, root := ~, start := 40507, stop := 40520 ), 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 := 40596, root := ~, start := 40577, stop := 40595 ), 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 := 40627, root := ~, start := 40613, stop := 40626 ), 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 := 40717, root := ~, start := 40703, stop := 40716 ), 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 := 40764, root := ~, start := 40749, stop := 40763 ), 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 := 40778, root := ~, start := 40764, stop := 40777 ), 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 := 40786, root := ~, start := 40778, stop := 40785 ), 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 := 41197, root := ~, start := 40870, stop := 41196 ), rec( content := "\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 41214, root := ~, start := 40486, stop := 41213 ) ], count := [ 3, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 41228, root := ~, start := 40386, stop := 41227 ), 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 := 41294, root := ~, start := 41249, stop := 41293 ), 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 := 41338, root := ~, start := 41327, stop := 41337 ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 41348, root := ~, start := 41297, stop := 41347 ), 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 := 41386, root := ~, start := 41372, stop := 41385 ), 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 := 41440, root := ~, start := 41417, stop := 41439 ), 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 := 41506, root := ~, start := 41492, stop := 41505 ), 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 := 41552, root := ~, start := 41513, stop := 41551 ), 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 := 41593, root := ~, start := 41569, stop := 41592 ), 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 := 41607, root := ~, start := 41593, stop := 41606 ), 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 := 41615, root := ~, start := 41607, stop := 41614 ), 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 := 41635, root := ~, start := 41624, stop := 41634 ), 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 := 41642, root := ~, start := 41638, stop := 41641 ), 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 := 41665, root := ~, start := 41651, stop := 41664 ), 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 := 41703, root := ~, start := 41695, stop := 41702 ), 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 := 41747, root := ~, start := 41723, stop := 41746 ), 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 := 41761, root := ~, start := 41747, stop := 41760 ), 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 := 41769, root := ~, start := 41761, stop := 41768 ), 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 := 41813, root := ~, start := 41805, stop := 41812 ), 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 := 41840, root := ~, start := 41832, stop := 41839 ), 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 := 41884, root := ~, start := 41856, stop := 41883 ), 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 := 41898, root := ~, start := 41886, stop := 41897 ), 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 := 41958, root := ~, start := 41950, stop := 41957 ), 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 := 41976, root := ~, start := 41962, stop := 41975 ), 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 := 42442, root := ~, start := 41983, stop := 42441 ), rec( content := "\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 42459, root := ~, start := 41351, stop := 42458 ) ], count := [ 3, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 42473, root := ~, start := 41234, stop := 42472 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 42487, root := ~, start := 36289, stop := 42486 ), 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 := 42540, root := ~, start := 42500, stop := 42539 ), 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 := 42609, root := ~, start := 42560, stop := 42608 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opy" ), content := 0, count := [ 3, 3, 1, 4 ], mathmode := "Text", name := "Oper", next := 42659, root := ~, start := 42612, stop := 42658 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopySame\ Mutability" ), content := 0, count := [ 3, 3, 1, 6 ], mathmode := "Text", name := "Oper", next := 42716, root := ~, start := 42662, stop := 42715 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopy" ), content := 0, count := [ 3, 3, 1, 8 ], mathmode := "Text", name := "Oper", next := 42759, root := ~, start := 42719, stop := 42758 ), 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 := 42791, root := ~, start := 42762, stop := 42790 ), 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 := 42873, root := ~, start := 42859, stop := 42872 ), 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 := 42978, root := ~, start := 42964, stop := 42977 ), 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 := 42984, root := ~, start := 42980, stop := 42983 ), 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 := 43008, root := ~, start := 42990, stop := 43007 ), 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 := 43058, root := ~, start := 43026, stop := 43057 ), 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 := 43312, root := ~, start := 43060, stop := 43311 ), rec( content := "\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Description", next := 43329, root := ~, start := 42794, stop := 43328 ) ], count := [ 3, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 43343, root := ~, start := 42545, stop := 43342 ), 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 := 43424, root := ~, start := 43364, stop := 43423 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfMutable" ), content := 0, count := [ 3, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 43485, root := ~, start := 43427, stop := 43484 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfMutable" ), content := 0, count := [ 3, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 43544, root := ~, start := 43488, stop := 43543 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfImmutable" ), content := 0, count := [ 3, 3, 2, 8 ], mathmode := "Text", name := "Oper", next := 43605, root := ~, start := 43547, stop := 43604 ), 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 := 43637, root := ~, start := 43608, stop := 43636 ), 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 := 43738, root := ~, start := 43724, stop := 43737 ), 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 := 43771, root := ~, start := 43757, stop := 43770 ), 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 := 43852, root := ~, start := 43838, stop := 43851 ), 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 := 45114, root := ~, start := 43854, stop := 45113 ), rec( content := "\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Description", next := 45131, root := ~, start := 43640, stop := 45130 ) ], count := [ 3, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 45145, root := ~, start := 43349, stop := 45144 ), 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 := 45219, root := ~, start := 45166, stop := 45218 ), 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 := 45251, root := ~, start := 45222, stop := 45250 ), 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 := 45289, root := ~, start := 45275, stop := 45288 ), 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 := 45320, root := ~, start := 45308, stop := 45319 ), 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 := 45370, root := ~, start := 45356, stop := 45369 ), 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 := 45445, root := ~, start := 45431, stop := 45444 ), 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 := 45503, root := ~, start := 45491, stop := 45502 ), 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 := 45591, root := ~, start := 45579, stop := 45590 ), 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 := 45597, root := ~, start := 45593, stop := 45596 ), 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 := 45675, root := ~, start := 45650, stop := 45674 ), 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 := 45764, root := ~, start := 45731, stop := 45763 ), 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 := 45770, root := ~, start := 45766, stop := 45769 ), 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 := 45793, root := ~, start := 45779, stop := 45792 ), 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 := 45835, root := ~, start := 45805, stop := 45834 ), 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 := 45860, root := ~, start := 45846, stop := 45859 ), 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 := 45900, root := ~, start := 45886, stop := 45899 ), 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 := 45950, root := ~, start := 45912, stop := 45949 ), 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 := 46046, root := ~, start := 46042, stop := 46045 ), 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 := 46524, root := ~, start := 46052, stop := 46523 ), rec( content := "\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 46541, root := ~, start := 45254, stop := 46540 ) ], count := [ 3, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 46555, root := ~, start := 45151, stop := 46554 ), 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 := 46619, root := ~, start := 46576, stop := 46618 ), rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraphA\ ttr" ), content := 0, count := [ 3, 3, 4, 4 ], mathmode := "Text", name := "Attr", next := 46669, root := ~, start := 46622, stop := 46668 ), 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 := 46701, root := ~, start := 46672, stop := 46700 ), 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 := 46804, root := ~, start := 46790, stop := 46803 ), 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 := 46902, root := ~, start := 46888, stop := 46901 ), 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 := 46967, root := ~, start := 46953, stop := 46966 ), 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 := 47008, root := ~, start := 46977, stop := 47007 ), 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 := 47018, root := ~, start := 47014, stop := 47017 ), 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 := 47080, root := ~, start := 47066, stop := 47079 ), 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 := 47195, root := ~, start := 47162, stop := 47194 ), 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 := 47231, root := ~, start := 47200, stop := 47230 ), 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 := 47346, root := ~, start := 47332, stop := 47345 ), 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 := 47356, root := ~, start := 47352, stop := 47355 ), 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 := 47379, root := ~, start := 47365, stop := 47378 ), 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 := 47510, root := ~, start := 47496, stop := 47509 ), 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 := 48203, root := ~, start := 47541, stop := 48202 ), rec( content := "\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Description", next := 48220, root := ~, start := 46704, stop := 48219 ) ], count := [ 3, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 48234, root := ~, start := 46561, stop := 48233 ), 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 := 48310, root := ~, start := 48255, stop := 48309 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphAttr" ), content := 0, count := [ 3, 3, 5, 4 ], mathmode := "Text", name := "Attr", next := 48372, root := ~, start := 48313, stop := 48371 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoops" ), content := 0, count := [ 3, 3, 5, 6 ], mathmode := "Text", name := "Oper", next := 48442, root := ~, start := 48375, stop := 48441 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoopsAttr" ), content := 0, count := [ 3, 3, 5, 8 ], mathmode := "Text", name := "Attr", next := 48516, root := ~, start := 48445, stop := 48515 ), 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 := 48548, root := ~, start := 48519, stop := 48547 ), 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 := 48586, root := ~, start := 48572, stop := 48585 ), 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 := 48639, root := ~, start := 48606, stop := 48638 ), 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 := 48746, root := ~, start := 48732, stop := 48745 ), 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 := 48801, root := ~, start := 48787, stop := 48800 ), 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 := 48879, root := ~, start := 48867, stop := 48878 ), 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 := 48931, root := ~, start := 48919, stop := 48930 ), 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 := 48941, root := ~, start := 48937, stop := 48940 ), 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 := 49016, root := ~, start := 48971, stop := 49015 ), 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 := 49068, root := ~, start := 49064, stop := 49067 ), 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 := 49095, root := ~, start := 49081, stop := 49094 ), 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 := 49201, root := ~, start := 49187, stop := 49200 ), 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 := 49266, root := ~, start := 49262, stop := 49265 ), 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 := 49308, root := ~, start := 49276, stop := 49307 ), 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 := 49338, root := ~, start := 49310, stop := 49337 ), 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 := 49377, root := ~, start := 49348, stop := 49376 ), 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 := 50079, root := ~, start := 49404, stop := 50078 ), rec( content := "\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Description", next := 50096, root := ~, start := 48551, stop := 50095 ) ], count := [ 3, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 50110, root := ~, start := 48240, stop := 50109 ), 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 := 50190, root := ~, start := 50131, stop := 50189 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraphAttr" ), content := 0, count := [ 3, 3, 6, 4 ], mathmode := "Text", name := "Attr", next := 50256, root := ~, start := 50193, stop := 50255 ), 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 := 50288, root := ~, start := 50259, stop := 50287 ), 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 := 50326, root := ~, start := 50312, stop := 50325 ), 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 := 50383, root := ~, start := 50346, stop := 50382 ), 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 := 50442, root := ~, start := 50428, stop := 50441 ), 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 := 50497, root := ~, start := 50483, stop := 50496 ), 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 := 50571, root := ~, start := 50559, stop := 50570 ), 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 := 50589, root := ~, start := 50575, stop := 50588 ), 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 := 50608, root := ~, start := 50596, stop := 50607 ), 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 := 50646, root := ~, start := 50634, stop := 50645 ), 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 := 50675, root := ~, start := 50661, stop := 50674 ), 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 := 50747, root := ~, start := 50735, stop := 50746 ), 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 := 50764, root := ~, start := 50752, stop := 50763 ), 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 := 50782, root := ~, start := 50768, stop := 50781 ), 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 := 50803, root := ~, start := 50795, stop := 50802 ), 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 := 50816, root := ~, start := 50808, stop := 50815 ), rec( content := " are dis\ tinct, it discards the 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 := 50888, root := ~, start := 50860, stop := 50887 ), 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 := 50898, root := ~, start := 50894, stop := 50897 ), 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 := 50921, root := ~, start := 50907, stop := 50920 ), 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 := 51027, root := ~, start := 51013, stop := 51026 ), 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 := 51053, root := ~, start := 51049, stop := 51052 ), 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 := 51099, root := ~, start := 51063, stop := 51098 ), 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 := 51829, root := ~, start := 51126, stop := 51828 ), rec( content := "\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Description", next := 51846, root := ~, start := 50291, stop := 51845 ) ], count := [ 3, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 51860, root := ~, start := 50116, stop := 51859 ), 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 := 51934, root := ~, start := 51881, stop := 51933 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForestAttr" ), content := 0, count := [ 3, 3, 7, 4 ], mathmode := "Text", name := "Attr", next := 51994, root := ~, start := 51937, stop := 51993 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTree" ), content := 0, count := [ 3, 3, 7, 6 ], mathmode := "Text", name := "Oper", next := 52048, root := ~, start := 51997, stop := 52047 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTreeAttr" ), content := 0, count := [ 3, 3, 7, 8 ], mathmode := "Text", name := "Attr", next := 52106, root := ~, start := 52051, stop := 52105 ), 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 := 52143, root := ~, start := 52132, stop := 52142 ), rec( content := ".", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Returns", next := 52154, root := ~, start := 52109, stop := 52153 ), 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 := 52192, root := ~, start := 52178, stop := 52191 ), 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 := 52272, root := ~, start := 52241, stop := 52271 ), 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 := 52332, root := ~, start := 52318, stop := 52331 ), 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 := 52378, root := ~, start := 52367, stop := 52377 ), 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 := 52431, root := ~, start := 52384, stop := 52430 ), 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 := 52492, root := ~, start := 52488, stop := 52491 ), 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 := 52515, root := ~, start := 52501, stop := 52514 ), 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 := 52610, root := ~, start := 52564, stop := 52609 ), 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 := 52668, root := ~, start := 52629, stop := 52667 ), 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 := 52705, root := ~, start := 52676, stop := 52704 ), 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 := 52763, root := ~, start := 52749, stop := 52762 ), 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 := 52813, root := ~, start := 52802, stop := 52812 ), 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 := 52859, root := ~, start := 52820, stop := 52858 ), 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 := 52918, root := ~, start := 52914, stop := 52917 ), 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 := 52941, root := ~, start := 52927, stop := 52940 ), 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 := 53044, root := ~, start := 53030, stop := 53043 ), 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 := 53119, root := ~, start := 53105, stop := 53118 ), 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 := 53124, root := ~, start := 53120, stop := 53123 ), 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 := 53253, root := ~, start := 53224, stop := 53252 ), 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 := 53331, root := ~, start := 53300, stop := 53330 ), 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 := 54493, root := ~, start := 53338, stop := 54492 ), rec( content := "\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Description", next := 54510, root := ~, start := 52157, stop := 54509 ) ], count := [ 3, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 54524, root := ~, start := 51866, stop := 54523 ), 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 := 54592, root := ~, start := 54545, stop := 54591 ), 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 := 54624, root := ~, start := 54595, stop := 54623 ), 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 := 54662, root := ~, start := 54648, stop := 54661 ), 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 := 54689, root := ~, start := 54681, stop := 54688 ), 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 := 54742, root := ~, start := 54728, stop := 54741 ), 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 := 54844, root := ~, start := 54830, stop := 54843 ), 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 := 54887, root := ~, start := 54879, stop := 54886 ), 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 := 54897, root := ~, start := 54893, stop := 54896 ), 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 := 54948, root := ~, start := 54934, stop := 54947 ), 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 := 55077, root := ~, start := 55063, stop := 55076 ), 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 := 55157, root := ~, start := 55153, stop := 55156 ), 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 := 55193, root := ~, start := 55179, stop := 55192 ), 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 := 55210, root := ~, start := 55202, stop := 55209 ), 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 := 55222, root := ~, start := 55214, stop := 55221 ), 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 := 55254, root := ~, start := 55246, stop := 55253 ), 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 := 55316, root := ~, start := 55302, stop := 55315 ), 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 := 55349, root := ~, start := 55341, stop := 55348 ), 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 := 55376, root := ~, start := 55368, stop := 55375 ), 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 := 55426, root := ~, start := 55418, stop := 55425 ), 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 := 55438, root := ~, start := 55430, stop := 55437 ), 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 := 55626, root := ~, start := 55622, stop := 55625 ), 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 := 55649, root := ~, start := 55635, stop := 55648 ), 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 := 55691, root := ~, start := 55661, stop := 55690 ), 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 := 55716, root := ~, start := 55702, stop := 55715 ), 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 := 55756, root := ~, start := 55742, stop := 55755 ), 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 := 55806, root := ~, start := 55768, stop := 55805 ), 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 := 55878, root := ~, start := 55874, stop := 55877 ), 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 := 56492, root := ~, start := 55884, stop := 56491 ), rec( content := "\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 56509, root := ~, start := 54627, stop := 56508 ) ], count := [ 3, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 56523, root := ~, start := 54530, stop := 56522 ), 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 := 56587, root := ~, start := 56544, stop := 56586 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReverseA\ ttr" ), content := 0, count := [ 3, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 56637, root := ~, start := 56590, stop := 56636 ), 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 := 56669, root := ~, start := 56640, stop := 56668 ), 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 := 56825, root := ~, start := 56812, stop := 56824 ), 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 := 56901, root := ~, start := 56888, stop := 56900 ), 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 := 56906, root := ~, start := 56902, stop := 56905 ), 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 := 56933, root := ~, start := 56912, stop := 56932 ), 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 := 56983, root := ~, start := 56969, stop := 56982 ), 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 := 57006, root := ~, start := 56992, stop := 57005 ), 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 := 57112, root := ~, start := 57098, stop := 57111 ), 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 := 57976, root := ~, start := 57155, stop := 57975 ), rec( content := "\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 57993, root := ~, start := 56672, stop := 57992 ) ], count := [ 3, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 58007, root := ~, start := 56529, stop := 58006 ), 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 := 58068, root := ~, start := 58028, stop := 58067 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphDualAttr\ " ), content := 0, count := [ 3, 3, 10, 4 ], mathmode := "Text", name := "Attr", next := 58115, root := ~, start := 58071, stop := 58114 ), 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 := 58147, root := ~, start := 58118, stop := 58146 ), 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 := 58183, root := ~, start := 58172, stop := 58182 ), 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 := 58201, root := ~, start := 58187, stop := 58200 ), 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 := 58241, root := ~, start := 58227, stop := 58240 ), 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 := 58293, root := ~, start := 58285, stop := 58292 ), 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 := 58305, root := ~, start := 58297, stop := 58304 ), 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 := 58349, root := ~, start := 58341, stop := 58348 ), 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 := 58361, root := ~, start := 58353, stop := 58360 ), 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 := 58379, root := ~, start := 58365, stop := 58378 ), 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 := 58400, root := ~, start := 58389, stop := 58399 ), 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 := 58442, root := ~, start := 58425, stop := 58441 ), 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 := 58447, root := ~, start := 58443, stop := 58446 ), 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 := 58471, root := ~, start := 58453, stop := 58470 ), 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 := 58518, root := ~, start := 58504, stop := 58517 ), 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 := 58541, root := ~, start := 58527, stop := 58540 ), 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 := 58658, root := ~, start := 58644, stop := 58657 ), 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 := 59124, root := ~, start := 58702, stop := 59123 ), rec( content := "\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Description", next := 59141, root := ~, start := 58150, stop := 59140 ) ], count := [ 3, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 59155, root := ~, start := 58013, stop := 59154 ), 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 := 59228, root := ~, start := 59176, stop := 59227 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosureAttr" ), content := 0, count := [ 3, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 59287, root := ~, start := 59231, stop := 59286 ), 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 := 59319, root := ~, start := 59290, stop := 59318 ), 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 := 59357, root := ~, start := 59343, stop := 59356 ), 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 := 59508, root := ~, start := 59494, stop := 59507 ), 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 := 59513, root := ~, start := 59509, stop := 59512 ), 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 := 59552, root := ~, start := 59536, stop := 59551 ), 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 := 59611, root := ~, start := 59582, stop := 59610 ), 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 := 59741, root := ~, start := 59733, stop := 59740 ), 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 := 59762, root := ~, start := 59754, stop := 59761 ), 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 := 59789, root := ~, start := 59781, stop := 59788 ), 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 := 59801, root := ~, start := 59793, stop := 59800 ), 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 := 59844, root := ~, start := 59812, stop := 59843 ), 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 := 59849, root := ~, start := 59845, stop := 59848 ), 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 := 59872, root := ~, start := 59858, stop := 59871 ), 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 := 59978, root := ~, start := 59964, stop := 59977 ), 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 := 60819, root := ~, start := 60031, stop := 60818 ), rec( content := "\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 60836, root := ~, start := 59322, stop := 60835 ) ], count := [ 3, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 60850, root := ~, start := 59161, stop := 60849 ), 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 := 60924, root := ~, start := 60871, stop := 60923 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosureAttr" ), content := 0, count := [ 3, 3, 12, 4 ], mathmode := "Text", name := "Attr", next := 60984, root := ~, start := 60927, stop := 60983 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosure" ), content := 0, count := [ 3, 3, 12, 6 ], mathmode := "Text", name := "Oper", next := 61049, root := ~, start := 60987, stop := 61048 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosureAttr" ), content := 0, count := [ 3, 3, 12, 8 ], mathmode := "Text", name := "Attr", next := 61118, root := ~, start := 61052, stop := 61117 ), 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 := 61150, root := ~, start := 61121, stop := 61149 ), 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 := 61188, root := ~, start := 61174, stop := 61187 ), 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 := 61311, root := ~, start := 61297, stop := 61310 ), 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 := 61316, root := ~, start := 61312, stop := 61315 ), 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 := 61351, root := ~, start := 61335, stop := 61350 ), 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 := 61417, root := ~, start := 61400, stop := 61416 ), 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 := 61442, root := ~, start := 61430, stop := 61441 ), 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 := 61459, root := ~, start := 61447, stop := 61458 ), 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 := 61491, root := ~, start := 61477, stop := 61490 ), 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 := 61505, root := ~, start := 61493, stop := 61504 ), 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 := 61571, root := ~, start := 61528, stop := 61570 ), 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 := 61599, root := ~, start := 61585, stop := 61598 ), 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 := 61677, root := ~, start := 61663, stop := 61676 ), 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 := 61682, root := ~, start := 61678, stop := 61681 ), 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 := 61705, root := ~, start := 61691, stop := 61704 ), 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 := 61811, root := ~, start := 61797, stop := 61810 ), 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 := 61880, root := ~, start := 61876, stop := 61879 ), 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 := 61898, root := ~, start := 61890, stop := 61897 ), 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 := 61942, root := ~, start := 61928, stop := 61941 ), 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 := 61964, root := ~, start := 61956, stop := 61963 ), 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 := 62122, root := ~, start := 62109, stop := 62121 ), 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 := 62222, root := ~, start := 62180, stop := 62221 ), 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 := 62306, root := ~, start := 62279, stop := 62305 ), 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 := 62331, root := ~, start := 62327, stop := 62330 ), 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 := 63327, root := ~, start := 62337, stop := 63326 ), rec( content := "\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Description", next := 63344, root := ~, start := 61153, stop := 63343 ) ], count := [ 3, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 63358, root := ~, start := 60856, stop := 63357 ), 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 := 63434, root := ~, start := 63379, stop := 63433 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReductionAttr" ), content := 0, count := [ 3, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 63496, root := ~, start := 63437, stop := 63495 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReduction" ), content := 0, count := [ 3, 3, 13, 6 ], mathmode := "Text", name := "Oper", next := 63563, root := ~, start := 63499, stop := 63562 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReductionAttr" ), content := 0, count := [ 3, 3, 13, 8 ], mathmode := "Text", name := "Attr", next := 63634, root := ~, start := 63566, stop := 63633 ), 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 := 63666, root := ~, start := 63637, stop := 63665 ), 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 := 63704, root := ~, start := 63690, stop := 63703 ), 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 := 63786, root := ~, start := 63750, stop := 63785 ), 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 := 63904, root := ~, start := 63890, stop := 63903 ), 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 := 63909, root := ~, start := 63905, stop := 63908 ), 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 := 64132, root := ~, start := 64118, stop := 64131 ), 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 := 64189, root := ~, start := 64142, stop := 64188 ), 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 := 64254, root := ~, start := 64240, stop := 64253 ), 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 := 64312, root := ~, start := 64298, stop := 64311 ), 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 := 64317, root := ~, start := 64313, stop := 64316 ), 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 := 64340, root := ~, start := 64326, stop := 64339 ), 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 := 64446, root := ~, start := 64432, stop := 64445 ), 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 := 64517, root := ~, start := 64513, stop := 64516 ), 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 := 64535, root := ~, start := 64527, stop := 64534 ), 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 := 64611, root := ~, start := 64603, stop := 64610 ), 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 := 64721, root := ~, start := 64694, stop := 64720 ), 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 := 64731, root := ~, start := 64727, stop := 64730 ), 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 := 65591, root := ~, start := 64736, stop := 65590 ), rec( content := "\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Description", next := 65608, root := ~, start := 63669, stop := 65607 ) ], count := [ 3, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 65622, root := ~, start := 63364, stop := 65621 ), 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 := 65698, root := ~, start := 65643, stop := 65697 ), 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 := 65730, root := ~, start := 65701, stop := 65729 ), 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 := 65814, root := ~, start := 65800, stop := 65813 ), 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 := 65872, root := ~, start := 65868, stop := 65871 ), 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 := 65922, root := ~, start := 65910, stop := 65921 ), 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 := 65995, root := ~, start := 65983, stop := 65994 ), 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 := 66005, root := ~, start := 66001, stop := 66004 ), 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 := 66028, root := ~, start := 66014, stop := 66027 ), 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 := 66070, root := ~, start := 66040, stop := 66069 ), 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 := 66127, root := ~, start := 66113, stop := 66126 ), 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 := 66146, root := ~, start := 66132, stop := 66145 ), 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 := 66194, root := ~, start := 66162, stop := 66193 ), 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 := 66231, root := ~, start := 66217, stop := 66230 ), 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 := 66279, root := ~, start := 66275, stop := 66278 ), 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 := 66884, root := ~, start := 66285, stop := 66883 ), rec( content := "\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 66901, root := ~, start := 65733, stop := 66900 ) ], count := [ 3, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 66915, root := ~, start := 65628, stop := 66914 ), 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 := 67027, root := ~, start := 66936, stop := 67026 ), 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 := 67132, root := ~, start := 67030, stop := 67131 ), 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 := 67300, root := ~, start := 67271, stop := 67299 ), 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 := 67356, root := ~, start := 67348, stop := 67355 ), 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 := 67426, root := ~, start := 67412, stop := 67425 ), 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 := 67445, root := ~, start := 67437, stop := 67444 ), 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 := 67468, root := ~, start := 67464, stop := 67467 ), 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 := 67501, root := ~, start := 67488, stop := 67500 ), 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 := 67538, root := ~, start := 67530, stop := 67537 ), 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 := 67627, root := ~, start := 67613, stop := 67626 ), 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 := 67646, root := ~, start := 67638, stop := 67645 ), 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 := 67716, root := ~, start := 67712, stop := 67715 ), 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 := 67739, root := ~, start := 67725, stop := 67738 ), 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 := 67781, root := ~, start := 67751, stop := 67780 ), 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 := 67841, root := ~, start := 67827, stop := 67840 ), 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 := 67891, root := ~, start := 67877, stop := 67890 ), 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 := 67935, root := ~, start := 67903, stop := 67934 ), rec( content := ", then t\ he\n result is returned as an 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 := 68002, root := ~, start := 67998, stop := 68001 ), 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 := 68651, root := ~, start := 68008, stop := 68650 ), rec( content := "\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Description", next := 68668, root := ~, start := 67303, stop := 68667 ) ], count := [ 3, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 68682, root := ~, start := 66921, stop := 68681 ), 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 := 68790, root := ~, start := 68703, stop := 68789 ), 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 := 68891, root := ~, start := 68793, stop := 68890 ), 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 := 68923, root := ~, start := 68894, stop := 68922 ), 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 := 68958, root := ~, start := 68947, stop := 68957 ), 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 := 68998, root := ~, start := 68984, stop := 68997 ), 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 := 69013, root := ~, start := 69003, stop := 69012 ), 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 := 69032, root := ~, start := 69022, stop := 69031 ), 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 := 69063, root := ~, start := 69049, stop := 69062 ), 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 := 69138, root := ~, start := 69124, stop := 69137 ), 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 := 69187, root := ~, start := 69176, stop := 69186 ), 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 := 69197, root := ~, start := 69187, stop := 69196 ), 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 := 69209, root := ~, start := 69199, stop := 69208 ), 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 := 69236, root := ~, start := 69225, stop := 69235 ), 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 := 69246, root := ~, start := 69236, stop := 69245 ), 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 := 69258, root := ~, start := 69248, stop := 69257 ), 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 := 69269, root := ~, start := 69265, stop := 69268 ), 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 := 69296, root := ~, start := 69282, stop := 69295 ), 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 := 69338, root := ~, start := 69308, stop := 69337 ), 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 := 69393, root := ~, start := 69379, stop := 69392 ), 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 := 69412, root := ~, start := 69398, stop := 69411 ), 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 := 69462, root := ~, start := 69424, stop := 69461 ), 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 := 69504, root := ~, start := 69490, stop := 69503 ), 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 := 69552, root := ~, start := 69548, stop := 69551 ), 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 := 70278, root := ~, start := 69558, stop := 70277 ), rec( content := "\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 70295, root := ~, start := 68926, stop := 70294 ) ], count := [ 3, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 70309, root := ~, start := 68688, stop := 70308 ), 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 := 70384, root := ~, start := 70330, stop := 70383 ), 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 := 70428, root := ~, start := 70387, stop := 70427 ), 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 := 70540, root := ~, start := 70526, stop := 70539 ), 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 := 70618, root := ~, start := 70607, stop := 70617 ), 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 := 70669, root := ~, start := 70643, stop := 70668 ), 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 := 70691, root := ~, start := 70677, stop := 70690 ), 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 := 70708, root := ~, start := 70697, stop := 70707 ), 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 := 70745, root := ~, start := 70731, stop := 70744 ), 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 := 70770, root := ~, start := 70756, stop := 70769 ), 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 := 70821, root := ~, start := 70807, stop := 70820 ), 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 := 70864, root := ~, start := 70860, stop := 70863 ), 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 := 70924, root := ~, start := 70910, stop := 70923 ), 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, 8 ], [ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 \ ], [ 6, 2 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \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 := 71900, root := ~, start := 70931, stop := 71899 ), rec( content := "\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 71917, root := ~, start := 70431, stop := 71916 ) ], count := [ 3, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 71931, root := ~, start := 70315, stop := 71930 ), 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 := 72003, root := ~, start := 71952, stop := 72002 ), 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 := 72035, root := ~, start := 72006, stop := 72034 ), 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 := 72071, root := ~, start := 72059, stop := 72070 ), 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 := 72141, root := ~, start := 72127, stop := 72140 ), 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 := 72216, root := ~, start := 72202, stop := 72215 ), 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 := 72262, root := ~, start := 72250, stop := 72261 ), 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 := 72306, root := ~, start := 72295, stop := 72305 ), 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 := 72322, root := ~, start := 72310, stop := 72321 ), 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 := 72379, root := ~, start := 72365, stop := 72378 ), 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 := 72405, root := ~, start := 72390, stop := 72404 ), 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 := 72415, root := ~, start := 72411, stop := 72414 ), 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 := 72459, root := ~, start := 72447, stop := 72458 ), 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 := 72486, root := ~, start := 72478, stop := 72485 ), 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 := 72522, root := ~, start := 72514, stop := 72521 ), 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 := 72552, root := ~, start := 72538, stop := 72551 ), 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 := 72594, root := ~, start := 72564, stop := 72593 ), 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 := 72651, root := ~, start := 72637, stop := 72650 ), 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 := 72670, root := ~, start := 72656, stop := 72669 ), 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 := 72720, root := ~, start := 72682, stop := 72719 ), 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 := 72787, root := ~, start := 72783, stop := 72786 ), 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 := 73524, root := ~, start := 72793, stop := 73523 ), rec( content := "\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 73541, root := ~, start := 72038, stop := 73540 ) ], count := [ 3, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 73555, root := ~, start := 71937, stop := 73554 ), 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 := 73627, root := ~, start := 73576, stop := 73626 ), 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 := 73659, root := ~, start := 73630, stop := 73658 ), 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 := 73691, root := ~, start := 73683, stop := 73690 ), 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 := 73721, root := ~, start := 73707, stop := 73720 ), 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 := 73796, root := ~, start := 73782, stop := 73795 ), 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 := 73824, root := ~, start := 73816, stop := 73823 ), 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 := 73890, root := ~, start := 73882, stop := 73889 ), 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 := 73895, root := ~, start := 73891, stop := 73894 ), 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 := 73918, root := ~, start := 73904, stop := 73917 ), 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 := 73931, root := ~, start := 73923, stop := 73930 ), 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 := 74007, root := ~, start := 73992, stop := 74006 ), 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 := 74090, root := ~, start := 74057, stop := 74089 ), 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 := 74096, root := ~, start := 74092, stop := 74095 ), 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 := 74119, root := ~, start := 74105, stop := 74118 ), 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 := 74161, root := ~, start := 74131, stop := 74160 ), 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 := 74222, root := ~, start := 74208, stop := 74221 ), 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 := 74241, root := ~, start := 74227, stop := 74240 ), 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 := 74289, root := ~, start := 74257, stop := 74288 ), 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 := 74326, root := ~, start := 74312, stop := 74325 ), 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 := 74366, root := ~, start := 74362, stop := 74365 ), 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 := 75060, root := ~, start := 74372, stop := 75059 ), rec( content := "\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 75077, root := ~, start := 73662, stop := 75076 ) ], count := [ 3, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 75091, root := ~, start := 73561, stop := 75090 ), 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 := 75169, root := ~, start := 75112, stop := 75168 ), 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 := 75201, root := ~, start := 75172, stop := 75200 ), 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 := 75237, root := ~, start := 75225, stop := 75236 ), 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 := 75313, root := ~, start := 75299, stop := 75312 ), 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 := 75388, root := ~, start := 75374, stop := 75387 ), 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 := 75429, root := ~, start := 75417, stop := 75428 ), 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 := 75502, root := ~, start := 75490, stop := 75501 ), 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 := 75507, root := ~, start := 75503, stop := 75506 ), 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 := 75530, root := ~, start := 75516, stop := 75529 ), 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 := 75543, root := ~, start := 75535, stop := 75542 ), 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 := 75629, root := ~, start := 75617, stop := 75628 ) , rec( content := "\ )]", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 75635, root := ~, start := 75599, stop := 75634 ), 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 := 75718, root := ~, start := 75685, stop := 75717 ), 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 := 75724, root := ~, start := 75720, stop := 75723 ), 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 := 75747, root := ~, start := 75733, stop := 75746 ), 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 := 75789, root := ~, start := 75759, stop := 75788 ), 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 := 75853, root := ~, start := 75839, stop := 75852 ), 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 := 75872, root := ~, start := 75858, stop := 75871 ), 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 := 75920, root := ~, start := 75888, stop := 75919 ), 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 := 75957, root := ~, start := 75943, stop := 75956 ), 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 := 75999, root := ~, start := 75995, stop := 75998 ), 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 := 76582, root := ~, start := 76005, stop := 76581 ), rec( content := "\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 76599, root := ~, start := 75204, stop := 76598 ) ], count := [ 3, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 76613, root := ~, start := 75097, stop := 76612 ), 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 := 76724, root := ~, start := 76634, stop := 76723 ), 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 := 76828, root := ~, start := 76727, stop := 76827 ), 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 := 76860, root := ~, start := 76831, stop := 76859 ), 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 := 76898, root := ~, start := 76884, stop := 76897 ), 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 := 76950, root := ~, start := 76939, stop := 76949 ), 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 := 76994, root := ~, start := 76980, stop := 76993 ), 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 := 77009, root := ~, start := 76999, stop := 77008 ), 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 := 77024, root := ~, start := 77014, stop := 77023 ), 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 := 77059, root := ~, start := 77045, stop := 77058 ), 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 := 77134, root := ~, start := 77120, stop := 77133 ), 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 := 77184, root := ~, start := 77173, stop := 77183 ), 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 := 77205, root := ~, start := 77188, stop := 77204 ), 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 := 77211, root := ~, start := 77207, stop := 77210 ), 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 := 77236, root := ~, start := 77222, stop := 77235 ), 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 := 77284, root := ~, start := 77248, stop := 77283 ), 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 := 77343, root := ~, start := 77329, stop := 77342 ), 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 := 77362, root := ~, start := 77348, stop := 77361 ), 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 := 77412, root := ~, start := 77374, stop := 77411 ), 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 := 77449, root := ~, start := 77435, stop := 77448 ), 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 := 77492, root := ~, start := 77488, stop := 77491 ), 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 := 77525, root := ~, start := 77511, stop := 77524 ), 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 := 77569, root := ~, start := 77537, stop := 77568 ), 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 := 77608, root := ~, start := 77594, stop := 77607 ), 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 := 77645, root := ~, start := 77634, stop := 77644 ), 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 := 77671, root := ~, start := 77654, stop := 77670 ), 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 := 77713, root := ~, start := 77699, stop := 77712 ), 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 := 77718, root := ~, start := 77714, stop := 77717 ), 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 := 78148, root := ~, start := 77724, stop := 78147 ), rec( content := "\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Description", next := 78165, root := ~, start := 76863, stop := 78164 ) ], count := [ 3, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 78179, root := ~, start := 76619, stop := 78178 ), 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 := 78257, root := ~, start := 78200, stop := 78256 ), 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 := 78301, root := ~, start := 78260, stop := 78300 ), 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 := 78403, root := ~, start := 78389, stop := 78402 ), 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 := 78446, root := ~, start := 78435, stop := 78445 ), 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 := 78502, root := ~, start := 78476, stop := 78501 ), 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 := 78520, root := ~, start := 78506, stop := 78519 ), 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 := 78549, root := ~, start := 78538, stop := 78548 ), 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 := 78582, root := ~, start := 78568, stop := 78581 ), 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 := 78603, root := ~, start := 78589, stop := 78602 ), 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 := 78658, root := ~, start := 78644, stop := 78657 ), 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 := 78701, root := ~, start := 78697, stop := 78700 ), 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 := 78761, root := ~, start := 78747, stop := 78760 ), 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, 8 ], [ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 \ ], [ 6, 2 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \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 := 79736, root := ~, start := 78767, stop := 79735 ), rec( content := "\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 79753, root := ~, start := 78304, stop := 79752 ) ], count := [ 3, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 79767, root := ~, start := 78185, stop := 79766 ), 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 := 79842, root := ~, start := 79788, stop := 79841 ), 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 := 79874, root := ~, start := 79845, stop := 79873 ), 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 := 79972, root := ~, start := 79958, stop := 79971 ), 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 := 80034, root := ~, start := 80022, stop :=\ 80033 ), 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 := 80083, root := ~, start := 80069, stop :=\ 80082 ), rec( content := ", or\n ", count := [ 3, 3, 23, 5 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], name := "Ite\ m", next := 80101, root := ~, start := 79943, stop := 80100 ), 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 := 80129, root := ~, start := 80115, stop := 80128 ), 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\ := 80159, root := ~, start := 80147, stop := 80158 ), rec( content := " is an\ empty list ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 3, 23, 7 ], name := "Item", next := 80184, root := ~, start := 801\ 08, stop := 80183 ) ], count := [ 3, 3, 23, 4 ], mathmode := "Text", name := "List", next := 80196, root := ~, start := 79930, stop := 80195 ), 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 := 80271, root := ~, start := 80257, stop := 80270 ), 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 := 80330, root := ~, start := 80318, stop := 80329 ), 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 := 80405, root := ~, start := 80340, stop := 80404 ), 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 := 80416, root := ~, start := 80412, stop := 80415 ), 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 := 80439, root := ~, start := 80425, stop := 80438 ), 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 := 80481, root := ~, start := 80451, stop := 80480 ), 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 := 80540, root := ~, start := 80526, stop := 80539 ), 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 := 80559, root := ~, start := 80545, stop := 80558 ), 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 := 80609, root := ~, start := 80571, stop := 80608 ), 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 := 80675, root := ~, start := 80661, stop := 80674 ), 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 := 80713, root := ~, start := 80709, stop := 80712 ), 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 := 80744, root := ~, start := 80732, stop := 80743 ), 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 := 80812, root := ~, start := 80798, stop := 80811 ), 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 := 80873, root := ~, start := 80861, stop := 80872 ), 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 := 80932, root := ~, start := 80918, stop := 80931 ), 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 := 81420, root := ~, start := 80985, stop := 81419 ), rec( content := "\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 81437, root := ~, start := 79877, stop := 81436 ) ], count := [ 3, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 81451, root := ~, start := 79773, stop := 81450 ), 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 := 81519, root := ~, start := 81472, stop := 81518 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ opsAttr" ), content := 0, count := [ 3, 3, 24, 4 ], mathmode := "Text", name := "Attr", next := 81573, root := ~, start := 81522, stop := 81572 ), 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 := 81605, root := ~, start := 81576, stop := 81604 ), 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 := 81643, root := ~, start := 81629, stop := 81642 ), 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 := 81732, root := ~, start := 81718, stop := 81731 ), 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 := 81811, root := ~, start := 81807, stop := 81810 ), 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 := 81834, root := ~, start := 81820, stop := 81833 ), 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 := 81967, root := ~, start := 81953, stop := 81966 ), 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 := 82421, root := ~, start := 81986, stop := 82420 ), rec( content := "\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Description", next := 82438, root := ~, start := 81608, stop := 82437 ) ], count := [ 3, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 82452, root := ~, start := 81457, stop := 82451 ), 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 := 82531, root := ~, start := 82473, stop := 82530 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdgesAttr" ), content := 0, count := [ 3, 3, 25, 4 ], mathmode := "Text", name := "Attr", next := 82596, root := ~, start := 82534, stop := 82595 ), 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 := 82628, root := ~, start := 82599, stop := 82627 ), 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 := 82666, root := ~, start := 82652, stop := 82665 ), 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 := 82755, root := ~, start := 82741, stop := 82754 ), 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 := 82846, root := ~, start := 82832, stop := 82845 ), 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 := 82893, root := ~, start := 82889, stop := 82892 ), 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 := 82916, root := ~, start := 82902, stop := 82915 ), 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 := 83044, root := ~, start := 83030, stop := 83043 ), 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 := 83597, root := ~, start := 83075, stop := 83596 ), rec( content := "\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Description", next := 83614, root := ~, start := 82631, stop := 83613 ) ], count := [ 3, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 83628, root := ~, start := 82458, stop := 83627 ), 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 := 83750, root := ~, start := 83649, stop := 83749 ), 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 := 83844, root := ~, start := 83753, stop := 83843 ), 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 := 83949, root := ~, start := 83847, stop := 83948 ), 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 := 83981, root := ~, start := 83952, stop := 83980 ), 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 := 84019, root := ~, start := 84005, stop := 84018 ), 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 := 84073, root := ~, start := 84061, stop := 84072 ), 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 := 84126, root := ~, start := 84112, stop := 84125 ), 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 := 84261, root := ~, start := 84235, stop := 84260 ), 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 := 84315, root := ~, start := 84301, stop := 84314 ), 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 := 84388, root := ~, start := 84376, stop := 84387 ), 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 := 84460, root := ~, start := 84435, stop := 84459 ), 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 := 84655, root := ~, start := 84651, stop := 84654 ), 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 := 84697, root := ~, start := 84683, stop := 84696 ), 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 := 84770, root := ~, start := 84766, stop := 84769 ), 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 := 84793, root := ~, start := 84779, stop := 84792 ), 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 := 84835, root := ~, start := 84805, stop := 84834 ), 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 := 84886, root := ~, start := 84872, stop := 84885 ), 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 := 84905, root := ~, start := 84891, stop := 84904 ), 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 := 84955, root := ~, start := 84917, stop := 84954 ), 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 := 84992, root := ~, start := 84978, stop := 84991 ), 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 := 85047, root := ~, start := 85043, stop := 85046 ), 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 := 85630, root := ~, start := 85052, stop := 85629 ), rec( content := "\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Description", next := 85647, root := ~, start := 83984, stop := 85646 ) ], count := [ 3, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 85661, root := ~, start := 83634, stop := 85660 ), 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 := 85783, root := ~, start := 85682, stop := 85782 ), 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 := 85863, root := ~, start := 85786, stop := 85862 ), 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 := 85895, root := ~, start := 85866, stop := 85894 ), 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 := 85947, root := ~, start := 85938, stop := 85946 ), 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 := 85958, root := ~, start := 85949, stop := 85957 ), 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 := 86015, root := ~, start := 85988, stop := 86014 ), 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 := 86085, root := ~, start := 86074, stop := 86084 ), 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 := 86155, root := ~, start := 86128, stop := 86154 ), 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 := 86231, root := ~, start := 86227, stop := 86230 ), 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 := 86399, root := ~, start := 86395, stop := 86398 ), 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 := 86462, root := ~, start := 86453, stop := 86461 ), 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 := 86473, root := ~, start := 86464, stop := 86472 ), 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 := 86489, root := ~, start := 86479, stop := 86488 ), 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 := 86549, root := ~, start := 86524, stop := 86548 ), 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 := 86558, root := ~, start := 86549, stop := 86557 ), 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 := 86566, root := ~, start := 86558, stop := 86565 ), 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 := 86575, root := ~, start := 86567, stop := 86574 ), 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 := 86605, root := ~, start := 86580, stop := 86604 ), 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 := 86614, root := ~, start := 86605, stop := 86613 ), 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 := 86622, root := ~, start := 86614, stop := 86621 ), 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 := 86631, root := ~, start := 86623, stop := 86630 ), 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 := 86642, root := ~, start := 86632, stop := 86641 ), 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 := 86679, root := ~, start := 86670, stop := 86678 ), 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 := 86739, root := ~, start := 86731, stop := 86738 ), 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 := 86759, root := ~, start := 86751, stop := 86758 ), 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 := 86769, root := ~, start := 86759, stop := 86768 ), 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 := 86878, root := ~, start := 86870, stop := 86877 ), 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 := 86888, root := ~, start := 86878, stop := 86887 ), 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 := 86927, root := ~, start := 86919, stop := 86926 ), 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 := 86937, root := ~, start := 86927, stop := 86936 ), 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 := 87017, root := ~, start := 86992, stop := 87016 ), 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 := 87026, root := ~, start := 87017, stop := 87025 ), 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 := 87034, root := ~, start := 87026, stop := 87033 ), 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 := 87043, root := ~, start := 87035, stop := 87042 ), 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 := 87054, root := ~, start := 87044, stop := 87053 ), 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 := 87063, root := ~, start := 87055, stop := 87062 ), 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 := 87093, root := ~, start := 87068, stop := 87092 ), 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 := 87101, root := ~, start := 87093, stop := 87100 ), 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 := 87114, root := ~, start := 87101, stop := 87113 ), 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 := 87125, root := ~, start := 87121, stop := 87124 ), 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 := 87189, root := ~, start := 87156, stop := 87188 ), 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 := 87207, root := ~, start := 87203, stop := 87206 ), 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 := 87243, root := ~, start := 87234, stop := 87242 ), 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 := 87259, root := ~, start := 87245, stop := 87258 ), 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 := 87306, root := ~, start := 87272, stop := 87305 ), 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 := 87322, root := ~, start := 87313, stop := 87321 ), 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 := 87338, root := ~, start := 87324, stop := 87337 ), 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 := 87429, root := ~, start := 87415, stop := 87428 ), 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 := 87477, root := ~, start := 87445, stop := 87476 ), 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 := 88019, root := ~, start := 87566, stop := 88018 ), rec( content := "\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Description", next := 88036, root := ~, start := 85898, stop := 88035 ) ], count := [ 3, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 88050, root := ~, start := 85667, stop := 88049 ), 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 := 88166, root := ~, start := 88071, stop := 88165 ), 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 := 88242, root := ~, start := 88169, stop := 88241 ), 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 := 88274, root := ~, start := 88245, stop := 88273 ), 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 := 88326, root := ~, start := 88317, stop := 88325 ), 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 := 88337, root := ~, start := 88328, stop := 88336 ), 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 := 88390, root := ~, start := 88367, stop := 88389 ), 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 := 88456, root := ~, start := 88445, stop := 88455 ), 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 := 88522, root := ~, start := 88499, stop := 88521 ), 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 := 88594, root := ~, start := 88590, stop := 88593 ), 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 := 88684, root := ~, start := 88672, stop := 88683 ), 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 := 88775, root := ~, start := 88755, stop := 88774 ), 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 := 88871, root := ~, start := 88857, stop := 88870 ), 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 := 88993, root := ~, start := 88983, stop := 88992 ), 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 := 89038, root := ~, start := 89034, stop := 89037 ), 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 := 89102, root := ~, start := 89069, stop := 89101 ), 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 := 89125, root := ~, start := 89121, stop := 89124 ), 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 := 89161, root := ~, start := 89152, stop := 89160 ), 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 := 89177, root := ~, start := 89163, stop := 89176 ), 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 := 89224, root := ~, start := 89190, stop := 89223 ), 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 := 89240, root := ~, start := 89231, stop := 89239 ), 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 := 89256, root := ~, start := 89242, stop := 89255 ), 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 := 89347, root := ~, start := 89333, stop := 89346 ), 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 := 89395, root := ~, start := 89363, stop := 89394 ), 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 := 90068, root := ~, start := 89484, stop := 90067 ), rec( content := "\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Description", next := 90085, root := ~, start := 88277, stop := 90084 ) ], count := [ 3, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 90099, root := ~, start := 88056, stop := 90098 ), 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 := 90210, root := ~, start := 90120, stop := 90209 ), 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 := 90281, root := ~, start := 90213, stop := 90280 ), 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 := 90313, root := ~, start := 90284, stop := 90312 ), 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 := 90365, root := ~, start := 90356, stop := 90364 ), 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 := 90376, root := ~, start := 90367, stop := 90375 ), 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 := 90424, root := ~, start := 90406, stop := 90423 ), 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 := 90484, root := ~, start := 90473, stop := 90483 ), 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 := 90545, root := ~, start := 90527, stop := 90544 ), 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 := 90610, root := ~, start := 90606, stop := 90609 ), 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 := 90662, root := ~, start := 90653, stop := 90661 ), 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 := 90673, root := ~, start := 90664, stop := 90672 ), 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 := 90685, root := ~, start := 90675, stop := 90684 ), 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 := 90791, root := ~, start := 90720, stop := 90790 ), 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 := 90857, root := ~, start := 90842, stop := 90856 ), 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 := 90937, root := ~, start := 90929, stop := 90936 ), 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 := 90947, root := ~, start := 90937, stop := 90946 ), 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 := 90964, root := ~, start := 90956, stop := 90963 ), 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 := 90974, root := ~, start := 90964, stop := 90973 ), 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 := 91050, root := ~, start := 91046, stop := 91049 ), 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 := 91140, root := ~, start := 91136, stop := 91139 ), 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 := 91204, root := ~, start := 91171, stop := 91203 ), 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 := 91227, root := ~, start := 91223, stop := 91226 ), 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 := 91263, root := ~, start := 91254, stop := 91262 ), 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 := 91279, root := ~, start := 91265, stop := 91278 ), 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 := 91326, root := ~, start := 91292, stop := 91325 ), 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 := 91342, root := ~, start := 91333, stop := 91341 ), 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 := 91358, root := ~, start := 91344, stop := 91357 ), 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 := 91449, root := ~, start := 91435, stop := 91448 ), 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 := 91497, root := ~, start := 91465, stop := 91496 ), 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 := 91882, root := ~, start := 91586, stop := 91881 ), rec( content := "\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Description", next := 91899, root := ~, start := 90316, stop := 91898 ) ], count := [ 3, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 91913, root := ~, start := 90105, stop := 91912 ), 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 := 92038, root := ~, start := 91934, stop := 92037 ), 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 := 92125, root := ~, start := 92041, stop := 92124 ), 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 := 92157, root := ~, start := 92128, stop := 92156 ), 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 := 92210, root := ~, start := 92200, stop := 92209 ), 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 := 92222, root := ~, start := 92212, stop := 92221 ), 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 := 92282, root := ~, start := 92252, stop := 92281 ), 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 := 92352, root := ~, start := 92348, stop := 92351 ), 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 := 92392, root := ~, start := 92381, stop := 92391 ), 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 := 92465, root := ~, start := 92435, stop := 92464 ), 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 := 92572, root := ~, start := 92568, stop := 92571 ), 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 := 92641, root := ~, start := 92633, stop := 92640 ), 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 := 92651, root := ~, start := 92643, stop := 92650 ), 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 := 92745, root := ~, start := 92685, stop := 92744 ), 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 := 92792, root := ~, start := 92778, stop := 92791 ), 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 := 92814, root := ~, start := 92800, stop := 92813 ), 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 := 92833, root := ~, start := 92819, stop := 92832 ), 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 := 92869, root := ~, start := 92860, stop := 92868 ), 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 := 92886, root := ~, start := 92877, stop := 92885 ), 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 := 92898, root := ~, start := 92890, stop := 92897 ), 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 := 92917, root := ~, start := 92902, stop := 92916 ), 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 := 92956, root := ~, start := 92948, stop := 92955 ), 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 := 92968, root := ~, start := 92960, stop := 92967 ), 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 := 92980, root := ~, start := 92972, stop := 92979 ), 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 := 92990, root := ~, start := 92986, stop := 92989 ), 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 := 93042, root := ~, start := 93034, stop := 93041 ), 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 := 93089, root := ~, start := 93059, stop := 93088 ), 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 := 93165, root := ~, start := 93109, stop := 93164 ), 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 := 93393, root := ~, start := 93345, stop := 93392 ), 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 := 93445, root := ~, start := 93412, stop := 93444 ), 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 := 93493, root := ~, start := 93480, stop := 93492 ), 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 := 93517, root := ~, start := 93509, stop := 93516 ), 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 := 93551, root := ~, start := 93543, stop := 93550 ), 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 := 93601, root := ~, start := 93593, stop := 93600 ), 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 := 93615, root := ~, start := 93606, stop := 93614 ), 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 := 93715, root := ~, start := 93625, stop := 93714 ), 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 := 93725, root := ~, start := 93721, stop := 93724 ), 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 := 93837, root := ~, start := 93827, stop := 93836 ), 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 := 93849, root := ~, start := 93839, stop := 93848 ), 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 := 93861, root := ~, start := 93851, stop := 93860 ), 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 := 93912, root := ~, start := 93908, stop := 93911 ), 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 := 94346, root := ~, start := 93914, stop := 94345 ), rec( content := "\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 94363, root := ~, start := 92160, stop := 94362 ) ], count := [ 3, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 94377, root := ~, start := 91919, stop := 94376 ), 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 := 94499, root := ~, start := 94398, stop := 94498 ), 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 := 94583, root := ~, start := 94502, stop := 94582 ), 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 := 94615, root := ~, start := 94586, stop := 94614 ), 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 := 94668, root := ~, start := 94658, stop := 94667 ), 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 := 94680, root := ~, start := 94670, stop := 94679 ), 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 := 94737, root := ~, start := 94710, stop := 94736 ), 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 := 94804, root := ~, start := 94800, stop := 94803 ), 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 := 94844, root := ~, start := 94833, stop := 94843 ), 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 := 94914, root := ~, start := 94887, stop := 94913 ), 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 := 95017, root := ~, start := 95013, stop := 95016 ), 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 := 95082, root := ~, start := 95074, stop := 95081 ), 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 := 95092, root := ~, start := 95084, stop := 95091 ), 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 := 95186, root := ~, start := 95126, stop := 95185 ), 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 := 95233, root := ~, start := 95219, stop := 95232 ), 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 := 95255, root := ~, start := 95241, stop := 95254 ), 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 := 95290, root := ~, start := 95282, stop := 95289 ), 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 := 95302, root := ~, start := 95294, stop := 95301 ), 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 := 95318, root := ~, start := 95310, stop := 95317 ), 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 := 95345, root := ~, start := 95336, stop := 95344 ), 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 := 95358, root := ~, start := 95349, stop := 95357 ), 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 := 95370, root := ~, start := 95362, stop := 95369 ), 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 := 95381, root := ~, start := 95377, stop := 95380 ), 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 := 95434, root := ~, start := 95426, stop := 95433 ), 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 := 95478, root := ~, start := 95451, stop := 95477 ), 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 := 95554, root := ~, start := 95498, stop := 95553 ), 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 := 95774, root := ~, start := 95729, stop := 95773 ), 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 := 95825, root := ~, start := 95792, stop := 95824 ), 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 := 95873, root := ~, start := 95860, stop := 95872 ), 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 := 95897, root := ~, start := 95889, stop := 95896 ), 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 := 95931, root := ~, start := 95923, stop := 95930 ), 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 := 95981, root := ~, start := 95973, stop := 95980 ), 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 := 95995, root := ~, start := 95986, stop := 95994 ), 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 := 96093, root := ~, start := 96005, stop := 96092 ), 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 := 96103, root := ~, start := 96099, stop := 96102 ), 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 := 96209, root := ~, start := 96199, stop := 96208 ), 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 := 96221, root := ~, start := 96211, stop := 96220 ), 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 := 96233, root := ~, start := 96223, stop := 96232 ), 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 := 96284, root := ~, start := 96280, stop := 96283 ), 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 := 96681, root := ~, start := 96286, stop := 96680 ), rec( content := "\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 96698, root := ~, start := 94618, stop := 96697 ) ], count := [ 3, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 96712, root := ~, start := 94383, stop := 96711 ), 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 := 96796, root := ~, start := 96733, stop := 96795 ), 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 := 96844, root := ~, start := 96799, stop := 96843 ), 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 := 96882, root := ~, start := 96868, stop := 96881 ), 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 := 96978, root := ~, start := 96920, stop := 96977 ), 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 := 97030, root := ~, start := 96989, stop := 97029 ), 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 := 97057, root := ~, start := 97046, stop := 97056 ), 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 := 97086, root := ~, start := 97072, stop := 97085 ), 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 := 97122, root := ~, start := 97114, stop := 97121 ), 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 := 97165, root := ~, start := 97161, stop := 97164 ), 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 := 97231, root := ~, start := 97217, stop := 97230 ), 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 := 97260, root := ~, start := 97240, stop := 97259 ), 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 := 97338, root := ~, start := 97324, stop := 97337 ), 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 := 97369, root := ~, start := 97355, stop := 97368 ), 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 := 97403, root := ~, start := 97392, stop := 97402 ), 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 := 97413, root := ~, start := 97409, stop := 97412 ), 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 := 97530, root := ~, start := 97500, stop := 97529 ), 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 := 97602, root := ~, start := 97598, stop := 97601 ), 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 := 97705, root := ~, start := 97637, stop := 97704 ), 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 := 98148, root := ~, start := 97712, stop := 98147 ), rec( content := "\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Description", next := 98165, root := ~, start := 96847, stop := 98164 ) ], count := [ 3, 3, 32, 1 ], mathmode := "Text", name := "ManSection", next := 98179, root := ~, start := 96718, stop := 98178 ), 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 := 98260, root := ~, start := 98200, stop := 98259 ), 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 := 98308, root := ~, start := 98263, stop := 98307 ), 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 := 98346, root := ~, start := 98332, stop := 98345 ), 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 := 98436, root := ~, start := 98381, stop := 98435 ), 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 := 98485, root := ~, start := 98447, stop := 98484 ), 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 := 98512, root := ~, start := 98501, stop := 98511 ), 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 := 98541, root := ~, start := 98527, stop := 98540 ), 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 := 98577, root := ~, start := 98569, stop := 98576 ), 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 := 98620, root := ~, start := 98616, stop := 98619 ), 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 := 98686, root := ~, start := 98672, stop := 98685 ), 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 := 98715, root := ~, start := 98695, stop := 98714 ), 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 := 98780, root := ~, start := 98766, stop := 98779 ), 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 := 98801, root := ~, start := 98787, stop := 98800 ), 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 := 98830, root := ~, start := 98819, stop := 98829 ), 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 := 98840, root := ~, start := 98836, stop := 98839 ), 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 := 98953, root := ~, start := 98926, stop := 98952 ), 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 := 99025, root := ~, start := 99021, stop := 99024 ), 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 := 99126, root := ~, start := 99061, stop := 99125 ), 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 := 99578, root := ~, start := 99133, stop := 99577 ), rec( content := "\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Description", next := 99595, root := ~, start := 98311, stop := 99594 ) ], count := [ 3, 3, 33, 1 ], mathmode := "Text", name := "ManSection", next := 99609, root := ~, start := 98185, stop := 99608 ), 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 := 99670, root := ~, start := 99630, stop := 99669 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeDigraph" ), content := 0, count := [ 3, 3, 34, 4 ], mathmode := "Text", name := "Oper", next := 99713, root := ~, start := 99673, stop := 99712 ), 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 := 99745, root := ~, start := 99716, stop := 99744 ), 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 := 99796, root := ~, start := 99782, stop := 99795 ), 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 := 99901, root := ~, start := 99887, stop := 99900 ), 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 := 99950, root := ~, start := 99942, stop := 99949 ), 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 := 99971, root := ~, start := 99963, stop := 99970 ), 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 := 100047, root := ~, start := 100039, stop := 100046 ), 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 := 100108, root := ~, start := 100100, stop := 100107 ), 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 := 100113, root := ~, start := 100109, stop := 100112 ), 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 := 100220, root := ~, start := 100206, stop := 100219 ), 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 := 100439, root := ~, start := 100244, stop := 100438 ), rec( content := "\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Description", next := 100456, root := ~, start := 99748, stop := 100455 ) ], count := [ 3, 3, 34, 1 ], mathmode := "Text", name := "ManSection", next := 100470, root := ~, start := 99615, stop := 100469 ), 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 := 100541, root := ~, start := 100491, stop := 100540 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 4 ], mathmode := "Text", name := "Oper", next := 100594, root := ~, start := 100544, stop := 100593 ), 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 := 100626, root := ~, start := 100597, stop := 100625 ), 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 := 100687, root := ~, start := 100673, stop := 100686 ), 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 := 100806, root := ~, start := 100792, stop := 100805 ), 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 := 100962, root := ~, start := 100958, stop := 100961 ), 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 := 101069, root := ~, start := 101055, stop := 101068 ), 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 := 101333, root := ~, start := 101093, stop := 101332 ), rec( content := "\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Description", next := 101350, root := ~, start := 100629, stop := 101349 ) ], count := [ 3, 3, 35, 1 ], mathmode := "Text", name := "ManSection", next := 101364, root := ~, start := 100476, stop := 101363 ), 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 := 101427, root := ~, start := 101385, stop := 101426 ), 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 := 101459, root := ~, start := 101430, stop := 101458 ), 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 := 101498, root := ~, start := 101484, stop := 101497 ), 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 := 101536, root := ~, start := 101528, stop := 101535 ), 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 := 101600, root := ~, start := 101586, stop := 101599 ), 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 := 101731, root := ~, start := 101714, stop := 101730 ), 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 := 101753, root := ~, start := 101736, stop := 101752 ), 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 := 101782, root := ~, start := 101769, stop := 101781 ), 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 := 101815, root := ~, start := 101801, stop := 101814 ), 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 := 101879, root := ~, start := 101875, stop := 101878 ), 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 := 101902, root := ~, start := 101888, stop := 101901 ), 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 := 101934, root := ~, start := 101920, stop := 101933 ), 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 := 101978, root := ~, start := 101964, stop := 101977 ), 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 := 102264, root := ~, start := 102074, stop := 102263 ), rec( content := "\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Description", next := 102281, root := ~, start := 101462, stop := 102280 ) ], count := [ 3, 3, 36, 1 ], mathmode := "Text", name := "ManSection", next := 102295, root := ~, start := 101370, stop := 102294 ), 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 := 102367, root := ~, start := 102316, stop := 102366 ), 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 := 102399, root := ~, start := 102370, stop := 102398 ), 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 := 102438, root := ~, start := 102424, stop := 102437 ), 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 := 102476, root := ~, start := 102468, stop := 102475 ), 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 := 102550, root := ~, start := 102536, stop := 102549 ), 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 := 102682, root := ~, start := 102665, stop := 102681 ), 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 := 102704, root := ~, start := 102687, stop := 102703 ), 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 := 102739, root := ~, start := 102720, stop := 102738 ), 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 := 102768, root := ~, start := 102754, stop := 102767 ), 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 := 102878, root := ~, start := 102874, stop := 102877 ), 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 := 102901, root := ~, start := 102887, stop := 102900 ), 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 := 102933, root := ~, start := 102919, stop := 102932 ), 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 := 102977, root := ~, start := 102963, stop := 102976 ), 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 := 103271, root := ~, start := 103073, stop := 103270 ), rec( content := "\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Description", next := 103288, root := ~, start := 102402, stop := 103287 ) ], count := [ 3, 3, 37, 1 ], mathmode := "Text", name := "ManSection", next := 103302, root := ~, start := 102301, stop := 103301 ), 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 := 103370, root := ~, start := 103323, stop := 103369 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ opsAttr" ), content := 0, count := [ 3, 3, 38, 4 ], mathmode := "Text", name := "Attr", next := 103424, root := ~, start := 103373, stop := 103423 ), 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 := 103456, root := ~, start := 103427, stop := 103455 ), 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 := 103505, root := ~, start := 103491, stop := 103504 ), 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 := 103578, root := ~, start := 103564, stop := 103577 ), 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 := 103670, root := ~, start := 103656, stop := 103669 ), 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 := 103675, root := ~, start := 103671, stop := 103674 ), 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 := 103698, root := ~, start := 103684, stop := 103697 ), 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 := 103856, root := ~, start := 103842, stop := 103855 ), 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 := 104638, root := ~, start := 103871, stop := 104637 ), rec( content := "\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Description", next := 104655, root := ~, start := 103459, stop := 104654 ) ], count := [ 3, 3, 38, 1 ], mathmode := "Text", name := "ManSection", next := 104669, root := ~, start := 103308, stop := 104668 ), 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 := 104773, root := ~, start := 104690, stop := 104772 ), 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 := 104863, root := ~, start := 104776, stop := 104862 ), 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 := 104895, root := ~, start := 104866, stop := 104894 ), 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 := 105113, root := ~, start := 105099, stop := 105112 ), 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 := 105210, root := ~, start := 105196, stop := 105209 ), 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 := 105226, root := ~, start := 105218, stop := 105225 ), 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 := 105260, root := ~, start := 105249, stop := 105259 ), 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 := 105347, root := ~, start := 105267, stop := 105346 ), 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 := 105353, root := ~, start := 105349, stop := 105352 ), 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 := 105376, root := ~, start := 105362, stop := 105375 ), 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 := 105408, root := ~, start := 105394, stop := 105407 ), 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 := 105452, root := ~, start := 105438, stop := 105451 ), 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> D\ istanceDigraph(digraph, 1);\n\ng\ ap> DistanceDigraph(digraph, [1, 2]);\n\n", count := [ 3, 3, 39, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 7 ], mathmode := "Text", name := "Example", next := 105892, root := ~, start := 105548, stop := 105891 ), rec( content := "\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Description", next := 105909, root := ~, start := 104898, stop := 105908 ) ], count := [ 3, 3, 39, 1 ], mathmode := "Text", name := "ManSection", next := 105923, root := ~, start := 104675, stop := 105922 ), 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 := 105990, root := ~, start := 105944, stop := 105989 ), 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 := 106022, root := ~, start := 105993, stop := 106021 ), 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 := 106115, root := ~, start := 106101, stop := 106114 ), 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 := 106136, root := ~, start := 106128, stop := 10613\ 5 ), 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 := 1061\ 62, root := ~, start := 106148, stop := 106161 ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "E", next := 106166, root := ~, start := 106125, stop := 106165 ), 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 := 106243, root := ~, start := 106235, stop := 106242 ), 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 := 106304, root := ~, start := 106290, stop := 106303 ), 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 := 106350, root := ~, start := 106336, stop := 106349 ), 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 := 106461, root := ~, start := 106453, stop := 106460 ), 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 := 106483, root := ~, start := 106475, stop := 106482 ), 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 := 106525, root := ~, start := 106489, stop := 106524 ), 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 := 106556, root := ~, start := 106527, stop := 106555 ), 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 := 106590, root := ~, start := 106558, stop := 106589 ), 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 := 106627, root := ~, start := 106596, stop := 106626 ), 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 := 106633, root := ~, start := 106629, stop := 106632 ), 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 := 106674, root := ~, start := 106653, stop := 106673 ), 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 := 106695, root := ~, start := 106687, stop := 106694 ), 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 := 106725, root := ~, start := 106711, stop := 106724 ), 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 := 107057, root := ~, start := 106731, stop := 107056 ), rec( content := "\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Description", next := 107074, root := ~, start := 106025, stop := 107073 ) ], count := [ 3, 3, 40, 1 ], mathmode := "Text", name := "ManSection", next := 107088, root := ~, start := 105929, stop := 107087 ), 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 := 107156, root := ~, start := 107109, stop := 107155 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ianAttr" ), content := 0, count := [ 3, 3, 41, 4 ], mathmode := "Text", name := "Attr", next := 107210, root := ~, start := 107159, stop := 107209 ), 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 := 107242, root := ~, start := 107213, stop := 107241 ), 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 := 107280, root := ~, start := 107266, stop := 107279 ), 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 := 107339, root := ~, start := 107314, stop := 107338 ), 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 := 107381, root := ~, start := 107367, stop := 107380 ), 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 := 107386, root := ~, start := 107382, stop := 107385 ), 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 := 107612, root := ~, start := 107560, stop := 107611 ), 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 := 107617, root := ~, start := 107613, stop := 107616 ), 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 := 107640, root := ~, start := 107626, stop := 107639 ), 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 := 107746, root := ~, start := 107732, stop := 107745 ), 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 := 107796, root := ~, start := 107792, stop := 107795 ), 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 := 108366, root := ~, start := 107802, stop := 108365 ), rec( content := "\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Description", next := 108383, root := ~, start := 107245, stop := 108382 ) ], count := [ 3, 3, 41, 1 ], mathmode := "Text", name := "ManSection", next := 108397, root := ~, start := 107094, stop := 108396 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 108411, root := ~, start := 42491, stop := 108410 ), 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 := 108458, root := ~, start := 108424, stop := 108457 ), 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 := 108527, root := ~, start := 108478, stop := 108526 ), 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 := 108559, root := ~, start := 108530, stop := 108558 ), 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 := 108622, root := ~, start := 108611, stop := 108621 ), 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 := 108768, root := ~, start := 108757, stop := 108767 ), 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 := 108802, root := ~, start := 108772, stop := 108801 ), 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 := 108870, root := ~, start := 108859, stop := 108869 ), 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 := 108910, root := ~, start := 108874, stop := 108909 ), 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 := 108995, root := ~, start := 108984, stop := 108994 ), 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 := 109055, root := ~, start := 109017, stop := 109054 ), 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 := 109079, root := ~, start := 109075, stop := 109078 ), 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 := 109096, root := ~, start := 109088, stop := 109095 ), 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 := 109181, root := ~, start := 109173, stop := 109180 ), 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 := 109264, root := ~, start := 109260, stop := 109263 ), 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 := 109310, root := ~, start := 109302, stop := 109309 ), 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 := 109351, root := ~, start := 109337, stop := 109350 ), 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 := 109362, root := ~, start := 109352, stop := 109361 ), 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 := 109377, root := ~, start := 109363, stop := 109376 ), 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 := 109476, root := ~, start := 109468, stop := 109475 ), 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 := 109493, root := ~, start := 109485, stop := 109492 ), 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 := 109891, root := ~, start := 109597, stop := 109890 ), rec( content := "\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 109908, root := ~, start := 108562, stop := 109907 ) ], count := [ 3, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 109922, root := ~, start := 108463, stop := 109921 ), 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 := 109989, root := ~, start := 109943, stop := 109988 ), 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 := 110021, root := ~, start := 109992, stop := 110020 ), 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 := 110053, root := ~, start := 110045, stop := 110052 ), 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 := 110138, root := ~, start := 110130, stop := 110137 ), 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 := 110193, root := ~, start := 110185, stop := 110192 ), 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 := 110252, root := ~, start := 110244, stop := 110251 ), 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 := 110275, root := ~, start := 110267, stop := 110274 ), 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 := 110406, root := ~, start := 110393, stop := 110405 ), 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 := 110426, root := ~, start := 110406, stop := 110425 ), 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 := 110434, root := ~, start := 110426, stop := 110433 ), 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 := 110444, root := ~, start := 110440, stop := 110443 ), 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 := 110748, root := ~, start := 110744, stop := 110747 ), 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 := 110957, root := ~, start := 110753, stop := 110956 ), rec( content := "\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 110974, root := ~, start := 110024, stop := 110973 ) ], count := [ 3, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 110988, root := ~, start := 109928, stop := 110987 ), 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 := 111056, root := ~, start := 111009, stop := 111055 ), 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 := 111088, root := ~, start := 111059, stop := 111087 ), 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 := 111151, root := ~, start := 111140, stop := 111150 ), 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 := 111297, root := ~, start := 111286, stop := 111296 ), 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 := 111331, root := ~, start := 111301, stop := 111330 ), 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 := 111399, root := ~, start := 111388, stop := 111398 ), 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 := 111439, root := ~, start := 111403, stop := 111438 ), 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 := 111524, root := ~, start := 111513, stop := 111523 ), 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 := 111584, root := ~, start := 111546, stop := 111583 ), 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 := 111608, root := ~, start := 111604, stop := 111607 ), 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 := 111625, root := ~, start := 111617, stop := 111624 ), 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 := 111712, root := ~, start := 111704, stop := 111711 ), 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 := 111753, root := ~, start := 111727, stop := 111752 ), 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 := 111759, root := ~, start := 111755, stop := 111758 ), 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 := 111951, root := ~, start := 111764, stop := 111950 ), rec( content := "\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Description", next := 111968, root := ~, start := 111091, stop := 111967 ) ], count := [ 3, 4, 3, 1 ], mathmode := "Text", name := "ManSection", next := 111982, root := ~, start := 110994, stop := 111981 ), 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 := 112039, root := ~, start := 112003, stop := 112038 ), 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 := 112071, root := ~, start := 112042, stop := 112070 ), 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 := 112134, root := ~, start := 112123, stop := 112133 ), 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 := 112280, root := ~, start := 112269, stop := 112279 ), 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 := 112314, root := ~, start := 112284, stop := 112313 ), 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 := 112382, root := ~, start := 112371, stop := 112381 ), 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 := 112422, root := ~, start := 112386, stop := 112421 ), 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 := 112507, root := ~, start := 112496, stop := 112506 ), 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 := 112567, root := ~, start := 112529, stop := 112566 ), 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 := 112591, root := ~, start := 112587, stop := 112590 ), 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 := 112608, root := ~, start := 112600, stop := 112607 ), 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 := 112687, root := ~, start := 112679, stop := 112686 ), 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 := 112734, root := ~, start := 112726, stop := 112733 ), 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 := 112758, root := ~, start := 112750, stop := 112757 ), 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 := 112778, root := ~, start := 112770, stop := 11277\ 7 ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 112782, root := ~, start := 112763, stop := 112781 ), 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 := 112818, root := ~, start := 112788, stop := 112817 ), 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 := 112824, root := ~, start := 112820, stop := 112823 ), 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 := 113019, root := ~, start := 112829, stop := 113018 ), rec( content := "\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Description", next := 113036, root := ~, start := 112074, stop := 113035 ) ], count := [ 3, 4, 4, 1 ], mathmode := "Text", name := "ManSection", next := 113050, root := ~, start := 111988, stop := 113049 ), rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 113064, root := ~, start := 108415, stop := 113063 ) , 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 := 113113, root := ~, start := 113077, stop := 113112 ), 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 := 113176, root := ~, start := 113133, stop := 113175 ), 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 := 113208, root := ~, start := 113179, stop := 113207 ), 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 := 113240, root := ~, start := 113232, stop := 113239 ), 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 := 113311, root := ~, start := 113303, stop := 113310 ), 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 := 113336, root := ~, start := 113328, stop := 11333\ 5 ), rec( content := " - 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 113344, root := ~, start := 113325, stop := 113343 ), 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 := 113398, root := ~, start := 113390, stop := 113397 ), 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 := 113413, root := ~, start := 113405, stop := 113412 ), 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 := 113427, root := ~, start := 113419, stop := 113426 ), 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 := 113479, root := ~, start := 113471, stop := 113478 ), 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 := 113502, root := ~, start := 113490, stop := 113501 ), 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 := 113508, root := ~, start := 113504, stop := 113507 ), 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 := 113556, root := ~, start := 113545, stop := 113555 ), 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 := 113702, root := ~, start := 113691, stop := 113701 ), 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 := 113736, root := ~, start := 113706, stop := 113735 ), 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 := 113804, root := ~, start := 113793, stop := 113803 ), 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 := 113844, root := ~, start := 113808, stop := 113843 ), 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 := 113929, root := ~, start := 113918, stop := 113928 ), 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 := 113989, root := ~, start := 113951, stop := 113988 ), 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 := 114013, root := ~, start := 114009, stop := 114012 ), 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 := 114070, root := ~, start := 114023, stop := 114069 ), 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 := 114299, root := ~, start := 114116, stop := 114298 ), rec( content := "\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 114316, root := ~, start := 113211, stop := 114315 ) ], count := [ 3, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 114330, root := ~, start := 113118, stop := 114329 ), 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 := 114397, root := ~, start := 114351, stop := 114396 ), 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 := 114429, root := ~, start := 114400, stop := 114428 ), 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 := 114461, root := ~, start := 114453, stop := 114460 ), 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 := 114549, root := ~, start := 114541, stop := 114548 ), 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 := 114595, root := ~, start := 114564, stop := 114594 ), 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 := 114601, root := ~, start := 114597, stop := 114600 ), 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 := 114649, root := ~, start := 114638, stop := 114648 ), 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 := 114795, root := ~, start := 114784, stop := 114794 ), 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 := 114829, root := ~, start := 114799, stop := 114828 ), 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 := 114897, root := ~, start := 114886, stop := 114896 ), 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 := 114937, root := ~, start := 114901, stop := 114936 ), 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 := 115022, root := ~, start := 115011, stop := 115021 ), 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 := 115082, root := ~, start := 115044, stop := 115081 ), 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 := 115300, root := ~, start := 115107, stop := 115299 ), rec( content := "\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 115317, root := ~, start := 114432, stop := 115316 ) ], count := [ 3, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 115331, root := ~, start := 114336, stop := 115330 ), 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 := 115410, root := ~, start := 115352, stop := 115409 ), 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 := 115442, root := ~, start := 115413, stop := 115441 ), 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 := 115631, root := ~, start := 115623, stop := 115630 ), 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 := 115650, root := ~, start := 115642, stop := 115649 ), 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 := 115678, root := ~, start := 115670, stop := 115677 ), 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 := 115691, root := ~, start := 115683, stop := 115690 ), 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 := 115726, root := ~, start := 115722, stop := 115725 ), 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 := 115743, root := ~, start := 115735, stop := 115742 ), 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 := 115756, root := ~, start := 115748, stop := 115755 ), 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 := 115875, root := ~, start := 115867, stop := 115874 ), 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 := 115916, root := ~, start := 115901, stop := 115915 ), 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 := 115934, root := ~, start := 115926, stop := 115933 ), 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 := 115983, root := ~, start := 115960, stop := 115982 ), 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 := 115994, root := ~, start := 115990, stop := 115993 ), 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 := 116042, root := ~, start := 116031, stop := 116041 ), 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 := 116188, root := ~, start := 116177, stop := 116187 ), 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 := 116222, root := ~, start := 116192, stop := 116221 ), 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 := 116290, root := ~, start := 116279, stop := 116289 ), 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 := 116330, root := ~, start := 116294, stop := 116329 ), 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 := 116415, root := ~, start := 116404, stop := 116414 ), 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 := 116475, root := ~, start := 116437, stop := 116474 ), 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 := 116739, root := ~, start := 116501, stop := 116738 ), rec( content := "\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 116756, root := ~, start := 115445, stop := 116755 ) ], count := [ 3, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 116770, root := ~, start := 115337, stop := 116769 ), 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 := 116860, root := ~, start := 116795, stop := 116859 ), 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 := 116894, root := ~, start := 116865, stop := 116893 ), 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 := 116943, root := ~, start := 116930, stop := 116942 ), 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 := 116955, root := ~, start := 116947, stop := 116954 ), 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 := 117034, root := ~, start := 117026, stop := 117033 ), 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 := 117092, root := ~, start := 117084, stop := 11709\ 1 ), 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 := 117107, root := ~, start := 117099, stop := 11710\ 6 ), rec( content := "[n]]", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 117115, root := ~, start := 117080, stop := 117114 ), 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 := 117231, root := ~, start := 117227, stop := 117230 ), 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 := 117281, root := ~, start := 117270, stop := 117280 ), 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 := 117431, root := ~, start := 117420, stop := 117430 ), 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 := 117465, root := ~, start := 117435, stop := 117464 ), 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 := 117535, root := ~, start := 117524, stop := 117534 ), 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 := 117577, root := ~, start := 117539, stop := 117576 ), 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 := 117664, root := ~, start := 117653, stop := 117663 ), 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 := 117726, root := ~, start := 117686, stop := 117725 ), 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 := 118007, root := ~, start := 117753, stop := 118006 ), rec( content := "\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Description", next := 118024, root := ~, start := 116899, stop := 118023 ) ], count := [ 3, 5, 4, 1 ], mathmode := "Text", name := "ManSection", next := 118038, root := ~, start := 116778, stop := 118037 ), 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 := 118102, root := ~, start := 118059, stop := 118101 ), 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 := 118134, root := ~, start := 118105, stop := 118133 ), 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 := 118166, root := ~, start := 118158, stop := 118165 ), 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 := 118226, root := ~, start := 118214, stop := 118225 ), 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 := 118252, root := ~, start := 118244, stop := 118251 ), 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 := 118274, root := ~, start := 118266, stop := 118273 ), 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 := 118324, root := ~, start := 118316, stop := 118323 ), 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 := 118339, root := ~, start := 118331, stop := 118338 ), 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 := 118353, root := ~, start := 118345, stop := 118352 ), 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 := 118405, root := ~, start := 118397, stop := 118404 ), 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 := 118428, root := ~, start := 118416, stop := 118427 ), 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 := 118484, root := ~, start := 118476, stop := 118483 ), 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 := 118503, root := ~, start := 118495, stop := 118502 ), 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 := 118509, root := ~, start := 118505, stop := 118508 ), 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 := 118557, root := ~, start := 118546, stop := 118556 ), 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 := 118703, root := ~, start := 118692, stop := 118702 ), 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 := 118737, root := ~, start := 118707, stop := 118736 ), 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 := 118805, root := ~, start := 118794, stop := 118804 ), 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 := 118845, root := ~, start := 118809, stop := 118844 ), 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 := 118930, root := ~, start := 118919, stop := 118929 ), 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 := 118988, root := ~, start := 118952, stop := 118987 ), 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 := 119261, root := ~, start := 119011, stop := 119260 ), rec( content := "\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Description", next := 119278, root := ~, start := 118137, stop := 119277 ) ], count := [ 3, 5, 5, 1 ], mathmode := "Text", name := "ManSection", next := 119292, root := ~, start := 118044, stop := 119291 ), 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 := 119356, root := ~, start := 119313, stop := 119355 ), rec( attributes := rec( Arg := "[filt, ]n", Name := "NullDigraph" ), content := 0, count := [ 3, 5, 6, 4 ], mathmode := "Text", name := "Oper", next := 119401, root := ~, start := 119359, stop := 119400 ), 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 := 119433, root := ~, start := 119404, stop := 119432 ), 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 := 119465, root := ~, start := 119457, stop := 119464 ), 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 := 119535, root := ~, start := 119523, stop := 119534 ), 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 := 119550, root := ~, start := 119539, stop := 119549 ), 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 := 119572, root := ~, start := 119564, stop := 119571 ), 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 := 119634, root := ~, start := 119630, stop := 119633 ), 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 := 119688, root := ~, start := 119677, stop := 119687 ), 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 := 119834, root := ~, start := 119823, stop := 119833 ), 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 := 119868, root := ~, start := 119838, stop := 119867 ), 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 := 119936, root := ~, start := 119925, stop := 119935 ), 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 := 119976, root := ~, start := 119940, stop := 119975 ), 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 := 120061, root := ~, start := 120050, stop := 120060 ), 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 := 120121, root := ~, start := 120083, stop := 120120 ), 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 := 120145, root := ~, start := 120141, stop := 120144 ), 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 := 120169, root := ~, start := 120151, stop := 120168 ), 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 := 120206, root := ~, start := 120187, stop := 120205 ), 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 := 120457, root := ~, start := 120213, stop := 120456 ), rec( content := "\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Description", next := 120474, root := ~, start := 119436, stop := 120473 ) ], count := [ 3, 5, 6, 1 ], mathmode := "Text", name := "ManSection", next := 120488, root := ~, start := 119298, stop := 120487 ), 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 := 120557, root := ~, start := 120509, stop := 120556 ), 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 := 120589, root := ~, start := 120560, stop := 120588 ), 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 := 120621, root := ~, start := 120613, stop := 120620 ), 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 := 120634, root := ~, start := 120626, stop := 120633 ), 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 := 120774, root := ~, start := 120750, stop := 120773 ), 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 := 120789, root := ~, start := 120775, stop := 120788 ), 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 := 120795, root := ~, start := 120791, stop := 120794 ), 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 := 120843, root := ~, start := 120832, stop := 120842 ), 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 := 120989, root := ~, start := 120978, stop := 120988 ), 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 := 121023, root := ~, start := 120993, stop := 121022 ), 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 := 121091, root := ~, start := 121080, stop := 121090 ), 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 := 121131, root := ~, start := 121095, stop := 121130 ), 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 := 121216, root := ~, start := 121205, stop := 121215 ), 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 := 121276, root := ~, start := 121238, stop := 121275 ), 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 := 121300, root := ~, start := 121296, stop := 121299 ), 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 := 121330, root := ~, start := 121310, stop := 121329 ), 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 := 121345, root := ~, start := 121331, stop := 121344 ), 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 := 121388, root := ~, start := 121380, stop := 121387 ), 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 := 121427, root := ~, start := 121419, stop := 12142\ 6 ), rec( content := "]", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "C", next := 121432, root := ~, start := 121410, stop := 121431 ), 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 := 121525, root := ~, start := 121517, stop := 12152\ 4 ), rec( content := " - 1", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 121533, root := ~, start := 121514, stop := 121532 ), 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 := 122082, root := ~, start := 121540, stop := 122081 ), rec( content := "\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Description", next := 122099, root := ~, start := 120592, stop := 122098 ) ], count := [ 3, 5, 7, 1 ], mathmode := "Text", name := "ManSection", next := 122113, root := ~, start := 120494, stop := 122112 ), 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 := 122175, root := ~, start := 122134, stop := 122174 ), 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 := 122207, root := ~, start := 122178, stop := 122206 ), 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 := 122289, root := ~, start := 122234, stop := 122288 ), 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 := 122294, root := ~, start := 122290, stop := 122293 ), 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 := 122650, root := ~, start := 122300, stop := 122649 ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "P", next := 122654, root := ~, start := 122650, stop := 122653 ), 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 := 122706, root := ~, start := 122695, stop := 122705 ), 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 := 122852, root := ~, start := 122841, stop := 122851 ), 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 := 122886, root := ~, start := 122856, stop := 122885 ), 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 := 122954, root := ~, start := 122943, stop := 122953 ), 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 := 122994, root := ~, start := 122958, stop := 122993 ), 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 := 123079, root := ~, start := 123068, stop := 123078 ), 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 := 123139, root := ~, start := 123101, stop := 123138 ), 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 := 123163, root := ~, start := 123159, stop := 123162 ), 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 := 123216, root := ~, start := 123178, stop := 123215 ), 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 := 123375, root := ~, start := 123219, stop := 123374 ), rec( content := "\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Description", next := 123392, root := ~, start := 122210, stop := 123391 ) ], count := [ 3, 5, 8, 1 ], mathmode := "Text", name := "ManSection", next := 123406, root := ~, start := 122119, stop := 123405 ), 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 := 123485, root := ~, start := 123427, stop := 123484 ), 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 := 123517, root := ~, start := 123488, stop := 123516 ), 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 := 123549, root := ~, start := 123541, stop := 123548 ), 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 := 123584, root := ~, start := 123576, stop := 123583 ), 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 := 123636, root := ~, start := 123628, stop := 12363\ 5 ), rec( content := " / 2", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "C", next := 123644, root := ~, start := 123625, stop := 123643 ), 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 := 123715, root := ~, start := 123682, stop := 123714 ), 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 := 123731, root := ~, start := 123723, stop := 12373\ 0 ), rec( content := ", k)", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "M", next := 123739, root := ~, start := 123716, stop := 123738 ), 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 := 123745, root := ~, start := 123741, stop := 123744 ), 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 := 123827, root := ~, start := 123760, stop := 123826 ), 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 := 123837, root := ~, start := 123833, stop := 123836 ), 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 := 124262, root := ~, start := 123843, stop := 124261 ), 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 := 124271, root := ~, start := 124267, stop := 124270 ), 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 := 124319, root := ~, start := 124308, stop := 124318 ), 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 := 124465, root := ~, start := 124454, stop := 124464 ), 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 := 124499, root := ~, start := 124469, stop := 124498 ), 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 := 124567, root := ~, start := 124556, stop := 124566 ), 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 := 124607, root := ~, start := 124571, stop := 124606 ), 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 := 124692, root := ~, start := 124681, stop := 124691 ), 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 := 124752, root := ~, start := 124714, stop := 124751 ), 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 := 124776, root := ~, start := 124772, stop := 124775 ), 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 := 124818, root := ~, start := 124791, stop := 124817 ), 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 := 125293, root := ~, start := 124821, stop := 125292 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Description", next := 125310, root := ~, start := 123520, stop := 125309 ) ], count := [ 3, 5, 9, 1 ], mathmode := "Text", name := "ManSection", next := 125324, root := ~, start := 123412, stop := 125323 ), rec( content := "\n\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 125338, root := ~, start := 113068, stop := 125337 ) , rec( content := "\n\n", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 125350, root := ~, start := 18066, stop := 125349 ), 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 := 125427, root := ~, start := 125399, stop := 125426 ) , 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 := 125481, root := ~, start := 125440, stop := 125480 ), 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 := 125522, root := ~, start := 125507, stop := 12552\ 1 ), 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 := 1255\ 40, root := ~, start := 125525, stop := 125539 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "C", next := 125544, root := ~, start := 125504, stop := 125543 ) ], count := [ 4, 1, 0, 4 ], name := "Mark", next := 125551, root := ~, start := 125498, stop := 125550 ), 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 := 125594, root := ~, start := 125586, stop := 12559\ 3 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 5 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "Index", next := 125617, root := ~, start := 125571, stop := 125616 ), 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 := 125644, root := ~, start := 125633, stop := 125643 ), 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 := 125663, root := ~, start := 125648, stop := 125662 ), 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 := 125683, root := ~, start := 125668, stop := 125682 ), 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 := 125750, root := ~, start := 125735, stop := 12574\ 9 ), 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 := 1257\ 91, root := ~, start := 125776, stop := 125790 ), rec( content := ")", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "C", next := 125796, root := ~, start := 125719, stop := 125795 ), 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 := 125851, root := ~, start := 125847, stop := 125850 ), 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 := 125944, root := ~, start := 125929, stop := 125943 ), 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 := 125964, root := ~, start := 125949, stop := 125963 ), rec( content := ".\n \ ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], name := "Item", next := 125978, root := ~, start := 125557, stop := 125977 ), 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 := 126009, root := ~, start := 125994, stop := 12600\ 8 ), 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 := 1260\ 30, root := ~, start := 126015, stop := 126029 ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "C", next := 126034, root := ~, start := 125991, stop := 126033 ) ], count := [ 4, 1, 0, 7 ], name := "Mark", next := 126041, root := ~, start := 125985, stop := 126040 ), 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 := 126090, root := ~, start := 126079, stop := 12608\ 9 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "Index", next := 126113, root := ~, start := 126061, stop := 126112 ), 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 := 126155, root := ~, start := 126144, stop := 126154 ), 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 := 126250, root := ~, start := 126240, stop \ := 126249 ), 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 := 126281, root := ~, start := 126266, stop := 126280 ), 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 := 126327, root := ~, start := 126317, stop := 126326 ), 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\ := 126358, root := ~, start := 126343, stop := 126357 ), rec( content := ";\n\ ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 10 ], name := "Item", next := 126376, root := ~, start := 126\ 211, stop := 126375 ), 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 := \ 126420, root := ~, start := 126404, stop := 126419 ), 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 := 126447, root :=\ ~, start := 126437, stop := 126446 ), 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 := 126486, root :=\ ~, start := 126471, stop := 126485 ), 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 := 126521, root := ~, star\ t := 126511, stop := 126520 ), 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 := 126560, root := ~, start := \ 126545, stop := 126559 ), rec( content := ";\n ", count := [ 4, 1, 0, \ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 12 ], name := "Ite\ m", next := 126578, root := ~, start := 126386, stop := 126577 ), 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 := 126622, root := ~, start := 126606, stop \ := 126621 ), 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 := 126640, root := ~, start := 126624, stop := 126639 ), 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 \ := 126688, root := ~, start := 126673, stop := 126687 ), rec( content := ")", \ count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "C", next := 126693, root :=\ ~, start := 126657, stop := 126692 ), 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 := 126749, root := ~,\ start := 126734, stop := 126748 ), rec( content := ")", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, \ 14 ], mathmode := "Text", name := "C", next := 126754, root := ~, start := 126\ 718, stop := 126753 ), 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 := 126874, root := ~, start := 126588, stop := 126873 ) ], count := [ 4, 1, 0, 9 ], mathmode := "Text", name := "List", next := 126889, root := ~, start := 126195, stop := 126888 ), rec( content := "\n " , count := [ 4, 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], name := "Item", next := 126902, root := ~, start := 126047, stop := 126901 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "List", next := 126913, root := ~, start := 125486, stop := 126912 ), 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 := 126977, root := ~, start := 126933, stop := 126976 ), 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 := 127000, root := ~, start := 126989, stop := 126999 ), 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 := 127016, root := ~, start := 127004, stop := 127015 ), rec( content := ".", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 127027, root := ~, start := 126980, stop := 127026 ), 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 := 127063, root := ~, start := 127051, stop := 127062 ), 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 := 127078, root := ~, start := 127068, stop := 127077 ), 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 := 127136, root := ~, start := 127125, stop := 127135 ), 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 := 127150, root := ~, start := 127140, stop := 127149 ), 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 := 127182, root := ~, start := 127170, stop := 127181 ), 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 := 127200, root := ~, start := 127188, stop := 127199 ), 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 := 127223, root := ~, start := 127219, stop := 127222 ), 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 := 127249, root := ~, start := 127239, stop := 127248 ), 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 := 127272, root := ~, start := 127255, stop := 127271 ), 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 := 127298, root := ~, start := 127286, stop := 127297 ), 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 := 127316, root := ~, start := 127306, stop := 127315 ), 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 := 127333, root := ~, start := 127321, stop := 127332 ), 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 := 127416, root := ~, start := 127404, stop := 127415 ), 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 := 127487, root := ~, start := 127477, stop := 127486 ), 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 := 127514, root := ~, start := 127510, stop := 127513 ), 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 := 127546, root := ~, start := 127536, stop := 127545 ), 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 := 127578, root := ~, start := 127566, stop := 127577 ), 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 := 127629, root := ~, start := 127619, stop := 12762\ 8 ), 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 := 1276\ 63, root := ~, start := 127651, stop := 127662 ), rec( content := ")", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 127668, root := ~, start := 127598, stop := 127667 ), 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 := 127712, root := ~, start := 127704, stop := 127711 ), 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 := 127725, root := ~, start := 127717, stop := 127724 ), 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 := 127789, root := ~, start := 127776, stop := 127788 ), 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 := 127805, root := ~, start := 127793, stop := 127804 ), 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 := 127832, root := ~, start := 127822, stop := 127831 ), 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 := 127842, root := ~, start := 127838, stop := 127841 ), 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 := 128450, root := ~, start := 127848, stop := 128449 ), rec( content := "\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 128467, root := ~, start := 127030, stop := 128466 ) ], count := [ 4, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 128481, root := ~, start := 126918, stop := 128480 ), 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 := 128557, root := ~, start := 128501, stop := 128556 ), rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningForest" ), content := 0, count := [ 4, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 128618, root := ~, start := 128560, stop := 128617 ), 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 := 128641, root := ~, start := 128630, stop := 128640 ), 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 := 128657, root := ~, start := 128645, stop := 128656 ), rec( content := ".", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 128668, root := ~, start := 128621, stop := 128667 ), 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 := 128734, root := ~, start := 128703, stop := 128733 ), 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 := 128754, root := ~, start := 128743, stop := 128753 ), 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 := 128784, root := ~, start := 128774, stop := 128783 ), 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 := 128847, root := ~, start := 128835, stop := 128846 ), 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 := 128900, root := ~, start := 128867, stop := 128899 ), 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 := 128924, root := ~, start := 128913, stop := 128923 ), 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 := 128950, root := ~, start := 128940, stop := 128949 ), 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 := 129015, root := ~, start := 129003, stop := 129014 ), 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 := 129021, root := ~, start := 129017, stop := 129020 ), 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 := 129061, root := ~, start := 129030, stop := 129060 ), 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 := 129087, root := ~, start := 129075, stop := 129086 ), 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 := 129123, root := ~, start := 129111, stop := 129122 ), 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 := 129183, root := ~, start := 129156, stop := 129182 ), 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 := 129223, root := ~, start := 129192, stop := 129222 ), 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 := 129298, root := ~, start := 129252, stop := 129297 ), 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 := 129390, root := ~, start := 129358, stop := 129389 ), 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 := 129397, root := ~, start := 129393, stop := 129396 ), 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 := 129439, root := ~, start := 129406, stop := 129438 ), 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 := 129465, root := ~, start := 129453, stop := 129464 ), 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 := 129501, root := ~, start := 129489, stop := 129500 ), 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 := 129568, root := ~, start := 129536, stop := 129567 ), 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 := 129605, root := ~, start := 129573, stop := 129604 ), 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 := 129678, root := ~, start := 129666, stop := 129677 ), 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 := 129760, root := ~, start := 129748, stop := 129759 ), 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 := 129865, root := ~, start := 129819, stop := 129864 ), 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 := 129881, root := ~, start := 129869, stop := 129880 ), 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 := 129933, root := ~, start := 129887, stop := 129932 ), 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 := 129940, root := ~, start := 129936, stop := 129939 ), 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 := 130792, root := ~, start := 130045, stop := 130791 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 130809, root := ~, start := 128671, stop := 130808 ) ], count := [ 4, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 130823, root := ~, start := 128486, stop := 130822 ), rec( content := "\n\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 130837, root := ~, start := 125431, stop := 130836 ) , rec( content := "\n", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 130848, root := ~, start := 125372, stop := 130847 ), 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 := 130964, root := ~, start := 130920, stop := 130963 ), 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 := 131013, root := ~, start := 130976, stop := 131012 ), 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 := 131077, root := ~, start := 131033, stop := 131076 ), 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 := 131118, root := ~, start := 131080, stop := 131117 ), 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 := 131189, root := ~, start := 131175, stop := 131188 ), 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 := 131195, root := ~, start := 131191, stop := 131194 ), 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 := 131294, root := ~, start := 131286, stop := 131293 ), 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 := 131935, root := ~, start := 131339, stop := 131934 ), rec( content := "\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 131952, root := ~, start := 131121, stop := 131951 ) ], count := [ 5, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 131966, root := ~, start := 131018, stop := 131965 ), 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 := 132033, root := ~, start := 131987, stop := 132032 ), 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 := 132066, root := ~, start := 132036, stop := 132065 ), 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 := 132147, root := ~, start := 132133, stop := 132146 ), 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 := 132153, root := ~, start := 132149, stop := 132152 ), 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 := 132727, root := ~, start := 132159, stop := 132726 ), rec( content := "\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 132744, root := ~, start := 132069, stop := 132743 ) ], count := [ 5, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 132758, root := ~, start := 131972, stop := 132757 ), 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 := 132820, root := ~, start := 132779, stop := 132819 ), 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 := 132858, root := ~, start := 132823, stop := 132857 ), 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 := 132898, root := ~, start := 132879, stop := 132897 ), 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 := 132952, root := ~, start := 132938, stop := 132951 ), 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 := 133028, root := ~, start := 132999, stop := 133027 ), 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 := 133066, root := ~, start := 133045, stop := 133065 ), 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 := 133076, root := ~, start := 133072, stop := 133075 ), 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 := 133117, root := ~, start := 133097, stop := 133116 ), 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 := 133131, root := ~, start := 133117, stop := 133130 ), 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 := 133139, root := ~, start := 133131, stop := 133138 ), rec( content := " are in \ one-to-one\n corresponence 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 := 133208, root := ~, start := 133194, stop := 133207 ), 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 := 133241, root := ~, start := 133221, stop := 133240 ), 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 := 133255, root := ~, start := 133241, stop := 133254 ), 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 := 133263, root := ~, start := 133255, stop := 133262 ), 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 := 133315, root := ~, start := 133301, stop := 133314 ), 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 := 133348, root := ~, start := 133344, stop := 133347 ), 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 := 133388, root := ~, start := 133369, stop := 133387 ), 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 := 134078, root := ~, start := 133555, stop := 134077 ), rec( content := "\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 134095, root := ~, start := 132861, stop := 134094 ) ], count := [ 5, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 134109, root := ~, start := 132764, stop := 134108 ), 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 := 134173, root := ~, start := 134130, stop := 134172 ), 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 := 134206, root := ~, start := 134176, stop := 134205 ), 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 := 134298, root := ~, start := 134284, stop := 134297 ), 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 := 134745, root := ~, start := 134304, stop := 134744 ), rec( content := "\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 134762, root := ~, start := 134209, stop := 134761 ) ], count := [ 5, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 134776, root := ~, start := 134115, stop := 134775 ), rec( content := "\n\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSinks" ) , content := 0, count := [ 5, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 134838, root := ~, start := 134797, stop := 134837 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 134879, root := ~, start := 134841, stop := 134878 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a list of the sinks of the digraph\n ", 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 := 134975, root := ~, start := 134961, stop := 134974 ), rec( content := ".\n A\ sink of a digraph is a vertex with out-degree zero.\n See ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Ref", next := 135074, root := ~, start := 135043, stop := 135073 ), rec( content := ".\n " , count := [ 5, 1, 5, 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, 5, 4 ], name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 5, 4 ], mathmode := "Text", name := "Example", next := 135372, root := ~, start := 135080, stop := 135371 ), rec( content := "\n ", count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 135389, root := ~, start := 134882, stop := 135388 ) ], count := [ 5, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 135403, root := ~, start := 134782, stop := 135402 ), rec( content := "\n\n ", count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSources"\ ), content := 0, count := [ 5, 1, 6, 2 ], mathmode := "Text", name := "Attr", next := 135467, root := ~, start := 135424, stop := 135466 ), 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 := 135508, root := ~, start := 135470, stop := 135507 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns an immutable list of the sources 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 := 135617, root := ~, start := 135603, stop := 135616 ), rec( content := ".\n A\ source of a digraph is a vertex with in-degree zero.\n See ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Ref", next := 135716, root := ~, start := 135686, stop := 135715 ), 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> DigraphSources(gr);\n[ 1, 4 ]\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSources(D);\n[ ]\n", count := [ 5, 1, 6, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 6, 4 ], mathmode := "Text", name := "Example", next := 136018, root := ~, start := 135722, stop := 136017 ), rec( content := "\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 136035, root := ~, start := 135511, stop := 136034 ) ], count := [ 5, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 136049, root := ~, start := 135409, stop := 136048 ), rec( content := "\n\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTopologi\ calSort" ), content := 0, count := [ 5, 1, 7, 2 ], mathmode := "Text", name := "Attr", next := 136121, root := ~, start := 136070, stop := 136120 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers, or ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "K", next := 136176, root := ~, start := 136165, stop := 136175 ), rec( content := ".", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Returns", next := 136187, root := ~, start := 136124, stop := 136186 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", 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 := 136225, root := ~, start := 136211, stop := 136224 ), rec( content := " is a di\ graph whose only directed cycles are loops, then\n ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphTopologicalSort", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "C", next := 136315, root := ~, start := 136286, stop := 136314 ), rec( content := " returns\ the vertices of ", 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 := 136354, root := ~, start := 136340, stop := 136353 ), 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, 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 := 136475, root := ~, start := 136461, stop := 136474 ), rec( content := " contain\ s directed cycles of length greater\n than ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "M", next := 136536, root := ~, start := 136528, stop := 136535 ), rec( content := ", then t\ his operation returns ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "K", next := 136577, root := ~, start := 136566, stop := 136576 ), rec( content := ".\n " , count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "P", next := 136587, root := ~, start := 136583, stop := 136586 ), rec( content := "\n\n \ See section ", count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "Ref", next := 136649, root := ~, start := 136605, stop := 136648 ), rec( content := " for the\ definition\n of a directed cycle, and the definition of a loop.\n\n ", count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "P", next := 136733, root := ~, start := 136729, stop := 136732 ), rec( content := "\n\n \ The method used for this attribute has complexity ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "M", next := 136802, root := ~, start := 136789, stop := 136801 ), rec( content := " where\n\ ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "M", next := 136821, root := ~, start := 136813, stop := 136820 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "M", next := 136894, root := ~, start := 136886, stop := 136893 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 7, 6 ], mathmode := "Text", name := "P", next := 136941, root := ~, start := 136937, stop := 136940 ), rec( content := "\n ", count := [ 5, 1, 7, 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, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 7 ], mathmode := "Text", name := "Example", next := 137312, root := ~, start := 136946, stop := 137311 ), rec( content := "\n ", count := [ 5, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Description", next := 137329, root := ~, start := 136190, stop := 137328 ) ], count := [ 5, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 137343, root := ~, start := 136055, stop := 137342 ), rec( content := "\n\n ", count := [ 5, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Name := "DigraphVertexLa\ bel" ), content := 0, count := [ 5, 1, 8, 2 ], mathmode := "Text", name := "Oper", next := 137416, root := ~, start := 137364, stop := 137415 ), rec( attributes := rec( Arg := "digraph, i, obj" , Name := "SetDigraphVerte\ xLabel" ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "Oper", next := 137478, root := ~, start := 137419, stop := 137477 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 137516, root := ~, start := 137502, stop := 137515 ), rec( content := " is a di\ graph, then the first operation returns the label\n of the vertex ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 137600, root := ~, start := 137592, stop := 137599 ), rec( content := ". The se\ cond operation can be used to set the label\n of the vertex ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 137678, root := ~, start := 137670, stop := 137677 ), rec( content := " in ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 137696, root := ~, start := 137682, stop := 137695 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 137741, root := ~, start := 137731, stop := 137740 ), rec( content := ". ", 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 := 137748, root := ~, start := 137744, stop := 137747 ), 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, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "A", next := 137873, root := ~, start := 137865, stop := 137872 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "A", next := 137913, root := ~, start := 137905, stop := 137912 ), rec( content := ". ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "P", next := 137919, root := ~, start := 137915, stop := 137918 ), rec( content := "\n\n \ If ", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "A", next := 137942, root := ~, start := 137928, stop := 137941 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertices", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "C", next := 138014, root := ~, start := 137999, stop := 138013 ), rec( content := ", then t\ he labels of the vertices are set to the value of\n this component.", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "P", next := 138095, root := ~, start := 138091, stop := 138094 ), 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, 8, 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, 8, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 8, 9 ], mathmode := "Text", name := "Example", next := 139031, root := ~, start := 138239, stop := 139030 ), rec( content := "\n ", count := [ 5, 1, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "Description", next := 139048, root := ~, start := 137481, stop := 139047 ) ], count := [ 5, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 139062, root := ~, start := 137349, stop := 139061 ), rec( content := "\n\n ", count := [ 5, 1, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertexLa\ bels" ), content := 0, count := [ 5, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 139131, root := ~, start := 139083, stop := 139130 ), rec( attributes := rec( Arg := "digraph, list", Name := "SetDigraphVerte\ xLabels" ), content := 0, count := [ 5, 1, 9, 4 ], mathmode := "Text", name := "Oper", next := 139191, root := ~, start := 139134, stop := 139190 ), 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 := 139229, root := ~, start := 139215, stop := 139228 ), rec( content := " is a di\ graph, then ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabels", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "C", next := 139275, root := ~, start := 139249, stop := 139274 ), rec( content := " returns\ a\n copy of the labels of the vertices 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 := 139342, root := ~, start := 139328, stop := 139341 ), rec( content := ".\n " , count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphVertexLabels", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "C", next := 139377, root := ~, start := 139348, stop := 139376 ), rec( content := " can be \ used to set the labels of the vertices\n 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 := 139445, root := ~, start := 139431, stop := 139444 ), rec( content := " to the \ list of\n 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 := " objects\ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 139500, root := ~, start := 139489, stop := 139499 ), 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 := 139506, root := ~, start := 139502, stop := 139505 ), 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 := 139631, root := ~, start := 139623, stop := 139630 ), 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 := 139671, root := ~, start := 139663, stop := 139670 ), 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 := 139677, root := ~, start := 139673, stop := 139676 ), 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 := 139700, root := ~, start := 139686, stop := 139699 ), 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 := "v\ ertices", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "C", next := 139772, root := ~, start := 139757, stop := 139771 ), rec( content := ", then t\ he labels of the vertices are set to the value of\n 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 := 139853, root := ~, start := 139849, stop := 139852 ), 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, 9, 8 ], 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, 9, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 9 ], mathmode := "Text", name := "Example", next := 140816, root := ~, start := 139992, stop := 140815 ), rec( content := "\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Description", next := 140833, root := ~, start := 139194, stop := 140832 ) ], count := [ 5, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 140847, root := ~, start := 139068, stop := 140846 ), rec( content := "\n\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i, j", Name := "DigraphEdgeLabe\ l" ), content := 0, count := [ 5, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 140921, root := ~, start := 140868, stop := 140920 ), rec( attributes := rec( Arg := "digraph, i, j, o\ bj", Name := "SetDigraphEdgeLabel" ), content := 0, count := [ 5, 1, 10, 4 ], mathmode := "Text", name := "Oper", next := 140985, root := ~, start := 140924, stop := 140984 ), 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 := 141023, root := ~, start := 141009, stop := 141022 ), 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, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141140, root := ~, start := 141132, stop := 141139 ), rec( content := " to vert\ ex\n ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141163, root := ~, start := 141155, stop := 141162 ), rec( content := ". The s\ econd operation can be used to set the label of the edge\n between vertex " , count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141255, root := ~, start := 141247, stop := 141254 ), rec( content := " and ver\ tex ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141275, root := ~, start := 141267, stop := 141274 ), rec( content := " to the \ 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 := " object\\ n ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141320, root := ~, start := 141310, stop := 141319 ), 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 := 141326, root := ~, start := 141322, stop := 141325 ), 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, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 141475, root := ~, start := 141467, stop := 141474 ), 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 := 141481, root := ~, start := 141477, stop := 141480 ), 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, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "Ref", next := 141670, root := ~, start := 141639, stop := 141669 ), rec( content := ".\n\n \ ", count := [ 5, 1, 10, 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, 10, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "Example", next := 142365, root := ~, start := 141677, stop := 142364 ), rec( content := "\n ", count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Description", next := 142382, root := ~, start := 140988, stop := 142381 ) ], count := [ 5, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 142396, root := ~, start := 140853, stop := 142395 ), rec( content := "\n\n ", count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdgeLabe\ ls" ), content := 0, count := [ 5, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 142463, root := ~, start := 142417, stop := 142462 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of lists", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 142569, root := ~, start := 142466, stop := 142568 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and a function", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "Oper", next := 142668, root := ~, start := 142572, stop := 142667 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 142706, root := ~, start := 142692, stop := 142705 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeLabels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 142777, root := ~, start := 142753, stop := 142776 ), rec( content := " returns\ a copy of the labels of the edges\n in ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 142841, root := ~, start := 142827, stop := 142840 ), rec( content := " as a li\ st of lists ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 142874, root := ~, start := 142861, stop := 142873 ), rec( content := " such th\ at\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 142908, root := ~, start := 142889, stop := 142907 ), rec( content := " is the \ label on the edge from vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 142954, root := ~, start := 142946, stop := 142953 ), rec( content := "\n to\ vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph)[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143004, root := ~, start := 142969, stop := 143003 ), rec( content := ".\n\n \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143038, root := ~, start := 143011, stop := 143037 ), rec( content := " can be \ used to set the labels of the edges in\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 143103, root := ~, start := 143089, stop := 143102 ), rec( content := " without\ multiple edges to the list ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 143152, root := ~, start := 143139, stop := 143151 ), rec( content := " of list\ s of\n arbitrary ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ such that ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143220, root := ~, start := 143203, stop := 143219 ), rec( content := " is the \ label on the edge\n from vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143270, root := ~, start := 143262, stop := 143269 ), rec( content := " to the \ vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph>[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143320, root := ~, start := 143285, stop := 143319 ), rec( content := ".\n\n \ Alternatively ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143368, root := ~, start := 143341, stop := 143367 ), rec( content := " can be \ called with binary function\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 143419, root := ~, start := 143408, stop := 143418 ), rec( content := " that as\ its second argument that when passed two vertices ", count := [ 5, 1, 11, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143486, root := ~, start := 143478, stop := 143485 ), rec( content := "\n an\ d ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143503, root := ~, start := 143495, stop := 143502 ), rec( content := " returns\ the label for the edge between vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143558, root := ~, start := 143550, stop := 143557 ), rec( content := " and ver\ tex\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143582, root := ~, start := 143574, stop := 143581 ), rec( content := ". ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "P", next := 143588, root := ~, start := 143584, stop := 143587 ), 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, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "C", next := 143736, root := ~, start := 143728, stop := 143735 ), rec( content := ". ", count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "P", next := 143742, root := ~, start := 143738, stop := 143741 ), 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, 11, 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, 11, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 10 ], mathmode := "Text", name := "Example", next := 144953, root := ~, start := 143886, stop := 144952 ), rec( content := "\n ", count := [ 5, 1, 11, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Description", next := 144970, root := ~, start := 142671, stop := 144969 ) ], count := [ 5, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 144984, root := ~, start := 142402, stop := 144983 ), rec( content := "\n\n ", count := [ 5, 1, 11, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphInEdges"\ ), content := 0, count := [ 5, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 145056, root := ~, start := 145005, stop := 145055 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 145094, root := ~, start := 145059, stop := 145093 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphInEdges", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "C", next := 145136, root := ~, start := 145115, stop := 145135 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "A", next := 145184, root := ~, start := 145170, stop := 145183 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "A", next := 145213, root := ~, start := 145200, stop := 145212 ), rec( content := " as thei\ r range.\n ", count := [ 5, 1, 12, 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, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 4 ], mathmode := "Text", name := "Example", next := 145419, root := ~, start := 145234, stop := 145418 ), rec( content := "\n ", count := [ 5, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 145436, root := ~, start := 145097, stop := 145435 ) ], count := [ 5, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 145450, root := ~, start := 144990, stop := 145449 ), rec( content := "\n\n ", count := [ 5, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphOutEdges\ " ), content := 0, count := [ 5, 1, 13, 2 ], mathmode := "Text", name := "Oper", next := 145523, root := ~, start := 145471, stop := 145522 ), 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 := 145561, root := ~, start := 145526, stop := 145560 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOutEdges", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "C", next := 145604, root := ~, start := 145582, stop := 145603 ), 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 := 145652, root := ~, start := 145638, stop := 145651 ), 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 := 145681, root := ~, start := 145668, stop := 145680 ), rec( content := " as thei\ r source.\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> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] \ ]\n", count := [ 5, 1, 13, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 4 ], mathmode := "Text", name := "Example", next := 145889, root := ~, start := 145703, stop := 145888 ), rec( content := "\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 145906, root := ~, start := 145564, stop := 145905 ) ], count := [ 5, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 145920, root := ~, start := 145456, stop := 145919 ), rec( content := "\n\n ", count := [ 5, 1, 13, 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, 14, 2 ], mathmode := "Text", name := "Oper", next := 146026, root := ~, start := 145941, stop := 146025 ), rec( attributes := rec( Arg := "digraph, u, v", Label := "for digraph an\ d two pos ints", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 14, 4 ], mathmode := "Text", name := "Oper", next := 146122, root := ~, start := 146029, stop := 146121 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 146145, root := ~, start := 146134, stop := 146144 ), rec( content := " or ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 146161, root := ~, start := 146149, stop := 146160 ), rec( content := ".", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "Returns", next := 146172, root := ~, start := 146125, stop := 146171 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, this function returns ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 146245, root := ~, start := 146234, stop := 146244 ), rec( content := " if and \ only if the list\n ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 146285, root := ~, start := 146274, stop := 146284 ), rec( content := " specifi\ es an edge in the digraph ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 146333, root := ~, start := 146319, stop := 146332 ), rec( content := ". Speci\ fically,\n this operation returns ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 146388, root := ~, start := 146377, stop := 146387 ), rec( content := " if ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 146403, root := ~, start := 146392, stop := 146402 ), rec( content := " is a pa\ ir of positive\n integers where ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 146456, root := ~, start := 146445, stop := 146455 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "C", next := 146466, root := ~, start := 146456, stop := 146465 ), rec( content := " is the \ source and ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 146496, root := ~, start := 146485, stop := 146495 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "C", next := 146506, root := ~, start := 146496, stop := 146505 ), rec( content := "\n is\ the range of an edge in ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 146552, root := ~, start := 146538, stop := 146551 ), rec( content := ", and ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 146570, root := ~, start := 146558, stop := 146569 ), rec( content := " otherwi\ se. ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "P", next := 146586, root := ~, start := 146582, stop := 146585 ), rec( content := "\n\n \ The second form simply returns ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "K", next := 146634, root := ~, start := 146623, stop := 146633 ), rec( content := " if ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "u", count := [ 5, 1, 14, 6 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := \ "Text", name := "A", next := 146650, root := ~, start := 146642, stop := 14664\ 9 ), rec( content := ", ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := \ "Text", name := "A", next := 146660, root := ~, start := 146652, stop := 14665\ 9 ), rec( content := "]", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "C", next := 146665, root := ~, start := 146638, stop := 146664 ), rec( content := " is\n \ an edge in ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "A", next := 146698, root := ~, start := 146684, stop := 146697 ), rec( content := ", and ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "K", next := 146716, root := ~, start := 146704, stop := 146715 ), rec( content := " otherwi\ se.\n\n", count := [ 5, 1, 14, 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, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 14, 7 ], mathmode := "Text", name := "Example", next := 146973, root := ~, start := 146729, stop := 146972 ), rec( content := "\n ", count := [ 5, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "Description", next := 146990, root := ~, start := 146175, stop := 146989 ) ], count := [ 5, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 147004, root := ~, start := 145926, stop := 147003 ), rec( content := "\n\n ", count := [ 5, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsMatching" ), content := 0, count := [ 5, 1, 15, 2 ], mathmode := "Text", name := "Oper", next := 147070, root := ~, start := 147025, stop := 147069 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximalMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 4 ], mathmode := "Text", name := "Oper", next := 147125, root := ~, start := 147073, stop := 147124 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsPerfectMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "Oper", next := 147180, root := ~, start := 147128, stop := 147179 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "K", next := 147203, root := ~, start := 147192, stop := 147202 ), rec( content := " or ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "K", next := 147219, root := ~, start := 147207, stop := 147218 ), rec( content := ".", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "Returns", next := 147230, root := ~, start := 147183, stop := 147229 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147268, root := ~, start := 147254, stop := 147267 ), rec( content := " is a di\ graph and ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147297, root := ~, start := 147286, stop := 147296 ), rec( content := " is a li\ st of pairs of\n vertices of ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147350, root := ~, start := 147336, stop := 147349 ), rec( content := ", then " , count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMatching", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "C", next := 147374, root := ~, start := 147357, stop := 147373 ), rec( content := " returns\ ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "K", next := 147394, root := ~, start := 147383, stop := 147393 ), rec( content := " if\n \ ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147413, root := ~, start := 147402, stop := 147412 ), rec( content := " is a ma\ tching of ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147445, root := ~, start := 147431, stop := 147444 ), rec( content := ". The o\ perations\n ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalMatching", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "C", next := 147491, root := ~, start := 147467, stop := 147490 ), rec( content := " and ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPerfectMatching", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "C", next := 147520, root := ~, start := 147496, stop := 147519 ), rec( content := " return \ ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "K", next := 147539, root := ~, start := 147528, stop := 147538 ), rec( content := " if\n \ ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147558, root := ~, start := 147547, stop := 147557 ), rec( content := " is a ma\ ximal, or perfect, matching of ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "A", next := 147611, root := ~, start := 147597, stop := 147610 ), rec( content := ",\n r\ espectively. Otherwise, these operations return ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "K", next := 147679, root := ~, start := 147667, stop := 147678 ), rec( content := ". ", count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "P", next := 147685, root := ~, start := 147681, stop := 147684 ), rec( content := "\n\n \ A ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ atching", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "E", next := 147708, root := ~, start := 147693, stop := 147707 ), rec( content := " ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "C", next := 147717, root := ~, start := 147709, stop := 147716 ), rec( content := " of a di\ graph ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "A", next := 147745, root := ~, start := 147731, stop := 147744 ), rec( content := " is a su\ bset of the\n edges of ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "A", next := 147792, root := ~, start := 147778, stop := 147791 ), rec( content := ", i.e. " , count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 1, 15, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := \ "Text", name := "A", next := 147829, root := ~, start := 147815, stop := 14782\ 8 ), rec( content := ")", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "C", next := 147834, root := ~, start := 147799, stop := 147833 ), rec( content := ", such\n\ that no pair of distinct edges in ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "C", next := 147887, root := ~, start := 147879, stop := 147886 ), rec( content := " are inc\ ident to the same vertex\n of ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "A", next := 147941, root := ~, start := 147927, stop := 147940 ), rec( content := ". Note \ that this definition allows a matching to contain\n loops. See ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "Ref", next := 148045, root := ~, start := 148015, stop := 148044 ), rec( content := ". The m\ atching ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "C", next := 148069, root := ~, start := 148061, stop := 148068 ), rec( content := " is\n \ ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "E", next := 148091, root := ~, start := 148077, stop := 148090 ), rec( content := " if it i\ s contained in no larger matching of the digraph, and\n is ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erfect", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "E", next := 148174, root := ~, start := 148160, stop := 148173 ), rec( content := " if ever\ y vertex of the digraph is incident to an edge in\n the matching. Every pe\ rfect matching is maximal.\n\n ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [1], [2, 3, 4], [3, 5], [1]]);\n\ngap> IsMatching(D, [[2, 1], [3, 2]]);\nfalse\ngap> ed\ ges := [[3, 2]];;\ngap> IsMatching(D, edges);\ntrue\ngap> IsMaximalMatching(D,\ edges);\nfalse\ngap> edges := [[5, 1], [3, 3]];;\ngap> IsMaximalMatching(D, e\ dges);\ntrue\ngap> IsPerfectMatching(D, edges);\nfalse\ngap> edges := [[1, 2],\ [3, 3], [4, 5]];;\ngap> IsPerfectMatching(D, edges);\ntrue\n", count := [ 5, 1, 15, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "Example", next := 148756, root := ~, start := 148291, stop := 148755 ), rec( content := "\n ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "Description", next := 148773, root := ~, start := 147233, stop := 148772 ) ], count := [ 5, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 148787, root := ~, start := 147010, stop := 148786 ), rec( content := "\n\n ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 148801, root := ~, start := 130967, stop := 148800 ) , rec( content := "\n\n ", count := [ 5, 1, 15, 10 ], 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 := 148854, root := ~, start := 148814, stop := 148853 ), 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 := 148918, root := ~, start := 148874, stop := 148917 ), rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ MutableCopy" ), content := 0, count := [ 5, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 148976, root := ~, start := 148921, stop := 148975 ), 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 := 149039, root := ~, start := 148979, stop := 149038 ), 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 := 149113, root := ~, start := 149103, stop := 149112 ), 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 := 149147, root := ~, start := 149133, stop := 149146 ), 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 := 149199, root := ~, start := 149183, stop := 149198 ), 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 := 149244, root := ~, start := 149230, stop := 149243 ), 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 := 149265, root := ~, start := 149257, stop := 149264 ), 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 := 149284, root := ~, start := 149276, stop := 149283 ), 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 := 149303, root := ~, start := 149289, stop := 149302 ), 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 := 149362, root := ~, start := 149358, stop := 149361 ), 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 := 149403, root := ~, start := 149381, stop := 149402 ), 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 := 149498, root := ~, start := 149465, stop := 149497 ), 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 := 149543, root := ~, start := 149521, stop := 149542 ), 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 := 149589, root := ~, start := 149585, stop := 149588 ), 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 := 150559, root := ~, start := 149595, stop := 150558 ), rec( content := "\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 150576, root := ~, start := 149042, stop := 150575 ) ], count := [ 5, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 150590, root := ~, start := 148859, stop := 150589 ), 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 := 150664, root := ~, start := 150611, stop := 150663 ), 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 := 150725, root := ~, start := 150667, stop := 150724 ), 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 := 150832, root := ~, start := 150818, stop := 150831 ), 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 := 150942, root := ~, start := 150928, stop := 150941 ), 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 := 151608, root := ~, start := 150947, stop := 151607 ), rec( content := "\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 151625, root := ~, start := 150728, stop := 151624 ) ], count := [ 5, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 151639, root := ~, start := 150596, stop := 151638 ), 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 := 151711, root := ~, start := 151660, stop := 151710 ), rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrixMutableCopy" ), content := 0, count := [ 5, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 151776, root := ~, start := 151714, stop := 151775 ), 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 := 151826, root := ~, start := 151779, stop := 151825 ), 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 := 151864, root := ~, start := 151850, stop := 151863 ), 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 := 151925, root := ~, start := 151917, stop := 151924 ), 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 := 151962, root := ~, start := 151932, stop := 151961 ), 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 := 151976, root := ~, start := 151962, stop := 151975 ), 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 := 151984, root := ~, start := 151976, stop := 151983 ), 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 := 152036, root := ~, start := 152026, stop := 152035 ), 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 := 152054, root := ~, start := 152040, stop := 152053 ), 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 := 152107, root := ~, start := 152091, stop := 152106 ), 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 := 152122, root := ~, start := 152111, stop := 152121 ), 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 := 152180, root := ~, start := 152166, stop := 152179 ), 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 := 152201, root := ~, start := 152193, stop := 152200 ), 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 := 152224, root := ~, start := 152216, stop := 152223 ), 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 := 152244, root := ~, start := 152230, stop := 152243 ), 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 := 152303, root := ~, start := 152299, stop := 152302 ), 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 := 152397, root := ~, start := 152393, stop := 152396 ), 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 := 152446, root := ~, start := 152417, stop := 152445 ), 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 := 152562, root := ~, start := 152522, stop := 152561 ), 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 := 152618, root := ~, start := 152589, stop := 152617 ), 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 := 152664, root := ~, start := 152660, stop := 152663 ), 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 := 153499, root := ~, start := 152669, stop := 153498 ), rec( content := "\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Description", next := 153516, root := ~, start := 151829, stop := 153515 ) ], count := [ 5, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 153530, root := ~, start := 151645, stop := 153529 ), 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 := 153604, root := ~, start := 153551, stop := 153603 ), 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 := 153637, root := ~, start := 153607, stop := 153636 ), 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 := 153675, root := ~, start := 153661, stop := 153674 ), 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 := 153726, root := ~, start := 153695, stop := 153725 ), 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 := 153796, root := ~, start := 153785, stop := 153795 ), 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 := 153824, root := ~, start := 153813, stop := 153823 ), 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 := 153869, root := ~, start := 153861, stop := 153868 ), 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 := 153888, root := ~, start := 153880, stop := 153887 ), 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 := 153910, root := ~, start := 153896, stop := 153909 ), 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 := 153927, root := ~, start := 153915, stop := 153926 ), 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 := 154708, root := ~, start := 153941, stop := 154707 ), rec( content := "\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 154725, root := ~, start := 153640, stop := 154724 ) ], count := [ 5, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 154739, root := ~, start := 153536, stop := 154738 ), 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 := 154801, root := ~, start := 154760, stop := 154800 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSource" ), content := 0, count := [ 5, 2, 5, 4 ], mathmode := "Text", name := "Attr", next := 154846, root := ~, start := 154804, stop := 154845 ), 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 := 154896, root := ~, start := 154849, stop := 154895 ), 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 := 154936, root := ~, start := 154917, stop := 154935 ), 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 := 154961, root := ~, start := 154941, stop := 154960 ), 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 := 155023, root := ~, start := 155009, stop := 155022 ), 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 := 155058, root := ~, start := 155050, stop := 155057 ), 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 := 155096, root := ~, start := 155082, stop := 15509\ 5 ), rec( content := ")", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 155101, root := ~, start := 155066, stop := 155100 ), 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 := 155130, root := ~, start := 155122, stop := 155129 ), 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 := 155159, root := ~, start := 155145, stop := 155158 ), 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 := 155787, root := ~, start := 155166, stop := 155786 ), rec( content := "\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Description", next := 155804, root := ~, start := 154899, stop := 155803 ) ], count := [ 5, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 155818, root := ~, start := 154745, stop := 155817 ), 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 := 155881, root := ~, start := 155839, stop := 155880 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighbors" ) , content := 0, count := [ 5, 2, 6, 4 ], mathmode := "Text", name := "Attr", next := 155925, root := ~, start := 155884, stop := 155924 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighboursMu\ tableCopy" ), content := 0, count := [ 5, 2, 6, 6 ], mathmode := "Text", name := "Oper", next := 155981, root := ~, start := 155928, stop := 155980 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighborsMut\ ableCopy" ), content := 0, count := [ 5, 2, 6, 8 ], mathmode := "Text", name := "Oper", next := 156036, root := ~, start := 155984, stop := 156035 ), 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 := 156087, root := ~, start := 156039, stop := 156086 ), 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 := 156149, root := ~, start := 156139, stop := 156148 ), 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 := 156216, root := ~, start := 156202, stop := 156215 ), 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 := 156258, root := ~, start := 156250, stop := 156257 ), 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 := 156283, root := ~, start := 156270, stop := 156282 ), 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 := 156339, root := ~, start := 156331, stop := 156338 ), 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 := 156358, root := ~, start := 156350, stop := 156357 ), 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 := 156380, root := ~, start := 156366, stop := 156379 ), 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 := 156386, root := ~, start := 156382, stop := 156385 ), 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 := 156425, root := ~, start := 156405, stop := 156424 ), 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 := 156518, root := ~, start := 156487, stop := 156517 ), 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 := 156561, root := ~, start := 156541, stop := 156560 ), 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 := 156608, root := ~, start := 156604, stop := 156607 ), 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 := 157396, root := ~, start := 156614, stop := 157395 ), rec( content := "\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Description", next := 157413, root := ~, start := 156090, stop := 157412 ) ], count := [ 5, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 157427, root := ~, start := 155824, stop := 157426 ), 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 := 157489, root := ~, start := 157448, stop := 157488 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighbors" ), content := 0, count := [ 5, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 157532, root := ~, start := 157492, stop := 157531 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighboursMut\ ableCopy" ), content := 0, count := [ 5, 2, 7, 6 ], mathmode := "Text", name := "Oper", next := 157587, root := ~, start := 157535, stop := 157586 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighborsMuta\ bleCopy" ), content := 0, count := [ 5, 2, 7, 8 ], mathmode := "Text", name := "Oper", next := 157641, root := ~, start := 157590, stop := 157640 ), 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 := 157691, root := ~, start := 157644, stop := 157690 ), 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 := 157753, root := ~, start := 157743, stop := 157752 ), 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 := 157819, root := ~, start := 157805, stop := 157818 ), 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 := 157861, root := ~, start := 157853, stop := 157860 ), 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 := 157886, root := ~, start := 157873, stop := 157885 ), 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 := 157942, root := ~, start := 157934, stop := 157941 ), 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 := 157961, root := ~, start := 157953, stop := 157960 ), 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 := 157983, root := ~, start := 157969, stop := 157982 ), 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 := 157989, root := ~, start := 157985, stop := 157988 ), 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 := 158027, root := ~, start := 158008, stop := 158026 ), 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 := 158119, root := ~, start := 158089, stop := 158118 ), 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 := 158161, root := ~, start := 158142, stop := 158160 ), 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 := 158208, root := ~, start := 158204, stop := 158207 ), 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 := 158248, root := ~, start := 158238, stop := 158247 ), 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 := 158285, root := ~, start := 158281, stop := 158284 ), 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 := 159066, root := ~, start := 158291, stop := 159065 ), rec( content := "\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Description", next := 159083, root := ~, start := 157694, stop := 159082 ) ], count := [ 5, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 159097, root := ~, start := 157433, stop := 159096 ), 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 := 159157, root := ~, start := 159118, stop := 159156 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSequen\ ce" ), content := 0, count := [ 5, 2, 8, 4 ], mathmode := "Text", name := "Attr", next := 159206, root := ~, start := 159160, stop := 159205 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSet" ) , content := 0, count := [ 5, 2, 8, 6 ], mathmode := "Text", name := "Attr", next := 159250, root := ~, start := 159209, stop := 159249 ), 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 := 159304, root := ~, start := 159253, stop := 159303 ), 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 := 159356, root := ~, start := 159342, stop := 159355 ), 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 := 159370, root := ~, start := 159362, stop := 159369 ), 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 := 159415, root := ~, start := 159398, stop := 159414 ), 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 := 159452, root := ~, start := 159442, stop := 159451 ), 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 := 159471, root := ~, start := 159463, stop := 159470 ), 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 := 159508, root := ~, start := 159500, stop := 159507 ), 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 := 159526, root := ~, start := 159512, stop := 159525 ), 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 := 159554, root := ~, start := 159541, stop := 159553 ), 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 := 159595, root := ~, start := 159587, stop := 159594 ), 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 := 159636, root := ~, start := 159605, stop := 159635 ), 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 := 159642, root := ~, start := 159638, stop := 159641 ), 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 := 159685, root := ~, start := 159661, stop := 159684 ), 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 := 159769, root := ~, start := 159765, stop := 159768 ), 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 := 159807, root := ~, start := 159788, stop := 159806 ), 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 := 159900, root := ~, start := 159896, stop := 159899 ), 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 := 160270, root := ~, start := 159906, stop := 160269 ), rec( content := "\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Description", next := 160287, root := ~, start := 159307, stop := 160286 ) ], count := [ 5, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 160301, root := ~, start := 159103, stop := 160300 ), 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 := 160360, root := ~, start := 160322, stop := 160359 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSequenc\ e" ), content := 0, count := [ 5, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 160408, root := ~, start := 160363, stop := 160407 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSet" ), content := 0, count := [ 5, 2, 9, 6 ], mathmode := "Text", name := "Attr", next := 160451, root := ~, start := 160411, stop := 160450 ), 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 := 160505, root := ~, start := 160454, stop := 160504 ), 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 := 160557, root := ~, start := 160543, stop := 160556 ), 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 := 160571, root := ~, start := 160563, stop := 160570 ), 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 := 160615, root := ~, start := 160599, stop := 160614 ), 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 := 160652, root := ~, start := 160642, stop := 160651 ), 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 := 160671, root := ~, start := 160663, stop := 160670 ), 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 := 160708, root := ~, start := 160700, stop := 160707 ), 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 := 160726, root := ~, start := 160712, stop := 160725 ), 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 := 160754, root := ~, start := 160741, stop := 160753 ), 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 := 160794, root := ~, start := 160786, stop := 160793 ), 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 := 160834, root := ~, start := 160804, stop := 160833 ), 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 := 160840, root := ~, start := 160836, stop := 160839 ), 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 := 160882, root := ~, start := 160859, stop := 160881 ), 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 := 160966, root := ~, start := 160962, stop := 160965 ), 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 := 161003, root := ~, start := 160985, stop := 161002 ), 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 := 161096, root := ~, start := 161092, stop := 161095 ), 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 := 161468, root := ~, start := 161102, stop := 161467 ), rec( content := "\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Description", next := 161485, root := ~, start := 160508, stop := 161484 ) ], count := [ 5, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 161499, root := ~, start := 160307, stop := 161498 ), 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 := 161574, root := ~, start := 161520, stop := 161573 ), 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 := 161621, root := ~, start := 161577, stop := 161620 ), 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 := 161707, root := ~, start := 161694, stop := 161706 ), 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 := 161741, root := ~, start := 161727, stop := 161740 ), 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 := 161778, root := ~, start := 161765, stop := 161777 ), 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 := 161819, root := ~, start := 161805, stop := 161818 ), 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 := 161853, root := ~, start := 161840, stop := 161852 ), 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 := 161863, root := ~, start := 161859, stop := 161862 ), 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 := 162159, root := ~, start := 161868, stop := 162158 ), rec( content := "\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 162176, root := ~, start := 161624, stop := 162175 ) ], count := [ 5, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 162190, root := ~, start := 161505, stop := 162189 ), 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 := 162269, root := ~, start := 162211, stop := 162268 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighborsOfV\ ertex" ), content := 0, count := [ 5, 2, 11, 4 ], mathmode := "Text", name := "Oper", next := 162329, root := ~, start := 162272, stop := 162328 ), 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 := 162370, root := ~, start := 162332, stop := 162369 ), 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 := 162434, root := ~, start := 162424, stop := 162433 ), 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 := 162480, root := ~, start := 162466, stop := 162479 ), 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 := 162503, root := ~, start := 162495, stop := 162502 ), 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 := 162534, root := ~, start := 162524, stop := 162533 ), 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 := 162596, root := ~, start := 162583, stop := 162595 ), 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 := 162615, root := ~, start := 162607, stop := 162614 ), 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 := 162633, root := ~, start := 162619, stop := 162632 ), 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 := 162680, root := ~, start := 162670, stop := 162679 ), 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 := 162708, root := ~, start := 162704, stop := 162707 ), 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 := 162968, root := ~, start := 162713, stop := 162967 ), rec( content := "\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Description", next := 162985, root := ~, start := 162373, stop := 162984 ) ], count := [ 5, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 162999, root := ~, start := 162196, stop := 162998 ), 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 := 163073, root := ~, start := 163020, stop := 163072 ), 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 := 163118, root := ~, start := 163076, stop := 163117 ), 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 := 163203, root := ~, start := 163190, stop := 163202 ), 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 := 163237, root := ~, start := 163223, stop := 163236 ), 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 := 163273, root := ~, start := 163260, stop := 163272 ), 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 := 163314, root := ~, start := 163300, stop := 163313 ), 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 := 163347, root := ~, start := 163334, stop := 163346 ), 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 := 163640, root := ~, start := 163353, stop := 163639 ), rec( content := "\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 163657, root := ~, start := 163121, stop := 163656 ) ], count := [ 5, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 163671, root := ~, start := 163005, stop := 163670 ), 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 := 163749, root := ~, start := 163692, stop := 163748 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighborsOfVe\ rtex" ), content := 0, count := [ 5, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 163808, root := ~, start := 163752, stop := 163807 ), 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 := 163859, root := ~, start := 163811, stop := 163858 ), 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 := 163923, root := ~, start := 163913, stop := 163922 ), 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 := 163969, root := ~, start := 163955, stop := 163968 ), 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 := 163992, root := ~, start := 163984, stop := 163991 ), 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 := 164023, root := ~, start := 164013, stop := 164022 ), 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 := 164080, root := ~, start := 164072, stop := 164079 ), 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 := 164104, root := ~, start := 164091, stop := 164103 ), 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 := 164122, root := ~, start := 164108, stop := 164121 ), 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 := 164169, root := ~, start := 164159, stop := 164168 ), 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 := 164198, root := ~, start := 164194, stop := 164197 ), 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 := 164475, root := ~, start := 164204, stop := 164474 ), rec( content := "\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 164492, root := ~, start := 163862, stop := 164491 ) ], count := [ 5, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 164506, root := ~, start := 163677, stop := 164505 ), 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 := 164568, root := ~, start := 164527, stop := 164567 ), 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 := 164609, root := ~, start := 164571, stop := 164608 ), 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 := 164647, root := ~, start := 164633, stop := 164646 ), 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 := 164686, root := ~, start := 164667, stop := 164685 ), 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 := 164755, root := ~, start := 164726, stop := 164754 ), 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 := 164773, root := ~, start := 164759, stop := 164772 ), 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 := 164837, root := ~, start := 164808, stop := 164836 ), 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 := 164843, root := ~, start := 164839, stop := 164842 ), 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 := 165273, root := ~, start := 164849, stop := 165272 ), rec( content := "\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 165290, root := ~, start := 164612, stop := 165289 ) ], count := [ 5, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 165304, root := ~, start := 164512, stop := 165303 ), 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 := 165397, root := ~, start := 165325, stop := 165396 ), rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphJoinOfVertices" ), content := 0, count := [ 5, 2, 15, 4 ], mathmode := "Text", name := "Oper", next := 165472, root := ~, start := 165400, stop := 165471 ), 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 := 165517, root := ~, start := 165506, stop := 165516 ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Returns", next := 165527, root := ~, start := 165475, stop := 165526 ), 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 := 165638, root := ~, start := 165599, stop := 165637 ), 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 := 165782, root := ~, start := 165771, stop := 165781 ), 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 := 165918, root := ~, start := 165880, stop := 165917 ), 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 := 165966, root := ~, start := 165928, stop := 165965 ), 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 := 166719, root := ~, start := 166068, stop := 166718 ), rec( content := "\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Description", next := 166736, root := ~, start := 165532, stop := 166735 ) ], count := [ 5, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 166750, root := ~, start := 165310, stop := 166749 ), 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 := 166812, root := ~, start := 166771, stop := 166811 ), 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 := 166875, root := ~, start := 166815, stop := 166874 ), 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 := 166949, root := ~, start := 166939, stop := 166948 ), 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 := 166983, root := ~, start := 166969, stop := 166982 ), 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 := 167010, root := ~, start := 167002, stop := 167009 ), 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 := 167078, root := ~, start := 167070, stop := 167077 ), 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 := 167096, root := ~, start := 167082, stop := 167095 ), 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 := 167140, root := ~, start := 167132, stop := 167139 ), 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 := 167163, root := ~, start := 167149, stop := 167162 ), 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 := 167218, root := ~, start := 167214, stop := 167217 ), 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 := 167252, root := ~, start := 167228, stop := 167251 ), 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 := 167939, root := ~, start := 167279, stop := 167938 ), rec( content := "\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 167956, root := ~, start := 166878, stop := 167955 ) ], count := [ 5, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 167970, root := ~, start := 166756, stop := 167969 ), 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 := 168035, root := ~, start := 167991, stop := 168034 ), 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 := 168085, root := ~, start := 168038, stop := 168084 ), 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 := 168169, root := ~, start := 168159, stop := 168168 ), 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 := 168203, root := ~, start := 168189, stop := 168202 ), 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 := 168309, root := ~, start := 168254, stop := 168308 ), 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 := 168333, root := ~, start := 168319, stop := 168332 ), 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 := 168388, root := ~, start := 168384, stop := 168387 ), 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 := 168424, root := ~, start := 168398, stop := 168423 ), 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 := 168458, root := ~, start := 168429, stop := 168457 ), 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 := 168992, root := ~, start := 168489, stop := 168991 ), rec( content := "\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 169009, root := ~, start := 168088, stop := 169008 ) ], count := [ 5, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 169023, root := ~, start := 167976, stop := 169022 ), rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 169037, root := ~, start := 148805, stop := 169036 ) , 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 := 169098, root := ~, start := 169050, stop := 169097 ), 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 := 169162, root := ~, start := 169118, stop := 169161 ), 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 := 169199, root := ~, start := 169188, stop := 169198 ), rec( content := ".", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 169210, root := ~, start := 169165, stop := 169209 ), 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 := 169295, root := ~, start := 169281, stop := 169294 ), 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 := 169305, root := ~, start := 169301, stop := 169304 ), 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 := 169338, root := ~, start := 169324, stop := 169337 ), 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 := 169416, root := ~, start := 169401, stop := 169415 ), 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 := 169542, root := ~, start := 169528, stop := 169541 ), 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 := 169608, root := ~, start := 169597, stop := 169607 ), 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 := 169614, root := ~, start := 169610, stop := 169613 ), 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 := 169662, root := ~, start := 169624, stop := 169661 ), 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 := 169668, root := ~, start := 169664, stop := 169667 ), 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 := 170112, root := ~, start := 169673, stop := 170111 ), rec( content := "\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 170129, root := ~, start := 169213, stop := 170128 ) ], count := [ 5, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 170143, root := ~, start := 169103, stop := 170142 ), 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 := 170269, root := ~, start := 170164, stop := 170268 ), 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 := 170371, root := ~, start := 170272, stop := 170370 ), 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 := 170490, root := ~, start := 170374, stop := 170489 ), 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 := 170527, root := ~, start := 170516, stop := 170526 ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Returns", next := 170537, root := ~, start := 170493, stop := 170536 ), 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 := 170615, root := ~, start := 170601, stop := 170614 ), 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 := 170643, root := ~, start := 170635, stop := 170642 ), 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 := 170663, root := ~, start := 170655, stop := 170662 ), 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 := 170821, root := ~, start := 170810, stop := 170820 ), 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 := 170885, root := ~, start := 170835, stop := 170884 ), 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 := 170929, root := ~, start := 170925, stop := 170928 ), 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 := 170979, root := ~, start := 170968, stop := 170978 ), 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 := 171090, root := ~, start := 171082, stop := 171089 ), 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 := 171107, root := ~, start := 171099, stop := 171106 ), 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 := 171113, root := ~, start := 171109, stop := 171112 ), 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 := 171203, root := ~, start := 171199, stop := 171202 ), 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 := 171253, root := ~, start := 171241, stop := 171252 ), 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 := 171270, root := ~, start := 171258, stop := 171269 ), 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 := 171350, root := ~, start := 171338, stop := 171349 ), 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 := 171371, root := ~, start := 171359, stop := 171370 ), 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 := 171467, root := ~, start := 171455, stop := 171466 ), 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 := 171510, root := ~, start := 171498, stop := 171509 ), 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 := 171561, root := ~, start := 171549, stop := 171560 ), 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 := 171578, root := ~, start := 171566, stop := 171577 ), 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 := 171642, root := ~, start := 171634, stop := 171641 ), 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 := 171647, root := ~, start := 171643, stop := 171646 ), 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 := 172011, root := ~, start := 171652, stop := 172010 ), rec( content := "\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Description", next := 172028, root := ~, start := 170540, stop := 172027 ) ], count := [ 5, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 172042, root := ~, start := 170149, stop := 172041 ), 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 := 172116, root := ~, start := 172063, stop := 172115 ), 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 := 172154, root := ~, start := 172119, stop := 172153 ), 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 := 172192, root := ~, start := 172178, stop := 172191 ), 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 := 172219, root := ~, start := 172211, stop := 172218 ), 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 := 172281, root := ~, start := 172264, stop := 172280 ), 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 := 172299, root := ~, start := 172289, stop := 172298 ), 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 := 172370, root := ~, start := 172359, stop := 172369 ), 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 := 172388, root := ~, start := 172376, stop := 172387 ), 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 := 172430, root := ~, start := 172426, stop := 172429 ), 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 := 172484, root := ~, start := 172476, stop := 172483 ), 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 := 172503, root := ~, start := 172495, stop := 172502 ), 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 := 172543, root := ~, start := 172527, stop := 172542 ), 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 := 172665, root := ~, start := 172649, stop := 172664 ), 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 := 172684, root := ~, start := 172673, stop := 172683 ), 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 := 172770, root := ~, start := 172762, stop := 172769 ), 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 := 172797, root := ~, start := 172777, stop := 172796 ), 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 := 172823, root := ~, start := 172815, stop := 172822 ), 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 := 172833, root := ~, start := 172829, stop := 172832 ), 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 := 172954, root := ~, start := 172941, stop := 172953 ), 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 := 173450, root := ~, start := 172961, stop := 173449 ), rec( content := "\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 173467, root := ~, start := 172157, stop := 173466 ) ], count := [ 5, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 173481, root := ~, start := 172048, stop := 173480 ), 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 := 173566, root := ~, start := 173502, stop := 173565 ), 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 := 173608, root := ~, start := 173593, stop := 173607 ), rec( content := ".", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 173619, root := ~, start := 173569, stop := 173618 ), 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 := 173657, root := ~, start := 173643, stop := 173656 ), 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 := 173683, root := ~, start := 173675, stop := 173682 ), 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 := 173713, root := ~, start := 173699, stop := 173712 ), 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 := 173808, root := ~, start := 173794, stop := 173807 ), 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 := 173840, root := ~, start := 173832, stop := 173839 ), 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 := 173905, root := ~, start := 173855, stop := 173904 ), 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 := 173982, root := ~, start := 173978, stop := 173981 ), 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 := 1740\ 75, root := ~, start := 174067, stop := 174074 ), 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\ := 174204, root := ~, start := 174196, stop := 174203 ), 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 := \ 174318, root := ~, start := 174303, stop := 174317 ), 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 := 174409, root := ~, start := 174377, stop \ := 174408 ), 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 := 174414, root := ~, star\ t := 174410, stop := 174413 ), 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 := 174493\ , root := ~, start := 174479, stop := 174492 ), 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 := 174502, root := ~, start := 174494, sto\ p := 174501 ), rec( content := "))", count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "C", next := 174508, root := ~, start := 174424, stop := 174507 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 3, 4, 6 ], name := "Item", next := 174523, root := ~, start := 174\ 001, stop := 174522 ), 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 := 174614, root := ~, start := 174606, stop \ := 174613 ), 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 := 174710, root := ~, start := 174531, stop := 174709 ) ], count := [ 5, 3, 4, 5 ], mathmode := "Text", name := "List", next := 174722, root := ~, start := 173988, stop := 174721 ), 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 := 174760, root := ~, start := 174752, stop := 174759 ), 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 := 174784, root := ~, start := 174776, stop := 174783 ), 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 := 174816, root := ~, start := 174802, stop := 174815 ), 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 := 174849, root := ~, start := 174823, stop := 174848 ), 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 := 175161, root := ~, start := 174852, stop := 175160 ), rec( content := "\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 175178, root := ~, start := 173622, stop := 175177 ) ], count := [ 5, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 175192, root := ~, start := 173487, stop := 175191 ), 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 := 175323, root := ~, start := 175213, stop := 175322 ), 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 := 175437, root := ~, start := 175326, stop := 175436 ), 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 := 175477, root := ~, start := 175440, stop := 175476 ), 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 := 175571, root := ~, start := 175557, stop := 175570 ), 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 := 175633, root := ~, start := 175620, stop := 175632 ), 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 := 175656, root := ~, start := 175641, stop := 175655 ), 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 := 175691, root := ~, start := 175675, stop := 175690 ), 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 := 175697, root := ~, start := 175693, stop := 175696 ), 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 := 175717, root := ~, start := 175703, stop := 175716 ), 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 := 175752, root := ~, start := 175739, stop := 175751 ), 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 := 175802, root := ~, start := 175787, stop := 175801 ), 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 := 175861, root := ~, start := 175845, stop := 175860 ), 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 := 176168, root := ~, start := 175910, stop := 176167 ), rec( content := "\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 176185, root := ~, start := 175480, stop := 176184 ) ], count := [ 5, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 176199, root := ~, start := 175198, stop := 176198 ), 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 := 176261, root := ~, start := 176220, stop := 176260 ), 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 := 176303, root := ~, start := 176288, stop := 176302 ), rec( content := ".", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 176314, root := ~, start := 176264, stop := 176313 ), 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 := 176374, root := ~, start := 176362, stop := 176373 ), 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 := 176404, root := ~, start := 176390, stop := 176403 ), 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 := 176426, root := ~, start := 176414, stop := 176425 ), 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 := 176546, root := ~, start := 176502, stop := 176545 ), 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 := 176624, root := ~, start := 176620, stop := 176623 ), 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 := 176647, root := ~, start := 176633, stop := 176646 ), 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 := 176722, root := ~, start := 176707, stop := 176721 ), 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 := 176742, root := ~, start := 176728, stop := 176741 ), 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 := 176803, root := ~, start := 176795, stop := 176802 ), 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 := 176813, root := ~, start := 176809, stop := 176812 ), 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 := 176953, root := ~, start := 176945, stop := 17695\ 2 ), rec( content := " ^ 3)", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "C", next := 176962, root := ~, start := 176940, stop := 176961 ), 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 := 176982, root := ~, start := 176974, stop := 176981 ), 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 := 177026, root := ~, start := 177012, stop := 177025 ), 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 := 177103, root := ~, start := 177077, stop := 177102 ), 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 := 177163, root := ~, start := 177159, stop := 177162 ), 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 := 177238, root := ~, start := 177202, stop := 177237 ), 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 := 177244, root := ~, start := 177240, stop := 177243 ), 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 := 177742, root := ~, start := 177249, stop := 177741 ), rec( content := "\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 177759, root := ~, start := 176317, stop := 177758 ) ], count := [ 5, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 177773, root := ~, start := 176205, stop := 177772 ), 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 := 177838, root := ~, start := 177794, stop := 177837 ), 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 := 177880, root := ~, start := 177865, stop := 177879 ), rec( content := ".", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 177891, root := ~, start := 177841, stop := 177890 ), 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 := 177955, root := ~, start := 177939, stop := 177954 ), 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 := 177985, root := ~, start := 177971, stop := 177984 ), 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 := 178011, root := ~, start := 177995, stop := 178010 ), 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 := 178150, root := ~, start := 178102, stop := 178149 ), 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 := 178228, root := ~, start := 178224, stop := 178227 ), 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 := 178251, root := ~, start := 178237, stop := 178250 ), 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 := 178340, root := ~, start := 178325, stop := 178339 ), 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 := 178364, root := ~, start := 178350, stop := 178363 ), 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 := 178424, root := ~, start := 178410, stop := 178423 ), 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 := 178485, root := ~, start := 178477, stop := 178484 ), 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 := 178495, root := ~, start := 178491, stop := 178494 ), 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 := 178536, root := ~, start := 178510, stop := 178535 ), 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 := 178542, root := ~, start := 178538, stop := 178541 ), 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 := 179122, root := ~, start := 178547, stop := 179121 ), rec( content := "\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 179139, root := ~, start := 177894, stop := 179138 ) ], count := [ 5, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 179153, root := ~, start := 177779, stop := 179152 ), 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 := 179225, root := ~, start := 179174, stop := 179224 ), 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 := 179266, root := ~, start := 179251, stop := 179265 ), rec( content := ".", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 179277, root := ~, start := 179228, stop := 179276 ), 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 := 179315, root := ~, start := 179301, stop := 179314 ), 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 := 179398, root := ~, start := 179384, stop := 179397 ), 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 := 179477, root := ~, start := 179464, stop := 179476 ), 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 := 179495, root := ~, start := 179482, stop := 179494 ), 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 := 179544, root := ~, start := 179536, stop := 179543 ), 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 := 179557, root := ~, start := 179549, stop := 179556 ), 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 := 179564, root := ~, start := 179560, stop := 179563 ), 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 := 179586, root := ~, start := 179574, stop := 179585 ), 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 := 179794, root := ~, start := 179790, stop := 179793 ), 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 := 179817, root := ~, start := 179803, stop := 179816 ), 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 := 179883, root := ~, start := 179868, stop := 179882 ), 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 := 179889, root := ~, start := 179885, stop := 179888 ), 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 := 180441, root := ~, start := 179894, stop := 180440 ), rec( content := "\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 180458, root := ~, start := 179280, stop := 180457 ) ], count := [ 5, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 180472, root := ~, start := 179159, stop := 180471 ), 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 := 180548, root := ~, start := 180493, stop := 180547 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrConnec\ tedComponents" ), content := 0, count := [ 5, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 180608, root := ~, start := 180551, stop := 180607 ), 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 := 180639, root := ~, start := 180611, stop := 180638 ), 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 := 180703, root := ~, start := 180693, stop := 180702 ), 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 := 180786, root := ~, start := 180772, stop := 180785 ), 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 := 180823, root := ~, start := 180809, stop := 180822 ), 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 := 181075, root := ~, start := 181061, stop := 181074 ), 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 := 181198, root := ~, start := 181150, stop := 181197 ), 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 := 181248, root := ~, start := 181211, stop := 181247 ), 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 := 181266, root := ~, start := 181252, stop := 181265 ), 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 := 181273, root := ~, start := 181269, stop := 181272 ), 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 := 181372, root := ~, start := 181358, stop := 181371 ), 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 := 181378, root := ~, start := 181374, stop := 181377 ), 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 := 181405, root := ~, start := 181395, stop := 181404 ), 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 := 181436, root := ~, start := 181424, stop := 181435 ), 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 := 181450, root := ~, start := 181441, stop := 181449 ), 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 := 181482, root := ~, start := 181470, stop := 181481 ), 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 := 181550, root := ~, start := 181536, stop := 181549 ), 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 := 181617, root := ~, start := 181608, stop := 181616 ), 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 := 181657, root := ~, start := 181649, stop := 181656 ), 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 := 181729, root := ~, start := 181710, stop := 181728 ), 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 := 181735, root := ~, start := 181731, stop := 181734 ), 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 := 181803, root := ~, start := 181790, stop := 181802 ), 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 := 181823, root := ~, start := 181815, stop := 181822 ), 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 := 181863, root := ~, start := 181855, stop := 181862 ), 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 := 181910, root := ~, start := 181906, stop := 181909 ), 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 := 1819\ 62, root := ~, start := 181948, stop := 181961 ), rec( content := ")", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 181967, root := ~, start := 181916, stop := 181966 ), 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 := 1820\ 48, root := ~, start := 182034, stop := 182047 ), rec( content := ").comps)", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 182060, root := ~, start := 181997, stop := 182059 ), 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 := 182994, root := ~, start := 182097, stop := 182993 ), rec( content := "\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 183011, root := ~, start := 180642, stop := 183010 ) ], count := [ 5, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 183025, root := ~, start := 180478, stop := 183024 ), 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 := 183108, root := ~, start := 183046, stop := 183107 ), 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 := 183149, root := ~, start := 183111, stop := 183148 ), 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 := 183186, root := ~, start := 183173, stop := 183185 ), 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 := 183228, root := ~, start := 183214, stop := 183227 ), 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 := 183302, root := ~, start := 183289, stop := 183301 ), 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 := 183324, root := ~, start := 183310, stop := 183323 ), 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 := 183374, root := ~, start := 183334, stop := 183373 ), 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 := 183666, root := ~, start := 183401, stop := 183665 ), rec( content := "\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 183683, root := ~, start := 183152, stop := 183682 ) ], count := [ 5, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 183697, root := ~, start := 183031, stop := 183696 ), 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 := 183781, root := ~, start := 183718, stop := 183780 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrStrong\ lyConnectedComponents" ), content := 0, count := [ 5, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 183849, root := ~, start := 183784, stop := 183848 ), 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 := 183880, root := ~, start := 183852, stop := 183879 ), 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 := 183944, root := ~, start := 183934, stop := 183943 ), 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 := 184029, root := ~, start := 184015, stop := 184028 ), 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 := 184066, root := ~, start := 184052, stop := 184065 ), 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 := 184300, root := ~, start := 184286, stop := 184299 ), 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 := 184310, root := ~, start := 184306, stop := 184309 ), 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 := 184337, root := ~, start := 184327, stop := 184336 ), 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 := 184368, root := ~, start := 184356, stop := 184367 ), 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 := 184382, root := ~, start := 184373, stop := 184381 ), 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 := 184414, root := ~, start := 184402, stop := 184413 ), 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 := 184483, root := ~, start := 184469, stop := 184482 ), 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 := 184550, root := ~, start := 184541, stop := 184549 ), 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 := 184590, root := ~, start := 184582, stop := 184589 ), 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 := 184664, root := ~, start := 184645, stop := 184663 ), 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 := 184670, root := ~, start := 184666, stop := 184669 ), 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 := 184780, root := ~, start := 184761, stop := 184779 ), 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 := 184813, root := ~, start := 184800, stop := 184812 ), 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 := 184832, root := ~, start := 184824, stop := 184831 ), 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 := 184905, root := ~, start := 184897, stop := 184904 ), 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 := 184952, root := ~, start := 184948, stop := 184951 ), 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 := 1850\ 12, root := ~, start := 184998, stop := 185011 ), rec( content := ")", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 185017, root := ~, start := 184958, stop := 185016 ), 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 := 1851\ 10, root := ~, start := 185096, stop := 185109 ), rec( content := ").comps)", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 185122, root := ~, start := 185051, stop := 185121 ), 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 := 186065, root := ~, start := 185159, stop := 186064 ), rec( content := "\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 186082, root := ~, start := 183883, stop := 186081 ) ], count := [ 5, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 186096, root := ~, start := 183703, stop := 186095 ), 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 := 186187, root := ~, start := 186117, stop := 186186 ), 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 := 186228, root := ~, start := 186190, stop := 186227 ), 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 := 186265, root := ~, start := 186252, stop := 186264 ), 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 := 186307, root := ~, start := 186293, stop := 186306 ), 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 := 186390, root := ~, start := 186377, stop := 186389 ), 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 := 186412, root := ~, start := 186398, stop := 186411 ), 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 := 186470, root := ~, start := 186422, stop := 186469 ), 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 := 186777, root := ~, start := 186497, stop := 186776 ), rec( content := "\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 186794, root := ~, start := 186231, stop := 186793 ) ], count := [ 5, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 186808, root := ~, start := 186102, stop := 186807 ), 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 := 186877, root := ~, start := 186829, stop := 186876 ), 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 := 186932, root := ~, start := 186921, stop := 186931 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Returns", next := 186943, root := ~, start := 186880, stop := 186942 ), 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 := 186981, root := ~, start := 186967, stop := 186980 ), 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 := 187065, root := ~, start := 187027, stop := 187064 ), 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 := 187098, root := ~, start := 187072, stop := 187097 ), 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 := 187151, root := ~, start := 187137, stop := 187150 ), 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 := 187194, root := ~, start := 187168, stop := 187193 ), 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 := 187214, root := ~, start := 187203, stop := 187213 ), 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 := 187219, root := ~, start := 187215, stop := 187218 ), 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 := 187437, root := ~, start := 187418, stop := 187436 ), 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 := 187455, root := ~, start := 187441, stop := 187454 ), 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 := 187515, root := ~, start := 187501, stop := 187514 ), 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 := 187583, root := ~, start := 187569, stop := 187582 ), 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 := 187589, root := ~, start := 187585, stop := 187588 ), 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 := 187784, root := ~, start := 187746, stop := 187783 ), 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 := 188377, root := ~, start := 187811, stop := 188376 ), rec( content := "\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Description", next := 188394, root := ~, start := 186946, stop := 188393 ) ], count := [ 5, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 188408, root := ~, start := 186814, stop := 188407 ), 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 := 188476, root := ~, start := 188429, stop := 188475 ), 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 := 188517, root := ~, start := 188479, stop := 188516 ), rec( attributes := rec( ), content := [ rec( content := "\n\n A\ connected 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 := 188580, root := ~, start := 188562, stop := 188579 ), 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 := 188659, root := ~, start := 188627, stop := 188658 ), 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 := 188723, root := ~, start := 188709, stop := 188722 ), 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 := 188789, root := ~, start := 188781, stop := 188788 ), 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 := 188807, root := ~, start := 188793, stop := 188806 ), 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 := 188905, root := ~, start := 188880, stop := 188904 ), 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 := 188910, root := ~, start := 188906, stop := 188909 ), 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 := 188941, root := ~, start := 188916, stop := 188940 ), 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 := 189005, root := ~, start := 188991, stop := 189004 ), 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 := 189079, root := ~, start := 189065, stop := 189078 ), 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 := 189102, root := ~, start := 189098, stop := 189101 ), rec( content := "\n\n \ Multiple edges and loops 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 := 189165, root := ~, start := 189161, stop := 189164 ), 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 := 189233, root := ~, start := 189220, stop := 189232 ), 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 := 189252, root := ~, start := 189244, stop := 189251 ), rec( content := " is the \ number of edges (counting multiple edges as one, and not\n counting loops) \ 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 := 189349, root := ~, start := 189341, stop := 189348 ), rec( content := " is the \ number of vertices in the digraph.\n\n See also ", count := [ 5, 3, 14, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "Ref", next := 189440, root := ~, start := 189406, stop := 189439 ), rec( content := ".\n", count := [ 5, 3, 14, 6 ], 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, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 189845, root := ~, start := 189442, stop := 189844 ), rec( content := "\n ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 189862, root := ~, start := 188520, stop := 189861 ) ], count := [ 5, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 189876, root := ~, start := 188414, stop := 189875 ), rec( content := "\n\n ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphPeriod" ), content := 0, count := [ 5, 3, 15, 2 ], mathmode := "Text", name := "Attr", next := 189939, root := ~, start := 189897, stop := 189938 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Returns", next := 189972, root := ~, start := 189942, stop := 189971 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the period of the digraph ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 190055, root := ~, start := 190041, stop := 190054 ), 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 := 190060, root := ~, start := 190056, stop := 190059 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 190093, root := ~, start := 190079, stop := 190092 ), 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, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 190250, root := ~, start := 190236, stop := 190249 ), rec( content := ". If ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 190270, root := ~, start := 190256, stop := 190269 ), rec( content := " has no \ directed cycles, then\n this function returns ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "M", next := 190334, root := ~, start := 190326, stop := 190333 ), rec( content := ". See s\ ection ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "Ref", next := 190399, root := ~, start := 190349, stop := 190398 ), rec( content := " for the\ definition of a directed cycle. ", 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 := 190444, root := ~, start := 190440, stop := 190443 ), rec( content := "\n\n \ A digraph with a period of ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "M", next := 190485, root := ~, start := 190477, stop := 190484 ), rec( content := " is said\ to be ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ periodic", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "E", next := 190516, root := ~, start := 190500, stop := 190515 ), rec( content := ". See\n\ ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAperi\ odicDigraph" ), content := 0, count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "Ref", next := 190559, root := ~, start := 190527, stop := 190558 ), rec( content := ". ", 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 := 190565, root := ~, start := 190561, stop := 190564 ), rec( content := "\n ", count := [ 5, 3, 15, 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, 15, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "Example", next := 191108, root := ~, start := 190570, stop := 191107 ), rec( content := "\n ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Description", next := 191125, root := ~, start := 189975, stop := 191124 ) ], count := [ 5, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 191139, root := ~, start := 189882, stop := 191138 ), rec( content := "\n\n ", count := [ 5, 3, 15, 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, 16, 2 ], mathmode := "Text", name := "Oper", next := 191229, root := ~, start := 191160, stop := 191228 ), rec( attributes := rec( ), content := [ rec( content := "A matrix.\ ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "Returns", next := 191260, root := ~, start := 191232, stop := 191259 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "A", next := 191298, root := ~, start := 191284, stop := 191297 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "M", next := 191325, root := ~, start := 191317, stop := 191324 ), rec( content := " vertice\ s, then\n this operation returns an ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "M", next := 191388, root := ~, start := 191371, stop := 191387 ), rec( content := " matrix \ ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "C", next := 191406, root := ~, start := 191396, stop := 191405 ), rec( content := " contain\ ing\n the output of a generalised version of the Floyd-Warshall algorithm,\ \n applied to ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "A", next := 191520, root := ~, start := 191506, stop := 191519 ), rec( content := ". ", count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "P", next := 191526, root := ~, start := 191522, stop := 191525 ), rec( content := "\n\n \ The operation ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "C", next := 191573, root := ~, start := 191546, stop := 191572 ), rec( content := " is cust\ omised by the arguments\n ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191620, root := ~, start := 191609, stop := 191619 ), rec( content := ", ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191635, root := ~, start := 191622, stop := 191634 ), rec( content := ", and ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191652, root := ~, start := 191641, stop := 191651 ), rec( content := ".\n T\ he arguments ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191685, root := ~, start := 191672, stop := 191684 ), rec( content := " and ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191701, root := ~, start := 191690, stop := 191700 ), rec( content := " can be \ arbitrary ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ .\n The argument ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191762, root := ~, start := 191751, stop := 191761 ), rec( content := " must be\ a function which accepts 4 arguments:\n the matrix ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "C", next := 191834, root := ~, start := 191824, stop := 191833 ), rec( content := ", follow\ ed by 3 postive integers. The function\n ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "A", next := 191896, root := ~, start := 191885, stop := 191895 ), rec( content := " is wher\ e the work to calculate the desired outcome must be\n performed. ", count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "P", next := 191975, root := ~, start := 191971, stop := 191974 ), rec( content := "\n\n \ This method initialises the matrix ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 192026, root := ~, start := 192016, stop := 192025 ), rec( content := " by sett\ ing entry\n ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 192064, root := ~, start := 192048, stop := 192063 ), rec( content := " to equa\ l ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 192085, root := ~, start := 192074, stop := 192084 ), rec( content := " if ther\ e is an edge with source\n ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 192130, root := ~, start := 192122, stop := 192129 ), rec( content := " and ran\ ge ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 192149, root := ~, start := 192141, stop := 192148 ), rec( content := ", and by\ setting entry ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 192188, root := ~, start := 192172, stop := 192187 ), rec( content := " to equa\ l\n ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 192215, root := ~, start := 192202, stop := 192214 ), rec( content := " otherwi\ se.\n The final part of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 192276, root := ~, start := 192249, stop := 192275 ), rec( content := " then ca\ lls the function\n ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 192316, root := ~, start := 192305, stop := 192315 ), rec( content := " inside \ three nested for loops, over the vertices of\n ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 192387, root := ~, start := 192373, stop := 192386 ), rec( content := ": ", 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 := 192393, root := ~, start := 192389, stop := 192392 ), rec( content := "\n\n", count := [ 5, 3, 16, 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, 16, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "Listing", next := 192591, root := ~, start := 192395, stop := 192590 ), rec( content := "\n\n \ The matrix ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "C", next := 192618, root := ~, start := 192608, stop := 192617 ), rec( content := " is then\ returned as the result. An example of using\n ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "C", next := 192702, root := ~, start := 192675, stop := 192701 ), rec( content := " to calc\ ulate the shortest (non-zero) distances\n between the vertices of a digraph\ is shown below: ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 9 ], mathmode := "Text", name := "P", next := 192808, root := ~, start := 192804, stop := 192807 ), rec( content := "\n\n \ ", count := [ 5, 3, 16, 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, 16, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "Example", next := 193514, root := ~, start := 192814, stop := 193513 ), rec( content := "\n ", count := [ 5, 3, 16, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 3 ], mathmode := "Text", name := "Description", next := 193531, root := ~, start := 191263, stop := 193530 ) ], count := [ 5, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 193545, root := ~, start := 191145, stop := 193544 ), rec( content := "\n\n ", count := [ 5, 3, 16, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IsReachable" ), content := 0, count := [ 5, 3, 17, 2 ], mathmode := "Text", name := "Oper", next := 193612, root := ~, start := 193566, stop := 193611 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "K", next := 193635, root := ~, start := 193624, stop := 193634 ), rec( content := " or ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "K", next := 193651, root := ~, start := 193639, stop := 193650 ), rec( content := ".", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 193662, root := ~, start := 193615, stop := 193661 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "K", next := 193717, root := ~, start := 193706, stop := 193716 ), rec( content := " if ther\ e exists a non-trivial directed\n walk from vertex ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "A", next := 193786, root := ~, start := 193778, stop := 193785 ), rec( content := " to vert\ ex ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "A", next := 193805, root := ~, start := 193797, stop := 193804 ), rec( content := " in 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 := 193835, root := ~, start := 193821, stop := 193834 ), rec( content := ",\n a\ nd ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "K", next := 193857, root := ~, start := 193845, stop := 193856 ), rec( content := " if ther\ e does not exist such a directed walk. See section\n ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 193965, root := ~, start := 193921, stop := 193964 ), rec( content := " for the\ definition of a\n non-trivial directed walk.\n ", 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 := 194029, root := ~, start := 194025, stop := 194028 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sReachable", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "C", next := 194068, root := ~, start := 194050, stop := 194067 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "M", next := 194119, root := ~, start := 194098, stop := 194118 ), rec( content := " where " , count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "M", next := 194134, root := ~, start := 194126, stop := 194133 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "M", next := 194170, root := ~, start := 194162, stop := 194169 ), rec( content := " the num\ ber of\n vertices in ", 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 := 194215, root := ~, start := 194201, stop := 194214 ), rec( content := ".\n\n", count := [ 5, 3, 17, 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, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 194464, root := ~, start := 194218, stop := 194463 ), rec( content := "\n ", count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 194481, root := ~, start := 193665, stop := 194480 ) ], count := [ 5, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 194495, root := ~, start := 193551, stop := 194494 ), rec( content := "\n\n ", count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphPath" ), content := 0, count := [ 5, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 194562, root := ~, start := 194516, stop := 194561 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "K", next := 194605, root := ~, start := 194594, stop := 194604 ), rec( content := ".", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 194616, root := ~, start := 194565, stop := 194615 ), 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, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 194735, root := ~, start := 194727, stop := 194734 ), rec( content := " ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "C", next := 194744, root := ~, start := 194736, stop := 194743 ), rec( content := " ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 194753, root := ~, start := 194745, stop := 194752 ), rec( content := ") from v\ ertex ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 194775, root := ~, start := 194767, stop := 194774 ), rec( content := " to vert\ ex\n ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 194798, root := ~, start := 194790, stop := 194797 ), rec( content := " in the \ digraph ", 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 := 194828, root := ~, start := 194814, stop := 194827 ), rec( content := ", then t\ his operation returns such a\n directed path (or directed cycle). Otherwise\ , this operation returns\n ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "K", next := 194953, root := ~, start := 194942, stop := 194952 ), rec( content := ". See S\ ection ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Ref", next := 195009, root := ~, start := 194968, stop := 195008 ), rec( content := " for the\ \n definition of a directed path and a directed cycle.\n ", 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 := 195082, root := ~, start := 195078, stop := 195081 ), rec( content := "\n\n \ A directed path (or directed cycle) of non-zero length ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 195153, root := ~, start := 195143, stop := 195152 ), rec( content := ",\n " , count := [ 5, 3, 18, 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, 18, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "M", next := 195205, root := ~, start := 195159, stop := 195204 ), rec( content := ",\n i\ s represented by a pair of lists ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,a]", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 195257, root := ~, start := 195245, stop := 195256 ), rec( content := " as foll\ ows:\n\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 5, 3, 18, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "v", count := [ 5, 3, 18, 6 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 6 ], mathmode := "Text", name\ := "C", next := 195311, root := ~, start := 195303, stop := 195310 ), rec( cont\ ent := " is the list ", count := [ 5, 3, 18, 6 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "[v_1, v_2, ..., v_n]", count := [ 5, 3, 18, 6 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 6 ], mathmode := "Text", name \ := "M", next := 195351, root := ~, start := 195324, stop := 195350 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 6 ], name := "Item", next := 195366, root := ~, start := 195\ 288, stop := 195365 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "a", count :\ = [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 195396, root := ~ , star\ t := 195388, stop := 195395 ), rec( content := " is the list of positive integ\ ers ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "[a_1, a_2, ...\ , a_{n-1}]", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "M", next \ := 195461, root := ~, start := 195430, stop := 195460 ), rec( content := "\n \ where for each each ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i < n", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "M", next\ := 195505, root := ~, start := 195490, stop := 195504 ), rec( content := ", " , coun\ t := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "a_i", count := [ 5, 3, 18, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 8\ ], mathmode := "Text", name := "M", next := 195517, root := ~, start := 19550\ 7, stop := 195516 ), rec( content := " is the position of\n ", count :=\ [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 5, 3, 18\ , 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1\ 8, 8 ], mathmode := "Text", name := "M", next := 195559, root := ~, start := 1\ 95545, stop := 195558 ), rec( content := " in\n ", count := [ 5, 3, 18,\ 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "OutNeighboursOfVertex(", count := [ 5, 3,\ 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3\ , 18, 8 ], mathmode := "Text", name := "C", next := 195600, root := ~, start :\ = 195571, stop := 195599 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "A", next\ := 195614, root := ~, start := 195600, stop := 195613 ), rec( attributes := re\ c( ), content := [ rec( content := ",", count := [ 5, 3, 18, 8 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode :\ = "Text", name := "C", next := 195622, root := ~, start := 195614, stop := 195\ 621 ), rec( attributes := rec( ), content := [ rec( content := "v_i", count :\ = [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 18, 8 ], mathmode := "Text", name := "M", next := 195632, root := ~ , star\ t := 195622, stop := 195631 ), rec( attributes := rec( ), content := [ rec( c\ ontent := ")", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next\ := 195640, root := ~, start := 195632, stop := 195639 ), rec( content := "\n \ corresponding to the edge ", count := [ 5, 3, 18, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "e_i", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "M\ ", next := 195685, root := ~, start := 195675, stop := 195684 ), rec( content \ := ". This is can be useful if the\n position of a vertex in a list of\ out-neighours is significant, for\n example in orbit digraphs.\n \ ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], name := "Ite\ m", next := 195841, root := ~, start := 195373, stop := 195840 ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "List", next := 195853, root := ~, start := 195275, stop := 195852 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPath", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "C", next := 195892, root := ~, start := 195874, stop := 195891 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "M", next := 195943, root := ~, start := 195922, stop := 195942 ), rec( content := " where " , count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "M", next := 195958, root := ~, start := 195950, stop := 195957 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "M", next := 195994, root := ~, start := 195986, stop := 195993 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "A", next := 196039, root := ~, start := 196025, stop := 196038 ), rec( content := ".\n\n", count := [ 5, 3, 18, 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, 18, 11 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "Example", next := 196322, root := ~, start := 196042, stop := 196321 ), rec( content := "\n ", count := [ 5, 3, 18, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 196339, root := ~, start := 194619, stop := 196338 ) ], count := [ 5, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 196353, root := ~, start := 194501, stop := 196352 ), rec( content := "\n\n ", count := [ 5, 3, 18, 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, 19, 2 ], mathmode := "Text", name := "Oper", next := 196428, root := ~, start := 196374, stop := 196427 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 196471, root := ~, start := 196460, stop := 196470 ), rec( content := ".", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 196482, root := ~, start := 196431, stop := 196481 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the shortest directed path in the digraph digraph from the vertex\n ", 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 := 196589, root := ~, start := 196581, stop := 196588 ), rec( content := " to the \ vertex ", 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 := 196612, root := ~, start := 196604, stop := 196611 ), rec( content := ", if suc\ h a path exists. If ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ 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 := 196651, root := ~, start := 196643, stop := 19665\ 0 ), rec( content := " =\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCD\ ATA", 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 := 196666, root := ~, start := 196658, stop := 19666\ 5 ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "C", next := 196670, root := ~, start := 196640, stop := 196669 ), rec( content := ", then t\ he shortest non-trivial cycle is returned, again, if it\n exists. Otherwise\ , this operation returns ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 196791, root := ~, start := 196780, stop := 196790 ), rec( content := ". See S\ ection\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Ref", next := 196851, root := ~, start := 196810, stop := 196850 ), rec( content := " for the\ definition of a directed\n path and a directed cycle.\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 := 196924, root := ~, start := 196920, stop := 196923 ), rec( content := "\n\n \ See ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "Ref", next := 196959, root := ~, start := 196934, stop := 196958 ), rec( content := " for det\ ails on the output.\n\n The method for ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphShortestPath", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "C", next := 197033, root := ~, start := 197007, stop := 197032 ), rec( content := " has wor\ st case complexity of\n ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n)", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 197082, root := ~, start := 197067, stop := 197081 ), 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 := 197097, root := ~, start := 197089, stop := 197096 ), 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 := 197133, root := ~, start := 197125, stop := 197132 ), rec( content := " the\n \ number of 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 := 197178, root := ~, start := 197164, stop := 197177 ), rec( content := ".\n\n", count := [ 5, 3, 19, 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, 19, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 5 ], mathmode := "Text", name := "Example", next := 197530, root := ~, start := 197181, stop := 197529 ), rec( content := "\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 197547, root := ~, start := 196485, stop := 197546 ) ], count := [ 5, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 197561, root := ~, start := 196359, stop := 197560 ), rec( content := "\n\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IteratorOfPaths\ " ), content := 0, count := [ 5, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 197632, root := ~, start := 197582, stop := 197631 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 197666, root := ~, start := 197635, stop := 197665 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", 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 := 197704, root := ~, start := 197690, stop := 197703 ), rec( content := " is a di\ graph or a list of adjacencies which defines a\n digraph - see ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Ref", next := 197804, root := ~, start := 197777, stop := 197803 ), 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, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 197935, root := ~, start := 197927, stop := 197934 ), rec( content := " ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "C", next := 197944, root := ~, start := 197936, stop := 197943 ), rec( content := " ", 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 := 197953, root := ~, start := 197945, stop := 197952 ), rec( content := ") in ", 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 := 197972, root := ~, start := 197958, stop := 197971 ), rec( content := " from th\ e vertex ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 197997, root := ~, start := 197989, stop := 197996 ), rec( content := "\n to\ the vertex ", 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 := 198024, root := ~, start := 198016, stop := 198023 ), rec( content := ".\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 := 198034, root := ~, start := 198030, stop := 198033 ), rec( content := "\n\n \ See ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "Ref", next := 198069, root := ~, start := 198044, stop := 198068 ), 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, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Sect := "Iterators" ), content := 0, count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "Ref", next := 198223, root := ~, start := 198179, stop := 198222 ), rec( content := " for mor\ e information about iterators.\n See Section ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "Ref", next := 198319, root := ~, start := 198278, stop := 198318 ), rec( content := " for the\ definition of\n a directed path and a directed cycle.\n ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 20, 5 ], mathmode := "Text", name := "P", next := 198392, root := ~, start := 198388, stop := 198391 ), rec( content := "\n\n", count := [ 5, 3, 20, 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, 20, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "Example", next := 198877, root := ~, start := 198394, stop := 198876 ), rec( content := "\n ", count := [ 5, 3, 20, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 198894, root := ~, start := 197669, stop := 198893 ) ], count := [ 5, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 198908, root := ~, start := 197567, stop := 198907 ), rec( content := "\n\n ", count := [ 5, 3, 20, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAllSimpl\ eCircuits" ), content := 0, count := [ 5, 3, 21, 2 ], mathmode := "Text", name := "Attr", next := 198982, root := ~, start := 198929, stop := 198981 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Returns", next := 199032, root := ~, start := 198985, stop := 199031 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", 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 := 199070, root := ~, start := 199056, stop := 199069 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "C", next := 199121, root := ~, start := 199090, stop := 199120 ), rec( content := "\n re\ turns a list of the ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuits", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "E", next := 199170, root := ~, start := 199148, stop := 199169 ), rec( content := " in ", 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 := 199188, root := ~, start := 199174, stop := 199187 ), rec( content := ". ", 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 := 199194, root := ~, start := 199190, stop := 199193 ), rec( content := "\n\n \ See section ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "Ref", next := 199256, root := ~, start := 199212, stop := 199255 ), 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, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "P", next := 199368, root := ~, start := 199364, stop := 199367 ), 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, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "C", next := 199587, root := ~, start := 199556, stop := 199586 ), 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, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e, v)", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "M", next := 199750, root := ~, start := 199734, stop := 199749 ), rec( content := " and\n \ ", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e', v)", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "M", next := 199776, root := ~, start := 199759, stop := 199775 ), rec( content := " (for di\ stinct edges ", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "M", next := 199805, root := ~, start := 199797, stop := 199804 ), rec( content := " and ", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ '", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "M", next := 199819, root := ~, start := 199810, stop := 199818 ), rec( content := ") are\n \ mathematically distinct, ", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "C", next := 199885, root := ~, start := 199854, stop := 199884 ), rec( content := " conside\ rs them to\n be the same. ", count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "P", next := 199925, root := ~, start := 199921, stop := 199924 ), rec( content := "\n\n \ With this approach, a directed circuit of length ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "C", next := 199988, root := ~, start := 199980, stop := 199987 ), rec( content := " can be \ determined\n by a list of its first ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "C", next := 200042, root := ~, start := 200034, stop := 200041 ), rec( content := " vertice\ s. Thus a simple circuit ", count := [ 5, 3, 21, 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, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "M", next := 200141, root := ~, start := 200075, stop := 200140 ), rec( content := " can be \ represented as\n the list ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v_1, \\ldots, v_n]", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "M", next := 200202, root := ~, start := 200177, stop := 200201 ), rec( content := ", or any\ cyclic permutation thereof. For\n each simple circuit of ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "A", next := 200285, root := ~, start := 200271, stop := 200284 ), rec( content := ",\n " , count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "A", next := 2003\ 33, root := ~, start := 200319, stop := 200332 ), rec( content := ")", count := [ 5, 3, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "C", next := 200338, root := ~, start := 200291, stop := 200337 ), rec( content := " include\ s precisely one such\n list to represent the circuit. ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 21, 7 ], mathmode := "Text", name := "P", next := 200407, root := ~, start := 200403, stop := 200406 ), rec( content := "\n\n \ ", count := [ 5, 3, 21, 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, 21, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "Example", next := 201060, root := ~, start := 200413, stop := 201059 ), rec( content := "\n ", count := [ 5, 3, 21, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Description", next := 201077, root := ~, start := 199035, stop := 201076 ) ], count := [ 5, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 201091, root := ~, start := 198914, stop := 201090 ), rec( content := "\n\n ", count := [ 5, 3, 21, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLongestS\ impleCircuit" ), content := 0, count := [ 5, 3, 22, 2 ], mathmode := "Text", name := "Attr", next := 201168, root := ~, start := 201112, stop := 201167 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices, 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 := 201214, root := ~, start := 201203, stop := 201213 ), rec( content := ".", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 201225, root := ~, start := 201171, stop := 201224 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 201263, root := ~, start := 201249, stop := 201262 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLongestSimpleCircuit", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "C", next := 201317, root := ~, start := 201283, stop := 201316 ), rec( content := "\n re\ turns the longest ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuit", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "E", next := 201363, root := ~, start := 201342, stop := 201362 ), rec( content := " in ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 201381, root := ~, start := 201367, stop := 201380 ), rec( content := ". See se\ ction\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 201443, root := ~, start := 201399, stop := 201442 ), rec( content := " for the\ definition of simple\n circuit, and the definition of length for a simple \ circuit.", 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 := 201540, root := ~, start := 201536, stop := 201539 ), rec( content := "\n\n \ This attribute computes\n ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "C", next := 201606, root := ~, start := 201574, stop := 201605 ), 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 := 201620, root := ~, start := 201606, stop := 201619 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "C", next := 201628, root := ~, start := 201620, stop := 201627 ), rec( content := " to find\ all the\n simple circuits of ", 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 := 201682, root := ~, start := 201668, stop := 201681 ), 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, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "Ref", next := 201853, root := ~, start := 201815, stop := 201852 ), rec( content := ".", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "P", next := 201858, root := ~, start := 201854, stop := 201857 ), rec( content := "\n\n \ If ", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "A", next := 201881, root := ~, start := 201867, stop := 201880 ), rec( content := " has no \ simple circuits, then this attribute returns\n ", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "K", next := 201949, root := ~, start := 201938, stop := 201948 ), rec( content := ". If ", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "A", next := 201969, root := ~, start := 201955, stop := 201968 ), rec( content := " has mul\ tiple simple circuits of maximal\n length, then this attribute returns one \ of them.", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 22, 6 ], mathmode := "Text", name := "P", next := 202066, root := ~, start := 202062, stop := 202065 ), rec( content := "\n\n \ ", count := [ 5, 3, 22, 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, 22, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 22, 7 ], mathmode := "Text", name := "Example", next := 202538, root := ~, start := 202072, stop := 202537 ), rec( content := "\n ", count := [ 5, 3, 22, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 202555, root := ~, start := 201228, stop := 202554 ) ], count := [ 5, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 202569, root := ~, start := 201097, stop := 202568 ), rec( content := "\n\n ", count := [ 5, 3, 22, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphLayers" ), content := 0, count := [ 5, 3, 23, 2 ], mathmode := "Text", name := "Oper", next := 202640, root := ~, start := 202590, stop := 202639 ), rec( attributes := rec( ), content := [ rec( content := "A list.", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 202669, root := ~, start := 202643, stop := 202668 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a list ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "K", next := 202731, root := ~, start := 202720, stop := 202730 ), rec( content := " such th\ at ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i]", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "K", next := 202756, root := ~, start := 202742, stop := 202755 ), rec( content := " is the\\ n list of vertices whose minimum distance from the vertex ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 202837, root := ~, start := 202824, stop := 202836 ), rec( content := " in\n \ ", 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 := 202859, root := ~, start := 202845, stop := 202858 ), rec( content := " is ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "K", next := 202875, root := ~, start := 202863, stop := 202874 ), rec( content := ". Vertex\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 202897, root := ~, start := 202884, stop := 202896 ), rec( content := " is\n \ assumed to be at distance ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "K", next := 202939, root := ~, start := 202931, stop := 202938 ), rec( content := " from it\ self.\n ", count := [ 5, 3, 23, 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, 23, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Example", next := 203069, root := ~, start := 202957, stop := 203068 ), rec( content := "\n ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 203086, root := ~, start := 202672, stop := 203085 ) ], count := [ 5, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 203100, root := ~, start := 202575, stop := 203099 ), rec( content := "\n\n ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cy" ), content := 0, count := [ 5, 3, 24, 2 ], mathmode := "Text", name := "Attr", next := 203167, root := ~, start := 203121, stop := 203166 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer, or ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "K", next := 203217, root := ~, start := 203206, stop := 203216 ), rec( content := ".", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Returns", next := 203228, root := ~, start := 203170, stop := 203227 ), 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 := 203266, root := ~, start := 203252, stop := 203265 ), rec( content := " is a sy\ mmetric digraph without multiple edges - see\n ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Ref", next := 203355, root := ~, start := 203323, stop := 203354 ), rec( content := " and ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Ref", next := 203388, root := ~, start := 203360, stop := 203387 ), rec( content := " - then\\ n this attribute returns the degeneracy of ", 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 := 203455, root := ~, start := 203441, stop := 203454 ), 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 := 203461, root := ~, start := 203457, stop := 203460 ), rec( content := "\n\n \ The degeneracy of a digraph is the least integer ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "C", next := 203524, root := ~, start := 203516, stop := 203523 ), rec( content := " such\n \ that every induced of ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "A", next := 203570, root := ~, start := 203556, stop := 203569 ), rec( content := " contain\ s a vertex whose number of\n neighbours (excluding itself) is at most ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "C", next := 203658, root := ~, start := 203650, stop := 203657 ), rec( content := ". Note t\ hat this means\n that loops are ignored.", 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 := 203712, root := ~, start := 203708, stop := 203711 ), rec( content := "\n\n \ If ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "A", next := 203735, root := ~, start := 203721, stop := 203734 ), rec( content := " is not \ symmetric or has multiple edges then this\n attribute returns ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "K", next := 203818, root := ~, start := 203807, stop := 203817 ), rec( content := ".\n " , count := [ 5, 3, 24, 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, 24, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "Example", next := 204273, root := ~, start := 203824, stop := 204272 ), rec( content := "\n ", count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Description", next := 204290, root := ~, start := 203231, stop := 204289 ) ], count := [ 5, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 204304, root := ~, start := 203106, stop := 204303 ), rec( content := "\n\n ", count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cyOrdering" ), content := 0, count := [ 5, 3, 25, 2 ], mathmode := "Text", name := "Attr", next := 204379, root := ~, start := 204325, stop := 204378 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers, 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 := 204425, root := ~, start := 204414, stop := 204424 ), rec( content := ".", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Returns", next := 204436, root := ~, start := 204382, stop := 204435 ), 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 := 204474, root := ~, start := 204460, stop := 204473 ), rec( content := " is a di\ graph for which\n ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "C", next := 204527, root := ~, start := 204502, stop := 204526 ), 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 := 204541, root := ~, start := 204527, stop := 204540 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "C", next := 204549, root := ~, start := 204541, stop := 204548 ), rec( content := " is a no\ n-negative integer\n ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "C", next := 204588, root := ~, start := 204580, stop := 204587 ), rec( content := " - see " , count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Degeneracy" ), content := 0, count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Ref", next := 204626, root := ~, start := 204595, stop := 204625 ), rec( content := " - then \ this attribute\n returns a degeneracy ordering of the vertices of the verti\ ces of\n ", 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 := 204736, root := ~, start := 204722, stop := 204735 ), rec( content := ".", 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 := 204741, root := ~, start := 204737, stop := 204740 ), rec( content := "\n\n \ A degeneracy ordering 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 := 204786, root := ~, start := 204772, stop := 204785 ), rec( content := " is a li\ st ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 204812, root := ~, start := 204797, stop := 204811 ), rec( content := " of the\\ n vertices 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 := 204850, root := ~, start := 204836, stop := 204849 ), rec( content := " ordered\ such that for any\n position ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 204898, root := ~, start := 204890, stop := 204897 ), rec( content := " of the \ list, the vertex ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering[i]", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 204941, root := ~, start := 204923, stop := 204940 ), rec( content := " has at \ most\n ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 204966, root := ~, start := 204958, stop := 204965 ), rec( content := " neighbo\ urs in later position of the list.", 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 := 205012, root := ~, start := 205008, stop := 205011 ), rec( content := "\n\n \ If ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "C", next := 205046, root := ~, start := 205021, stop := 205045 ), 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 := 205060, root := ~, start := 205046, stop := 205059 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "C", next := 205068, root := ~, start := 205060, stop := 205067 ), rec( content := " returns\ ", 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 := 205088, root := ~, start := 205077, stop := 205087 ), rec( content := ",\n t\ hen this attribute returns ", 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 := 205133, root := ~, start := 205122, stop := 205132 ), rec( content := ".\n " , count := [ 5, 3, 25, 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, 25, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 6 ], mathmode := "Text", name := "Example", next := 205688, root := ~, start := 205139, stop := 205687 ), rec( content := "\n ", count := [ 5, 3, 25, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Description", next := 205705, root := ~, start := 204439, stop := 205704 ) ], count := [ 5, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 205719, root := ~, start := 204310, stop := 205718 ), rec( content := "\n\n ", count := [ 5, 3, 25, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "HamiltonianPath\ " ), content := 0, count := [ 5, 3, 26, 2 ], mathmode := "Text", name := "Attr", next := 205784, root := ~, start := 205740, stop := 205783 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 205817, root := ~, start := 205806, stop := 205816 ), rec( content := ".", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Returns", next := 205828, root := ~, start := 205787, stop := 205827 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a Hamiltonian path if one exists, ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 205902, root := ~, start := 205891, stop := 205901 ), rec( content := " if not.\ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "P", next := 205914, root := ~, start := 205910, stop := 205913 ), rec( content := "\n\n \ A ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian path", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "E", next := 205945, root := ~, start := 205922, stop := 205944 ), rec( content := " of a di\ graph with ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 205972, root := ~, start := 205964, stop := 205971 ), rec( content := " vertice\ s is directed\n cycle of length ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 206022, root := ~, start := 206014, stop := 206021 ), rec( content := ". If ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "A", next := 206041, root := ~, start := 206027, stop := 206040 ), 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, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "Ref", next := 206197, root := ~, start := 206159, stop := 206196 ), rec( content := ". Note i\ f ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "A", next := 206221, root := ~, start := 206207, stop := 206220 ), rec( content := " has\n \ 0 or 1 vertices, then ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonianPath", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 206274, root := ~, start := 206252, stop := 206273 ), rec( content := " returns\ ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ]", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 206292, root := ~, start := 206283, stop := 206291 ), rec( content := " or\n \ ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 206310, root := ~, start := 206300, stop := 206309 ), rec( content := ", respec\ tively.", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "P", next := 206329, root := ~, start := 206325, stop := 206328 ), rec( content := "\n\n \ The method used in this attribute has the same worst case complexity as\n " , count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "Ref", next := 206444, root := ~, start := 206411, stop := 206443 ), rec( content := ".\n\n \ ", count := [ 5, 3, 26, 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, 26, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 26, 6 ], mathmode := "Text", name := "Example", next := 207052, root := ~, start := 206451, stop := 207051 ), rec( content := "\n ", count := [ 5, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Description", next := 207069, root := ~, start := 205831, stop := 207068 ) ], count := [ 5, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 207083, root := ~, start := 205725, stop := 207082 ), rec( content := "\n\n ", count := [ 5, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "NrSpanningTrees\ " ), content := 0, count := [ 5, 3, 27, 2 ], mathmode := "Text", name := "Attr", next := 207148, root := ~, start := 207104, stop := 207147 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Returns", next := 207181, root := ~, start := 207151, stop := 207180 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of spanning trees of the symmetric digraph\n ", 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 := 207282, root := ~, start := 207268, stop := 207281 ), rec( content := ".\n " , count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ rSpanningTrees", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "C", next := 207310, root := ~, start := 207288, stop := 207309 ), rec( content := " will re\ turn an error 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 := 207349, root := ~, start := 207335, stop := 207348 ), rec( content := " is not \ a\n symmetric digraph. ", 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 := 207386, root := ~, start := 207382, stop := 207385 ), rec( content := " \n\n \ See ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "Ref", next := 207429, root := ~, start := 207397, stop := 207428 ), rec( content := " and\n \ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "Ref", next := 207477, root := ~, start := 207438, stop := 207476 ), rec( content := " for mor\ e information. \n\n ", count := [ 5, 3, 27, 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, 27, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "Example", next := 207903, root := ~, start := 207506, stop := 207902 ), rec( content := "\n ", count := [ 5, 3, 27, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Description", next := 207920, root := ~, start := 207184, stop := 207919 ) ], count := [ 5, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 207934, root := ~, start := 207089, stop := 207933 ), rec( content := "\n\n ", count := [ 5, 3, 27, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 207948, root := ~, start := 169041, stop := 207947 ) , rec( content := "\n\n ", count := [ 5, 3, 27, 6 ], 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 := 208003, root := ~, start := 207961, stop := 208002 ), 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 := 208072, root := ~, start := 208023, stop := 208071 ), rec( attributes := rec( Arg := "digraph", Name := "SemigroupOfCayl\ eyDigraph" ), content := 0, count := [ 5, 4, 1, 4 ], mathmode := "Text", name := "Attr", next := 208128, root := ~, start := 208075, stop := 208127 ), 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 := 208171, root := ~, start := 208131, stop := 208170 ), 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 := 208209, root := ~, start := 208195, stop := 208208 ), 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 := 208258, root := ~, start := 208250, stop := 208257 ), 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 := 208281, root := ~, start := 208267, stop := 208280 ), 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 := 208335, root := ~, start := 208306, stop := 208334 ), 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 := 208373, root := ~, start := 208346, stop := 208372 ), 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 := 208390, root := ~, start := 208382, stop := 208389 ), 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 := 208400, root := ~, start := 208396, stop := 208399 ), 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 := 208423, root := ~, start := 208409, stop := 208422 ), 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 := 208465, root := ~, start := 208457, stop := 208464 ), 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 := 208488, root := ~, start := 208474, stop := 208487 ), 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 := 208542, root := ~, start := 208513, stop := 208541 ), 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 := 208584, root := ~, start := 208553, stop := 208583 ), 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 := 208601, root := ~, start := 208593, stop := 208600 ), 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 := 208611, root := ~, start := 208607, stop := 208610 ), 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 := 208665, root := ~, start := 208626, stop := 208664 ), 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 := 208897, root := ~, start := 208671, stop := 208896 ), rec( content := "\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Description", next := 208914, root := ~, start := 208174, stop := 208913 ) ], count := [ 5, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 208928, root := ~, start := 208008, stop := 208927 ), 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 := 209003, root := ~, start := 208949, stop := 209002 ), 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 := 209046, root := ~, start := 209006, stop := 209045 ), 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 := 209084, root := ~, start := 209070, stop := 209083 ), 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 := 209189, root := ~, start := 209178, stop := 209188 ), 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 := 209208, root := ~, start := 209194, stop := 209207 ), 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 := 209266, root := ~, start := 209237, stop := 209265 ), 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 := 209309, root := ~, start := 209277, stop := 209308 ), 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 := 209351, root := ~, start := 209340, stop := 209350 ), 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 := 209381, root := ~, start := 209367, stop := 209380 ), 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 := 209402, root := ~, start := 209398, stop := 209401 ), 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 := 209451, root := ~, start := 209417, stop := 209450 ), 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 := 209498, root := ~, start := 209460, stop := 209497 ), 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 := 209891, root := ~, start := 209504, stop := 209890 ), rec( content := "\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 209908, root := ~, start := 209049, stop := 209907 ) ], count := [ 5, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 209922, root := ~, start := 208934, stop := 209921 ), rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 209936, root := ~, start := 207952, stop := 209935 ) , 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 := 209989, root := ~, start := 209949, stop := 209988 ), 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 := 210100, root := ~, start := 210013, stop := 210099 ), rec( attributes := rec( Arg := "filt, digraph", Name := "AsMonoid" ), content := 0, count := [ 5, 5, 1, 4 ], mathmode := "Text", name := "Oper", next := 210148, root := ~, start := 210105, stop := 210147 ), 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 := 210203, root := ~, start := 210153, stop := 210202 ), 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 := 210260, root := ~, start := 210242, stop := 210259 ), 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 := 210286, root := ~, start := 210275, stop := 210285 ), 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 := 210356, root := ~, start := 210305, stop := 210355 ), 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 := 210381, root := ~, start := 210367, stop := 210380 ), 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 := 210431, root := ~, start := 210393, stop := 210430 ), 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 := 210465, root := ~, start := 210435, stop := 210464 ), 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 := 210495, root := ~, start := 210477, stop := 210494 ), 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 := 210631, root := ~, start := 210617, stop := 210630 ), 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 := 210711, root := ~, start := 210664, stop := 210710 ), 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 := 210736, root := ~, start := 210722, stop := 210735 ), 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 := 210792, root := ~, start := 210754, stop := 210791 ), 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 := 210845, root := ~, start := 210807, stop := 210844 ), 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 := 210875, root := ~, start := 210857, stop := 210874 ), 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 := 211011, root := ~, start := 210997, stop := 211010 ), 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 := 211091, root := ~, start := 211044, stop := 211090 ), 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 := 211103, root := ~, start := 211099, stop := 211102 ), 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 := 211140, root := ~, start := 211125, stop := 211139 ), 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 := 211180, root := ~, start := 211162, stop := 211179 ), 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 := 211210, root := ~, start := 211199, stop := 211209 ), 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 := 211286, root := ~, start := 211238, stop := 211285 ), 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 := 211308, root := ~, start := 211294, stop := 211307 ), 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 := 211352, root := ~, start := 211322, stop := 211351 ), 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 := 211422, root := ~, start := 211385, stop := 211421 ), 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 := 211428, root := ~, start := 211424, stop := 211427 ), 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 := 211581, root := ~, start := 211524, stop := 211580 ), 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 := 211656, root := ~, start := 211606, stop := 211655 ), 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 := 212288, root := ~, start := 211746, stop := 212287 ), rec( content := "\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Description", next := 212307, root := ~, start := 210208, stop := 212306 ) ], count := [ 5, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 212323, root := ~, start := 209996, stop := 212322 ), 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 := 212462, root := ~, start := 212348, stop := 212461 ), 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 := 212526, root := ~, start := 212467, stop := 212525 ), 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 := 212583, root := ~, start := 212565, stop := 212582 ), 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 := 212609, root := ~, start := 212598, stop := 212608 ), 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 := 212679, root := ~, start := 212628, stop := 212678 ), 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 := 212698, root := ~, start := 212690, stop := 212697 ), 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 := 212742, root := ~, start := 212704, stop := 212741 ), 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 := 212790, root := ~, start := 212752, stop := 212789 ), 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 := 212808, root := ~, start := 212798, stop := 212807 ), 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 := 212880, root := ~, start := 212869, stop := 212879 ), 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 := 212929, root := ~, start := 212916, stop := 212928 ), 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 := 212981, root := ~, start := 212967, stop := 212980 ), 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 := 213009, root := ~, start := 212991, stop := 213008 ), 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 := 213032, root := ~, start := 213022, stop := 213031 ), 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 := 213075, root := ~, start := 213062, stop := 213074 ), 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 := 213098, root := ~, start := 213085, stop := 213097 ), 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 := 213172, root := ~, start := 213154, stop := 213171 ), 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 := 213294, root := ~, start := 213272, stop := 213293 ), 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 := 213871, root := ~, start := 213302, stop := 213870 ), rec( content := "\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 213890, root := ~, start := 212531, stop := 213889 ) ], count := [ 5, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 213906, root := ~, start := 212331, stop := 213905 ), rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 213920, root := ~, start := 209940, stop := 213919 ) , 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 := 213961, root := ~, start := 213933, stop := 213960 ), 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 := 214036, root := ~, start := 213981, stop := 214035 ), 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 := 214069, root := ~, start := 214058, stop := 214068 ), rec( content := ".", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Returns", next := 214080, root := ~, start := 214039, stop := 214079 ), 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 := 214134, root := ~, start := 214101, stop := 214133 ), 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 := 214268, root := ~, start := 214254, stop := 214267 ), 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 := 214312, root := ~, start := 214298, stop := 214311 ), 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 := 214346, root := ~, start := 214335, stop := 214345 ), 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 := 214364, root := ~, start := 214350, stop := 214363 ), 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 := 214430, root := ~, start := 214397, stop := 214429 ), 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 := 214495, root := ~, start := 214481, stop := 214494 ), 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 := 214556, root := ~, start := 214548, stop := 214555 ), 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 := 214640, root := ~, start := 214632, stop := 214639 ), 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 := 214653, root := ~, start := 214645, stop := 214652 ), 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 := 214659, root := ~, start := 214655, stop := 214658 ), 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 := 214733, root := ~, start := 214719, stop := 214732 ), 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 := 214796, root := ~, start := 214782, stop := 214795 ), 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 := 214812, root := ~, start := 214808, stop := 214811 ), 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 := 214856, root := ~, start := 214827, stop := 214855 ), 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 := 214906, root := ~, start := 214865, stop := 214905 ), 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 := 214916, root := ~, start := 214912, stop := 214915 ), 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 := 215078, root := ~, start := 215060, stop := 215077 ), 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 := 216380, root := ~, start := 215081, stop := 216379 ), rec( content := "\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Description", next := 216397, root := ~, start := 214083, stop := 216396 ) ], count := [ 5, 6, 1, 1 ], mathmode := "Text", name := "ManSection", next := 216411, root := ~, start := 213966, stop := 216410 ), 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 := 216492, root := ~, start := 216432, stop := 216491 ), 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 := 216525, root := ~, start := 216514, stop := 216524 ), rec( content := ".", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Returns", next := 216536, root := ~, start := 216495, stop := 216535 ), 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 := 216595, root := ~, start := 216557, stop := 216594 ), 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 := 216739, root := ~, start := 216725, stop := 216738 ), 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 := 216787, root := ~, start := 216773, stop := 216786 ), 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 := 216823, root := ~, start := 216812, stop := 216822 ), 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 := 216845, root := ~, start := 216831, stop := 216844 ), 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 := 216922, root := ~, start := 216884, stop := 216921 ), 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 := 216987, root := ~, start := 216973, stop := 216986 ), 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 := 217048, root := ~, start := 217040, stop := 217047 ), 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 := 217132, root := ~, start := 217124, stop := 217131 ), 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 := 217145, root := ~, start := 217137, stop := 217144 ), 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 := 217151, root := ~, start := 217147, stop := 217150 ), 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 := 217225, root := ~, start := 217211, stop := 217224 ), 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 := 217288, root := ~, start := 217274, stop := 217287 ), 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 := 217315, root := ~, start := 217311, stop := 217314 ), 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 := 217369, root := ~, start := 217335, stop := 217368 ), 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 := 217415, root := ~, start := 217375, stop := 217414 ), 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 := 217467, root := ~, start := 217426, stop := 217466 ), 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 := 217477, root := ~, start := 217473, stop := 217476 ), 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 := 217635, root := ~, start := 217617, stop := 217634 ), 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 := 219128, root := ~, start := 217638, stop := 219127 ), rec( content := "\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Description", next := 219145, root := ~, start := 216539, stop := 219144 ) ], count := [ 5, 6, 2, 1 ], mathmode := "Text", name := "ManSection", next := 219159, root := ~, start := 216417, stop := 219158 ), 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 := 219224, root := ~, start := 219180, stop := 219223 ), 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 := 219257, root := ~, start := 219246, stop := 219256 ), rec( content := ".", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Returns", next := 219268, root := ~, start := 219227, stop := 219267 ), 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 := 219306, root := ~, start := 219292, stop := 219305 ), 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 := 219360, root := ~, start := 219338, stop := 219359 ), 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 := 219452, root := ~, start := 219438, stop := 219451 ), 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 := 219539, root := ~, start := 219525, stop := 219538 ), 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 := 219575, root := ~, start := 219564, stop := 219574 ), 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 := 219593, root := ~, start := 219589, stop := 219592 ), 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 := 219667, root := ~, start := 219653, stop := 219666 ), 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 := 219709, root := ~, start := 219687, stop := 219708 ), 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 := 219719, root := ~, start := 219715, stop := 219718 ), 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 := 219768, root := ~, start := 219739, stop := 219767 ), 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 := 219778, root := ~, start := 219774, stop := 219777 ), 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 := 219936, root := ~, start := 219918, stop := 219935 ), 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 := 221154, root := ~, start := 219939, stop := 221153 ), rec( content := "\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Description", next := 221171, root := ~, start := 219271, stop := 221170 ) ], count := [ 5, 6, 3, 1 ], mathmode := "Text", name := "ManSection", next := 221185, root := ~, start := 219165, stop := 221184 ), 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 := 221255, root := ~, start := 221206, stop := 221254 ), 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 := 221288, root := ~, start := 221277, stop := 221287 ), rec( content := ".", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Returns", next := 221299, root := ~, start := 221258, stop := 221298 ), 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 := 221337, root := ~, start := 221323, stop := 221336 ), 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 := 221402, root := ~, start := 221375, stop := 221401 ), 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 := 221494, root := ~, start := 221480, stop := 221493 ), 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 := 221581, root := ~, start := 221567, stop := 221580 ), 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 := 221623, root := ~, start := 221612, stop := 221622 ), 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 := 221641, root := ~, start := 221637, stop := 221640 ), 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 := 221715, root := ~, start := 221701, stop := 221714 ), 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 := 221762, root := ~, start := 221735, stop := 221761 ), 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 := 221772, root := ~, start := 221768, stop := 221771 ), 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 := 221821, root := ~, start := 221787, stop := 221820 ), 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 := 221831, root := ~, start := 221827, stop := 221830 ), 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 := 221989, root := ~, start := 221971, stop := 221988 ), 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 := 223227, root := ~, start := 221992, stop := 223226 ), rec( content := "\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Description", next := 223244, root := ~, start := 221302, stop := 223243 ) ], count := [ 5, 6, 4, 1 ], mathmode := "Text", name := "ManSection", next := 223258, root := ~, start := 221191, stop := 223257 ), 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 := 223335, root := ~, start := 223279, stop := 223334 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK33" ), content := 0, count := [ 5, 6, 5, 4 ], mathmode := "Text", name := "Attr", next := 223394, root := ~, start := 223338, stop := 223393 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK4" ), content := 0, count := [ 5, 6, 5, 6 ], mathmode := "Text", name := "Attr", next := 223452, root := ~, start := 223397, stop := 223451 ), 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 := 223485, root := ~, start := 223474, stop := 223484 ), rec( content := ".", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Returns", next := 223496, root := ~, start := 223455, stop := 223495 ), 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 := 223636, root := ~, start := 223622, stop := 223635 ), 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 := 223755, root := ~, start := 223747, stop := 223754 ), 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 := 223772, root := ~, start := 223764, stop := 223771 ), 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 := 223837, root := ~, start := 223829, stop := 223836 ), 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 := 223854, root := ~, start := 223846, stop := 223853 ), 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 := 223891, root := ~, start := 223883, stop := 223890 ), 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 := 223923, root := ~, start := 223909, stop := 223922 ), 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 := 223965, root := ~, start := 223954, stop := 223964 ), 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 := 223988, root := ~, start := 223984, stop := 223987 ), 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 := 224032, root := ~, start := 224003, stop := 224031 ), 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 := 224077, root := ~, start := 224037, stop := 224076 ), 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 := 224099, root := ~, start := 224095, stop := 224098 ), 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 := 224257, root := ~, start := 224239, stop := 224256 ), 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 := 226569, root := ~, start := 224260, stop := 226568 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Description", next := 226586, root := ~, start := 223499, stop := 226585 ) ], count := [ 5, 6, 5, 1 ], mathmode := "Text", name := "ManSection", next := 226600, root := ~, start := 223264, stop := 226599 ), rec( content := "\n\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 1 ], mathmode := "Text", name := "Section", next := 226614, root := ~, start := 213924, stop := 226613 ) , rec( content := "\n\n", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 226626, root := ~, start := 130877, stop := 226625 ), 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 := 226743, root := ~, start := 226702, stop := 226742 ) , 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 := 226790, root := ~, start := 226756, stop := 226789 ), 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 := 226854, root := ~, start := 226810, stop := 226853 ), 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 := 226877, root := ~, start := 226866, stop := 226876 ), 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 := 226893, root := ~, start := 226881, stop := 226892 ), rec( content := ".", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 226904, root := ~, start := 226857, stop := 226903 ), 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 := 226944, root := ~, start := 226933, stop := 226943 ), 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 := 226974, root := ~, start := 226960, stop := 226973 ), 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 := 227006, root := ~, start := 226994, stop := 227005 ), 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 := 227078, root := ~, start := 227074, stop := 227077 ), 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 := 227120, root := ~, start := 227106, stop := 227119 ), 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 := 227715, root := ~, start := 227217, stop := 227714 ), rec( content := "\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 227732, root := ~, start := 226907, stop := 227731 ) ], count := [ 6, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 227746, root := ~, start := 226795, stop := 227745 ), 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 := 227818, root := ~, start := 227767, stop := 227817 ), rec( attributes := rec( Arg := "digraph", Name := "IsAntisymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 4 ], mathmode := "Text", name := "Prop", next := 227872, root := ~, start := 227821, stop := 227871 ), 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 := 227895, root := ~, start := 227884, stop := 227894 ), 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 := 227911, root := ~, start := 227899, stop := 227910 ), rec( content := ".", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 227922, root := ~, start := 227875, stop := 227921 ), 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 := 227971, root := ~, start := 227960, stop := 227970 ), 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 := 228001, root := ~, start := 227987, stop := 228000 ), 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 := 228040, root := ~, start := 228028, stop := 228039 ), 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 := 228063, root := ~, start := 228059, stop := 228062 ), 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 := 228102, root := ~, start := 228082, stop := 228101 ), 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 := 228156, root := ~, start := 228148, stop := 228155 ), 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 := 228175, root := ~, start := 228167, stop := 228174 ), 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 := 228209, root := ~, start := 228201, stop := 228208 ), 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 := 228232, root := ~, start := 228224, stop := 228231 ), 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 := 228260, root := ~, start := 228252, stop := 228259 ), 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 := 228273, root := ~, start := 228265, stop := 228272 ), 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 := 228293, root := ~, start := 228289, stop := 228292 ), 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 := 228335, root := ~, start := 228321, stop := 228334 ), 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 := 228862, root := ~, start := 228436, stop := 228861 ), rec( content := "\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 228879, root := ~, start := 227925, stop := 228878 ) ], count := [ 6, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 228893, root := ~, start := 227752, stop := 228892 ), 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 := 228961, root := ~, start := 228914, stop := 228960 ), 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 := 228984, root := ~, start := 228973, stop := 228983 ), 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 := 229000, root := ~, start := 228988, stop := 228999 ), rec( content := ".", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 229011, root := ~, start := 228964, stop := 229010 ), 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 := 229058, root := ~, start := 229047, stop := 229057 ), 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 := 229088, root := ~, start := 229074, stop := 229087 ), 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 := 229121, root := ~, start := 229109, stop := 229120 ), 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 := 229207, root := ~, start := 229193, stop := 229206 ), 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 := 229313, root := ~, start := 229299, stop := 229312 ), 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 := 229473, root := ~, start := 229415, stop := 229472 ), 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 := 229479, root := ~, start := 229475, stop := 229478 ), 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 := 229525, root := ~, start := 229492, stop := 229524 ), 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 := 229535, root := ~, start := 229531, stop := 229534 ), 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 := 229577, root := ~, start := 229563, stop := 229576 ), 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 := 230050, root := ~, start := 229677, stop := 230049 ), rec( content := "\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 230067, root := ~, start := 229014, stop := 230066 ) ], count := [ 6, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 230081, root := ~, start := 228899, stop := 230080 ), 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 := 230157, root := ~, start := 230102, stop := 230156 ), 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 := 230180, root := ~, start := 230169, stop := 230179 ), 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 := 230196, root := ~, start := 230184, stop := 230195 ), rec( content := ".", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 230207, root := ~, start := 230160, stop := 230206 ), 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 := 230247, root := ~, start := 230236, stop := 230246 ), 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 := 230277, root := ~, start := 230263, stop := 230276 ), 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 := 230331, root := ~, start := 230319, stop := 230330 ), 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 := 230350, root := ~, start := 230346, stop := 230349 ), 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 := 230404, root := ~, start := 230371, stop := 230403 ), 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 := 230465, root := ~, start := 230433, stop := 230464 ), 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 := 230522, root := ~, start := 230514, stop := 230521 ), 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 := 230541, root := ~, start := 230533, stop := 230540 ), 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 := 230565, root := ~, start := 230557, stop := 230564 ), 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 := 230578, root := ~, start := 230570, stop := 230577 ), 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 := 230630, root := ~, start := 230616, stop := 230629 ), 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 := 230675, root := ~, start := 230636, stop := 230674 ), 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 := 230681, root := ~, start := 230677, stop := 230680 ), 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 := 230755, root := ~, start := 230747, stop := 230754 ), 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 := 230772, root := ~, start := 230764, stop := 230771 ), 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 := 230817, root := ~, start := 230807, stop := 230816 ), 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 := 230867, root := ~, start := 230863, stop := 230866 ), 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 := 230920, root := ~, start := 230882, stop := 230919 ), 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 := 230930, root := ~, start := 230926, stop := 230929 ), 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 := 230972, root := ~, start := 230958, stop := 230971 ), 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 := 231503, root := ~, start := 231073, stop := 231502 ), rec( content := "\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 231520, root := ~, start := 230210, stop := 231519 ) ], count := [ 6, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 231534, root := ~, start := 230087, stop := 231533 ), 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 := 231601, root := ~, start := 231555, stop := 231600 ), 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 := 231624, root := ~, start := 231613, stop := 231623 ), 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 := 231640, root := ~, start := 231628, stop := 231639 ), rec( content := ".", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 231651, root := ~, start := 231604, stop := 231650 ), 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 := 231691, root := ~, start := 231680, stop := 231690 ), 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 := 231721, root := ~, start := 231707, stop := 231720 ), 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 := 231755, root := ~, start := 231743, stop := 231754 ), 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 := 231774, root := ~, start := 231770, stop := 231773 ), 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 := 231808, root := ~, start := 231793, stop := 231807 ), 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 := 231860, root := ~, start := 231845, stop := 231859 ), 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 := 231878, root := ~, start := 231870, stop := 231877 ), 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 := 231891, root := ~, start := 231883, stop := 231890 ), 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 := 231943, root := ~, start := 231935, stop := 231942 ), 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 := 231962, root := ~, start := 231954, stop := 231961 ), 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 := 232006, root := ~, start := 231998, stop := 232005 ), 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 := 232069, root := ~, start := 232054, stop := 232068 ), 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 := 232118, root := ~, start := 232114, stop := 232117 ), 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 := 232160, root := ~, start := 232146, stop := 232159 ), 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 := 232668, root := ~, start := 232261, stop := 232667 ), rec( content := "\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 232685, root := ~, start := 231654, stop := 232684 ) ], count := [ 6, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 232699, root := ~, start := 231540, stop := 232698 ), 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 := 232778, root := ~, start := 232720, stop := 232777 ), 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 := 232801, root := ~, start := 232790, stop := 232800 ), 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 := 232817, root := ~, start := 232805, stop := 232816 ), rec( content := ".", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 232828, root := ~, start := 232781, stop := 232827 ), 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 := 232882, root := ~, start := 232871, stop := 232881 ), 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 := 232900, root := ~, start := 232886, stop := 232899 ), 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 := 232957, root := ~, start := 232945, stop := 232956 ), 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 := 232970, root := ~, start := 232966, stop := 232969 ), 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 := 233027, root := ~, start := 232991, stop := 233026 ), 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 := 233232, root := ~, start := 233228, stop := 233231 ), 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 := 233274, root := ~, start := 233260, stop := 233273 ), 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 := 233717, root := ~, start := 233375, stop := 233716 ), rec( content := "\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 233734, root := ~, start := 232831, stop := 233733 ) ], count := [ 6, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 233748, root := ~, start := 232705, stop := 233747 ), 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 := 233812, root := ~, start := 233769, stop := 233811 ), rec( attributes := rec( Arg := "digraph", Name := "IsNullDigraph" ), content := 0, count := [ 6, 1, 7, 4 ], mathmode := "Text", name := "Prop", next := 233857, root := ~, start := 233815, stop := 233856 ), 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 := 233880, root := ~, start := 233869, stop := 233879 ), 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 := 233896, root := ~, start := 233884, stop := 233895 ), rec( content := ".", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Returns", next := 233907, root := ~, start := 233860, stop := 233906 ), 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 := 233947, root := ~, start := 233936, stop := 233946 ), 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 := 233977, root := ~, start := 233963, stop := 233976 ), 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 := 234008, root := ~, start := 233996, stop := 234007 ), 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 := 234048, root := ~, start := 234036, stop := 234047 ), 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 := 234076, root := ~, start := 234072, stop := 234075 ), 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 := 234102, root := ~, start := 234082, stop := 234101 ), 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 := 234141, root := ~, start := 234120, stop := 234140 ), 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 := 234151, root := ~, start := 234147, stop := 234150 ), 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 := 234193, root := ~, start := 234179, stop := 234192 ), 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 := 234586, root := ~, start := 234294, stop := 234585 ), rec( content := "\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Description", next := 234603, root := ~, start := 233910, stop := 234602 ) ], count := [ 6, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 234617, root := ~, start := 233754, stop := 234616 ), 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 := 234687, root := ~, start := 234638, stop := 234686 ), 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 := 234710, root := ~, start := 234699, stop := 234709 ), 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 := 234726, root := ~, start := 234714, stop := 234725 ), rec( content := ".", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 234737, root := ~, start := 234690, stop := 234736 ), 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 := 234875, root := ~, start := 234843, stop := 234874 ), 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 := 234913, root := ~, start := 234881, stop := 234912 ), 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 := 234951, root := ~, start := 234918, stop := 234950 ), 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 := 234987, root := ~, start := 234973, stop := 234986 ), 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 := 235036, root := ~, start := 235032, stop := 235035 ), 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 := 235078, root := ~, start := 235064, stop := 235077 ), 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 := 235329, root := ~, start := 235175, stop := 235328 ), rec( content := "\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 235346, root := ~, start := 234740, stop := 235345 ) ], count := [ 6, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 235360, root := ~, start := 234623, stop := 235359 ), 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 := 235429, root := ~, start := 235381, stop := 235428 ), 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 := 235452, root := ~, start := 235441, stop := 235451 ), 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 := 235468, root := ~, start := 235456, stop := 235467 ), rec( content := ".", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 235479, root := ~, start := 235432, stop := 235478 ), 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 := 235528, root := ~, start := 235517, stop := 235527 ), 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 := 235558, root := ~, start := 235544, stop := 235557 ), 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 := 235582, root := ~, start := 235578, stop := 235581 ), 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 := 235618, root := ~, start := 235601, stop := 235617 ), 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 := 235679, root := ~, start := 235675, stop := 235678 ), 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 := 235721, root := ~, start := 235707, stop := 235720 ), 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 := 236232, root := ~, start := 235821, stop := 236231 ), rec( content := "\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 236249, root := ~, start := 235482, stop := 236248 ) ], count := [ 6, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 236263, root := ~, start := 235366, stop := 236262 ), 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 := 236327, root := ~, start := 236284, stop := 236326 ), 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 := 236350, root := ~, start := 236339, stop := 236349 ), 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 := 236366, root := ~, start := 236354, stop := 236365 ), rec( content := ".", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 236377, root := ~, start := 236330, stop := 236376 ), 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 := 236419, root := ~, start := 236400, stop := 236418 ), 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 := 236491, root := ~, start := 236487, stop := 236490 ), 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 := 236527, root := ~, start := 236513, stop := 236526 ), 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 := 237251, root := ~, start := 236628, stop := 237250 ), rec( content := "\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 237268, root := ~, start := 236380, stop := 237267 ) ], count := [ 6, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 237282, root := ~, start := 236269, stop := 237281 ), 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 := 237350, root := ~, start := 237303, stop := 237349 ), 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 := 237373, root := ~, start := 237362, stop := 237372 ), 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 := 237389, root := ~, start := 237377, stop := 237388 ), rec( content := ".", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Returns", next := 237400, root := ~, start := 237353, stop := 237399 ), 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 := 237449, root := ~, start := 237438, stop := 237448 ), 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 := 237479, root := ~, start := 237465, stop := 237478 ), 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 := 237514, root := ~, start := 237502, stop := 237513 ), 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 := 237562, root := ~, start := 237546, stop := 237561 ), 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 := 237601, root := ~, start := 237597, stop := 237600 ), 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 := 237610, root := ~, start := 237606, stop := 237609 ), 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 := 237652, root := ~, start := 237638, stop := 237651 ), 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 := 238025, root := ~, start := 237752, stop := 238024 ), rec( content := "\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Description", next := 238042, root := ~, start := 237403, stop := 238041 ) ], count := [ 6, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 238056, root := ~, start := 237288, stop := 238055 ), 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 := 238124, root := ~, start := 238077, stop := 238123 ), 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 := 238147, root := ~, start := 238136, stop := 238146 ), 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 := 238163, root := ~, start := 238151, stop := 238162 ), rec( content := ".", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 238174, root := ~, start := 238127, stop := 238173 ), 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 := 238223, root := ~, start := 238212, stop := 238222 ), 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 := 238253, root := ~, start := 238239, stop := 238252 ), 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 := 238288, root := ~, start := 238276, stop := 238287 ), 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 := 238306, root := ~, start := 238302, stop := 238305 ), 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 := 238338, root := ~, start := 238314, stop := 238337 ), 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 := 238402, root := ~, start := 238394, stop := 238401 ), 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 := 238421, root := ~, start := 238413, stop := 238420 ), 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 := 238477, root := ~, start := 238469, stop := 238476 ), 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 := 238496, root := ~, start := 238488, stop := 238495 ), 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 := 238520, root := ~, start := 238512, stop := 238519 ), 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 := 238551, root := ~, start := 238543, stop := 238550 ), 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 := 238570, root := ~, start := 238562, stop := 238569 ), 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 := 238609, root := ~, start := 238601, stop := 238608 ), 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 := 238640, root := ~, start := 238632, stop := 238639 ), 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 := 238659, root := ~, start := 238651, stop := 238658 ), 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 := 238764, root := ~, start := 238735, stop := 238763 ), 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 := 238774, root := ~, start := 238770, stop := 238773 ), 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 := 238816, root := ~, start := 238802, stop := 238815 ), 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 := 239536, root := ~, start := 238916, stop := 239535 ), rec( content := "\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 239553, root := ~, start := 238177, stop := 239552 ) ], count := [ 6, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 239567, root := ~, start := 238062, stop := 239566 ), 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 := 239629, root := ~, start := 239588, stop := 239628 ), 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 := 239652, root := ~, start := 239641, stop := 239651 ), 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 := 239668, root := ~, start := 239656, stop := 239667 ), rec( content := ".", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 239679, root := ~, start := 239632, stop := 239678 ), 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 := 239728, root := ~, start := 239717, stop := 239727 ), 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 := 239758, root := ~, start := 239744, stop := 239757 ), 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 := 239796, root := ~, start := 239784, stop := 239795 ), 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 := 239815, root := ~, start := 239811, stop := 239814 ), 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 := 240050, root := ~, start := 240046, stop := 240049 ), 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 := 240092, root := ~, start := 240078, stop := 240091 ), 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 := 240673, root := ~, start := 240192, stop := 240672 ), rec( content := "\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 240690, root := ~, start := 239682, stop := 240689 ) ], count := [ 6, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 240704, root := ~, start := 239573, stop := 240703 ), 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 := 240773, root := ~, start := 240725, stop := 240772 ), 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 := 240796, root := ~, start := 240785, stop := 240795 ), 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 := 240812, root := ~, start := 240800, stop := 240811 ), rec( content := ".", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 240823, root := ~, start := 240776, stop := 240822 ), 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 := 240872, root := ~, start := 240861, stop := 240871 ), 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 := 240902, root := ~, start := 240888, stop := 240901 ), 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 := 240938, root := ~, start := 240926, stop := 240937 ), 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 := 240988, root := ~, start := 240971, stop := 240987 ), 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 := 241016, root := ~, start := 241001, stop := 241015 ), 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 := 241040, root := ~, start := 241025, stop := 241039 ), 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 := 241087, root := ~, start := 241072, stop := 241086 ), 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 := 241128, root := ~, start := 241124, stop := 241127 ), 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 := 241146, root := ~, start := 241138, stop := 241145 ), 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 := 241222, root := ~, start := 241214, stop := 241221 ), 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 := 241392, root := ~, start := 241379, stop := 241391 ), 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 := 241494, root := ~, start := 241458, stop := 241493 ), 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 := 241557, root := ~, start := 241530, stop := 241556 ), 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 := 241597, root := ~, start := 241593, stop := 241596 ), 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 := 241639, root := ~, start := 241625, stop := 241638 ), 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 := 241743, root := ~, start := 241739, stop := 241742 ), 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 := 242205, root := ~, start := 241748, stop := 242204 ), rec( content := "\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 242222, root := ~, start := 240826, stop := 242221 ) ], count := [ 6, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 242236, root := ~, start := 240710, stop := 242235 ), 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 := 242303, root := ~, start := 242257, stop := 242302 ), rec( attributes := rec( Arg := "digraph", Name := "IsQuasiorderDig\ raph" ), content := 0, count := [ 6, 1, 15, 4 ], mathmode := "Text", name := "Prop", next := 242354, root := ~, start := 242306, stop := 242353 ), 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 := 242377, root := ~, start := 242366, stop := 242376 ), 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 := 242393, root := ~, start := 242381, stop := 242392 ), rec( content := ".", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 242404, root := ~, start := 242357, stop := 242403 ), 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 := 242536, root := ~, start := 242504, stop := 242535 ), 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 := 242574, root := ~, start := 242541, stop := 242573 ), 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 := 242637, root := ~, start := 242623, stop := 242636 ), 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 := 242690, root := ~, start := 242679, stop := 242689 ), 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 := 242717, root := ~, start := 242702, stop := 242716 ), 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 := 242750, root := ~, start := 242737, stop := 242749 ), 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 := 242779, root := ~, start := 242765, stop := 242778 ), 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 := 242789, root := ~, start := 242785, stop := 242788 ), 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 := 242831, root := ~, start := 242817, stop := 242830 ), 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 := 243464, root := ~, start := 242928, stop := 243463 ), rec( content := "\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 243481, root := ~, start := 242407, stop := 243480 ) ], count := [ 6, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 243495, root := ~, start := 242242, stop := 243494 ), 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 := 243566, root := ~, start := 243516, stop := 243565 ), 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 := 243589, root := ~, start := 243578, stop := 243588 ), 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 := 243605, root := ~, start := 243593, stop := 243604 ), rec( content := ".", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Returns", next := 243616, root := ~, start := 243569, stop := 243615 ), 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 := 243755, root := ~, start := 243723, stop := 243754 ), 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 := 243797, root := ~, start := 243761, stop := 243796 ), 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 := 243835, root := ~, start := 243802, stop := 243834 ), 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 := 243871, root := ~, start := 243857, stop := 243870 ), 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 := 243929, root := ~, start := 243918, stop := 243928 ), 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 := 243956, root := ~, start := 243941, stop := 243955 ), 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 := 243989, root := ~, start := 243976, stop := 243988 ), 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 := 244018, root := ~, start := 244004, stop := 244017 ), 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 := 244028, root := ~, start := 244024, stop := 244027 ), 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 := 244070, root := ~, start := 244056, stop := 244069 ), 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 := 244574, root := ~, start := 244167, stop := 244573 ), rec( content := "\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 244591, root := ~, start := 243619, stop := 244590 ) ], count := [ 6, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 244605, root := ~, start := 243501, stop := 244604 ), 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 := 244679, root := ~, start := 244626, stop := 244678 ), rec( attributes := rec( Arg := "digraph", Name := "IsJoinSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 4 ], mathmode := "Text", name := "Prop", next := 244735, root := ~, start := 244682, stop := 244734 ), rec( attributes := rec( Arg := "digraph", Name := "IsLatticeDigrap\ h" ), content := 0, count := [ 6, 1, 17, 6 ], mathmode := "Text", name := "Prop", next := 244783, root := ~, start := 244738, stop := 244782 ), 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 := 244806, root := ~, start := 244795, stop := 244805 ), 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 := 244822, root := ~, start := 244810, stop := 244821 ), rec( content := ".", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Returns", next := 244833, root := ~, start := 244786, stop := 244832 ), 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 := 244885, root := ~, start := 244854, stop := 244884 ), 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 := 244905, root := ~, start := 244894, stop := 244904 ), 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 := 244939, root := ~, start := 244925, stop := 244938 ), 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 := 244994, root := ~, start := 244963, stop := 244993 ), 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 := 245018, root := ~, start := 245007, stop := 245017 ), 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 := 245048, root := ~, start := 245034, stop := 245047 ), 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 := 245104, root := ~, start := 245081, stop := 245103 ), 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 := 245124, root := ~, start := 245113, stop := 245123 ), 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 := 245158, root := ~, start := 245144, stop := 245157 ), 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 := 245206, root := ~, start := 245202, stop := 245205 ), 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 := 245275, root := ~, start := 245240, stop := 245274 ), 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 := 245339, root := ~, start := 245328, stop := 245338 ), 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 := 245371, root := ~, start := 245356, stop := 245370 ), 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 := 245400, root := ~, start := 245387, stop := 245399 ), 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 := 245455, root := ~, start := 245432, stop := 245454 ), 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 := 245620, root := ~, start := 245597, stop := 245619 ), 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 := 245746, root := ~, start := 245742, stop := 245745 ), 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 := 245788, root := ~, start := 245774, stop := 245787 ), 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 := 246776, root := ~, start := 245889, stop := 246775 ), rec( content := "\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Description", next := 246793, root := ~, start := 244836, stop := 246792 ) ], count := [ 6, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 246807, root := ~, start := 244611, stop := 246806 ), rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 246821, root := ~, start := 226747, stop := 246820 ) , 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 := 246863, root := ~, start := 246834, stop := 246862 ), 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 := 246930, root := ~, start := 246883, stop := 246929 ), 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 := 246953, root := ~, start := 246942, stop := 246952 ), 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 := 246969, root := ~, start := 246957, stop := 246968 ), rec( content := ".", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 246980, root := ~, start := 246933, stop := 246979 ), 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 := 247029, root := ~, start := 247018, stop := 247028 ), 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 := 247061, root := ~, start := 247053, stop := 247060 ), 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 := 247101, root := ~, start := 247093, stop := 247100 ), 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 := 247127, root := ~, start := 247113, stop := 247126 ), 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 := 247158, root := ~, start := 247150, stop := 247157 ), 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 := 247188, root := ~, start := 247180, stop := 247187 ), 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 := 247237, root := ~, start := 247204, stop := 247236 ), 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 := 247276, root := ~, start := 247246, stop := 247275 ), 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 := 247286, root := ~, start := 247282, stop := 247285 ), 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 := 247328, root := ~, start := 247314, stop := 247327 ), 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 := 247558, root := ~, start := 247428, stop := 247557 ), rec( content := "\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 247575, root := ~, start := 246983, stop := 247574 ) ], count := [ 6, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 247589, root := ~, start := 246868, stop := 247588 ), 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 := 247658, root := ~, start := 247610, stop := 247657 ), 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 := 247681, root := ~, start := 247670, stop := 247680 ), 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 := 247697, root := ~, start := 247685, stop := 247696 ), rec( content := ".", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 247708, root := ~, start := 247661, stop := 247707 ), 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 := 247757, root := ~, start := 247746, stop := 247756 ), 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 := 247789, root := ~, start := 247781, stop := 247788 ), 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 := 247829, root := ~, start := 247821, stop := 247828 ), 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 := 247855, root := ~, start := 247841, stop := 247854 ), 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 := 247886, root := ~, start := 247878, stop := 247885 ), 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 := 247913, root := ~, start := 247905, stop := 247912 ), 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 := 247923, root := ~, start := 247919, stop := 247922 ), 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 := 247970, root := ~, start := 247938, stop := 247969 ), 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 := 248009, root := ~, start := 247979, stop := 248008 ), 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 := 248019, root := ~, start := 248015, stop := 248018 ), 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 := 248061, root := ~, start := 248047, stop := 248060 ), 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 := 248293, root := ~, start := 248161, stop := 248292 ), rec( content := "\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 248310, root := ~, start := 247711, stop := 248309 ) ], count := [ 6, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 248324, root := ~, start := 247595, stop := 248323 ), 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 := 248390, root := ~, start := 248345, stop := 248389 ), 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 := 248413, root := ~, start := 248402, stop := 248412 ), 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 := 248429, root := ~, start := 248417, stop := 248428 ), rec( content := ".", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 248440, root := ~, start := 248393, stop := 248439 ), 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 := 248489, root := ~, start := 248478, stop := 248488 ), 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 := 248521, root := ~, start := 248513, stop := 248520 ), 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 := 248561, root := ~, start := 248553, stop := 248560 ), 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 := 248587, root := ~, start := 248573, stop := 248586 ), 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 := 248618, root := ~, start := 248610, stop := 248617 ), 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 := 248661, root := ~, start := 248653, stop := 248660 ), 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 := 248710, root := ~, start := 248699, stop := 248709 ), 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 := 248728, root := ~, start := 248714, stop := 248727 ), 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 := 248819, root := ~, start := 248787, stop := 248818 ), 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 := 248861, root := ~, start := 248828, stop := 248860 ), 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 := 248871, root := ~, start := 248867, stop := 248870 ), 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 := 248913, root := ~, start := 248899, stop := 248912 ), 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 := 249139, root := ~, start := 249013, stop := 249138 ), rec( content := "\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 249156, root := ~, start := 248443, stop := 249155 ) ], count := [ 6, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 249170, root := ~, start := 248330, stop := 249169 ), 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 := 249244, root := ~, start := 249191, stop := 249243 ), 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 := 249267, root := ~, start := 249256, stop := 249266 ), 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 := 249283, root := ~, start := 249271, stop := 249282 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 249294, root := ~, start := 249247, stop := 249293 ), 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 := 249332, root := ~, start := 249318, stop := 249331 ), 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 := 249402, root := ~, start := 249391, stop := 249401 ), 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 := 249435, root := ~, start := 249427, stop := 249434 ), 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 := 249448, root := ~, start := 249440, stop := 249447 ), 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 := 249466, root := ~, start := 249452, stop := 249465 ), 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 := 249500, root := ~, start := 249492, stop := 249499 ), 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 := 249513, root := ~, start := 249505, stop := 249512 ), 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 := 249530, root := ~, start := 249522, stop := 249529 ), 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 := 249569, root := ~, start := 249555, stop := 249568 ), 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 := 249614, root := ~, start := 249606, stop := 249613 ), 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 := 249632, root := ~, start := 249624, stop := 249631 ), 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 := 249654, root := ~, start := 249646, stop := 249653 ), 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 := 249668, root := ~, start := 249660, stop := 249667 ), 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 := 249697, root := ~, start := 249689, stop := 249696 ), 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 := 249707, root := ~, start := 249699, stop := 249706 ), 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 := 249751, root := ~, start := 249743, stop := 249750 ), 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 := 249768, root := ~, start := 249760, stop := 249767 ), 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 := 249773, root := ~, start := 249769, stop := 249772 ), 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 := 249875, root := ~, start := 249865, stop := 249874 ), 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 := 249887, root := ~, start := 249877, stop := 249886 ), 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 := 249901, root := ~, start := 249893, stop := 249900 ), 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 := 249945, root := ~, start := 249937, stop := 249944 ), 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 := 249955, root := ~, start := 249947, stop := 249954 ), 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 := 249973, root := ~, start := 249959, stop := 249972 ), 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 := 250001, root := ~, start := 249993, stop := 250000 ), 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 := 250041, root := ~, start := 250031, stop := 250040 ), 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 := 250063, root := ~, start := 250055, stop := 250062 ), 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 := 250102, root := ~, start := 250090, stop := 250101 ), 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 := 250121, root := ~, start := 250113, stop := 250120 ), 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 := 250137, root := ~, start := 250127, stop := 250136 ), 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 := 250163, root := ~, start := 250155, stop := 250162 ), 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 := 250198, root := ~, start := 250186, stop := 250197 ), 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 := 250217, root := ~, start := 250209, stop := 250216 ), 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 := 250278, root := ~, start := 250264, stop := 250277 ), 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 := 250303, root := ~, start := 250299, stop := 250302 ), 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 := 250341, root := ~, start := 250327, stop := 250340 ), 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 := 250409, root := ~, start := 250397, stop := 250408 ), 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 := 250419, root := ~, start := 250415, stop := 250418 ), 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 := 250461, root := ~, start := 250447, stop := 250460 ), 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 := 250837, root := ~, start := 250562, stop := 250836 ), rec( content := "\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 250854, root := ~, start := 249297, stop := 250853 ) ], count := [ 6, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 250868, root := ~, start := 249176, stop := 250867 ), rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 250882, root := ~, start := 246825, stop := 250881 ) , 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 := 250937, root := ~, start := 250895, stop := 250936 ), 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 := 251002, root := ~, start := 250957, stop := 251001 ), 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 := 251025, root := ~, start := 251014, stop := 251024 ), 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 := 251041, root := ~, start := 251029, stop := 251040 ), rec( content := ".", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 251052, root := ~, start := 251005, stop := 251051 ), 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 := 251101, root := ~, start := 251090, stop := 251100 ), 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 := 251131, root := ~, start := 251117, stop := 251130 ), 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 := 251164, root := ~, start := 251152, stop := 251163 ), 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 := 251206, root := ~, start := 251192, stop := 251205 ), 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 := 251324, root := ~, start := 251274, stop := 251323 ), 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 := 251405, root := ~, start := 251401, stop := 251404 ), 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 := 251473, root := ~, start := 251460, stop := 251472 ), 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 := 251492, root := ~, start := 251484, stop := 251491 ), 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 := 251565, root := ~, start := 251557, stop := 251564 ), 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 := 251612, root := ~, start := 251608, stop := 251611 ), 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 := 251648, root := ~, start := 251634, stop := 251647 ), 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 := 252397, root := ~, start := 251749, stop := 252396 ), rec( content := "\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 252414, root := ~, start := 251055, stop := 252413 ) ], count := [ 6, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 252428, root := ~, start := 250942, stop := 252427 ), 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 := 252492, root := ~, start := 252449, stop := 252491 ), 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 := 252515, root := ~, start := 252504, stop := 252514 ), 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 := 252531, root := ~, start := 252519, stop := 252530 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 252542, root := ~, start := 252495, stop := 252541 ), 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 := 252584, root := ~, start := 252563, stop := 252583 ), 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 := 252604, root := ~, start := 252593, stop := 252603 ), 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 := 252634, root := ~, start := 252620, stop := 252633 ), 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 := 252728, root := ~, start := 252714, stop := 252727 ), 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 := 252746, root := ~, start := 252734, stop := 252745 ), 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 := 252797, root := ~, start := 252765, stop := 252796 ), 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 := 252802, root := ~, start := 252798, stop := 252801 ), 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 := 252835, root := ~, start := 252823, stop := 252834 ), 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 := 252960, root := ~, start := 252932, stop := 252959 ), 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 := 252993, root := ~, start := 252969, stop := 252992 ), 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 := 253003, root := ~, start := 252999, stop := 253002 ), 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 := 253045, root := ~, start := 253031, stop := 253044 ), 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 := 253704, root := ~, start := 253142, stop := 253703 ), rec( content := "\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 253721, root := ~, start := 252545, stop := 253720 ) ], count := [ 6, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 253735, root := ~, start := 252434, stop := 253734 ), 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 := 253803, root := ~, start := 253756, stop := 253802 ), 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 := 253826, root := ~, start := 253815, stop := 253825 ), 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 := 253842, root := ~, start := 253830, stop := 253841 ), rec( content := ".", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 253853, root := ~, start := 253806, stop := 253852 ), 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 := 253902, root := ~, start := 253891, stop := 253901 ), 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 := 253932, root := ~, start := 253918, stop := 253931 ), 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 := 253973, root := ~, start := 253961, stop := 253972 ), 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 := 254017, root := ~, start := 254003, stop := 254016 ), 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 := 254044, root := ~, start := 254021, stop := 254043 ), 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 := 254213, root := ~, start := 254209, stop := 254212 ), 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 := 254278, root := ~, start := 254267, stop := 254277 ), 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 := 254327, root := ~, start := 254300, stop := 254326 ), 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 := 254391, root := ~, start := 254378, stop := 254390 ), 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 := 254411, root := ~, start := 254403, stop := 254410 ), 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 := 254451, root := ~, start := 254443, stop := 254450 ), 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 := 254503, root := ~, start := 254499, stop := 254502 ), 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 := 254545, root := ~, start := 254531, stop := 254544 ), 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 := 255007, root := ~, start := 254646, stop := 255006 ), rec( content := "\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 255024, root := ~, start := 253856, stop := 255023 ) ], count := [ 6, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 255038, root := ~, start := 253741, stop := 255037 ), 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 := 255108, root := ~, start := 255059, stop := 255107 ), 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 := 255131, root := ~, start := 255120, stop := 255130 ), 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 := 255147, root := ~, start := 255135, stop := 255146 ), rec( content := ".", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 255158, root := ~, start := 255111, stop := 255157 ), 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 := 255220, root := ~, start := 255202, stop := 255219 ), 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 := 255299, root := ~, start := 255267, stop := 255298 ), rec( content := ") when a\ ny vertex is removed. \n ", 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 := 255361, root := ~, start := 255334, stop := 255360 ), rec( content := " returns\ ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , 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 := 255381, root := ~, start := 255370, stop := 255380 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "A", next := 255415, root := ~, start := 255401, stop := 255414 ), rec( content := " is bico\ nnected, and ", 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 := 255448, root := ~, start := 255436, stop := 255447 ), rec( content := " if it i\ s not. In\n particular, ", 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 := 255509, root := ~, start := 255482, stop := 255508 ), rec( content := " returns\ ", 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 := 255530, root := ~, start := 255518, stop := 255529 ), rec( content := " if\n \ ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "A", next := 255552, root := ~, start := 255538, stop := 255551 ), rec( content := " is not \ connected. ", 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 := 255575, root := ~, start := 255571, stop := 255574 ), rec( content := "\n\n \ Multiple edges and loops are ignored by this method. ", 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 := 255638, root := ~, start := 255634, stop := 255637 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "M", next := 255706, root := ~, start := 255693, stop := 255705 ), rec( content := " where\n\ ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "M", next := 255725, root := ~, start := 255717, stop := 255724 ), rec( content := " is the \ number of edges (counting multiple edges as one, and not\n counting loops) \ and ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "M", next := 255822, root := ~, start := 255814, stop := 255821 ), rec( content := " is the \ number of vertices in the digraph.\n ", 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 := 255873, root := ~, start := 255869, stop := 255872 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 255920, root := ~, start := 255888, stop := 255919 ), rec( content := ".\n " , 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 := 255930, root := ~, start := 255926, stop := 255929 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "A", next := 255972, root := ~, start := 255958, stop := 255971 ), 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, 4, 7 ], 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, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Example", next := 256448, root := ~, start := 256068, stop := 256447 ), rec( content := "\n ", count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 256465, root := ~, start := 255161, stop := 256464 ) ], count := [ 6, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 256479, root := ~, start := 255044, stop := 256478 ), rec( content := "\n\n ", count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsStronglyConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 5, 2 ], mathmode := "Text", name := "Prop", next := 256555, root := ~, start := 256500, stop := 256554 ), 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 := 256578, root := ~, start := 256567, stop := 256577 ), 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 := 256594, root := ~, start := 256582, stop := 256593 ), rec( content := ".", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Returns", next := 256605, root := ~, start := 256558, stop := 256604 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), 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 := 256654, root := ~, start := 256643, stop := 256653 ), rec( content := " if the \ digraph ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "A", next := 256684, root := ~, start := 256670, stop := 256683 ), rec( content := " is stro\ ngly\n connected and ", 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 := 256727, root := ~, start := 256715, stop := 256726 ), rec( content := " if it i\ s not. ", 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 := 256746, root := ~, start := 256742, stop := 256745 ), rec( content := "\n\n \ A digraph ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 256776, root := ~, start := 256762, stop := 256775 ), rec( content := " is ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trongly connected", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "E", next := 256805, root := ~, start := 256780, stop := 256804 ), rec( content := " if ther\ e is a directed\n path from every vertex to every other vertex. See sectio\ n ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "Ref", next := 256942, root := ~, start := 256892, stop := 256941 ), rec( content := " for the\ definition of a directed\n path. ", 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 := 256993, root := ~, start := 256989, stop := 256992 ), rec( content := "\n\n \ The method used in this operation is based on Gabow's Algorithm ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "Cite", next := 257088, root := ~, start := 257063, stop := 257087 ), rec( content := " and has\ complexity ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "M", next := 257121, root := ~, start := 257108, stop := 257120 ), rec( content := ", where \ ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "M", next := 257137, root := ~, start := 257129, stop := 257136 ), rec( content := " is\n \ the number of edges (counting multiple edges as one) and ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "M", next := 257210, root := ~, start := 257202, stop := 257209 ), rec( content := " is the\\ n number of vertices in the digraph.\n ", 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 := 257265, root := ~, start := 257261, stop := 257264 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "A", next := 257307, root := ~, start := 257293, stop := 257306 ), 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, 5, 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, 5, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "Example", next := 257984, root := ~, start := 257408, stop := 257983 ), rec( content := "\n ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Description", next := 258001, root := ~, start := 256608, stop := 258000 ) ], count := [ 6, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 258015, root := ~, start := 256485, stop := 258014 ), rec( content := "\n\n ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAperiodicDigr\ aph" ), content := 0, count := [ 6, 3, 6, 2 ], mathmode := "Text", name := "Prop", next := 258083, root := ~, start := 258036, stop := 258082 ), 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 := 258106, root := ~, start := 258095, stop := 258105 ), 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 := 258122, root := ~, start := 258110, stop := 258121 ), rec( content := ".", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 258133, root := ~, start := 258086, stop := 258132 ), 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 := 258182, root := ~, start := 258171, stop := 258181 ), 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 := 258212, root := ~, start := 258198, stop := 258211 ), rec( content := "\n is\ aperiodic, i.e. if its ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Period" ), content := 0, count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 258272, root := ~, start := 258243, stop := 258271 ), rec( content := " is equa\ l to 1.\n Otherwise, the property is ", 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 := 258331, root := ~, start := 258319, stop := 258330 ), rec( content := ".\n " , 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 := 258341, root := ~, start := 258337, stop := 258340 ), rec( content := "\n \ \n If the argument ", 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 := 258383, root := ~, start := 258369, stop := 258382 ), 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, 6, 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, 6, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "Example", next := 258928, root := ~, start := 258483, stop := 258927 ), rec( content := "\n ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 258945, root := ~, start := 258136, stop := 258944 ) ], count := [ 6, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 258959, root := ~, start := 258021, stop := 258958 ), rec( content := "\n\n ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDirectedTree"\ ), content := 0, count := [ 6, 3, 7, 2 ], mathmode := "Text", name := "Prop", next := 259023, root := ~, start := 258980, stop := 259022 ), 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 := 259046, root := ~, start := 259035, stop := 259045 ), 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 := 259062, root := ~, start := 259050, stop := 259061 ), rec( content := ".", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 259073, root := ~, start := 259026, stop := 259072 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", 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 := 259113, root := ~, start := 259102, stop := 259112 ), 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 := 259143, root := ~, start := 259129, stop := 259142 ), rec( content := " is a di\ rected tree, and\n ", 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 := 259184, root := ~, start := 259172, stop := 259183 ), rec( content := " if it i\ s not. ", 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 := 259203, root := ~, start := 259199, stop := 259202 ), rec( content := "\n\n \ A ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ irected tree", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "E", next := 259231, root := ~, start := 259211, stop := 259230 ), 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, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "P", next := 259423, root := ~, start := 259419, stop := 259422 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sources" ), content := 0, count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 259466, root := ~, start := 259438, stop := 259465 ), rec( content := ".\n " , count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "P", next := 259476, root := ~, start := 259472, stop := 259475 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "A", next := 259518, root := ~, start := 259504, stop := 259517 ), 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, 7, 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, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 7 ], mathmode := "Text", name := "Example", next := 260102, root := ~, start := 259619, stop := 260101 ), rec( content := "\n ", count := [ 6, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 260119, root := ~, start := 259076, stop := 260118 ) ], count := [ 6, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 260133, root := ~, start := 258965, stop := 260132 ), rec( content := "\n\n ", count := [ 6, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedTre\ e" ), content := 0, count := [ 6, 3, 8, 2 ], mathmode := "Text", name := "Prop", next := 260199, root := ~, start := 260154, stop := 260198 ), rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedFor\ est" ), content := 0, count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "Prop", next := 260249, root := ~, start := 260202, stop := 260248 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "K", next := 260272, root := ~, start := 260261, stop := 260271 ), rec( content := " or ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "K", next := 260288, root := ~, start := 260276, stop := 260287 ), rec( content := ".", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "Returns", next := 260299, root := ~, start := 260252, stop := 260298 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ property ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedTree", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "C", next := 260356, root := ~, start := 260333, stop := 260355 ), rec( content := " returns\ ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "K", next := 260376, root := ~, start := 260365, stop := 260375 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "A", next := 260410, root := ~, start := 260396, stop := 260409 ), rec( content := " is an u\ ndirected tree, and the property\n ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedForest", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "C", next := 260480, root := ~, start := 260455, stop := 260479 ), rec( content := " returns\ ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "K", next := 260500, root := ~, start := 260489, stop := 260499 ), rec( content := " if ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "A", next := 260518, root := ~, start := 260504, stop := 260517 ), rec( content := " is an\n\ undirected forest; otherwise, these properties return ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "K", next := 260595, root := ~, start := 260583, stop := 260594 ), rec( content := ". ", 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 := 260601, root := ~, start := 260597, stop := 260600 ), rec( content := "\n\n \ An ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected tree", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "E", next := 260632, root := ~, start := 260610, stop := 260631 ), rec( content := " is a sy\ mmetric digraph without loops, in which for\n any pair of distinct vertices\ ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "C", next := 260726, root := ~, start := 260718, stop := 260725 ), rec( content := " and ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "C", next := 260739, root := ~, start := 260731, stop := 260738 ), rec( content := ", there \ is exactly one\n directed path from ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "C", next := 260793, root := ~, start := 260785, stop := 260792 ), rec( content := " to ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "C", next := 260805, root := ~, start := 260797, stop := 260804 ), rec( content := ". See ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 260849, root := ~, start := 260811, stop := 260848 ), rec( content := " and ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 260883, root := ~, start := 260854, stop := 260882 ), rec( content := ", and se\ e\n section ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 260949, root := ~, start := 260905, stop := 260948 ), rec( content := " for the\ definition of\n directed path. This definition implies that an undirected \ tree has\n no multiple edges. ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "P", next := 261070, root := ~, start := 261066, stop := 261069 ), rec( content := "\n\n \ An ", count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected forest", count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "E", next := 261103, root := ~, start := 261079, stop := 261102 ), 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, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "Ref", next := 261327, root := ~, start := 261280, stop := 261326 ), rec( content := " and ", count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "Ref", next := 261404, root := ~, start := 261332, stop := 261403 ), rec( content := ". In par\ ticular, every\n undirected tree is an undirected forest. ", count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "P", next := 261476, root := ~, start := 261472, stop := 261475 ), 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, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 9 ], mathmode := "Text", name := "P", next := 261615, root := ~, start := 261611, stop := 261614 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 9 ], mathmode := "Text", name := "A", next := 261657, root := ~, start := 261643, stop := 261656 ), 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, 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, 8, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 8, 10 ], mathmode := "Text", name := "Example", next := 262358, root := ~, start := 261758, stop := 262357 ), rec( content := "\n ", count := [ 6, 3, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "Description", next := 262375, root := ~, start := 260302, stop := 262374 ) ], count := [ 6, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 262389, root := ~, start := 260139, stop := 262388 ), rec( content := "\n\n ", count := [ 6, 3, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEulerianDigra\ ph" ), content := 0, count := [ 6, 3, 9, 2 ], mathmode := "Text", name := "Prop", next := 262456, root := ~, start := 262410, stop := 262455 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "K", next := 262479, root := ~, start := 262468, stop := 262478 ), rec( content := " or ", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "K", next := 262495, root := ~, start := 262483, stop := 262494 ), rec( content := ".", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "Returns", next := 262506, root := ~, start := 262459, stop := 262505 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns true if the digraph ", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "A", next := 262583, root := ~, start := 262569, stop := 262582 ), rec( content := " is Eule\ rian.\n ", count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "P", next := 262605, root := ~, start := 262601, stop := 262604 ), rec( content := "\n\n \ A connected digraph is called ", count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ulerian", count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "E", next := 262656, root := ~, start := 262641, stop := 262655 ), 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, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "Ref", next := 262812, root := ~, start := 262768, stop := 262811 ), 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, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "P", next := 262953, root := ~, start := 262949, stop := 262952 ), rec( content := "\n \ \n If the argument ", 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 := 262995, root := ~, start := 262981, stop := 262994 ), 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, 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, 9, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Example", next := 263558, root := ~, start := 263096, stop := 263557 ), rec( content := "\n ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 3 ], mathmode := "Text", name := "Description", next := 263575, root := ~, start := 262509, stop := 263574 ) ], count := [ 6, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 263589, root := ~, start := 262395, stop := 263588 ), rec( content := "\n\n ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsHamiltonianDi\ graph" ), content := 0, count := [ 6, 3, 10, 2 ], mathmode := "Text", name := "Prop", next := 263659, root := ~, start := 263610, stop := 263658 ), 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 := 263682, root := ~, start := 263671, stop := 263681 ), 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 := 263698, root := ~, start := 263686, stop := 263697 ), rec( content := ".", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 263709, root := ~, start := 263662, stop := 263708 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", 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 := 263747, root := ~, start := 263733, stop := 263746 ), rec( content := " is Hami\ ltonian, then this property returns\n ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), 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 := 263806, root := ~, start := 263795, stop := 263805 ), rec( content := ", and ", 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 := 263824, root := ~, start := 263812, stop := 263823 ), rec( content := " if it i\ s not. ", 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 := 263843, root := ~, start := 263839, stop := 263842 ), rec( content := "\n\n \ A digraph with ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "C", next := 263872, root := ~, start := 263864, stop := 263871 ), rec( content := " vertice\ s is ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "E", next := 263903, root := ~, start := 263885, stop := 263902 ), rec( content := " if it h\ as a\n directed cycle of length ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "C", next := 263953, root := ~, start := 263945, stop := 263952 ), rec( content := ". See Se\ ction ", 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 := 264016, root := ~, start := 263967, stop := 264015 ), 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, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "P", next := 264146, root := ~, start := 264142, stop := 264145 ), rec( content := "\n\n \ The method used in this operation has the worst case complexity as\n ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "Ref", next := 264256, root := ~, start := 264223, stop := 264255 ), rec( content := ".\n " , count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "P", next := 264266, root := ~, start := 264262, stop := 264265 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "A", next := 264308, root := ~, start := 264294, stop := 264307 ), 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, 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, 10, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "Example", next := 264884, root := ~, start := 264409, stop := 264883 ), rec( content := "\n ", count := [ 6, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 264901, root := ~, start := 263712, stop := 264900 ) ], count := [ 6, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 264915, root := ~, start := 263595, stop := 264914 ), rec( content := "\n\n ", count := [ 6, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCycleDigraph"\ ), content := 0, count := [ 6, 3, 11, 2 ], mathmode := "Text", name := "Prop", next := 264979, root := ~, start := 264936, stop := 264978 ), 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 := 265002, root := ~, start := 264991, stop := 265001 ), 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 := 265018, root := ~, start := 265006, stop := 265017 ), rec( content := ".", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Returns", next := 265029, root := ~, start := 264982, stop := 265028 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCycleDigraph", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "C", next := 265071, root := ~, start := 265050, stop := 265070 ), rec( content := " returns\ ", 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 := 265091, root := ~, start := 265080, stop := 265090 ), rec( content := " if the \ digraph ", 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 := 265121, root := ~, start := 265107, stop := 265120 ), rec( content := " is\n \ isomorphic to the cycle digraph with the same number of vertices as\n ", 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 := 265215, root := ~, start := 265201, stop := 265214 ), 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 := 265233, root := ~, start := 265221, stop := 265232 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CycleDi\ graph" ), content := 0, count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Ref", next := 265284, root := ~, start := 265252, stop := 265283 ), rec( content := ".", 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 := 265289, root := ~, start := 265285, stop := 265288 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "E", next := 265322, root := ~, start := 265310, stop := 265321 ), 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, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "P", next := 265422, root := ~, start := 265418, stop := 265421 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "A", next := 265464, root := ~, start := 265450, stop := 265463 ), 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, 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, 11, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "Example", next := 265949, root := ~, start := 265561, stop := 265948 ), rec( content := "\n ", count := [ 6, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Description", next := 265966, root := ~, start := 265032, stop := 265965 ) ], count := [ 6, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 265980, root := ~, start := 264921, stop := 265979 ), rec( content := "\n\n ", count := [ 6, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 265994, root := ~, start := 250886, stop := 265993 ) , rec( content := "\n \n ", count := [ 6, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), 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 := 266037, root := ~, start := 266009, stop := 266036 ), 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 := 266101, root := ~, start := 266057, stop := 266100 ), 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 := 266124, root := ~, start := 266113, stop := 266123 ), 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 := 266140, root := ~, start := 266128, stop := 266139 ), rec( content := ".", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 266151, root := ~, start := 266104, stop := 266150 ), 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 := 266187, root := ~, start := 266174, stop := 266186 ), 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 := 266428, root := ~, start := 266420, stop := 266427 ), 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 := 266508, root := ~, start := 266500, stop := 266507 ), 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 := 266521, root := ~, start := 266513, stop := 266520 ), 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 := 266531, root := ~, start := 266527, stop := 266530 ), 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 := 266563, root := ~, start := 266541, stop := 266562 ), 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 := 266583, root := ~, start := 266572, stop := 266582 ), 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 := 266613, root := ~, start := 266599, stop := 266612 ), 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 := 266644, root := ~, start := 266632, stop := 266643 ), 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 := 266727, root := ~, start := 266713, stop := 266726 ), 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 := 266765, root := ~, start := 266743, stop := 266764 ), 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 := 266775, root := ~, start := 266771, stop := 266774 ), 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 := 266825, root := ~, start := 266791, stop := 266824 ), 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 := 266835, root := ~, start := 266831, stop := 266834 ), 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 := 266993, root := ~, start := 266975, stop := 266992 ), 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 := 267532, root := ~, start := 267000, stop := 267531 ), rec( content := "\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 267549, root := ~, start := 266154, stop := 267548 ) ], count := [ 6, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 267563, root := ~, start := 266042, stop := 267562 ), 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 := 267633, root := ~, start := 267584, stop := 267632 ), 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 := 267656, root := ~, start := 267645, stop := 267655 ), 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 := 267672, root := ~, start := 267660, stop := 267671 ), rec( content := ".", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 267683, root := ~, start := 267636, stop := 267682 ), 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 := 267726, root := ~, start := 267707, stop := 267725 ), 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 := 268042, root := ~, start := 268034, stop := 268041 ), 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 := 268122, root := ~, start := 268114, stop := 268121 ), 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 := 268135, root := ~, start := 268127, stop := 268134 ), 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 := 268145, root := ~, start := 268141, stop := 268144 ), 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 := 268182, root := ~, start := 268155, stop := 268181 ), 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 := 268202, root := ~, start := 268191, stop := 268201 ), 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 := 268236, root := ~, start := 268222, stop := 268235 ), 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 := 268269, root := ~, start := 268257, stop := 268268 ), 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 := 268352, root := ~, start := 268338, stop := 268351 ), 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 := 268394, root := ~, start := 268372, stop := 268393 ), 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 := 268401, root := ~, start := 268397, stop := 268400 ), 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 := 268445, root := ~, start := 268416, stop := 268444 ), 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 := 268604, root := ~, start := 268586, stop := 268603 ), 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 := 269404, root := ~, start := 268607, stop := 269403 ), rec( content := "\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 269421, root := ~, start := 267686, stop := 269420 ) ], count := [ 6, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 269435, root := ~, start := 267569, stop := 269434 ), rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 269449, root := ~, start := 266000, stop := 269448 ) , 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 := 269514, root := ~, start := 269462, stop := 269513 ), 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 := 269576, root := ~, start := 269534, stop := 269575 ), 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 := 269599, root := ~, start := 269588, stop := 269598 ), 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 := 269615, root := ~, start := 269603, stop := 269614 ), rec( content := ".", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 269626, root := ~, start := 269579, stop := 269625 ), 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 := 269680, root := ~, start := 269669, stop := 269679 ), 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 := 269698, root := ~, start := 269684, stop := 269697 ), 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 := 269730, root := ~, start := 269718, stop := 269729 ), 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 := 269748, root := ~, start := 269744, stop := 269747 ), 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 := 269772, root := ~, start := 269764, stop := 269771 ), 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 := 269789, root := ~, start := 269778, stop := 269788 ), 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 := 269846, root := ~, start := 269838, stop := 269845 ), 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 := 269898, root := ~, start := 269890, stop := 269897 ), 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 := 269914, root := ~, start := 269906, stop := 269913 ), 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 := 269950, root := ~, start := 269942, stop := 269949 ), 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 := 270010, root := ~, start := 270002, stop := 270009 ), 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 := 270041, root := ~, start := 270033, stop := 270040 ), 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 := 270051, root := ~, start := 270047, stop := 270050 ), 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 := 270093, root := ~, start := 270079, stop := 270092 ), 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 := 270554, root := ~, start := 270194, stop := 270553 ), rec( content := "\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 270571, root := ~, start := 269629, stop := 270570 ) ], count := [ 6, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 270585, root := ~, start := 269519, stop := 270584 ), 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 := 270651, root := ~, start := 270606, stop := 270650 ), 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 := 270674, root := ~, start := 270663, stop := 270673 ), 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 := 270690, root := ~, start := 270678, stop := 270689 ), rec( content := ".", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 270701, root := ~, start := 270654, stop := 270700 ), 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 := 270739, root := ~, start := 270725, stop := 270738 ), 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 := 270809, root := ~, start := 270786, stop := 270808 ), 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 := 270829, root := ~, start := 270818, stop := 270828 ), 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 := 270847, root := ~, start := 270833, stop := 270846 ), 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 := 270888, root := ~, start := 270876, stop := 270887 ), 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 := 270939, root := ~, start := 270917, stop := 270938 ), 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 := 271057, root := ~, start := 271021, stop := 271056 ), 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 := 271068, root := ~, start := 271064, stop := 271067 ), 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 := 271110, root := ~, start := 271096, stop := 271109 ), 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 := 271457, root := ~, start := 271211, stop := 271456 ), rec( content := "\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 271474, root := ~, start := 270704, stop := 271473 ) ], count := [ 6, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 271488, root := ~, start := 270591, stop := 271487 ), 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 := 271556, root := ~, start := 271509, stop := 271555 ), 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 := 271579, root := ~, start := 271568, stop := 271578 ), 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 := 271595, root := ~, start := 271583, stop := 271594 ), rec( content := ".", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 271606, root := ~, start := 271559, stop := 271605 ), 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 := 271644, root := ~, start := 271630, stop := 271643 ), 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 := 271689, root := ~, start := 271664, stop := 271688 ), 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 := 271713, root := ~, start := 271702, stop := 271712 ), 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 := 271731, root := ~, start := 271717, stop := 271730 ), 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 := 271770, root := ~, start := 271758, stop := 271769 ), 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 := 271823, root := ~, start := 271799, stop := 271822 ), 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 := 271897, root := ~, start := 271893, stop := 271896 ), 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 := 271939, root := ~, start := 271925, stop := 271938 ), 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 := 272170, root := ~, start := 272040, stop := 272169 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 272187, root := ~, start := 271609, stop := 272186 ) ], count := [ 6, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 272201, root := ~, start := 271494, stop := 272200 ), rec( content := "\n\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 272215, root := ~, start := 269453, stop := 272214 ) , 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 := 272230, root := ~, start := 226662, stop := 272229 ), 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 := 272317, root := ~, start := 272285, stop := 272316 ) , 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 := 272961, root := ~, start := 272924, stop := 272960 ), 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 := 273063, root := ~, start := 272981, stop := 273062 ), 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 := 273159, root := ~, start := 273066, stop := 273158 ), 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 := 273191, root := ~, start := 273162, stop := 273190 ), 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 := 273229, root := ~, start := 273215, stop := 273228 ), 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 := 273279, root := ~, start := 273268, stop := 273278 ), 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 := 273307, root := ~, start := 273289, stop := 273306 ), 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 := 273341, root := ~, start := 273327, stop := 273340 ), 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 := 273446, root := ~, start := 273432, stop := 273445 ), 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 := 273471, root := ~, start := 273460, stop := 273470 ), 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 := 273516, root := ~, start := 273508, stop := 273515 ), 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 := 273567, root := ~, start := 273553, stop := 27356\ 6 ), 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 := 273582, root := ~, start := 273539, stop := 273581 ), 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 := 273596, root := ~, start := 273582, stop := 273595 ), 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 := 273606, root := ~, start := 273602, stop := 273605 ), 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 := 273661, root := ~, start := 273640, stop := 273660 ), 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 := 273674, root := ~, start := 273662, stop := 273673 ), 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 := 273712, root := ~, start := 273698, stop := 273711 ), 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 := 273847, root := ~, start := 273835, stop := 273846 ), 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 := 273917, root := ~, start := 273905, stop := 273916 ), 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 := 274039, root := ~, start := 274025, stop := 274038 ), 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 := 274069, root := ~, start := 274057, stop := 274068 ), 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 := 274166, root := ~, start := 274152, stop := 27416\ 5 ), 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 := 2741\ 94, root := ~, start := 274182, stop := 274193 ), rec( content := "))", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "C", next := 274200, root := ~, start := 274138, stop := 274199 ), 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 := 274210, root := ~, start := 274206, stop := 274209 ), 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 := 274253, root := ~, start := 274220, stop := 274252 ), 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 := 274271, root := ~, start := 274257, stop := 274270 ), 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 := 274326, root := ~, start := 274322, stop := 274325 ), 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 := 274349, root := ~, start := 274335, stop := 274348 ), 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 := 274391, root := ~, start := 274361, stop := 274390 ), 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 := 274469, root := ~, start := 274455, stop := 274468 ), 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 := 274492, root := ~, start := 274478, stop := 274491 ), 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 := 274536, root := ~, start := 274504, stop := 274535 ), 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 := 274577, root := ~, start := 274563, stop := 274576 ), 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 := 274623, root := ~, start := 274619, stop := 274622 ), 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 := 275266, root := ~, start := 274629, stop := 275265 ), rec( content := "\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 275283, root := ~, start := 273194, stop := 275282 ) ], count := [ 7, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 275297, root := ~, start := 272966, stop := 275296 ), 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 := 275368, root := ~, start := 275318, stop := 275367 ), 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 := 275471, root := ~, start := 275371, stop := 275470 ), 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 := 275503, root := ~, start := 275474, stop := 275502 ), 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 := 275541, root := ~, start := 275527, stop := 275540 ), 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 := 275571, root := ~, start := 275560, stop := 275570 ), 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 := 275682, root := ~, start := 275668, stop := 275681 ), 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 := 275793, root := ~, start := 275779, stop := 275792 ), 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 := 275825, root := ~, start := 275811, stop := 275824 ), 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 := 275844, root := ~, start := 275830, stop := 275843 ), 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 := 275864, root := ~, start := 275860, stop := 275863 ), 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 := 275912, root := ~, start := 275890, stop := 275911 ), 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 := 275983, root := ~, start := 275971, stop := 275982 ), 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 := 276000, root := ~, start := 275988, stop := 275999 ), 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 := 276020, root := ~, start := 276016, stop := 276019 ), 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 := 276064, root := ~, start := 276050, stop := 27606\ 3 ), 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 := 2761\ 08, root := ~, start := 276094, stop := 276107 ), rec( content := ", [perm, ()\ ])", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 276125, root := ~, start := 276036, stop := 276124 ), 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 := 276143, root := ~, start := 276132, stop := 276142 ), 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 := 276198, root := ~, start := 276184, stop := 276197 ), 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 := 276330, root := ~, start := 276313, stop := 276329 ), 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 := 276364, root := ~, start := 276342, stop := 276363 ), 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 := 276388, root := ~, start := 276374, stop := 276387 ), 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 := 276430, root := ~, start := 276400, stop := 276429 ), 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 := 276508, root := ~, start := 276494, stop := 276507 ), 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 := 276531, root := ~, start := 276517, stop := 276530 ), 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 := 276575, root := ~, start := 276543, stop := 276574 ), 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 := 276616, root := ~, start := 276602, stop := 276615 ), 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 := 276662, root := ~, start := 276658, stop := 276661 ), 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 := 277107, root := ~, start := 276668, stop := 277106 ), rec( content := "\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 277124, root := ~, start := 275506, stop := 277123 ) ], count := [ 7, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 277138, root := ~, start := 275303, stop := 277137 ), rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 277152, root := ~, start := 272915, stop := 277151 ) , 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 := 277272, root := ~, start := 277216, stop := 277271 ), 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 := "http://pa\ llini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100, stop := 123 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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/sebasguts/NautyTrac\ esInterface", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ) , 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := 277489, root := ~, start := 277471, stop := 277488 ), 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 := 277518, root := ~, start := 277494, stop := 277517 ), 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 := "http://pa\ llini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100, stop := 123 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 277640, root := ~, start := 277602, stop := 277639 ), rec( attributes := rec( Arg := "", Name := "DigraphsUseBlis\ s" ), content := 0, count := [ 7, 2, 1, 4 ], mathmode := "Text", name := "Func", next := 277681, root := ~, start := 277643, stop := 277680 ), 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 := 277711, root := ~, start := 277684, stop := 277710 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 1, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := 277973, root := ~, start := 277950, stop := 277972 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 278077, root := ~, start := 278073, stop := 278076 ), 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 := 278368, root := ~, start := 278314, stop := 278367 ), 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 := 278456, root := ~, start := 278374, stop := 278455 ), 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 := 278516, root := ~, start := 278462, stop := 278515 ), 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 := 278608, root := ~, start := 278526, stop := 278607 ), 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 := 278628, root := ~, start := 278624, stop := 278627 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 278787, root := ~, start := 278751, stop := 278786 ), 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 := 278854, root := ~, start := 278796, stop := 278853 ), 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 := 278886, root := ~, start := 277714, stop := 278885 ) ], count := [ 7, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 278900, root := ~, start := 277587, stop := 278899 ), 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 := 278989, root := ~, start := 278921, stop := 278988 ), 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 := 279031, root := ~, start := 278992, stop := 279030 ), 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 := 279069, root := ~, start := 279055, stop := 279068 ), 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 := 279161, root := ~, start := 279147, stop := 279160 ), 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 := 279186, root := ~, start := 279167, stop := 279185 ), 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 := 279204, root := ~, start := 279190, stop := 279203 ), 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 := 279246, root := ~, start := 279232, stop := 279245 ), 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 := 279323, root := ~, start := 279262, stop := 279322 ), 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 := 279385, root := ~, start := 279381, stop := 279384 ), 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 := 279408, root := ~, start := 279394, stop := 279407 ), 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 := 279546, root := ~, start := 279532, stop := 279545 ), 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 := 279552, root := ~, start := 279548, stop := 279551 ), 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 := 279575, root := ~, start := 279561, stop := 279574 ), 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 := 279731, root := ~, start := 279717, stop := 279730 ), 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 := 279802, root := ~, start := 279788, stop := 279801 ), 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 := 279928, root := ~, start := 279839, stop := 279927 ), 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 := 279959, root := ~, start := 279955, stop := 279958 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 2, 6 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 280244, root := ~, start := 280186, stop := 280243 ), 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 := 280287, root := ~, start := 280251, stop := 280286 ), 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 := 280319, root := ~, start := 280289, stop := 280318 ), 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 := 280359, root := ~, start := 280329, stop := 280358 ), 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 := 280369, root := ~, start := 280365, stop := 280368 ), 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 := 280405, root := ~, start := 280391, stop := 280404 ), 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 := 280891, root := ~, start := 280507, stop := 280890 ), rec( content := "\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 280910, root := ~, start := 279034, stop := 280909 ) ], count := [ 7, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 280926, root := ~, start := 278906, stop := 280925 ), 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 := 281020, root := ~, start := 280947, stop := 281019 ), 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 := 281137, root := ~, start := 281023, stop := 281136 ), 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 := 281275, root := ~, start := 281140, stop := 281274 ), 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 := 281317, root := ~, start := 281278, stop := 281316 ), 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 := 281355, root := ~, start := 281341, stop := 281354 ), 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 := 281447, root := ~, start := 281433, stop := 281446 ), 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 := 281521, root := ~, start := 281517, stop := 281520 ), 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 := 281594, root := ~, start := 281541, stop := 281593 ), 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 := 281696, root := ~, start := 281613, stop := 281695 ), 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 := 281758, root := ~, start := 281729, stop := 281757 ), 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 := 281804, root := ~, start := 281762, stop := 281803 ), 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 := 281887, root := ~, start := 281883, stop := 281886 ), 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 := 281994, root := ~, start := 281906, stop := 281993 ), 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 := 282049, root := ~, start := 282020, stop := 282048 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 3, 9 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 282157, root := ~, start := 282136, stop := 282156 ), 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 := 282186, root := ~, start := 282167, stop := 282185 ), 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 := 282293, root := ~, start := 282205, stop := 282292 ), 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 := 282339, root := ~, start := 282309, stop := 282338 ), 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 := 282375, root := ~, start := 282345, stop := 282374 ), 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 := 282385, root := ~, start := 282381, stop := 282384 ), 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 := 282421, root := ~, start := 282407, stop := 282420 ), 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 := 282660, root := ~, start := 282523, stop := 282659 ), rec( content := "\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Description", next := 282679, root := ~, start := 281320, stop := 282678 ) ], count := [ 7, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 282695, root := ~, start := 280932, stop := 282694 ), 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 := 282783, root := ~, start := 282716, stop := 282782 ), 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 := 282825, root := ~, start := 282786, stop := 282824 ), 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 := 282863, root := ~, start := 282849, stop := 282862 ), 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 := 282955, root := ~, start := 282941, stop := 282954 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 4, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := 283190, root := ~, start := 283137, stop := 283189 ), 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 := 283292, root := ~, start := 283209, stop := 283291 ), 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 := 283354, root := ~, start := 283325, stop := 283353 ), 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 := 283423, root := ~, start := 283359, stop := 283422 ), 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 := 283505, root := ~, start := 283501, stop := 283504 ), 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 := 283550, root := ~, start := 283520, stop := 283549 ), 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 := 283586, root := ~, start := 283556, stop := 283585 ), 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 := 283596, root := ~, start := 283592, stop := 283595 ), 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 := 283632, root := ~, start := 283618, stop := 283631 ), 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 := 283888, root := ~, start := 283734, stop := 283887 ), rec( content := "\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 283907, root := ~, start := 282828, stop := 283906 ) ], count := [ 7, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 283923, root := ~, start := 282701, stop := 283922 ), 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 := 284053, root := ~, start := 283944, stop := 284052 ), 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 := 284095, root := ~, start := 284056, stop := 284094 ), 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 := 284285, root := ~, start := 284271, stop := 284284 ), 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 := 284323, root := ~, start := 284304, stop := 284322 ), 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 := 284337, root := ~, start := 284329, stop := 284336 ), 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 := 284385, root := ~, start := 284371, stop := 284384 ), 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 := 284420, root := ~, start := 284401, stop := 284419 ), 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 := 284517, root := ~, start := 284509, stop := 2\ 84516 ), 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 := 284553, root := ~, start := 284534, stop := 284552 ), 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 := 284563, root := ~, start := 284553, stop \ := 284562 ), 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 := 284604, root := ~, start := 284596, stop \ := 284603 ), 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 := 284639, root := ~, start := 284624, stop := 284638 ), 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\ := 284673, root := ~, start := 284657, stop := 284672 ), rec( content := "; o\ r\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 5 ], name := "Item", next := 284691, root := ~, start := 284\ 484, stop := 284690 ), 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 := 284804, root := ~, start := 284790\ , stop := 284803 ), rec( content := ")", count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "C", next := 284809, root := ~, start := 284771, stop := 28480\ 8 ), 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 := 284848, root := ~, start := 284829, stop := 284847 ), 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 := 284858, root := ~, start := 284848, stop \ := 284857 ), 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 := 284915, root := ~, start := 284907, stop \ := 284914 ), rec( content := ".\n ", count := [ 7, 2, 5, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], name := "Ite\ m", next := 284930, root := ~, start := 284698, stop := 284929 ) ], count := [ 7, 2, 5, 4 ], mathmode := "Text", name := "List", next := 284942, root := ~, start := 284471, stop := 284941 ), 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 := 284977, root := ~, start := 284952, stop := 284976 ), 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 := 285014, root := ~, start := 285000, stop := 285013 ), 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 := 285053, root := ~, start := 285034, stop := 285052 ), 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 := 285126, root := ~, start := 285107, stop := 285125 ), 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 := 285144, root := ~, start := 285130, stop := 285143 ), 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 := 285207, root := ~, start := 285193, stop := 285206 ), 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 := 2852\ 98, root := ~, start := 285284, stop := 285297 ), rec( content := ")", count := [ 7, 2, 5, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "C", next := 285303, root := ~, start := 285263, stop := 285302 ), 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 := 285402, root := ~, start := 285383, stop := 285401 ), 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 := 285466, root := ~, start := 285412, stop := 285465 ), 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 := 285559, root := ~, start := 285476, stop := 285558 ), 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 := 285630, root := ~, start := 285626, stop := 285629 ), 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 := 285653, root := ~, start := 285639, stop := 285652 ), 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 := 285791, root := ~, start := 285777, stop := 285790 ), 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 := 285797, root := ~, start := 285793, stop := 285796 ), 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 := 285820, root := ~, start := 285806, stop := 285819 ), 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 := 285976, root := ~, start := 285962, stop := 285975 ), 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 := 286047, root := ~, start := 286033, stop := 286046 ), 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 := 286173, root := ~, start := 286084, stop := 286172 ), 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 := 286204, root := ~, start := 286200, stop := 286203 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 5, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 286515, root := ~, start := 286431, stop := 286514 ), 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 := 286557, root := ~, start := 286521, stop := 286556 ), 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 := 286589, root := ~, start := 286559, stop := 286588 ), 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 := 286629, root := ~, start := 286599, stop := 286628 ), 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 := 287151, root := ~, start := 286634, stop := 287150 ), rec( content := "\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 287168, root := ~, start := 284098, stop := 287167 ) ], count := [ 7, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 287182, root := ~, start := 283929, stop := 287181 ), 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 := 287335, root := ~, start := 287203, stop := 287334 ), 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 := 287377, root := ~, start := 287338, stop := 287376 ), 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 := 287577, root := ~, start := 287563, stop := 287576 ), 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 := 287612, root := ~, start := 287593, stop := 287611 ), 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 := 287633, root := ~, start := 287614, stop := 287632 ), 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 := 287651, root := ~, start := 287643, stop := 287650 ), 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 := 287695, root := ~, start := 287681, stop := 287694 ), 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 := 287799, root := ~, start := 287780, stop := 28779\ 8 ), 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 :\ = 287819, root := ~, start := 287808, stop := 287818 ), 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 := 287841, root := ~, star\ t := 287833, stop := 287840 ), 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 := 287885, root := ~, start :\ = 287866, stop := 287884 ), 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 :\ = 287895, root := ~, start := 287885, stop := 287894 ), 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 := 287928, root\ := ~, start := 287920, stop := 287927 ), 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 := 287971, root := \ ~, start := 287956, stop := 287970 ), 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 := 287997, root := ~, start := 287981, stop \ := 287996 ), rec( content := ";\n ", count := [ 7, 2, 6, 5 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], name := "Ite\ m", next := 288012, root := ~, start := 287765, stop := 288011 ), 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 := 288053, root := ~, start := 288034, stop := 28805\ 2 ), 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 :\ = 288073, root := ~, start := 288062, stop := 288072 ), 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 := 288095, root := ~, star\ t := 288087, stop := 288094 ), 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 := 288172, root := ~, start := 288143, stop := 288171 ), 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\ := 288207, root := ~, start := 288188, stop := 288206 ), 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 := 288220, root := ~, start := 288207, stop := \ 288219 ), 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 := 288290, root := \ ~, start := 288255, stop := 288289 ), 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 := 288325, root := ~, start := 28\ 8310, stop := 288324 ), 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 := 288359, root := ~, start := 288343, stop \ := 288358 ), rec( content := ";\n ", count := [ 7, 2, 6, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], name := "Ite\ m", next := 288374, root := ~, start := 288019, stop := 288373 ) ], count := [ 7, 2, 6, 4 ], mathmode := "Text", name := "List", next := 288386, root := ~, start := 287752, stop := 288385 ), 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 := 288418, root := ~, start := 288399, stop := 288417 ), 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 := 288439, root := ~, start := 288420, stop := 288438 ), 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 := 288455, root := ~, start := 288444, stop := 288454 ), 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 := 288533, root := ~, start := 288529, stop := 288532 ), 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 := 288599, root := ~, start := 288546, stop := 288598 ), 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 := 288724, root := ~, start := 288695, stop := 288723 ), 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 := 288824, root := ~, start := 288799, stop := 288823 ), 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 := 288877, root := ~, start := 288863, stop := 288876 ), 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 := 288901, root := ~, start := 288893, stop := 288900 ), 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 := 289008, root := ~, start := 288989, stop := 289007 ), 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 := 289026, root := ~, start := 289012, stop := 289025 ), 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 := 289101, root := ~, start := 289087, stop := 289100 ), 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 := 2891\ 92, root := ~, start := 289178, stop := 289191 ), rec( content := ")", count := [ 7, 2, 6, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "C", next := 289197, root := ~, start := 289157, stop := 289196 ), 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 := 289291, root := ~, start := 289277, stop := 289290 ), 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 := 289355, root := ~, start := 289301, stop := 289354 ), 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 := 289448, root := ~, start := 289365, stop := 289447 ), 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 := 289519, root := ~, start := 289515, stop := 289518 ), 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 := 289542, root := ~, start := 289528, stop := 289541 ), 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 := 289680, root := ~, start := 289666, stop := 289679 ), 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 := 289686, root := ~, start := 289682, stop := 289685 ), 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 := 289709, root := ~, start := 289695, stop := 289708 ), 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 := 289865, root := ~, start := 289851, stop := 289864 ), 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 := 289936, root := ~, start := 289922, stop := 289935 ), 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 := 290062, root := ~, start := 289973, stop := 290061 ), 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 := 290093, root := ~, start := 290089, stop := 290092 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 6, 13 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 290413, root := ~, start := 290321, stop := 290412 ), 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 := 290456, root := ~, start := 290420, stop := 290455 ), 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 := 290488, root := ~, start := 290458, stop := 290487 ), 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 := 290529, root := ~, start := 290499, stop := 290528 ), 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 := 291076, root := ~, start := 290534, stop := 291075 ), rec( content := "\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 291093, root := ~, start := 287380, stop := 291092 ) ], count := [ 7, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 291107, root := ~, start := 287188, stop := 291106 ), 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 := 291202, root := ~, start := 291128, stop := 291201 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "NautyCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 291279, root := ~, start := 291205, stop := 291278 ), 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 := 291346, root := ~, start := 291282, stop := 291345 ), 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 := 291389, root := ~, start := 291378, stop := 291388 ), 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 := 291469, root := ~, start := 291428, stop := 291468 ), 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 := 291536, root := ~, start := 291528, stop := 291535 ), 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 := 291549, root := ~, start := 291541, stop := 291548 ), 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 := 291555, root := ~, start := 291551, stop := 291554 ), 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 := 291603, root := ~, start := 291589, stop := 291602 ), 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 := 291616, root := ~, star\ t := 291608, stop := 291615 ), 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 := 291662, root := ~, start := 291\ 574, stop := 291661 ), 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 := 2\ 91706, root := ~, start := 291684, stop := 291705 ), 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 := 291730, root := ~ , star\ t := 291722, stop := 291729 ), 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 := 291743, root := ~, start := 291735, stop := 29174\ 2 ), 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 := 291793, root := ~, start := 291669, stop := 291792 ) ], count := [ 7, 2, 7, 7 ], mathmode := "Text", name := "List", next := 291805, root := ~, start := 291561, stop := 291804 ), 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 := 291839, root := ~, start := 291813, stop := 291838 ), 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 := 291861, root := ~, start := 291853, stop := 291860 ), 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 := 291880, root := ~, start := 291869, stop := 291879 ), 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 := 291915, root := ~, start := 291907, stop := 291914 ), 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 := 291956, root := ~, start := 291925, stop := 291955 ), 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 := 291976, root := ~, start := 291962, stop := 291975 ), 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 := 292038, root := ~, start := 291983, stop := 292037 ), 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 := 292100, root := ~, start := 292096, stop := 292099 ), 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 := 292136, root := ~, start := 292106, stop := 292135 ), 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 := 292200, root := ~, start := 292186, stop := 292199 ), 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 := 292292, root := ~, start := 292262, stop := 292291 ), 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 := 292356, root := ~, start := 292342, stop := 292355 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 292555, root := ~, start := 292551, stop := 292554 ), 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 := 292591, root := ~, start := 292561, stop := 292590 ), 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 := 292630, root := ~, start := 292616, stop := 292629 ), 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 := 292691, root := ~, start := 292662, stop := 292690 ), 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 := 292697, root := ~, start := 292693, stop := 292696 ), 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 := 293193, root := ~, start := 292703, stop := 293192 ), rec( content := "\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Description", next := 293210, root := ~, start := 291349, stop := 293209 ) ], count := [ 7, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 293224, root := ~, start := 291113, stop := 293223 ), 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 := 293343, root := ~, start := 293245, stop := 293342 ), 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 := 293444, root := ~, start := 293346, stop := 293443 ), 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 := 293480, root := ~, start := 293447, stop := 293479 ), 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 := 293523, root := ~, start := 293512, stop := 293522 ), 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 := 293619, root := ~, start := 293584, stop := 293618 ), 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 := 293695, root := ~, start := 293687, stop := 293694 ), 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 := 293708, root := ~, start := 293700, stop := 293707 ), 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 := 293757, root := ~, start := 293743, stop := 293756 ), 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 := 293770, root := ~, star\ t := 293762, stop := 293769 ), 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 := 293825, root := ~, start := 293\ 728, stop := 293824 ), 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 := 293\ 869, root := ~, start := 293847, stop := 293868 ), 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 := 293893, root := ~, star\ t := 293885, stop := 293892 ), 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 := 293906, root := ~, start := 293898, stop := 293905 ), 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 := 293965, root := ~, start := 293832, stop := 293964 ) ], count := [ 7, 2, 8, 6 ], mathmode := "Text", name := "List", next := 293977, root := ~, start := 293715, stop := 293976 ), 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 := 294011, root := ~, start := 293985, stop := 294010 ), 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 := 294042, root := ~, start := 294034, stop := 294041 ), 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 := 294065, root := ~, start := 294054, stop := 294064 ), 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 := 294096, root := ~, start := 294088, stop := 294095 ), 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 := 294143, root := ~, start := 294106, stop := 294142 ), 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 := 294159, root := ~, start := 294145, stop := 294158 ), 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 := 294249, root := ~, start := 294166, stop := 294248 ), 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 := 294320, root := ~, start := 294316, stop := 294319 ), 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 := 294402, root := ~, start := 294388, stop := 294401 ), 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 := 294439, root := ~, start := 294425, stop := 294438 ), 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 := 294454, root := ~, start := 294446, stop := 294453 ), 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 := 294502, root := ~, start := 294488, stop := 294501 ), 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 := 294532, root := ~, start := 294518, stop := 294531 ), 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 := 294582, root := ~, start := 294578, stop := 294581 ), 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 := 294634, root := ~, start := 294626, stop :\ = 294633 ), 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 := 294665, root := ~, start := 294651, stop := 294664 ) , 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 := 294675, root := ~, start := 29466\ 5, stop := 294674 ), 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 := 294716, root := ~, start := 294708, stop \ := 294715 ), 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 := 294751, root := ~, start := 294736, stop := 294750 ), 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\ := 294785, root := ~, start := 294769, stop := 294784 ), rec( content := "; o\ r\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 15 ], name := "Item", next := 294803, root := ~, start := 294\ 601, stop := 294802 ), 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 := 294916, root := ~, start := 294\ 902, stop := 294915 ), rec( content := ")", count := [ 7, 2, 8, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmod\ e := "Text", name := "C", next := 294921, root := ~, start := 294883, stop := \ 294920 ), 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 := 294955, root := ~, start := 294941, stop := 294954 ) , 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 := 294965, root := ~, start := 29495\ 5, stop := 294964 ), 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 := 295022, root := ~ , star\ t := 295014, stop := 295021 ), rec( content := ".\n ", count := [ 7, 2, 8\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], name := "Ite\ m", next := 295037, root := ~, start := 294810, stop := 295036 ) ], count := [ 7, 2, 8, 14 ], mathmode := "Text", name := "List", next := 295049, root := ~, start := 294588, stop := 295048 ), 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 := 295072, root := ~, start := 295058, stop := 295071 ), 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 := 295091, root := ~, start := 295077, stop := 295090 ), 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 := 295160, root := ~, start := 295130, stop := 295159 ), 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 := 295224, root := ~, start := 295210, stop := 295223 ), 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 := 295327, root := ~, start := 295297, stop := 295326 ), 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 := 295391, root := ~, start := 295377, stop := 295390 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 295591, root := ~, start := 295587, stop := 295590 ), 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 := 295627, root := ~, start := 295597, stop := 295626 ), 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 := 295666, root := ~, start := 295652, stop := 295665 ), 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 := 295727, root := ~, start := 295698, stop := 295726 ), 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 := 295774, root := ~, start := 295760, stop := 295773 ), 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 := 295868, root := ~, start := 295854, stop := 295867 ), 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 := 295907, root := ~, start := 295893, stop := 295906 ), 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 := 295944, root := ~, start := 295936, stop := 295943 ), 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 := 296030, root := ~, start := 295972, stop := 296029 ), 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 := 296075, root := ~, start := 296036, stop := 296074 ), 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 := 296175, root := ~, start := 296167, stop := 296174 ), 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 := 296194, root := ~, start := 296180, stop := 296193 ), 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 := 296215, root := ~, start := 296207, stop := 296214 ), 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 := 296258, root := ~, start := 296246, stop := 296257 ), 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 := 296316, root := ~, start := 296308, stop := 296315 ), 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 := 296332, root := ~, start := 296324, stop := 296331 ), 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 := 296426, root := ~, start := 296412, stop := 296425 ), 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 := 296464, root := ~, start := 296450, stop := 296463 ), 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 := 296650, root := ~, start := 296582, stop := 296649 ), 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 := 296692, root := ~, start := 296678, stop := 296691 ), 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 := 296819, root := ~, start := 296778, stop := 296818 ), 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 := 296829, root := ~, start := 296825, stop := 296828 ), 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 := 297540, root := ~, start := 296835, stop := 297539 ), rec( content := "\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 297557, root := ~, start := 293483, stop := 297556 ) ], count := [ 7, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 297571, root := ~, start := 293230, stop := 297570 ), 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 := 297642, root := ~, start := 297592, stop := 297641 ), rec( attributes := rec( Arg := "digraph", Name := "NautyCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 297695, root := ~, start := 297645, stop := 297694 ), 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 := 297727, root := ~, start := 297698, stop := 297726 ), 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 := 297792, root := ~, start := 297762, stop := 297791 ), 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 := 297917, root := ~, start := 297852, stop := 297916 ), 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 := 298047, root := ~, start := 298033, stop := 298046 \ ), rec( content := ")", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ C", next := 298052, root := ~, start := 298008, stop := 298051 ), 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 := 298071, root := \ ~, start := 298057, stop := 298070 ), 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 := 2981\ 25, root := ~, start := 297993, stop := 298124 ), 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 := 29815\ 9, root := ~, start := 298150, stop := 298158 ), 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\ := 298212, root := ~, start := 298180, stop := 298211 ), 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 := 298264, ro\ ot := ~, start := 298250, stop := 298263 ), rec( content := ")", count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "C", next := 298269, root := ~, start \ := 298225, stop := 298268 ), 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 := 298312, root := ~, star\ t := 298303, stop := 298311 ), 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 := 298331, root := ~, start := 298317, stop := 298330 ), 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 := 298373, root := ~, start := 298132, stop := 298372 ) ], count := [ 7, 2, 9, 6 ], mathmode := "Text", name := "List", next := 298385, root := ~, start := 297980, stop := 298384 ), 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 := 298448, root := ~, start := 298418, stop := 298447 ), 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 := 298573, root := ~, start := 298508, stop := 298572 ), 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 := 298583, root := ~, start := 298579, stop := 298582 ), 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 := 298619, root := ~, start := 298605, stop := 298618 ), 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 := 299012, root := ~, start := 298721, stop := 299011 ), rec( content := "\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Description", next := 299029, root := ~, start := 297730, stop := 299028 ) ], count := [ 7, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 299043, root := ~, start := 297577, stop := 299042 ), 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 := 299105, root := ~, start := 299064, stop := 299104 ), 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 := 299147, root := ~, start := 299108, stop := 299146 ), 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 := 299185, root := ~, start := 299171, stop := 299184 ), 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 := 299331, root := ~, start := 299312, stop := 299330 ), 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 := 299350, root := ~, start := 299336, stop := 299349 ), 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 := 299456, root := ~, start := 299437, stop := 299455 ), 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 := 299516, root := ~, start := 299502, stop := 299515 ), 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 := 299553, root := ~, start := 299539, stop := 299552 ), 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 := 299656, root := ~, start := 299652, stop := 299655 ), 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 := 299751, root := ~, start := 299718, stop := 299750 ), 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 := 299789, root := ~, start := 299753, stop := 299788 ), 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 := 299828, root := ~, start := 299795, stop := 299827 ), 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 := 299979, root := ~, start := 299960, stop := 299978 ), 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)(3,8)(4,6)(5,7), (\ 1,3,4,7)(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 := 301449, root := ~, start := 300022, stop := 301448 ), rec( content := "\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 301466, root := ~, start := 299150, stop := 301465 ) ], count := [ 7, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 301480, root := ~, start := 299049, stop := 301479 ), 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 := 301543, root := ~, start := 301501, stop := 301542 ), 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 := 301612, root := ~, start := 301546, stop := 301611 ), 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 := 301653, root := ~, start := 301633, stop := 301652 ), 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 := 301724, root := ~, start := 301698, stop := 301723 ), 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 := 301765, root := ~, start := 301751, stop := 301764 ), 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 := 302154, root := ~, start := 301772, stop := 302153 ), rec( content := "\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 302171, root := ~, start := 301615, stop := 302170 ) ], count := [ 7, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 302185, root := ~, start := 301486, stop := 302184 ), 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 := 302251, root := ~, start := 302206, stop := 302250 ), 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 := 302311, root := ~, start := 302254, stop := 302310 ), 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 := 302355, root := ~, start := 302332, stop := 302354 ), 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 := 302447, root := ~, start := 302421, stop := 302446 ), 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 := 302492, root := ~, start := 302478, stop := 302491 ), 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 := 303013, root := ~, start := 302499, stop := 303012 ), rec( content := "\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 303030, root := ~, start := 302314, stop := 303029 ) ], count := [ 7, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 303044, root := ~, start := 302191, stop := 303043 ), 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 := 303115, root := ~, start := 303065, stop := 303114 ), 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 := 303175, root := ~, start := 303118, stop := 303174 ), 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 := 303224, root := ~, start := 303196, stop := 303223 ), 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 := 303269, root := ~, start := 303247, stop := 303268 ), 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 := 303321, root := ~, start := 303295, stop := 303320 ), 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 := 303366, root := ~, start := 303352, stop := 303365 ), 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 := 303408, root := ~, start := 303398, stop := 303407 ), 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 := 3034\ 72, root := ~, start := 303458, stop := 303471 ), rec( content := ")", count := [ 7, 2, 13, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 303477, root := ~, start := 303437, stop := 303476 ), 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\ := 303529, root := ~, start := 303508, stop := 303528 ) ], count := [ 7, 2, 13, 4 ], name := "Mar\ k", next := 303536, root := ~, start := 303502, stop := 303535 ), 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 := 303579, root := ~, start := 303571, stop \ := 303578 ), 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 := 303653, root := ~, start := 30\ 3639, stop := 303652 ), 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 := 303671, root := ~, start := 30\ 3619, stop := 303670 ), rec( content := ".\n ", count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], name := "Ite\ m", next := 303686, root := ~, start := 303543, stop := 303685 ), 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\ := 303721, root := ~, start := 303700, stop := 303720 ) ], count := [ 7, 2, 13, 6 ], name := "Mar\ k", next := 303728, root := ~, start := 303694, stop := 303727 ), 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 := 303786, root := ~, star\ t := 303763, stop := 303785 ), 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 := 303884, root := ~, start := 303873, stop \ := 303883 ), 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 := 303936, root := ~, start := 303922, stop := 303935 ), rec\ ( content := ")", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "C", next\ := 303941, root := ~, start := 303906, stop := 303940 ), rec( content := ".\n\ ", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 7 ], name := "Item", next := 303956, root := ~, start := 303\ 735, stop := 303955 ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "List", next := 303968, root := ~, start := 303489, stop := 303967 ), 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, 1, 1, 1, 2, 2, 2, 1 ]\ngap> DigraphOrbit\ Reps(D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,5,7)(2\ ,4,8)(3,6,9)(10,11,12), (1,2,3)(4,7,10)(5,9,11)(6,8,12) ]\ngap> 10 / gens[sch[\ 10]];\n7\ngap> 7 / gens[sch[7]];\n5\ngap> 5 / gens[sch[5]];\n1", count := [ 7, 2, 13, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 10 ], mathmode := "Text", name := "Example", next := 304745, root := ~, start := 303974, stop := 304744 ), rec( content := "\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Description", next := 304762, root := ~, start := 303178, stop := 304761 ) ], count := [ 7, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 304776, root := ~, start := 303050, stop := 304775 ), 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 := 304846, root := ~, start := 304797, stop := 304845 ), 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 := 304896, root := ~, start := 304849, stop := 304895 ), 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 := 304941, root := ~, start := 304917, stop := 304940 ), 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 := 304987, root := ~, start := 304979, stop := 304986 ), 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 := 305045, root := ~, start := 305019, stop := 305044 ), 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 := 305090, root := ~, start := 305076, stop := 305089 ), 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 := 305466, root := ~, start := 305097, stop := 305465 ), rec( content := "\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 305483, root := ~, start := 304899, stop := 305482 ) ], count := [ 7, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 305497, root := ~, start := 304782, stop := 305496 ), 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 := 305598, root := ~, start := 305518, stop := 305597 ), 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 := 305621, root := ~, start := 305610, stop := 305620 ), 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 := 305637, root := ~, start := 305625, stop := 305636 ), rec( content := ".", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Returns", next := 305648, root := ~, start := 305601, stop := 305647 ), 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 := 305703, root := ~, start := 305692, stop := 305702 ), 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 := 305771, root := ~, start := 305756, stop := 305770 ), 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 := 305802, root := ~, start := 305787, stop := 305801 ), 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 := 305870, root := ~, start := 305809, stop := 305869 ), 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 := 305932, root := ~, start := 305928, stop := 305931 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 15, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 306240, root := ~, start := 306210, stop := 306239 ), 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 := 306276, root := ~, start := 306246, stop := 306275 ), 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 := 306878, root := ~, start := 306283, stop := 306877 ), rec( content := "\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Description", next := 306895, root := ~, start := 305651, stop := 306894 ) ], count := [ 7, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 306909, root := ~, start := 305503, stop := 306908 ), 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 := 307056, root := ~, start := 306930, stop := 307055 ), 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 := 307079, root := ~, start := 307068, stop := 307078 ), 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 := 307095, root := ~, start := 307083, stop := 307094 ), rec( content := ".", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 307106, root := ~, start := 307059, stop := 307105 ), 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 := 307268, root := ~, start := 307253, stop := 307267 ), 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 := 307306, root := ~, start := 307291, stop := 307305 ), 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 := 307321, root := ~, start := 307313, stop := 307320 ), 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 := 307370, root := ~, start := 307355, stop := 307369 ), 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 := 307401, root := ~, start := 307386, stop := 307400 ), 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 := 307498, root := ~, start := 307490, stop :\ = 307497 ), 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 := 307529, root := ~, start := 307515, stop := 307528 ) , 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 := 307539, root := ~, start := 30752\ 9, stop := 307538 ), 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 := 307580, root := ~, start := 307572, stop \ := 307579 ), 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 := 307615, root := ~, start := 307600, stop := 307614 ), 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\ := 307649, root := ~, start := 307633, stop := 307648 ), rec( content := "; o\ r\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 5 ], name := "Item", next := 307667, root := ~, start := 307\ 465, stop := 307666 ), 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 := 307780, root := ~, start := 307\ 766, stop := 307779 ), rec( content := ")", count := [ 7, 2, 16, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmod\ e := "Text", name := "C", next := 307785, root := ~, start := 307747, stop := \ 307784 ), 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 := 307819, root := ~, start := 307805, stop := 307818 ) , 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 := 307829, root := ~, start := 30781\ 9, stop := 307828 ), 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 := 307886, root := ~ , star\ t := 307878, stop := 307885 ), rec( content := ".\n ", count := [ 7, 2, 1\ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], name := "Ite\ m", next := 307901, root := ~, start := 307674, stop := 307900 ) ], count := [ 7, 2, 16, 4 ], mathmode := "Text", name := "List", next := 307913, root := ~, start := 307452, stop := 307912 ), 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 := 307937, root := ~, start := 307922, stop := 307936 ), 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 := 307957, root := ~, start := 307942, stop := 307956 ), 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 := 308018, root := ~, start := 308003, stop := 308017 ), 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 := 308038, root := ~, start := 308023, stop := 308037 ), 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 := 308072, root := ~, start := 308057, stop := 308071 ), 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 := 308096, root := ~, start := 308081, stop := 308095 ), 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 := 308151, root := ~, start := 308140, stop := 308150 ), 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 := 308316, root := ~, start := 308233, stop := 308315 ), 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 := 308383, root := ~, start := 308379, stop := 308382 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 16, 10 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 308691, root := ~, start := 308661, stop := 308690 ), 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 := 308727, root := ~, start := 308697, stop := 308726 ), 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 := 309340, root := ~, start := 308734, stop := 309339 ), rec( content := "\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 309357, root := ~, start := 307109, stop := 309356 ) ], count := [ 7, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 309371, root := ~, start := 306915, stop := 309370 ), 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 := 309472, root := ~, start := 309392, stop := 309471 ), 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 := 309541, root := ~, start := 309530, stop := 309540 ), rec( content := ".", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 309552, root := ~, start := 309475, stop := 309551 ), 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 := 309647, root := ~, start := 309632, stop := 309646 ), 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 := 309671, root := ~, start := 309656, stop := 309670 ), 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 := 309726, root := ~, start := 309715, stop := 309725 ), 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 := 309736, root := ~, start := 309732, stop := 309735 ), 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 := 309763, root := ~, start := 309745, stop := 309762 ), 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 := 309794, root := ~, start := 309779, stop := 309793 ), 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 := 309827, root := ~, start := 309812, stop := 309826 ), 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 := 309851, root := ~, start := 309843, stop := 309850 ), 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 := 309892, root := ~, start := 309877, stop := 309891 ), 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 := 309927, root := ~, start := 309912, stop := 309926 ), 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 := 309986, root := ~, start := 309978, stop := 309985 ), 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 := 309999, root := ~, start := 309991, stop := 309998 ), 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 := 310018, root := ~, start := 310003, stop := 310017 ), 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 := 310037, root := ~, start := 310024, stop := 310036 ), 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 := 310067, root := ~, start := 310052, stop := 310066 ), 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 := 310110, root := ~, start := 310083, stop := 310109 ), 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 := 310140, root := ~, start := 310125, stop := 310139 ), 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 := 310146, root := ~, start := 310142, stop := 310145 ), 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 := 310280, root := ~, start := 310272, stop := 310279 ), 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 := 310319, root := ~, start := 310304, stop := 310318 ), 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 := 310334, root := ~, start := 310330, stop := 310333 ), 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 := 310369, root := ~, start := 310354, stop := 31036\ 8 ), rec( content := ", p) = digraph2", count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "C", next := 310388, root := ~, start := 310340, stop := 310387 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 17, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 310697, root := ~, start := 310667, stop := 310696 ), 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 := 310733, root := ~, start := 310703, stop := 310732 ), 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 := 311356, root := ~, start := 310740, stop := 311355 ), rec( content := "\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 311373, root := ~, start := 309555, stop := 311372 ) ], count := [ 7, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 311387, root := ~, start := 309377, stop := 311386 ), 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 := 311534, root := ~, start := 311408, stop := 311533 ), 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 := 311576, root := ~, start := 311565, stop := 311575 ), rec( content := ".", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Returns", next := 311587, root := ~, start := 311537, stop := 311586 ), 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 := 311760, root := ~, start := 311745, stop := 311759 ), 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 := 311798, root := ~, start := 311783, stop := 311797 ), 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 := 311813, root := ~, start := 311805, stop := 311812 ), 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 := 311862, root := ~, start := 311847, stop := 311861 ), 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 := 311893, root := ~, start := 311878, stop := 311892 ), 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 := 311990, root := ~, start := 311982, stop :\ = 311989 ), 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 := 312021, root := ~, start := 312007, stop := 312020 ) , 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 := 312031, root := ~, start := 31202\ 1, stop := 312030 ), 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 := 312072, root := ~, start := 312064, stop \ := 312071 ), 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 := 312107, root := ~, start := 312092, stop := 312106 ), 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\ := 312141, root := ~, start := 312125, stop := 312140 ), rec( content := "; o\ r\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 5 ], name := "Item", next := 312159, root := ~, start := 311\ 957, stop := 312158 ), 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 := 312272, root := ~, start := 312\ 258, stop := 312271 ), rec( content := ")", count := [ 7, 2, 18, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmod\ e := "Text", name := "C", next := 312277, root := ~, start := 312239, stop := \ 312276 ), 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 := 312311, root := ~, start := 312297, stop := 312310 ) , 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 := 312321, root := ~, start := 31231\ 1, stop := 312320 ), 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 := 312378, root := ~ , star\ t := 312370, stop := 312377 ), rec( content := ".\n ", count := [ 7, 2, 1\ 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], name := "Ite\ m", next := 312393, root := ~, start := 312166, stop := 312392 ) ], count := [ 7, 2, 18, 4 ], mathmode := "Text", name := "List", next := 312405, root := ~, start := 311944, stop := 312404 ), 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 := 312432, root := ~, start := 312414, stop := 312431 ), 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 := 312609, root := ~, start := 312549, stop := 312608 ), 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 := 312695, root := ~, start := 312687, stop := 312694 ), 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 := 312762, root := ~, start := 312747, stop := 312761 ), 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 := 312798, root := ~, start := 312783, stop := 312797 ), 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 := 312830, root := ~, start := 312815, stop := 312829 ), 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 := 312866, root := ~, start := 312851, stop := 312865 ), 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 := 312885, root := ~, start := 312877, stop := 312884 ), 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 := 312943, root := ~, start := 312928, stop := 312942 ), 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 := 312972, root := ~, start := 312964, stop := 312971 ), 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 := 312988, root := ~, start := 312980, stop := 312987 ), 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 := 313036, root := ~, start := 313021, stop := 313035 ), 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 := 313056, root := ~, start := 313041, stop := 313055 ), 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 := 313139, root := ~, start := 313124, stop := 31313\ 8 ), 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 := 3131\ 66, root := ~, start := 313151, stop := 313165 ), rec( content := "[i ^ p]", count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 313177, root := ~, start := 313121, stop := 313176 ), 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 := 313203, root := ~, start := 313195, stop := 313202 ), 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 := 313222, root := ~, start := 313207, stop := 313221 ), 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 := 313232, root := ~, start := 313228, stop := 313231 ), 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 := 313327, root := ~, start := 313316, stop := 313326 ), 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 := 313333, root := ~, start := 313329, stop := 313332 ), 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/sebasguts/NautyTracesInterface", count := [ 7, 2, 18, 11 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := \ "Text", name := "URL", next := 108, root := ~, start := 17, stop := 107 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 114, root := ~, start := 1, stop := 113 ), 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 := 173, root := ~, start := 134, stop := 172 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 179, root := ~, start := 114, stop := 178 ), 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 := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), 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 := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), 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 := 313641, root := ~, start := 313611, stop := 313640 ), 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 := 313677, root := ~, start := 313647, stop := 313676 ), 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 := 314660, root := ~, start := 313684, stop := 314659 ), rec( content := "\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Description", next := 314677, root := ~, start := 311590, stop := 314676 ) ], count := [ 7, 2, 18, 1 ], mathmode := "Text", name := "ManSection", next := 314691, root := ~, start := 311393, stop := 314690 ), 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 := 314768, root := ~, start := 314712, stop := 314767 ), 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 := 314817, root := ~, start := 314771, stop := 314816 ), 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 := 314879, root := ~, start := 314869, stop := 314878 ), 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 := 314904, root := ~, start := 314883, stop := 314903 ), 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 := 314989, root := ~, start := 314963, stop := 314988 ), 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 := 315041, root := ~, start := 315027, stop := 315040 ), 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 := 315047, root := ~, start := 315043, stop := 315046 ), 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 := 315086, root := ~, start := 315075, stop := 315085 ), 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 := 315151, root := ~, start := 315143, stop := 315150 ), 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 := 315176, root := ~, start := 315163, stop := 315175 ), 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 := 315238, root := ~, start := 315224, stop := 315237 ), 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 := 315257, root := ~, start := 315249, stop := 315256 ), 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 := 315279, root := ~, start := 315265, stop := 315278 ), 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 := 315289, root := ~, start := 315285, stop := 315288 ), 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 := 315324, root := ~, start := 315298, stop := 315323 ), 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 := 315369, root := ~, start := 315342, stop := 315368 ), 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 := 316262, root := ~, start := 315392, stop := 316261 ), rec( content := "\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Description", next := 316279, root := ~, start := 314820, stop := 316278 ) ], count := [ 7, 2, 19, 1 ], mathmode := "Text", name := "ManSection", next := 316293, root := ~, start := 314697, stop := 316292 ), 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 := 316431, root := ~, start := 316314, stop := 316430 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 4 ], mathmode := "Text", name := "Oper", next := 316499, root := ~, start := 316434, stop := 316498 ), 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 := 316622, root := ~, start := 316502, stop := 316621 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphAutomo\ rphism" ), content := 0, count := [ 7, 2, 20, 8 ], mathmode := "Text", name := "Oper", next := 316683, root := ~, start := 316625, stop := 316682 ), 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 := 316706, root := ~, start := 316695, stop := 316705 ), 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 := 316722, root := ~, start := 316710, stop := 316721 ), rec( content := ".", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Returns", next := 316733, root := ~, start := 316686, stop := 316732 ), 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 := 316781, root := ~, start := 316754, stop := 316780 ), 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 := 316801, root := ~, start := 316790, stop := 316800 ), 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 := 316851, root := ~, start := 316843, stop := 316850 ), 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 := 316897, root := ~, start := 316887, stop := 316896 ), 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 := 316927, root := ~, start := 316917, stop := 316926 ), 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 := 316937, root := ~, start := 316933, stop := 316936 ), 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 := 316971, root := ~, start := 316943, stop := 316970 ), 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 := 316991, root := ~, start := 316980, stop := 316990 ), 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 := 317041, root := ~, start := 317033, stop := 317040 ), 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 := 317090, root := ~, start := 317076, stop := 317089 ), 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 := 317100, root := ~, start := 317096, stop := 317099 ), 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 := 317146, root := ~, start := 317138, stop := 317145 ), 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 := 317171, root := ~, start := 317153, stop := 317170 ), 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 := 317201, root := ~, start := 317191, stop := 317200 ), 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 := 317225, root := ~, start := 317215, stop := 317224 ), 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 := 317289, root := ~, start := 317281, stop := 317288 ), 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 := 317336, root := ~, start := 317326, stop := 317335 ), 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\ := 317367, root := ~, start := 317357, stop := 317366 ), rec( content := ";\n\ ", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 13 ], name := "Item", next := 317382, root := ~, start := 317\ 266, stop := 317381 ), 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 := 317420, root := ~, s\ tart := 317412, stop := 317419 ), 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 := 317434, root := ~, start := 317426, stop :\ = 317433 ), rec( content := "]", count := [ 7, 2, 20, 15 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "C", next := 317439, root := ~, start := 317404, stop := 317438 ), 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 := 317472, root := ~, start := 317462, stop := 317471 ), 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\ := 317501, root := ~, start := 317488, stop := 317500 ), 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 :\ = 317534, root := ~, start := 317524, stop := 317533 ), rec( content := "; and\ \n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 15 ], name := "Item", next := 317553, root := ~, start := 317\ 389, stop := 317552 ), 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 := 317583, root :\ = ~, start := 317575, stop := 317582 ), 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 := 317604, root := ~, start := 31759\ 6, stop := 317603 ), 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 := 317640, root := ~, start := 31763\ 0, stop := 317639 ), rec( content := ".\n ", count := [ 7, 2, 20, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17 ], name := "Ite\ m", next := 317655, root := ~, start := 317560, stop := 317654 ) ], count := [ 7, 2, 20, 12 ], mathmode := "Text", name := "List", next := 317667, root := ~, start := 317253, stop := 317666 ), 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 := 317741, root := ~, start := 317681, stop := 317740 ), 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 := 317746, root := ~, start := 317742, stop := 317745 ), 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 := 317770, root := ~, start := 317759, stop := 317769 ), 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 := 317786, root := ~, start := 317775, stop := 317785 ), 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 := 317801, root := ~, start := 317791, stop := 317800 ), 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 := 317947, root := ~, start := 317871, stop := 317946 ), 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 := 318130, root := ~, start := 318102, stop \ := 318129 ), 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 := 318156, root := ~, start := 318148, stop := 318155 ), 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 := 318170, root :\ = ~, start := 318160, stop := 318169 ), 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 := 318211, root :\ = ~, start := 318184, stop := 318210 ), rec( content := ". ", count := [ 7, 2,\ 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ] , name := "Ite\ m", next := 318220, root := ~, start := 318087, stop := 318219 ), 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 := 318268, root := ~, start := 318242, stop \ := 318267 ), 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 := 318294, root := ~, start := 318286, stop := 318293 ), 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 := 318312, root :\ = ~, start := 318298, stop := 318311 ), 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 := 318354, root :\ = ~, start := 318326, stop := 318353 ), rec( content := ". ", count := [ 7, 2,\ 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ] , name := "Ite\ m", next := 318363, root := ~, start := 318227, stop := 318362 ) ], count := [ 7, 2, 20, 21 ], mathmode := "Text", name := "List", next := 318375, root := ~, start := 318074, stop := 318374 ), 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 := 318452, root := ~, start := 318424, stop := 318451 ), 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 := 318524, root := ~, start := 318510, stop := 318523 ), 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 := 319467, root := ~, start := 318531, stop := 319466 ), rec( content := "\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Description", next := 319484, root := ~, start := 316736, stop := 319483 ) ], count := [ 7, 2, 20, 1 ], mathmode := "Text", name := "ManSection", next := 319498, root := ~, start := 316299, stop := 319497 ), 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 := 319576, root := ~, start := 319523, stop := 319575 ), 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 := 319667, root := ~, start := 319581, stop := 319666 ), 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 := 319693, root := ~, start := 319682, stop := 319692 ), 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 := 319709, root := ~, start := 319697, stop := 319708 ), rec( content := ". ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Returns", next := 319721, root := ~, start := 319672, stop := 319720 ), 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 := 319785, root := ~, start := 319760, stop := 319784 ), 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 := 319837, root := ~, start := 319826, stop := 319836 ), 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 := 319903, root := ~, start := 319889, stop := 319902 ), 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 := 319916, root := ~, start := 319912, stop := 319915 ), 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 := 319942, root := ~, start := 319931, stop := 319941 ), 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 := 319978, root := ~, start := 319955, stop := 319977 ), 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 := 320012, root := ~, start := 319998, stop := 320011 ), 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 := 320027, root := ~, start := 320016, stop := 320026 ), 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 := 320090, root := ~, start := 320079, stop := 320089 ), 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 := 320138, root := ~, start := 320124, stop := 320137 ), 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 := 320178, root := ~, start := 320167, stop := 320177 ), 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 := 320196, root := ~, start := 320182, stop := 320195 ), 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 := 320208, root := ~, start := 320200, stop := 320207 ), 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 := 320221, root := ~, start := 320213, stop := 320220 ), 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 := 320261, root := ~, start := 320250, stop := 32026\ 0 ), 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 := 3202\ 85, root := ~, start := 320274, stop := 320284 ), rec( content := "[v]", count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 320292, root := ~, start := 320247, stop := 320291 ), 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 := 320304, root := ~, start := 320300, stop := 320303 ), 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 := 320337, root := ~, start := 320329, stop := 320336 ), 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 := 320400, root := ~, start := 320386, stop := 320399 ), 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 := 320442, root := ~, start := 320434, stop := 32044\ 1 ), 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 := 3204\ 82, root := ~, start := 320468, stop := 320481 ), rec( content := "))", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "C", next := 320488, root := ~, start := 320405, stop := 320487 ), 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 := 320535, root := ~, start := 320521, stop := 320534 ), 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 := 320541, root := ~, start := 320537, stop := 320540 ), 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 := 320593, root := ~, start := 320558, stop := 320592 ), 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 := 321029, root := ~, start := 320608, stop := 321028 ), rec( content := "\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Description", next := 321048, root := ~, start := 319726, stop := 321047 ) ], count := [ 7, 2, 21, 1 ], mathmode := "Text", name := "ManSection", next := 321064, root := ~, start := 319506, stop := 321063 ), rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 321078, root := ~, start := 277156, stop := 321077 ) , 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 := 321135, root := ~, start := 321091, stop := 321134 ), 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 := 321616, root := ~, start := 321458, stop := 321615 ), 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 := 321658, root := ~, start := 321641, stop := 321657 ), rec( content := ".", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 321669, root := ~, start := 321619, stop := 321668 ), 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 := 321750, root := ~, start := 321741, stop := 321749 ), 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 := 321779, root := ~, start := 321770, stop := 321778 ), 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 := 321864, root := ~, start := 321860, stop := 321863 ), 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 := 321881, root := ~, start := 321873, stop := 321880 ), 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 := 321894, root := ~, start := 321886, stop := 321893 ), 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 := 321959, root := ~, start := 321926, stop := 321958 ), 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 := 321974, root := ~, start := 321966, stop := 321973 ), 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 := 322011, root := ~, start := 322003, stop := 322010 ), 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 := 322064, root := ~, start := 322055, stop := 322063 ), 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 := 322086, root := ~, start := 322072, stop := 322085 ), 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 := 32\ 2123, root := ~, start := 322112, stop := 322122 ) ], count := [ 7, 3, 1, 5 ], name := "Mar\ k", next := 322130, root := ~, start := 322106, stop := 322129 ), 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 := 322201, root := \ ~, start := 322190, stop := 322200 ), 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\ := 322206, root := ~, start := 322202, stop := 322205 ), 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 := 322230, root := \ ~, start := 322219, stop := 322229 ), 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 := 322303, root := ~, start := 322286, stop := 322302 ), 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 := 322345, root := ~, start := 322337, stop := 322344 ), 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 := 322382, root := ~, start := 322371,\ stop := 322381 ), 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 := 322400, root := ~, start := 322383, stop := 322399 ), rec( \ content := ", t)", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 322408, root := ~, start := 322368, stop := 322407 ), 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 := 322465, root := ~, start := 322457, stop := 322464 ), 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 := 322519, root := ~, start := 322486, stop := 322518 ), 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\ := 322556, root := ~, start := 322545, stop := 322555 ), 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\ := 322604, root := ~, start := 322571, stop := 322603 ), 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 := 322871, root := \ ~, start := 322838, stop := 322870 ), 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 := 322885, root := ~, start := 322881, stop := 322884 ), 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\ 22909, root := ~, start := 322898, stop := 322908 ), 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 := 322924, root := ~, start := 322913, stop \ := 322923 ), 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 := 323057, root := ~, start :\ = 323024, stop := 323056 ), 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 := 323078, root := ~, start := 323061, stop := 323077 ), 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 := 323144, root := ~, start := 322137, stop := 323143 ), 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\ := 323175, root := ~, start := 323158, stop := 323174 ) ], count := [ 7, 3, 1, 9 ], name := "Mar\ k", next := 323182, root := ~, start := 323152, stop := 323181 ), 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 := 323218, root := ~, start := 323207, stop := 3\ 23217 ), 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 := 323256, root := ~, start := 323239, stop := 32325\ 5 ), 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\ := 323318, root := ~, start := 323301, stop := 323317 ), 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 := 323384, root := ~, start := 323373, stop := 32\ 3383 ), 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 := 323416, root := ~, start := 323399, stop := 323415 ), 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 :=\ 323480, root := ~, start := 323469, stop := 323479 ), 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 := 323498, root := ~, s\ tart := 323481, stop := 323497 ), rec( content := ", t)", count := [ 7, 3, 1, \ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1,\ 10 ], mathmode := "Text", name := "C", next := 323506, root := ~, start := 32\ 3466, stop := 323505 ), 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 := 323542, root := ~, start := 3235\ 25, stop := 323541 ), 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 := 323581, root := ~, start := 3235\ 64, stop := 323580 ), 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 := 323602, root := ~, start := 323594, stop := 32360\ 1 ), 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 := 323608, root := ~, star\ t := 323604, stop := 323607 ), 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 := 323645, root := ~, start := 3236\ 34, stop := 323644 ), 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 := 323660, root := ~, start := 323649, stop := 323659 ), 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 := 323705, root := ~, start := 323688, stop := 323704 ), 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 := 323743, root := ~, start := 323189, stop := 323742 ), 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\ := 323775, root := ~, start := 323757, stop := 323774 ) ], count := [ 7, 3, 1, 12 ], name := "Mar\ k", next := 323782, root := ~, start := 323751, stop := 323781 ), 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 :\ = 323865, root := ~, start := 323850, stop := 323864 ), 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\ := 323908, root := ~, start := 323875, stop := 323907 ), 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 := 323966, root := ~, start := 323948, stop := 323965 ), 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 := 324054, root := ~, start := 323\ 789, stop := 324053 ), 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 := 324079, root := ~, start := 324068, stop \ := 324078 ) ], count := [ 7, 3, 1, 14 ], name := "Mark", next := 324086, root := ~, start := 324\ 062, stop := 324085 ), 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 := 324165, root := ~, start := 324154, stop \ := 324164 ), 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 := 324171, root :=\ ~, start := 324167, stop := 324170 ), 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 := 324195, root := ~, start := 3241\ 84, stop := 324194 ), 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 \ := 324268, root := ~, start := 324257, stop := 324267 ), 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 := 324283, root := ~, start := 324279, stop \ := 324282 ), 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 := 324307, root := ~, start := 324296, stop := 324306 ), 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 := 324322, root :=\ ~, start := 324311, stop := 324321 ), 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 := 324408, root := ~, start := 324093, stop := 324407 ), 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\ := 324438, root := ~, start := 324422, stop := 324437 ) ], count := [ 7, 3, 1, 18 ], name := "Mar\ k", next := 324445, root := ~, start := 324416, stop := 324444 ), 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 := 324499, root := ~, start := 324491, stop \ := 324498 ), 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\ := 324509, root := ~, start := 324501, stop := 324508 ), 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 := 324522, root := ~, star\ t := 324514, stop := 324521 ), 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 := 324550, root := ~, start := 324542, stop \ := 324549 ), 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 := 324601, root := ~, start := 324593, stop \ := 324600 ), 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 := 324\ 677, root := ~, start := 324669, stop := 324676 ), 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 := 324750, root := ~, start := 324746, stop := 324749 ), 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 := 324805, root := ~, start := 324789, stop := 324804\ ), 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 := 324830, root := ~, start := 324818, stop := 324829 ), 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\ 24853, root := ~, start := 324842, stop := 324852 ), 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 := \ 324893, root := ~, start := 324885, stop := 324892 ), 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 := 324906, root := ~, start := 324898, stop \ := 324905 ), 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 := 3249\ 80, root := ~, start := 324452, stop := 324979 ), 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 := 325006, root := ~, start := 324\ 994, stop := 325005 ) ], count := [ 7, 3, 1, 21 ], name := "Mark", next := 3250\ 13, root := ~, start := 324988, stop := 325012 ), 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 := 325106, root := ~, star\ t := 325097, stop := 325105 ), 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 := 325149, root := ~, star\ t := 325116, stop := 325148 ), 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 := 325197, root := \ ~, start := 325188, stop := 325196 ), 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 := 325226, root := ~, start := 325217, stop \ := 325225 ), 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 := 325271, root := ~, start := 325259, stop \ := 325270 ), rec( content := ".\n ", count := [ 7, 3, 1, 22 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ], name := "Ite\ m", next := 325286, root := ~, start := 325020, stop := 325285 ), 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\ := 325318, root := ~, start := 325300, stop := 325317 ) ], count := [ 7, 3, 1, 23 ], name := "Mar\ k", next := 325325, root := ~, start := 325294, stop := 325324 ), 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 := 325399, root\ := ~, start := 325390, stop := 325398 ), 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 := 325412, root := ~, start := 325403, stop \ := 325411 ), 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 := 325506, root := ~, start := 325497, stop := 325505 ), 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 := 325585, root := ~, start := 325576, stop \ := 325584 ), 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 := 325620, root := ~, start := 325587, stop := 3\ 25619 ), 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 := 325682, root := ~, start \ := 325664, stop := 325681 ), rec( content := " (if any).\n ", count := [ \ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], name := "Ite\ m", next := 325706, root := ~, start := 325332, stop := 325705 ), 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\ := 325740, root := ~, start := 325726, stop := 325739 ) ], count := [ 7, 3, 1, 25 ], name := "Mar\ k", next := 325747, root := ~, start := 325720, stop := 325746 ), 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 := 325865, root := ~, start := 325856, stop \ := 325864 ), 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 := \ 325967, root := ~, start := 325880, stop := 325966 ), 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 := 326047, root := ~, start := 325754, stop := 326046 ), 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\ := 326081, root := ~, start := 326067, stop := 326080 ) ], count := [ 7, 3, 1, 27 ], name := "Mar\ k", next := 326088, root := ~, start := 326061, stop := 326087 ), 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 := 326206, root := ~, start := 326197, stop \ := 326205 ), 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 := \ 326308, root := ~, start := 326221, stop := 326307 ), 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 := 326388, root := ~, start := 326095, stop := 326387 ), 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 :\ = 326413, root := ~, start := 326401, stop := 326412 ) ], count := [ 7, 3, 1, 29 ], name := "Mar\ k", next := 326420, root := ~, start := 326395, stop := 326419 ), 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 := 326476, root := ~, start := 326\ 464, stop := 326475 ), 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 := 326530, root := \ ~, start := 326521, stop := 326529 ), 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 := 326777, root := ~, start := 326740, stop := 326776 ), 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 := 326853, root := ~, start := 326427, stop := 326852 ), 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 := 326911, root := ~, start := 3\ 26897, stop := 326910 ), 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 :=\ 326979, root := ~, start := 326970, stop := 326978 ), 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 := 327113, root := ~, start := 327099, stop := 327112 ), 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 := 32\ 7212, root := ~, start := 327203, stop := 327211 ), 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 := 3272\ 61, root := ~, start := 326860, stop := 327260 ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "List", next := 327273, root := ~, start := 322093, stop := 327272 ), 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 := 328828, root := ~, start := 327279, stop := 328827 ), rec( content := "\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 328845, root := ~, start := 321672, stop := 328844 ) ], count := [ 7, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 328859, root := ~, start := 321443, stop := 328858 ), 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 := 328939, root := ~, start := 328880, stop := 328938 ), 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 := 328984, root := ~, start := 328973, stop := 328983 ), rec( content := ".", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 328995, root := ~, start := 328942, stop := 328994 ), 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 := 329051, root := ~, start := 329036, stop := 329050 ), 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 := 329070, root := ~, start := 329055, stop := 329069 ), 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 := 329126, root := ~, start := 329111, stop := 329125 ), 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 := 329177, root := ~, start := 329162, stop := 329176 ), 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 := 329224, root := ~, start := 329212, stop := 329223 ), 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 := 329258, root := ~, start := 329247, stop := 329257 ), 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 := 329303, root := ~, start := 329291, stop := 329302 ), 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 := 329337, root := ~, start := 329326, stop := 329336 ), 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 := 329422, root := ~, start := 329418, stop := 329421 ), 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 := 329454, root := ~, start := 329428, stop := 329453 ), 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 := 329512, root := ~, start := 329497, stop := 329511 ), 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 := 329532, root := ~, start := 329517, stop := 329531 ), 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 := 329583, root := ~, start := 329572, stop := 329582 ), 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 := 329931, root := ~, start := 329585, stop := 329930 ), rec( content := "\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 329948, root := ~, start := 328998, stop := 329947 ) ], count := [ 7, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 329962, root := ~, start := 328865, stop := 329961 ), 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 := 330044, root := ~, start := 329983, stop := 330043 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 3, 4 ], mathmode := "Text", name := "Oper", next := 330123, root := ~, start := 330047, stop := 330122 ), 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 := 330172, root := ~, start := 330126, stop := 330171 ), 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 := 330236, root := ~, start := 330193, stop := 330235 ), 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 := 330286, root := ~, start := 330253, stop := 330285 ), 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 := 330310, root := ~, start := 330295, stop := 330309 ), 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 := 330334, root := ~, start := 330319, stop := 330333 ), 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 := 330441, root := ~, start := 330388, stop := 330440 ), 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 := 330460, root := ~, start := 330445, stop := 330459 ), 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 := 330509, root := ~, start := 330501, stop := 330508 ), 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 := 330533, root := ~, start := 330518, stop := 330532 ), 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 := 330557, root := ~, start := 330542, stop := 330556 ), 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 := 330608, root := ~, start := 330592, stop := 330607 ), 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 := 330628, root := ~, start := 330620, stop := 330627 ), 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 := 330686, root := ~, start := 330643, stop := 330685 ), 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 := 330703, root := ~, start := 330695, stop := 330702 ), 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 := 330741, root := ~, start := 330726, stop := 330740 ), 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 := 330751, root := ~, start := 330747, stop := 330750 ), 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 := 330785, root := ~, start := 330757, stop := 330784 ), 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 := 330839, root := ~, start := 330824, stop := 330838 ), 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 := 330859, root := ~, start := 330844, stop := 330858 ), 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 := 331465, root := ~, start := 330865, stop := 331464 ), rec( content := "\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Description", next := 331482, root := ~, start := 330175, stop := 331481 ) ], count := [ 7, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 331496, root := ~, start := 329968, stop := 331495 ), 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 := 331576, root := ~, start := 331517, stop := 331575 ), 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 := 331620, root := ~, start := 331609, stop := 331619 ), rec( content := ".", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 331631, root := ~, start := 331579, stop := 331630 ), 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 := 331678, root := ~, start := 331652, stop := 331677 ), 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 := 331712, root := ~, start := 331696, stop := 331711 ), 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 := 331752, root := ~, start := 331713, stop := 331751 ), 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 := 331776, root := ~, start := 331761, stop := 331775 ), 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 := 331796, root := ~, start := 331781, stop := 331795 ), 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 := 331848, root := ~, start := 331837, stop := 331847 ), 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 := 332145, root := ~, start := 331854, stop := 332144 ), rec( content := "\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 332162, root := ~, start := 331634, stop := 332161 ) ], count := [ 7, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 332176, root := ~, start := 331502, stop := 332175 ), 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 := 332258, root := ~, start := 332197, stop := 332257 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 332337, root := ~, start := 332261, stop := 332336 ), 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 := 332385, root := ~, start := 332340, stop := 332384 ), 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 := 332477, root := ~, start := 332442, stop := 332476 ), 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 := 332536, root := ~, start := 332486, stop := 332535 ), 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 := 332582, root := ~, start := 332566, stop := 332581 ), 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 := 332979, root := ~, start := 332602, stop := 332978 ), rec( content := "\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 332996, root := ~, start := 332388, stop := 332995 ) ], count := [ 7, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 333010, root := ~, start := 332182, stop := 333009 ), 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 := 333089, root := ~, start := 333031, stop := 333088 ), 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 := 333133, root := ~, start := 333122, stop := 333132 ), rec( content := ".", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 333144, root := ~, start := 333092, stop := 333143 ), 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 := 333190, root := ~, start := 333165, stop := 333189 ), 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 := 333225, root := ~, start := 333208, stop := 333224 ), 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 := 333265, root := ~, start := 333226, stop := 333264 ), 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 := 333289, root := ~, start := 333274, stop := 333288 ), 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 := 333309, root := ~, start := 333294, stop := 333308 ), 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 := 333361, root := ~, start := 333350, stop := 333360 ), 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 := 333701, root := ~, start := 333367, stop := 333700 ), rec( content := "\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 333718, root := ~, start := 333147, stop := 333717 ) ], count := [ 7, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 333732, root := ~, start := 333016, stop := 333731 ), 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 := 333813, root := ~, start := 333753, stop := 333812 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 4 ], mathmode := "Text", name := "Oper", next := 333891, root := ~, start := 333816, stop := 333890 ), 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 := 333939, root := ~, start := 333894, stop := 333938 ), 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 := 334031, root := ~, start := 333996, stop := 334030 ), 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 := 334090, root := ~, start := 334040, stop := 334089 ), 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 := 334137, root := ~, start := 334120, stop := 334136 ), 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 := 335224, root := ~, start := 334157, stop := 335223 ), rec( content := "\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Description", next := 335241, root := ~, start := 333942, stop := 335240 ) ], count := [ 7, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 335255, root := ~, start := 333738, stop := 335254 ), 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 := 335332, root := ~, start := 335276, stop := 335331 ), 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 := 335377, root := ~, start := 335366, stop := 335376 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 335388, root := ~, start := 335335, stop := 335387 ), 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 := 335450, root := ~, start := 335435, stop := 335449 ), 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 := 335491, root := ~, start := 335476, stop := 335490 ), 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 := 335530, root := ~, start := 335497, stop := 335529 ), 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 := 335551, root := ~, start := 335536, stop := 335550 ), 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 := 335574, root := ~, start := 335559, stop := 335573 ), 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 := 335650, root := ~, start := 335637, stop := 335649 ), 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 := 335683, root := ~, start := 335670, stop := 335682 ), 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 := 335702, root := ~, start := 335687, stop := 335701 ), 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 := 335754, root := ~, start := 335741, stop := 335753 ), 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 := 335785, root := ~, start := 335774, stop := 335784 ), 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 := 335808, root := ~, start := 335793, stop := 335807 ), 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 := 335813, root := ~, start := 335809, stop := 335812 ), 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 := 335856, root := ~, start := 335848, stop := 335855 ), 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 := 335899, root := ~, start := 335884, stop := 335898 ), 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 := 335938, root := ~, start := 335907, stop := 335937 ), 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 := 335957, root := ~, start := 335942, stop := 335956 ), 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 := 335986, root := ~, start := 335978, stop := 335985 ), 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 := 335992, root := ~, start := 335988, stop := 335991 ), 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 := 336021, root := ~, start := 335998, stop := 336020 ), 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 := 336100, root := ~, start := 336089, stop := 336099 ), 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 := 336354, root := ~, start := 336106, stop := 336353 ), rec( content := "\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 336371, root := ~, start := 335391, stop := 336370 ) ], count := [ 7, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 336385, root := ~, start := 335261, stop := 336384 ), 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 := 336452, root := ~, start := 336406, stop := 336451 ), rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phsRepresentatives" ), content := 0, count := [ 7, 3, 9, 4 ], mathmode := "Text", name := "Oper", next := 336516, root := ~, start := 336455, stop := 336515 ), 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 := 336564, root := ~, start := 336519, stop := 336563 ), 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 := 336656, root := ~, start := 336621, stop := 336655 ), 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 := 336715, root := ~, start := 336665, stop := 336714 ), 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 := 336768, root := ~, start := 336759, stop := 336767 ), 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 := 336783, root := ~, start := 336774, stop := 336782 ), 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 := 336788, root := ~, start := 336784, stop := 336787 ), 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 := 336835, root := ~, start := 336803, stop := 336834 ), 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 := 337549, root := ~, start := 336841, stop := 337548 ), rec( content := "\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 337566, root := ~, start := 336567, stop := 337565 ) ], count := [ 7, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 337580, root := ~, start := 336391, stop := 337579 ), 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 := 337721, root := ~, start := 337601, stop := 337720 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 4 ], mathmode := "Text", name := "Oper", next := 337790, root := ~, start := 337724, stop := 337789 ), 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 := 337911, root := ~, start := 337793, stop := 337910 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 8 ], mathmode := "Text", name := "Oper", next := 337979, root := ~, start := 337914, stop := 337978 ), 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 := 338101, root := ~, start := 337982, stop := 338100 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 12 ], mathmode := "Text", name := "Oper", next := 338170, root := ~, start := 338104, stop := 338169 ), 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 := 338291, root := ~, start := 338173, stop := 338290 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphEndomo\ rphism" ), content := 0, count := [ 7, 3, 10, 16 ], mathmode := "Text", name := "Oper", next := 338352, root := ~, start := 338294, stop := 338351 ), 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 := 338375, root := ~, start := 338364, stop := 338374 ), 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 := 338391, root := ~, start := 338379, stop := 338390 ), rec( content := ".", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Returns", next := 338402, root := ~, start := 338355, stop := 338401 ), 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 := 338451, root := ~, start := 338423, stop := 338450 ), 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 := 338471, root := ~, start := 338460, stop := 338470 ), 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 := 338521, root := ~, start := 338513, stop := 338520 ), 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 := 338571, root := ~, start := 338561, stop := 338570 ), 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 := 338597, root := ~, start := 338587, stop := 338596 ), 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 := 338607, root := ~, start := 338603, stop := 338606 ), 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 := 338640, root := ~, start := 338613, stop := 338639 ), 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 := 338660, root := ~, start := 338649, stop := 338659 ), 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 := 338710, root := ~, start := 338702, stop := 338709 ), 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 := 338771, root := ~, start := 338761, stop := 338770 ), 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 := 338797, root := ~, start := 338787, stop := 338796 ), 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 := 338807, root := ~, start := 338803, stop := 338806 ), 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 := 338841, root := ~, start := 338813, stop := 338840 ), 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 := 338861, root := ~, start := 338850, stop := 338860 ), 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 := 338911, root := ~, start := 338903, stop := 338910 ), 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 := 338972, root := ~, start := 338962, stop := 338971 ), 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 := 338998, root := ~, start := 338988, stop := 338997 ), 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 := 339008, root := ~, start := 339004, stop := 339007 ), 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 := 339042, root := ~, start := 339014, stop := 339041 ), 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 := 339062, root := ~, start := 339051, stop := 339061 ), 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 := 339112, root := ~, start := 339104, stop := 339111 ), 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 := 339165, root := ~, start := 339151, stop := 339164 ), 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 := 339175, root := ~, start := 339171, stop := 339174 ), 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 := 339221, root := ~, start := 339213, stop := 339220 ), 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 := 339246, root := ~, start := 339227, stop := 339245 ), 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 := 339276, root := ~, start := 339266, stop := 339275 ), 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 := 339300, root := ~, start := 339290, stop := 339299 ), 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 := 339372, root := ~, start := 339364, stop := 339371 ), 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 := 339386, root\ := ~, start := 339378, stop := 339385 ), rec( content := "]", count := [ 7, 3\ , 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 3, 10, 23 ], mathmode := "Text", name := "C", next := 339391, root := ~, star\ t := 339356, stop := 339390 ), 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 := 339424, root := ~, start := 339\ 414, stop := 339423 ), 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 := 339447, root := ~, start := 339434, stop := 339446 ), 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 := 339480, root := ~, start := 339470, stop := 339479 ), rec( conten\ t := "; and ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 10, 23 ], name := "Item", next := 339493, root := ~, start := 339\ 341, stop := 339492 ), 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 := 339523, root :\ = ~, start := 339515, stop := 339522 ), 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 := 339544, root := ~, start := 33953\ 6, stop := 339543 ), 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 := 339580, root := ~, start := 33957\ 0, stop := 339579 ), rec( content := ".\n ", count := [ 7, 3, 10, 25 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25 ], name := "Ite\ m", next := 339595, root := ~, start := 339500, stop := 339594 ) ], count := [ 7, 3, 10, 22 ], mathmode := "Text", name := "List", next := 339607, root := ~, start := 339328, stop := 339606 ), 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 := 339665, root := ~, start := 339621, stop := 339664 ), 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 := 339670, root := ~, start := 339666, stop := 339669 ), 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 := 339691, root := ~, start := 339680, stop := 339690 ), 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 := 339707, root := ~, start := 339696, stop := 339706 ), 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 := 339722, root := ~, start := 339712, stop := 339721 ), 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 := 339868, root := ~, start := 339792, stop := 339867 ), 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 := 340048, root := ~, start := 340022, stop \ := 340047 ), 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 := 340074, root := ~, start := 340066, stop := 340073 ), 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 := 340092, root :\ = ~, start := 340078, stop := 340091 ), 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\ := 340145, root := ~, start := 340117, stop := 340144 ), rec( content := ".", coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ] , name := "Ite\ m", next := 340153, root := ~, start := 340007, stop := 340152 ), 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 := 340203, root := ~, start := 340175, stop \ := 340202 ), 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 := 340229, root := ~, start := 340221, stop := 340228 ), 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 := 340243, root :\ = ~, start := 340233, stop := 340242 ), 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 := 3402\ 97, root := ~, start := 340160, stop := 340296 ) ], count := [ 7, 3, 10, 29 ], mathmode := "Text", name := "List", next := 340309, root := ~, start := 339994, stop := 340308 ), 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 := 340363, root := ~, start := 340324, stop := 340362 ), 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 := 341565, root := ~, start := 340374, stop := 341564 ), rec( content := "\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Description", next := 341582, root := ~, start := 338405, stop := 341581 ) ], count := [ 7, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 341596, root := ~, start := 337586, stop := 341595 ), 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 := 341733, root := ~, start := 341617, stop := 341732 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 4 ], mathmode := "Text", name := "Oper", next := 341799, root := ~, start := 341736, stop := 341798 ), 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 := 341822, root := ~, start := 341811, stop := 341821 ), 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 := 341838, root := ~, start := 341826, stop := 341837 ), rec( content := ".", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 341849, root := ~, start := 341802, stop := 341848 ), 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 := 341895, root := ~, start := 341870, stop := 341894 ), 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 := 341915, root := ~, start := 341904, stop := 341914 ), 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 := 341965, root := ~, start := 341957, stop := 341964 ), 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 := 342010, root := ~, start := 342000, stop := 342009 ), 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 := 342038, root := ~, start := 342028, stop := 342037 ), 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 := 342087, root := ~, start := 342076, stop := 342086 ), 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 := 342103, root := ~, start := 342092, stop := 342102 ), 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 := 342122, root := ~, start := 342118, stop := 342121 ), 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 := 342168, root := ~, start := 342160, stop := 342167 ), 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 := 342190, root := ~, start := 342174, stop := 342189 ), 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 := 342218, root := ~, start := 342208, stop := 342217 ), 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 := 342244, root := ~, start := 342234, stop := 342243 ), 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 := 342256, root := ~, start := 342248, stop := 342255 ), 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 := 342294, root := ~, start := 342284, stop := 342293 ), 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 := 342308, root := ~, start := 342298, stop := 342307 ), 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 := 342336, root := ~, start := 342328, stop := 342335 ), 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 := 342392, root := ~, start := 342382, stop := 342391 ), 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 := 342425, root := ~, start := 342417, stop := 342424 ), 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 := 342443, root := ~, start := 342433, stop := 342442 ), 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 := 342494, root := ~, start := 342459, stop := 342493 ), 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 := 342499, root := ~, start := 342495, stop := 342498 ), 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 := 343148, root := ~, start := 342505, stop := 343147 ), rec( content := "\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 343165, root := ~, start := 341852, stop := 343164 ) ], count := [ 7, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 343179, root := ~, start := 341602, stop := 343178 ), 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 := 343274, root := ~, start := 343204, stop := 343273 ), 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 := 343300, root := ~, start := 343289, stop := 343299 ), 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 := 343316, root := ~, start := 343304, stop := 343315 ), rec( content := ". ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 343328, root := ~, start := 343279, stop := 343327 ), 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 := 343399, root := ~, start := 343367, stop := 343398 ), 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 := 343473, root := ~, start := 343465, stop := 343472 ), 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 := 343522, root := ~, start := 343511, stop := 343521 ), 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 := 343538, root := ~, start := 343527, stop := 343537 ), 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 := 343565, root := ~, start := 343555, stop := 343564 ), 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 := 343582, root := ~, start := 343570, stop := 343581 ), 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 := 343631, root := ~, start := 343599, stop := 343630 ), 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 := 343651, root := ~, start := 343640, stop := 343650 ), 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 := 343698, root := ~, start := 343690, stop := 343697 ), 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 := 343712, root := ~, start := 343702, stop := 343711 ), 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 := 343742, root := ~, start := 343714, stop := 343741 ), 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 := 343754, root := ~, start := 343750, stop := 343753 ), 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 := 344113, root := ~, start := 343762, stop := 344112 ), rec( content := "\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 344132, root := ~, start := 343333, stop := 344131 ) ], count := [ 7, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 344148, root := ~, start := 343187, stop := 344147 ), 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 := 344247, root := ~, start := 344169, stop := 344246 ), rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfEnd\ omorphismMonoidAttr" ), content := 0, count := [ 7, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 344313, root := ~, start := 344250, stop := 344312 ), 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 := 344362, root := ~, start := 344316, stop := 344361 ), 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 := 344416, root := ~, start := 344402, stop := 344415 ), 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 := 344472, root := ~, start := 344439, stop := 344471 ), 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 := 344492, root := ~, start := 344478, stop := 344491 ), 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 := 344551, root := ~, start := 344514, stop := 344550 ), 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 := 344670, root := ~, start := 344656, stop := 344669 ), 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 := 344689, root := ~, start := 344675, stop := 344688 ), 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 := 344740, root := ~, start := 344702, stop := 344739 ), 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 := 344801, root := ~, start := 344764, stop := 344800 ), 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 := 344849, root := ~, start := 344845, stop := 344848 ), 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 := 344875, root := ~, start := 344862, stop := 344874 ), 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 := 344945, root := ~, start := 344908, stop := 344944 ), 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 := 345077, root := ~, start := 345064, stop := 345076 ), 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 := 345110, root := ~, start := 345106, stop := 345109 ), 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 := 345228, root := ~, start := 345214, stop := 345227 ), 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 := 345\ 249, root := ~, start := 345236, stop := 345248 ), 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 := 345259, root := ~, start := 345249, stop := 345258 ), 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 := 345300, root := ~, start := 345292, stop := 345299 ), rec( cont\ ent := ".\n ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], name := "Item", next := 345315, root := ~, start := 345\ 129, stop := 345314 ), 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 := 345377, root := ~, start := 345364, stop := 345376 ), 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 := 345387, root := ~, start := 345\ 377, stop := 345386 ), 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 := 345442, root :=\ ~, start := 345434, stop := 345441 ), rec( content := ".\n ", count := [ \ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ] , name := "Ite\ m", next := 345457, root := ~, start := 345322, stop := 345456 ) ], count := [ 7, 3, 13, 8 ], mathmode := "Text", name := "List", next := 345469, root := ~, start := 345116, stop := 345468 ), 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 := 345494, root := ~, start := 345482, stop := 345493 ), 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 := 345569, root := ~, start := 345557, stop := 345568 ), 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 := 345603, root := ~, start := 345591, stop := 345602 ), 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 := 345653, root := ~, start := 345638, stop := 345652 ), 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 := 345659, root := ~, start := 345655, stop := 345658 ), 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 := 346528, root := ~, start := 345665, stop := 346527 ), rec( content := "\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Description", next := 346545, root := ~, start := 344365, stop := 346544 ) ], count := [ 7, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 346559, root := ~, start := 344154, stop := 346558 ), 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 := 346678, root := ~, start := 346580, stop := 346677 ), 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 := 346723, root := ~, start := 346712, stop := 346722 ), rec( content := ".", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 346734, root := ~, start := 346681, stop := 346733 ), 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 := 346780, root := ~, start := 346757, stop := 346779 ), 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 := 346915, root := ~, start := 346907, stop := 34691\ 4 ), rec( content := "-colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 346929, root := ~, start := 346891, stop := 346928 ), 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 := 346978, root := ~, start := 346970, stop := 346977 ), 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 := 347025, root := ~, start := 347017, stop := 347024 ), 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 := 347109, root := ~, start := 347077, stop := 347108 ), 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 := 347169, root := ~, start := 347161, stop := 347168 ), 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 := 347220, root := ~, start := 347185, stop := 347219 ), 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 := 347290, root := ~, start := 347255, stop := 347289 ), 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 := 347323, root := ~, start := 347315, stop := 347322 ), 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 := 347360, root := ~, start := 347352, stop := 347359 ), 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 := 347370, root := ~, start := 347366, stop := 347369 ), 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 := 347393, root := ~, start := 347379, stop := 347392 ), 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 := 347419, root := ~, start := 347411, stop := 347418 ), 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 := 3474\ 90, root := ~, start := 347476, stop := 347489 ), 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 := 3475\ 00, root := ~, start := 347492, stop := 347499 ), rec( content := ")", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "C", next := 347505, root := ~, start := 347456, stop := 347504 ), 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 := 347552, root := ~, start := 347538, stop := 347551 ), 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 := 347592, root := ~, start := 347584, stop := 347591 ), 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 := 347648, root := ~, start := 347637, stop := 347647 ), 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 := 347654, root := ~, start := 347650, stop := 347653 ), 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 := 347727, root := ~, start := 347669, stop := 347726 ), 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 := 347741, root := ~, start := 347737, stop := 347740 ), 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 := 347884, root := ~, start := 347852, stop := 347883 ), 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 := 348257, root := ~, start := 347891, stop := 348256 ), rec( content := "\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 348274, root := ~, start := 346737, stop := 348273 ) ], count := [ 7, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 348288, root := ~, start := 346565, stop := 348287 ), 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 := 348424, root := ~, start := 348309, stop := 348423 ), 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 := 348550, root := ~, start := 348427, stop := 348549 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "DigraphGreedyCo\ louring" ), content := 0, count := [ 7, 3, 15, 6 ], mathmode := "Text", name := "Attr", next := 348644, root := ~, start := 348553, stop := 348643 ), 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 := 348689, root := ~, start := 348678, stop := 348688 ), rec( content := ".", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Returns", next := 348700, root := ~, start := 348647, stop := 348699 ), 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 := 348746, root := ~, start := 348723, stop := 348745 ), 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 := 348921, root := ~, start := 348892, stop := 348920 ), 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 := 348972, root := ~, start := 348968, stop := 348971 ), 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 := 348999, root := ~, start := 348985, stop := 348998 ), 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 := 349029, root := ~, start := 349017, stop := 349028 ), 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 := 349102, root := ~, start := 349088, stop := 349101 ), 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 := 349159, root := ~, start := 349130, stop := 349158 ), 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 := 349266, root := ~, start := 349252, stop := 349265 ), 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 := 349325, root := ~, start := 349321, stop := 349324 ), 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 := 349352, root := ~, start := 349338, stop := 349351 ), 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 := 349381, root := ~, start := 349370, stop := 349380 ), 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 := 349472, root := ~, start := 349460, stop := 349471 ), 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 := 3495\ 24, root := ~, start := 349510, stop := 349523 ), 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 := 3495\ 37, root := ~, start := 349526, stop := 349536 ), rec( content := ")", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 349542, root := ~, start := 349484, stop := 349541 ), 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 := 3495\ 95, root := ~, start := 349581, stop := 349594 ), 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 := 3496\ 08, root := ~, start := 349597, stop := 349607 ), 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 := 3496\ 23, root := ~, start := 349609, stop := 349622 ), rec( content := "))", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 349629, root := ~, start := 349555, stop := 349628 ), 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 := 349639, root := ~, start := 349635, stop := 349638 ), 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 := 349773, root := ~, start := 349736, stop := 349772 ), 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 := 349803, root := ~, start := 349799, stop := 349802 ), 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 := 349912, root := ~, start := 349826, stop := 349911 ), 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 := 349922, root := ~, start := 349918, stop := 349921 ), 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 := 350167, root := ~, start := 349928, stop := 350166 ), rec( content := "\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Description", next := 350184, root := ~, start := 348703, stop := 350183 ) ], count := [ 7, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 350198, root := ~, start := 348294, stop := 350197 ), 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 := 350271, root := ~, start := 350219, stop := 350270 ), 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 := 350317, root := ~, start := 350274, stop := 350316 ), 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 := 350368, root := ~, start := 350338, stop := 350367 ), 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 := 350441, root := ~, start := 350427, stop := 350440 ), 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 := 350550, root := ~, start := 350546, stop := 350549 ), 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 := 350800, root := ~, start := 350556, stop := 350799 ), rec( content := "\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Description", next := 350817, root := ~, start := 350320, stop := 350816 ) ], count := [ 7, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 350831, root := ~, start := 350204, stop := 350830 ), 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 := 350896, root := ~, start := 350852, stop := 350895 ), 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 := 350942, root := ~, start := 350899, stop := 350941 ), 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 := 350988, root := ~, start := 350965, stop := 350987 ), 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 := 351203, root := ~, start := 351165, stop := 351202 ), 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 := 351248, root := ~, start := 351244, stop := 351247 ), 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 := 351271, root := ~, start := 351257, stop := 351270 ), 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 := 351339, root := ~, start := 351304, stop := 351338 ), 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 := 351368, root := ~, start := 351346, stop := 351367 ), 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 := 351420, root := ~, start := 351412, stop := 351419 ), 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 := 351480, root := ~, start := 351466, stop := 351479 ), 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 := 351494, root := ~, start := 351486, stop := 351493 ), 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 := 351587, root := ~, start := 351565, stop := 351586 ), 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 := 351621, root := ~, start := 351613, stop := 351620 ), 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 := 3516\ 70, root := ~, start := 351656, stop := 351669 ), rec( content := ", n)", count := [ 7, 3, 17, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 351678, root := ~, start := 351636, stop := 351677 ), 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 := 351710, root := ~, start := 351699, stop := 351709 ), 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 := 351798, root := ~, start := 351716, stop := 351797 ), 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 := 352197, root := ~, start := 351805, stop := 352196 ), rec( content := "\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 352214, root := ~, start := 350945, stop := 352213 ) ], count := [ 7, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 352228, root := ~, start := 350837, stop := 352227 ), 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 := 352291, root := ~, start := 352253, stop := 352290 ), 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 := 352343, root := ~, start := 352296, stop := 352342 ), 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 := 352379, root := ~, start := 352371, stop := 352378 ), 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 := 352417, root := ~, start := 352399, stop := 352416 ), 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 := 352483, root := ~, start := 352472, stop := 352482 ), 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 := 352495, root := ~, start := 352487, stop := 352494 ), 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 := 352544, root := ~, start := 352536, stop := 352543 ), 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 := 352603, root := ~, start := 352595, stop := 352602 ), 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 := 352608, root := ~, start := 352604, stop := 352607 ), 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 := 352629, root := ~, start := 352618, stop := 352628 ), 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 := 352651, root := ~, start := 352643, stop := 352650 ), 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 := 352686, root := ~, start := 352678, stop := 352685 ), 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 := 352702, root := ~, start := 352694, stop := 352701 ), 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 := 352743, root := ~, start := 352735, stop := 352742 ), 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 := 353227, root := ~, start := 352807, stop := 353226 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 353246, root := ~, start := 352348, stop := 353245 ) ], count := [ 7, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 353262, root := ~, start := 352236, stop := 353261 ), rec( content := "\n\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 353276, root := ~, start := 321082, stop := 353275 ) , rec( content := "\n\n", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 353288, root := ~, start := 272254, stop := 353287 ), 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 := 353427, root := ~, start := 353372, stop := 353426 ) , 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 := 353463, root := ~, start := 353450, stop := 353462 ), 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 := 353565, root := ~, start := 353543, stop := 353564 ), 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 := 353652, root := ~, start := 353631, stop := 353651 ), 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 := 353762, root := ~, start := 353726, stop := 353761 ), 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 := 354070, root := ~, start := 354040, stop := 354069 ), 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 := 354080, root := ~, start := 354076, stop := 354079 ), 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 := 354718, root := ~, start := 354714, stop := 354717 ), 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 := 354974, root := ~, start := 354939, stop := 354973 ), 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 := 355007, root := ~, start := 354979, stop := 355006 ), 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 := 355102, root := ~, start := 355088, stop := 355101 ), 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 := 355154, root := ~, start := 355140, stop := 355153 ), 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 := 355310, root := ~, start := 355254, stop := 355309 ), 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 := 355324, root := ~, start := 355310, stop := 355323 ), 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 := 355333, root := ~, start := 355324, stop := 355332 ), rec( content := ".\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], name := "Item", next := 355348, root := ~, start := 355239, stop := 355347 ) ], count := [ 8, 0, 0, 6 ], mathmode := "Text", name := "List", next := 355360, root := ~, start := 355226, stop := 355359 ), 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 := 355402, root := ~, start := 355370, stop := 355401 ), 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 := 355456, root := ~, start := 355407, stop := 355455 ), 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 := 355632, root := ~, start := 355620, stop := 355631 ), 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 := 355690, root := ~, start := 355678, stop := 355689 ), 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 := 355761, root := ~, start := 355747, stop := 355760 ), 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 := 355896, root := ~, start := 355850, stop := 355895 ), 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 := 355910, root := ~, start := 355896, stop := 355909 ), 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 := 355918, root := ~, start := 355910, stop := 355917 ), rec( content := ".\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], name := "Item", next := 355933, root := ~, start := 355835, stop := 355932 ) ], count := [ 8, 0, 0, 10 ], mathmode := "Text", name := "List", next := 355945, root := ~, start := 355822, stop := 355944 ), 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 := 356007, root := ~, start := 355955, stop := 356006 ), 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 := 356109, root := ~, start := 356049, stop := 356108 ), 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 := 356189, root := ~, start := 356175, stop := 356188 ), 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 := 356467, root := ~, start := 356453, stop := 356466 ), 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 := 356622, root := ~, start := 356567, stop := 356621 ), 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 := 356647, root := ~, start := 356643, stop := 356646 ), 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 := 356769, root := ~, start := 356732, stop := 356768 ), 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 := 356834, root := ~, start := 356820, stop := 356833 ), 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 := 357021, root := ~, start := 356932, stop := 357020 ), 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 := 357035, root := ~, start := 357021, stop := 357034 ), 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 := 357045, root := ~, start := 357035, stop := 357044 ), rec( content := ".\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], name := "Item", next := 357060, root := ~, start := 356917, stop := 357059 ) ], count := [ 8, 0, 0, 15 ], mathmode := "Text", name := "List", next := 357072, root := ~, start := 356904, stop := 357071 ), 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 := 357187, root := ~, start := 357173, stop := 357186 ), 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 := 357294, root := ~, start := 357269, stop := 357293 ), 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 := 357534, root := ~, start := 357520, stop := 357533 ), 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 := 357667, root := ~, start := 357643, stop := 357666 ), 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 := 357735, root := ~, start := 357731, stop := 357734 ), 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 := 357897, root := ~, start := 357883, stop := 357896 ), 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 := 357980, root := ~, start := 357966, stop := 357979 ), 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 := 358072, root := ~, start := 358068, stop := 358071 ), 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 := 358119, root := ~, start := 358085, stop := 358118 ), 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 := 358177, root := ~, start := 358137, stop := 358176 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalClique\ " ), content := 0, count := [ 8, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 358225, root := ~, start := 358178, stop := 358224 ), 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 := 358246, root := ~, start := 358235, stop := 358245 ), 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 := 358262, root := ~, start := 358250, stop := 358261 ), rec( content := ".", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 358273, root := ~, start := 358226, stop := 358272 ), 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 := 358307, root := ~, start := 358293, stop := 358306 ), 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 := 358333, root := ~, start := 358325, stop := 358332 ), 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 := 358390, root := ~, start := 358376, stop := 358389 ), 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 := 358415, root := ~, start := 358399, stop := 358414 ), 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 := 358429, root := ~, start := 358415, stop := 358428 ), 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 := 358437, root := ~, start := 358429, stop := 358436 ), 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 := 358445, root := ~, start := 358437, stop := 358444 ), 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 := 358453, root := ~, start := 358445, stop := 358452 ), 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 := 358473, root := ~, start := 358462, stop := 358472 ), 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 := 358489, root := ~, start := 358481, stop := 358488 ), 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 := 358508, root := ~, start := 358495, stop := 358507 ), 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 := 358526, root := ~, start := 358512, stop := 358525 ), 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 := 358543, root := ~, start := 358531, stop := 358542 ), 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 := 358601, root := ~, start := 358578, stop := 358600 ), 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 := 358615, root := ~, start := 358601, stop := 358614 ), 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 := 358623, root := ~, start := 358615, stop := 358622 ), 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 := 358631, root := ~, start := 358623, stop := 358630 ), 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 := 358639, root := ~, start := 358631, stop := 358638 ), 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 := 358663, root := ~, start := 358652, stop := 358662 ), 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 := 358675, root := ~, start := 358667, stop := 358674 ), 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 := 358702, root := ~, start := 358681, stop := 358701 ), 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 := 358720, root := ~, start := 358706, stop := 358719 ), 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 := 358741, root := ~, start := 358729, stop := 358740 ), 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 := 358761, root := ~, start := 358757, stop := 358760 ), 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 := 358782, root := ~, start := 358769, stop := 358781 ), 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 := 358878, root := ~, start := 358857, stop := 358877 ), 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 := 359500, root := ~, start := 359040, stop := 359499 ), rec( content := "\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 359517, root := ~, start := 358274, stop := 359516 ) ], count := [ 8, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 359531, root := ~, start := 358124, stop := 359530 ), 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 := 359658, root := ~, start := 359552, stop := 359657 ), 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 := 359700, root := ~, start := 359683, stop := 359699 ), rec( content := ".", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 359711, root := ~, start := 359661, stop := 359710 ), 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 := 359789, root := ~, start := 359775, stop := 359788 ), 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 := 359977, root := ~, start := 359973, stop := 359976 ), 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 := 359995, root := ~, start := 359987, stop := 359994 ), 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 := 360083, root := ~, start := 360069, stop := 360082 ), 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 := 360162, root := ~, start := 360103, stop := 360161 ), 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 := 360225, root := ~, start := 360167, stop := 360224 ), 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 := 36\ 0263, root := ~, start := 360252, stop := 360262 ) ], count := [ 8, 1, 2, 5 ], name := "Mar\ k", next := 360270, root := ~, start := 360246, stop := 360269 ), 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 := 360341, root := \ ~, start := 360330, stop := 360340 ), 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\ := 360346, root := ~, start := 360342, stop := 360345 ), 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 := 360370, root := \ ~, start := 360359, stop := 360369 ), 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 := 360445, root := ~, start := 360428, stop := 360444 ), 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\ := 360479, root := ~, start := 360471, stop := 360478 ), 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 := 360516, root := ~, start := 360505, stop\ := 360515 ), 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 := 360534, root := ~, start := 360517, stop := 360533 ), 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 :\ = 360542, root := ~, start := 360502, stop := 360541 ), 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 := 360593, root := ~, start := 360585, stop := 360592 ), 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 \ := 360626, root := ~, start := 360606, stop := 360625 ), 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 := 360631, root := ~, start := 360627, stop := 360630 ), 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\ 60655, root := ~, start := 360644, stop := 360654 ), 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 := 360670, root := ~, start := 360659, stop \ := 360669 ), 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 := 360775, root := ~, start := 360755, stop := 360774 ), 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 \ := 360804, root := ~, start := 360787, stop := 360803 ), 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 := 360854, root := ~, start := 360277, stop := 360853 ), 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\ := 360885, root := ~, start := 360868, stop := 360884 ) ], count := [ 8, 1, 2, 9 ], name := "Mar\ k", next := 360892, root := ~, start := 360862, stop := 360891 ), 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 := 360928, root := ~, start := 360917, stop := 3\ 60927 ), 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 := 360966, root := ~, start := 360949, stop := 36096\ 5 ), 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\ := 361028, root := ~, start := 361011, stop := 361027 ), 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 := 361095, root := ~, start := 361084, stop := 3\ 61094 ), 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 := 361127, root := ~, start := 361110, stop := 361126 ) , 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 := 361171, root := ~, \ start := 361160, stop := 361170 ), 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 := 361189, root := ~, start := 361172, stop\ := 361188 ), rec( content := ", c)", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "C", next := 361197, root := ~, start := 361157, stop := 361196\ ), 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 := 361247, root := ~, start := 361239, stop \ := 361246 ), 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 := 361277, root := ~, start := 361260, stop := 361276\ ), 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 := 361283, root := ~, star\ t := 361279, stop := 361282 ), 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 := 361320, root := ~, start := 3613\ 09, stop := 361319 ), 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 := 361335, root := ~, start := 361324, stop := 361334 ), 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 := 361380, root := ~, start := 361363, stop := 361379 ), 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 := 3614\ 10, root := ~, start := 360899, stop := 361409 ), 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 := 361436, root := ~, start := 361\ 424, stop := 361435 ) ], count := [ 8, 1, 2, 12 ], name := "Mark", next := 3614\ 43, root := ~, start := 361418, stop := 361442 ), 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 := 361526, root := ~, start := 361\ 511, stop := 361525 ), 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 := 361556, root := ~, start := 361536, stop := 36\ 1555 ), 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 := 361608, root := ~, start := 361596, stop \ := 361607 ), 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 := 361657, root := ~, start := 361450, stop := 361656 ), 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\ := 361685, root := ~, start := 361671, stop := 361684 ), 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 := 361704, root := \ ~, start := 361690, stop := 361703 ) ], count := [ 8, 1, 2, 14 ], name := "Mar\ k", next := 361711, root := ~, start := 361665, stop := 361710 ), 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 := 361840, root := ~, start := 3\ 61826, stop := 361839 ), 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 := 361923, root := ~, start := 361909, stop := 361922 ), 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 := 361930, root := ~, start := 361926, stop \ := 361929 ), 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 := 361960, root := ~, start := 361940, stop := 3619\ 59 ), 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 := 36\ 2044, root := ~, start := 362030, stop := 362043 ), 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 := 362106, root := ~, start := 362092, stop := 362105 ), 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 := 362112, root := ~, start := 362108, stop \ := 362111 ), 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 := 362242, root := ~, start := 362234, stop \ := 362241 ), 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 := 362277, root := ~, start := 361718, stop := 362276 ), 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 := \ 362301, root := ~, start := 362291, stop := 362300 ) ], count := [ 8, 1, 2, 18 ], name := "Mar\ k", next := 362308, root := ~, start := 362285, stop := 362307 ), 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 := 362365, root := ~, start := 3623\ 54, stop := 362364 ), 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 := 362381, root := ~, start := 362369, stop := 362380 ), 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 := 362397, root :=\ ~, start := 362387, stop := 362396 ), 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 := 362439, root := ~, star\ t := 362419, stop := 362438 ), 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 := 362475, root := ~, start := 3\ 62461, stop := 362474 ), 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 := 362506, root := ~, start := 362496, stop \ := 362505 ), 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 := 362522, root := ~, start := 362510, stop := 362521 ), 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 := 362583, root := ~, start := 362315, stop := 362582 ), 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 :=\ 362608, root := ~, start := 362597, stop := 362607 ) ], count := [ 8, 1, 2, 20 ], name := "Mar\ k", next := 362615, root := ~, start := 362591, stop := 362614 ), 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 := 362672, root := ~, start := 3626\ 61, stop := 362671 ), 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 := 362718, root := ~, start := 3627\ 07, stop := 362717 ), 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 := 362766, root := ~, star\ t := 362746, stop := 362765 ), 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 := 362838, root := ~, start := 362827, stop := 362837 ), 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\ := 362871, root := ~, start := 362860, stop := 362870 ), 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 := 362886, root := ~, start := 3628\ 75, stop := 362885 ), 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 := 362939, root := ~, start := 362\ 622, stop := 362938 ), 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 := 362964, root := ~, start := 362953, stop \ := 362963 ) ], count := [ 8, 1, 2, 22 ], name := "Mark", next := 362971, root := ~, start := 362\ 947, stop := 362970 ), 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 := 363028, root := ~, start := 363017, stop := 363027 ), 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 := 363044, root :=\ ~, start := 363032, stop := 363043 ), 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\ := 363049, root := ~, start := 363045, stop := 363048 ), 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 := 363073, root :=\ ~, start := 363062, stop := 363072 ), 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 := 363088, root := ~, start := 363077, stop \ := 363087 ), 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 := 363122, root := ~, start := 363108, stop := 363121 ) , 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\ := 363149, root := ~, start := 363135, stop := 363148 ), 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 := 363220, root := ~, start := 363212, stop \ := 363219 ), 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 := 363277, root := ~, star\ t := 363257, stop := 363276 ), 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 := 363385, root := ~, start := 363377, stop \ := 363384 ), 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\ := 363471, root := ~, start := 363387, stop := 363470 ), 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 := 363495, root :=\ ~, start := 363484, stop := 363494 ), 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 := 363530, root := ~, star\ t := 363510, stop := 363529 ), 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 := 363581, root := ~, start :=\ 363577, stop := 363580 ), 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 := 363623, root := ~, star\ t := 363615, stop := 363622 ), 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 := 3637\ 67, root := ~, start := 362978, stop := 363766 ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "List", next := 363779, root := ~, start := 360233, stop := 363778 ), 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 := 365504, root := ~, start := 363850, stop := 365503 ), rec( content := "\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 365521, root := ~, start := 359714, stop := 365520 ) ], count := [ 8, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 365535, root := ~, start := 359537, stop := 365534 ), 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 := 365628, root := ~, start := 365556, stop := 365627 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalClique" ), content := 0, count := [ 8, 1, 3, 4 ], mathmode := "Text", name := "Func", next := 365714, root := ~, start := 365631, stop := 365713 ), 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 := 365780, root := ~, start := 365769, stop := 365779 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Returns", next := 365791, root := ~, start := 365717, stop := 365790 ), 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 := 365829, root := ~, start := 365815, stop := 365828 ), 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 := 365903, root := ~, start := 365889, stop := 365902 ), 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 := 365979, root := ~, start := 365968, stop := 365978 ), 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 := 366073, root := ~, start := 366051, stop := 366072 ), 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 := 366107, root := ~, start := 366078, stop := 366106 ), 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 := 366112, root := ~, start := 366108, stop := 366111 ), 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 := 366155, root := ~, start := 366141, stop := 366154 ), 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 := 366174, root := ~, start := 366160, stop := 366173 ), 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 := 366260, root := ~, start := 366246, stop := 366259 ), 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 := 366303, root := ~, start := 366292, stop := 366302 ), 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 := 366363, root := ~, start := 366349, stop := 366362 ), 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 := 366382, root := ~, start := 366368, stop := 366381 ), 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 := 366456, root := ~, start := 366442, stop := 366455 ), 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 := 366501, root := ~, start := 366487, stop := 366500 ), 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 := 366557, root := ~, start := 366543, stop := 366556 ), 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 := 366590, root := ~, start := 366579, stop := 366589 ), 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 := 366686, root := ~, start := 366675, stop := 366685 ), 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 := 366700, root := ~, start := 366696, stop := 366699 ), 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 := 366723, root := ~, start := 366709, stop := 366722 ), 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 := 366780, root := ~, start := 366769, stop := 366779 ), 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 := 366933, root := ~, start := 366900, stop := 366932 ), 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 :=\ 367018, root := ~, start := 366998, stop := 367017 ), 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 := \ 367058, root := ~, start := 367031, stop := 367057 ), 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 :=\ 367101, root := ~, start := 367087, stop := 367100 ), rec( content := "\n \ until it can not 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 := 367299, root :\ = ~, start := 367264, stop := 367298 ), 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 := 367334, root := ~, start := 366\ 940, stop := 367333 ), 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 := 367369 , root := ~, start := 367\ 341, stop := 367368 ), 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 := 367449, root := ~, start := 367429, stop := 36\ 7448 ), 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 := 367501, root := ~, start := 3674\ 87, stop := 367500 ), rec( content := " until it can not 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 := 367581, root\ := ~, start := 367577, stop := 367580 ), 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 := 367641, root := ~, start := 367614, stop := 367640 ), 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 := 367718, root := ~, start := \ 367689, stop := 367717 ), 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\ := 367732, root := ~, start := 367718, stop := 367731 ), 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 := 367749, root := ~, start := 367732, stop \ := 367748 ), 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 := 367763, root\ := ~, start := 367749, stop := 367762 ), 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 := 367772, root := ~, start := 367763, stop := 367771 ), 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 := 367786, root := ~, start := 3\ 67772, stop := 367785 ), 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 :\ = 367797, root := ~, start := 367786, stop := 367796 ), 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\ := 367837, root := ~, start := 367826, stop := 367836 ), rec( content := ".\n\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 3, 11 ], name := "Item", next := 367852, root := ~, start := 367\ 376, stop := 367851 ), 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 := 367886, root := ~, start := 367\ 859, stop := 367885 ), 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 := 367965, root := ~, start := 367945, stop := 36\ 7964 ), 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 := 368027, root := ~, start\ := 368005, stop := 368026 ), 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\ := 368041, root := ~, start := 368027, stop := 368040 ), 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 := 368060, root := ~, start := 368041, stop \ := 368059 ), 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 := 368074, root\ := ~, start := 368060, stop := 368073 ), 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 := 368083, root := ~, start := 368074, stop := 368082 ), 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 := 368097, root := ~, start := 3\ 68083, stop := 368096 ), 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 := 368117, root := ~, start := 368097, stop := 368116 ), 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 := 368128, root := ~, start := 368117, stop \ := 368127 ), 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 := 368136, root :=\ ~, start := 368128, stop := 368135 ), 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 := 368168, root := ~, start := 3681\ 57, stop := 368167 ), 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 := 368213, root := ~, start := 368209, stop :\ = 368212 ), 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 := 368263, root := ~, start := 368236, stop \ := 368262 ), 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\ := 368348, root := ~, start := 368319, stop := 368347 ), 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 := 368362, root := ~, start := 368348, stop \ := 368361 ), 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 := 368371, root :=\ ~, start := 368362, stop := 368370 ), 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 := 368385, root := ~, start := 368371, stop := 368384 ), 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 := 368402, root := ~, start \ := 368385, stop := 368401 ), 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\ := 368416, root := ~, start := 368402, stop := 368415 ), 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 := 368428, root := ~, start := 368416, stop :=\ 368427 ), 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 := 368439, root :=\ ~, start := 368428, stop := 368438 ), 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 := 368447, root := ~, start := 368439, stop := 368446 ), 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 := 368487, root := ~, start := 368476, stop := 368486 ), rec( conten\ t := ".\n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 3, 14 ], name := "Item", next := 368502, root := ~, start := 367\ 893, stop := 368501 ) ], count := [ 8, 1, 3, 8 ], mathmode := "Text", name := "List", next := 368514, root := ~, start := 366887, stop := 368513 ), 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 := 369219, root := ~, start := 368520, stop := 369218 ), rec( content := "\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Description", next := 369236, root := ~, start := 365794, stop := 369235 ) ], count := [ 8, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 369250, root := ~, start := 365541, stop := 369249 ), 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 := 369360, root := ~, start := 369271, stop := 369359 ), 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 := 369460, root := ~, start := 369363, stop := 369459 ), 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 := 369549, root := ~, start := 369463, stop := 369548 ), 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 := 369642, root := ~, start := 369552, stop := 369641 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesAttr" ), content := 0, count := [ 8, 1, 4, 10 ], mathmode := "Text", name := "Attr", next := 369699, root := ~, start := 369645, stop := 369698 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesRepsAttr" ), content := 0, count := [ 8, 1, 4, 12 ], mathmode := "Text", name := "Attr", next := 369760, root := ~, start := 369702, stop := 369759 ), 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 := 369830, root := ~, start := 369763, stop := 369829 ), 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 := 369868, root := ~, start := 369854, stop := 369867 ), 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 := 369954, root := ~, start := 369921, stop := 369953 ), 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 := 369990, root := ~, start := 369976, stop := 369989 ), 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 := 370089, root := ~, start := 370061, stop := 370088 ), 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 := 370123, root := ~, start := 370094, stop := 370122 ), 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 := 370128, root := ~, start := 370124, stop := 370127 ), 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 := 370171, root := ~, start := 370157, stop := 370170 ), 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 := 370190, root := ~, start := 370176, stop := 370189 ), 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 := 370261, root := ~, start := 370247, stop := 370260 ), 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 := 370301, root := ~, start := 370289, stop := 370300 ), 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 := 370354, root := ~, start := 370339, stop := 370353 ), 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 := 370397, root := ~, start := 370386, stop := 370396 ), 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 := 370468, root := ~, start := 370454, stop := 370467 ), 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 := 370487, root := ~, start := 370473, stop := 370486 ), 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 := 370538, root := ~, start := 370526, stop := 370537 ), 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 := 370564, root := ~, start := 370549, stop := 370563 ), 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 := 370570, root := ~, start := 370566, stop := 370569 ), 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 := 370673, root := ~, start := 370661, stop := 370672 ), 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 := 370764, root := ~, start := 370750, stop := 370763 ), 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 := 370825, root := ~, start := 370811, stop := 370824 ), 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 := 370852, root := ~, start := 370841, stop := 370851 ), 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 := 370929, root := ~, start := 370918, stop := 370928 ), 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 := 370991, root := ~, start := 370977, stop := 370990 ), 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 := 371108, root := ~, start := 371104, stop := 371107 ), 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 := 371126, root := ~, start := 371118, stop := 371125 ), 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 := 371210, root := ~, start := 371196, stop := 371209 ), 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 := 371289, root := ~, start := 371230, stop := 371288 ), 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 := 371352, root := ~, start := 371294, stop := 371351 ), 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 := 371402, root := ~, start := 371373, stop := 371401 ), 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 := 371452, root := ~, start := 371424, stop \ := 371451 ), 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 := 371478, root := ~, start := 371457, stop := 371477 ), 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 := 371544, root := ~, start := 371532, stop \ := 371543 ), 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 := 371570, root := ~, start := 371556, stop := 371569 ), 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\ 71610, root := ~, start := 371606, stop := 371609 ), 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 := 371681, root := ~, start := 371667, stop := 371680 ) , 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\ := 371708, root := ~, start := 371694, stop := 371707 ), 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\ := 371751, root := ~, start := 371743, stop := 371750 ), 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 := 371794, root := ~, start := 371409, stop := 371793 ), 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 := 371847, root := ~, start := 371802, stop := 371846 ), 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 := 371908, root := ~, star\ t := 371876, stop := 371907 ), 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 := 371937, root := ~, start := 371912, stop \ := 371936 ), 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 := 371975, root := ~, start := 371961, stop \ := 371974 ), 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 := 371994, root := ~, start := 371980, stop := 371993 ), 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 := 372054, root := ~, start := 372046, stop \ := 372053 ), 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\ 72079, root := ~, start := 372075, stop := 372078 ), 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 := 372147, root := ~, start := 372115, stop := 372146 ), 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 := 372185, root := ~, start := 372160, stop := 372184 ), 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 := 372251, root := ~, start := 372239, stop \ := 372250 ), 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 := 372304, root := ~, start := 372296, stop \ := 372303 ), 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 := 372359, root := ~, start := 3\ 72345, stop := 372358 ), 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 := 372399, root := ~, start := 372395, stop := 372398 ), 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 := 372515, root := ~, start := 372505, stop := 372514 ), 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 := 372642, root := ~, start := 371854, stop := 372641 ) ], count := [ 8, 1, 4, 17 ], mathmode := "Text", name := "List", next := 372654, root := ~, start := 371360, stop := 372653 ), 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 := 373669, root := ~, start := 372660, stop := 373668 ), rec( content := "\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Description", next := 373686, root := ~, start := 369833, stop := 373685 ) ], count := [ 8, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 373700, root := ~, start := 369256, stop := 373699 ), 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 := 373762, root := ~, start := 373721, stop := 373761 ), 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 := 373807, root := ~, start := 373765, stop := 373806 ), 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 := 373845, root := ~, start := 373831, stop := 373844 ), 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 := 373895, root := ~, start := 373881, stop := 37389\ 4 ), rec( content := ")", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 373900, root := ~, start := 373865, stop := 373899 ), 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 := 373941, root := ~, start := 373933, stop := 373940 ), 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 := 373966, root := ~, start := 373952, stop := 373965 ), 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 := 374002, root := ~, start := 373994, stop := 374001 ), 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 := 374046, root := ~, start := 374042, stop := 374045 ), 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 := 374067, root := ~, start := 374054, stop := 374066 ), 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 := 374545, root := ~, start := 374249, stop := 374544 ), rec( content := "\n", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 374560, root := ~, start := 373810, stop := 374559 ) ], count := [ 8, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 374574, root := ~, start := 373706, stop := 374573 ), rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 374588, root := ~, start := 358076, stop := 374587 ) , 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 := 374644, root := ~, start := 374601, stop := 374643 ), 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 := 374712, root := ~, start := 374664, stop := 374711 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalIndepe\ ndentSet" ), content := 0, count := [ 8, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 374770, root := ~, start := 374715, stop := 374769 ), 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 := 374793, root := ~, start := 374782, stop := 374792 ), 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 := 374809, root := ~, start := 374797, stop := 374808 ), rec( content := ".", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 374820, root := ~, start := 374773, stop := 374819 ), 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 := 374858, root := ~, start := 374844, stop := 374857 ), 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 := 374884, root := ~, start := 374876, stop := 374883 ), 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 := 374943, root := ~, start := 374929, stop := 374942 ), 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 := 374978, root := ~, start := 374954, stop := 374977 ), 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 := 374992, root := ~, start := 374978, stop := 374991 ), 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 := 375000, root := ~, start := 374992, stop := 374999 ), 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 := 375008, root := ~, start := 375000, stop := 375007 ), 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 := 375016, root := ~, start := 375008, stop := 375015 ), 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 := 375040, root := ~, start := 375029, stop := 375039 ), 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 := 375052, root := ~, start := 375044, stop := 375051 ), 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 := 375081, root := ~, start := 375059, stop := 375080 ), 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 := 375099, root := ~, start := 375085, stop := 375098 ), 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 := 375120, root := ~, start := 375108, stop := 375119 ), 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 := 375182, root := ~, start := 375151, stop := 375181 ), 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 := 375196, root := ~, start := 375182, stop := 375195 ), 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 := 375204, root := ~, start := 375196, stop := 375203 ), 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 := 375212, root := ~, start := 375204, stop := 375211 ), 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 := 375220, root := ~, start := 375212, stop := 375219 ), 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 := 375244, root := ~, start := 375233, stop := 375243 ), 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 := 375256, root := ~, start := 375248, stop := 375255 ), 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 := 375292, root := ~, start := 375262, stop := 375291 ), 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 := 375314, root := ~, start := 375300, stop := 375313 ), 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 := 375331, root := ~, start := 375319, stop := 375330 ), 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 := 375351, root := ~, start := 375347, stop := 375350 ), 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 := 375382, root := ~, start := 375360, stop := 375381 ), 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 := 375491, root := ~, start := 375461, stop := 375490 ), 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 := 376035, root := ~, start := 375686, stop := 376034 ), rec( content := "\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 376052, root := ~, start := 374823, stop := 376051 ) ], count := [ 8, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 376066, root := ~, start := 374649, stop := 376065 ), 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 := 376171, root := ~, start := 376087, stop := 376170 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalIndependentSet" ), content := 0, count := [ 8, 2, 2, 4 ], mathmode := "Text", name := "Func", next := 376269, root := ~, start := 376174, stop := 376268 ), 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 := 376335, root := ~, start := 376324, stop := 376334 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Returns", next := 376346, root := ~, start := 376272, stop := 376345 ), 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 := 376384, root := ~, start := 376370, stop := 376383 ), 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 := 376468, root := ~, start := 376454, stop := 376467 ), 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 := 376544, root := ~, start := 376533, stop := 376543 ), 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 := 376656, root := ~, start := 376626, stop := 376655 ), 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 := 376704, root := ~, start := 376661, stop := 376703 ), 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 := 376709, root := ~, start := 376705, stop := 376708 ), 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 := 376752, root := ~, start := 376738, stop := 376751 ), 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 := 376771, root := ~, start := 376757, stop := 376770 ), 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 := 376857, root := ~, start := 376843, stop := 376856 ), 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 := 376900, root := ~, start := 376889, stop := 376899 ), 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 := 376960, root := ~, start := 376946, stop := 376959 ), 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 := 376979, root := ~, start := 376965, stop := 376978 ), 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 := 377063, root := ~, start := 377049, stop := 377062 ), 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 := 377112, root := ~, start := 377098, stop := 377111 ), 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 := 377164, root := ~, start := 377150, stop := 377163 ), 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 := 377197, root := ~, start := 377186, stop := 377196 ), 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 := 377302, root := ~, start := 377291, stop := 377301 ), 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 := 377316, root := ~, start := 377312, stop := 377315 ), 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 := 377339, root := ~, start := 377325, stop := 377338 ), 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 := 377410, root := ~, start := 377399, stop := 377409 ), 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 := 377560, root := ~, start := 377527, stop := 377559 ), 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\ := 377653, root := ~, start := 377625, stop := 377652 ), 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\ := 377701, root := ~, start := 377666, stop := 377700 ), 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 := 377761, root := ~, start := 377747, stop := 377760 ), rec( cont\ ent := " until it can not continue. The result\n is guaranteed to be a\ maximal independent set. This may or may not\n return an answer more q\ uickly than using ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphMaximalIndependentS\ ets" ), content := 0, count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "Re\ f", next := 377978, root := ~, start := 377925, stop := 377977 ), rec( content\ := ". with a limit of 1.\n ", count := [ 8, 2, 2, 9 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], name := "Item", next := 3780\ 13, root := ~, start := 377567, stop := 378012 ), 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 := 3780\ 48, root := ~, start := 378020, stop := 378047 ), 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 := 378136, root := ~, start :=\ 378108, stop := 378135 ), 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 := 3781\ 97, root := ~, start := 378183, stop := 378196 ), rec( content := " until it c\ an not\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 := 378286, root := ~, start := 378282, stop := 378285 ), 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 := 378354, root := ~, start := 3783\ 19, stop := 378353 ), 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 := 378456, root := ~, start := 378419, stop := 3784\ 55 ), 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 := 378470, root :=\ ~, start := 378456, stop := 378469 ), 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 := 378487, root := ~, start := 378470, stop := 378486 ), 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 := 378501, root := ~, start := 3\ 78487, stop := 378500 ), 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 := \ 378510, root := ~, start := 378501, stop := 378509 ), 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 := 378524, root := ~, start := 378510, stop := \ 378523 ), 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 := 378535, root :=\ ~, start := 378524, stop := 378534 ), 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 := 378575, root :=\ ~, start := 378564, stop := 378574 ), rec( content := ".\n ", count := [ \ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 11 ], name := "Ite\ m", next := 378590, root := ~, start := 378055, stop := 378589 ), 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 := 378624, root := ~, start := 378597, stop := 378623 ), 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 \ := 378711, root := ~, start := 378683, stop := 378710 ), 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 := 378798, root := ~, start \ := 378768, stop := 378797 ), 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\ := 378812, root := ~, start := 378798, stop := 378811 ), 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 := 378821, root := ~, start := 378812, stop := 37\ 8820 ), 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 := 378835, root :=\ ~, start := 378821, stop := 378834 ), 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 := 378852, root := ~, start := 378835, stop := 378851 ), 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 := 378866, root := ~, start := 3\ 78852, stop := 378865 ), 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 \ := 378878, root := ~, start := 378866, stop := 378877 ), 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 := 378889, root := ~, start := 378878, stop := \ 378888 ), 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 := 378897, root := \ ~, start := 378889, stop := 378896 ), 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 := 378937, root :=\ ~, start := 378926, stop := 378936 ), 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 := 378983, root := ~, start \ := 378979, stop := 378982 ), 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 := 379049, root :=\ ~, start := 379014, stop := 379048 ), 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 := 379151, root := ~, start := 3\ 79114, stop := 379150 ), 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\ := 379165, root := ~, start := 379151, stop := 379164 ), 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 := 379184, root := ~, start := 379165, stop \ := 379183 ), 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 := 379198, root\ := ~, start := 379184, stop := 379197 ), 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 := 379207, root := ~, start := 379198, stop := 379206 ), 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 := 379221, root := ~, start := 3\ 79207, stop := 379220 ), 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 := 379241, root := ~, start := 379221, stop := 379240 ), 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 := 379252, root := ~, start := 379241, stop \ := 379251 ), 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 := 379260, root :=\ ~, start := 379252, stop := 379259 ), 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 := 379292, root := ~, start := 3792\ 81, stop := 379291 ), rec( content := ".\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], name := "Ite\ m", next := 379307, root := ~, start := 378631, stop := 379306 ) ], count := [ 8, 2, 2, 8 ], mathmode := "Text", name := "List", next := 379319, root := ~, start := 377514, stop := 379318 ), 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 := 380472, root := ~, start := 379324, stop := 380471 ), rec( content := "\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Description", next := 380489, root := ~, start := 376349, stop := 380488 ) ], count := [ 8, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 380503, root := ~, start := 376072, stop := 380502 ), 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 := 380625, root := ~, start := 380524, stop := 380624 ), 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 := 380733, root := ~, start := 380628, stop := 380732 ), 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 := 380830, root := ~, start := 380736, stop := 380829 ), 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 := 380931, root := ~, start := 380833, stop := 380930 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsAttr" ), content := 0, count := [ 8, 2, 3, 10 ], mathmode := "Text", name := "Attr", next := 380996, root := ~, start := 380934, stop := 380995 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsRepsAttr" ), content := 0, count := [ 8, 2, 3, 12 ], mathmode := "Text", name := "Attr", next := 381065, root := ~, start := 380999, stop := 381064 ), 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 := 381135, root := ~, start := 381068, stop := 381134 ), 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 := 381173, root := ~, start := 381159, stop := 381172 ), 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 := 381259, root := ~, start := 381226, stop := 381258 ), 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 := 381304, root := ~, start := 381290, stop := 381303 ), 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 := 381426, root := ~, start := 381389, stop := 381425 ), 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 := 381467, root := ~, start := 381431, stop := 381466 ), 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 := 381472, root := ~, start := 381468, stop := 381471 ), 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 := 381515, root := ~, start := 381501, stop := 381514 ), 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 := 381534, root := ~, start := 381520, stop := 381533 ), 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 := 381605, root := ~, start := 381591, stop := 381604 ), 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 := 381645, root := ~, start := 381633, stop := 381644 ), 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 := 381698, root := ~, start := 381683, stop := 381697 ), 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 := 381741, root := ~, start := 381730, stop := 381740 ), 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 := 381812, root := ~, start := 381798, stop := 381811 ), 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 := 381831, root := ~, start := 381817, stop := 381830 ), 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 := 381882, root := ~, start := 381870, stop := 381881 ), 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 := 381908, root := ~, start := 381893, stop := 381907 ), 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 := 381914, root := ~, start := 381910, stop := 381913 ), 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 := 382026, root := ~, start := 382014, stop := 382025 ), 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 := 382126, root := ~, start := 382112, stop := 382125 ), 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 := 382187, root := ~, start := 382173, stop := 382186 ), 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 := 382212, root := ~, start := 382201, stop := 382211 ), 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 := 382289, root := ~, start := 382278, stop := 382288 ), 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 := 382352, root := ~, start := 382338, stop := 382351 ), 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 := 382491, root := ~, start := 382487, stop := 382490 ), 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 := 382509, root := ~, start := 382501, stop := 382508 ), 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 := 382574, root := ~, start := 382521, stop := 382573 ), 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 := 382623, root := ~, start := 382586, stop := 382622 ), 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 := 382669, root := ~, start := 382655, stop := 382668 ), 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 := 382784, root := ~, start := 382746, stop := 382783 ), 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 := 382842, root := ~, start := 38\ 2806, stop := 382841 ), 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 := 382876, root := ~, start := 382847, stop \ := 382875 ), 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 := 382942, root :=\ ~, start := 382930, stop := 382941 ), 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 := 382985, root :=\ ~, start := 382971, stop := 382984 ), 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 := 383017, root := ~, start := 383013, stop \ := 383016 ), 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 := 383088, root :=\ ~, start := 383074, stop := 383087 ), 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 := 383115, root := ~, start := 3\ 83101, stop := 383114 ), 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 := 383158, root := ~, start := 383150, stop \ := 383157 ), 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 := 383201, root := ~, start := 382791, stop := 383200 ), 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 := 383276, root := ~, start := 383\ 209, stop := 383275 ), 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 := 383345, root := ~, start := 383305, stop := 383344\ ), 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 := 383390, root := ~, start := 383357, stop :=\ 383389 ), 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 := 383420, root := ~, start := 383406, stop := 383419 ) , 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\ := 383447, root := ~, start := 383433, stop := 383446 ), 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 := 383499, root := ~, start := 383491, stop := 383498 ), 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 := 383532, root\ := ~, start := 383528, stop := 383531 ), 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 := 383608, root := ~, start := 383568, stop := 383607 ), 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 := 383654, root := ~, start := 383621, stop := 383653 ), 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 := 383705, root := ~, start := 383693, stop := 383704 ), 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 := 383766, root := ~, start := 383758, stop \ := 383765 ), 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 := 383825, root := ~, start := 3\ 83811, stop := 383824 ), 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 := 383866, root := ~, start := 383862, stop :=\ 383865 ), 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 := 383971, root := ~, start := 383961, stop := 383970 ), 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 := 384014, root := ~, start := 384004, stop := 384013 ), 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 := 384158, root := ~, start := 383283, stop := 384157 ) ], count := [ 8, 2, 3, 17 ], mathmode := "Text", name := "List", next := 384170, root := ~, start := 382733, stop := 384169 ), 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 := 385198, root := ~, start := 384176, stop := 385197 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Description", next := 385215, root := ~, start := 381138, stop := 385214 ) ], count := [ 8, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 385229, root := ~, start := 380509, stop := 385228 ), rec( content := "\n\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 385243, root := ~, start := 374592, stop := 385242 ) , rec( content := "\n", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 385254, root := ~, start := 353315, stop := 385253 ), 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 := 385348, root := ~, start := 385311, stop := 385347 ) , 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 := 385401, root := ~, start := 385361, stop := 385400 ), 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 := 385464, root := ~, start := 385425, stop := 385463 ), 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 := 385496, root := ~, start := 385469, stop := 385495 ), 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 := 385576, root := ~, start := 385566, stop := 385575 ), 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 := 385671, root := ~, start := 385667, stop := 385670 ), 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 := 385700, root := ~, start := 385690, stop := 385699 ), 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 := 385738, root := ~, start := 385728, stop := 385737 ), 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 := 385760, root := ~, start := 385748, stop := 385759 ), 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 := 385809, root := ~, start := 385794, stop := 385808 ), 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 := 385835, root := ~, start := 385820, stop := 385834 ), 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\ ttp://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 := 385952, root := ~, start := 385913, stop := 385951 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 385997, root := ~, start := 385963, stop := 385996 ), 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 := 386002, root := ~, start := 385998, stop := 386001 ), 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 := 386146, root := ~, start := 386122, stop := 386145 ), 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 := 386151, root := ~, start := 386147, stop := 386150 ), 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 := 386199, root := ~, start := 386188, stop := 386198 ), 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 := 386327, root := ~, start := 386310, stop := 386326 ), 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 := 386454, root := ~, start := 386441, stop \ := 386453 ), 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 := 386528 , root\ := ~, start := 386517, stop := 386527 ), rec( content := ".\n ", count\ := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], name := "Ite\ m", next := 386545, root := ~, start := 386336, stop := 386544 ), 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 := 386577, root := ~, start := 386555, stop := 386576 ), 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 := 386690, root := ~, start := 386679, stop := 386689 ), 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\ := 386719, root := ~, start := 386706, stop := 386718 ), 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 := 386818, root := ~, start := 38\ 6814, stop := 386817 ), 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 := 386882, root\ := ~, start := 386866, stop := 386881 ), 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 := 386918, root :=\ ~, start := 386907, stop := 386917 ), 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 := 387006, root := ~, start := 386959, stop := 387005 ), 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 := 387041, root := ~, start := 386586, stop := 387040 ), 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 := 387072, root := ~, start := 387051, stop := 387071 ), 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 := 387182, root := ~, start := 387172, stop := 387181 ), 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 := 387266, root := ~, star\ t := 387247, stop := 387265 ), rec( content := ".\n ", count := [ 9, 1,\ 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], name := "Ite\ m", next := 387283, root := ~, start := 387081, stop := 387282 ), 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 := 387312, root := ~, start := 387293, stop := 387311 ), 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 := 387464, root :=\ ~, start := 387449, stop := 387463 ), 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 := 387483, root := ~, start := 387468, stop \ := 387482 ), rec( content := ".\n ", count := [ 9, 1, 1, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ], name := "Ite\ m", next := 387500, root := ~, start := 387321, stop := 387499 ), 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 := 387527, root := ~, start := 387510, stop := 387526 ), 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 \ := 387614, root := ~, start := 387604, stop := 387613 ), 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 := 387657, root := ~, start := 38764\ 7, stop := 387656 ), 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 := 387719, root :=\ ~, start := 387709, stop := 387718 ), 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 := 387769, root :=\ ~, start := 387754, stop := 387768 ), 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 := 387797, root := ~, start :=\ 387783, stop := 387796 ), 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 := 387896, root := ~, start := 387886, stop := 3\ 87895 ), 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 := 387918, root := ~, start := 387903, stop := 387917 ), rec( cont\ ent := ".\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := "Item", next := 38793\ 5, root := ~, start := 387536, stop := 387934 ), 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 := 387964, root := ~, start := 387\ 945, stop := 387963 ), 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 := \ 388084, root := ~, start := 388074, stop := 388083 ), 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\ := 388129, root := ~, start := 388117, stop := 388128 ), 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 := 388155, root :=\ ~, start := 388141, stop := 388154 ), 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 := 388171, root := ~, start := 388157, stop \ := 388170 ), 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 := 388187, root := ~, start := 388173, stop := 388186 ), 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 := 388201, root :=\ ~, start := 388189, stop := 388200 ), 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 := 388226, root := ~, start := \ 388213, stop := 388225 ), 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 := 388250, root := ~, start := 388232, stop := 38\ 8249 ), 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 := 388394, root :=\ ~, start := 388382, stop := 388393 ), 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\ := 388451, root := ~, start := 388434, stop := 388450 ), rec( content := ".\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], name := "Item", next := 388468, root := ~, start := 387\ 973, stop := 388467 ), 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 := 388499, root := ~, start := 388\ 478, stop := 388498 ), 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 := 388608, root :=\ ~, start := 388595, stop := 388607 ), 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\ := 388711, root := ~, start := 388699, stop := 388710 ), rec( content := " is\ used.\n ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Item", next := 388736, root := ~, start := 388\ 508, stop := 388735 ) ], count := [ 9, 1, 1, 7 ], mathmode := "Text", name := "List", next := 388750, root := ~, start := 386295, stop := 388749 ), 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 := 388875, root := ~, start := 388871, stop := 388874 ), rec( content := "\n\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 := 388950, root := ~, start := 388883, stop := 388949 ), rec( content := "\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 388969, root := ~, start := 385501, stop := 388968 ) ], count := [ 9, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 388985, root := ~, start := 385408, stop := 388984 ), 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 := 389049, root := ~, start := 389010, stop := 389048 ), rec( attributes := rec( Arg := "digraph", Name := "DotVertexLabell\ edDigraph" ), content := 0, count := [ 9, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 389107, root := ~, start := 389054, stop := 389106 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 389140, root := ~, start := 389112, stop := 389139 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "C", next := 389182, root := ~, start := 389165, stop := 389181 ), rec( content := " produce\ s a graphical representation of the digraph\n ", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "A", next := 389254, root := ~, start := 389240, stop := 389253 ), rec( content := ". Vertic\ es are displayed as circles, numbered consistently\n with ", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "A", next := 389338, root := ~, start := 389324, stop := 389337 ), 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, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "P", next := 389482, root := ~, start := 389478, stop := 389481 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "C", next := 389521, root := ~, start := 389490, stop := 389520 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "C", next := 389552, root := ~, start := 389535, stop := 389551 ), rec( content := " only in\ \n that the values in ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 389619, root := ~, start := 389586, stop := 389618 ), rec( content := " are use\ d to label\n the vertices in the produced picture rather than the numbers\ ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "C", next := 389713, root := ~, start := 389705, stop := 389712 ), rec( content := " to\n \ the number of vertices of the digraph. ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "P", next := 389766, root := ~, start := 389762, stop := 389765 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "C", next := 389801, root := ~, start := 389791, stop := 389800 ), rec( content := " format \ (also known as ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "C", next := 389839, root := ~, start := 389824, stop := 389838 ), 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, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "URL", next := 389992, root := ~, start := 389958, stop := 389991 ), rec( content := ". ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "P", next := 389998, root := ~, start := 389994, stop := 389997 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "C", next := 390046, root := ~, start := 390029, stop := 390045 ), rec( content := " or\n \ ", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "C", next := 390087, root := ~, start := 390056, stop := 390086 ), rec( content := " can be \ written to a file using\n the command ", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "Ref", next := 390179, root := ~, start := 390137, stop := 390178 ), rec( content := ".", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 9 ], mathmode := "Text", name := "P", next := 390184, root := ~, start := 390180, stop := 390183 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 9 ], 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> FileString(\"dot/k4.dot\", DotDigraph(gr));\n154", count := [ 9, 1, 2, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 10 ], mathmode := "Text", name := "Log", next := 390491, root := ~, start := 390192, stop := 390490 ), rec( content := "\n ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 390510, root := ~, start := 389145, stop := 390509 ) ], count := [ 9, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 390526, root := ~, start := 388993, stop := 390525 ), rec( content := "\n\n ", count := [ 9, 1, 2, 11 ], 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 := 390599, root := ~, start := 390551, stop := 390598 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 390632, root := ~, start := 390604, stop := 390631 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of the symmetric\n digra\ ph ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "A", next := 390752, root := ~, start := 390738, stop := 390751 ), rec( content := ". ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "C", next := 390780, root := ~, start := 390754, stop := 390779 ), rec( content := " will re\ turn an\n error if ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "A", next := 390825, root := ~, start := 390811, stop := 390824 ), rec( content := " is not \ a symmetric digraph. See\n ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 390896, root := ~, start := 390864, stop := 390895 ), rec( content := ".", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "P", next := 390901, root := ~, start := 390897, stop := 390900 ), rec( content := "\n\n \ Vertices are displayed as circles, numbered consistently with\n ", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "A", next := 390991, root := ~, start := 390977, stop := 390990 ), rec( content := ". Since \ ", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "A", next := 391013, root := ~, start := 390999, stop := 391012 ), 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, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "C", next := 391152, root := ~, start := 391126, stop := 391151 ), 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, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "P", next := 391271, root := ~, start := 391267, stop := 391270 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "C", next := 391306, root := ~, start := 391296, stop := 391305 ), rec( content := " format \ (also known as ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "C", next := 391344, root := ~, start := 391329, stop := 391343 ), 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, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "URL", next := 391503, root := ~, start := 391469, stop := 391502 ), rec( content := ". ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "P", next := 391509, root := ~, start := 391505, stop := 391508 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "C", next := 391566, root := ~, start := 391540, stop := 391565 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "Ref", next := 391658, root := ~, start := 391616, stop := 391657 ), rec( content := ".", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 7 ], mathmode := "Text", name := "P", next := 391663, root := ~, start := 391659, stop := 391662 ), rec( content := "\n\n \ ", count := [ 9, 1, 3, 7 ], 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));\n83", count := [ 9, 1, 3, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 8 ], mathmode := "Text", name := "Log", next := 391900, root := ~, start := 391671, stop := 391899 ), rec( content := "\n ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 391919, root := ~, start := 390637, stop := 391918 ) ], count := [ 9, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 391935, root := ~, start := 390534, stop := 391934 ), rec( content := "\n\n ", count := [ 9, 1, 3, 9 ], 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 := 392011, root := ~, start := 391960, stop := 392010 ), 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 := 392044, root := ~, start := 392016, stop := 392043 ), 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 := 392166, root := ~, start := 392152, stop := 392165 ), 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 := 392197, root := ~, start := 392168, stop := 392196 ), 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 := 392242, root := ~, start := 392228, stop := 392241 ), 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 := 392322, root := ~, start := 392279, stop := 392321 ), 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 := 392327, root := ~, start := 392323, stop := 392326 ), 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 := 392355, root := ~, start := 392341, stop := 392354 ), 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 := 392462, root := ~, start := 392433, stop := 392461 ), 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 := 392500, root := ~, start := 392476, stop := 392499 ), 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 := 392563, root := ~, start := 392514, stop := 392562 ), 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 := 392581, root := ~, start := 392567, stop := 392580 ), 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 := 392586, root := ~, start := 392582, stop := 392585 ), 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 := 392621, root := ~, start := 392611, stop := 392620 ), 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 := 392659, root := ~, start := 392644, stop := 392658 ), 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\ ttp://www.graphviz.org", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "URL", next := 392818, root := ~, start := 392784, stop := 392817 ), 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 := 392824, root := ~, start := 392820, stop := 392823 ), 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 := 392884, root := ~, start := 392855, stop := 392883 ), 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 := 392976, root := ~, start := 392934, stop := 392975 ), 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 := 392981, root := ~, start := 392977, stop := 392980 ), 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 := 393167, root := ~, start := 392989, stop := 393166 ), rec( content := "\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 393186, root := ~, start := 392049, stop := 393185 ) ], count := [ 9, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 393202, root := ~, start := 391943, stop := 393201 ), 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 := 393274, root := ~, start := 393227, stop := 393273 ), rec( attributes := rec( Arg := "digraph", Name := "DotQuasiorderDi\ graph" ), content := 0, count := [ 9, 1, 5, 4 ], mathmode := "Text", name := "Attr", next := 393328, root := ~, start := 393279, stop := 393327 ), 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 := 393361, root := ~, start := 393333, stop := 393360 ), 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 := 393478, root := ~, start := 393464, stop := 393477 ), 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 := 393505, root := ~, start := 393480, stop := 393504 ), 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 := 393550, root := ~, start := 393536, stop := 393549 ), 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 := 393621, root := ~, start := 393582, stop := 393620 ), 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 := 393626, root := ~, start := 393622, stop := 393625 ), 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 := 393842, root := ~, start := 393813, stop := 393841 ), 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 := 393857, root := ~, start := 393849, stop := 393856 ), 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 := 393908, root := ~, start := 393894, stop := 393907 ), 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 := 394038, root := ~, start := 394003, stop := 394037 ), 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 := 394075, root := ~, start := 394067, stop := 394074 ), 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 := 394139, root := ~, start := 394125, stop := 394138 ), 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 := 394186, root := ~, start := 394161, stop := 394185 ), 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 := 394256, root := ~, start := 394207, stop := 394255 ), 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 := 394268, root := ~, start := 394260, stop := 394267 ), 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 := 394376, root := ~, start := 394362, stop := 394375 ), 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 := 394432, root := ~, start := 394428, stop := 394431 ), 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 := 394467, root := ~, start := 394457, stop := 394466 ), 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 := 394505, root := ~, start := 394490, stop := 394504 ), 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\ ttp://www.graphviz.org", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "URL", next := 394664, root := ~, start := 394630, stop := 394663 ), 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 := 394670, root := ~, start := 394666, stop := 394669 ), 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 := 394726, root := ~, start := 394701, stop := 394725 ), 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 := 394818, root := ~, start := 394776, stop := 394817 ), 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 := 394823, root := ~, start := 394819, stop := 394822 ), 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 := 395028, root := ~, start := 394831, stop := 395027 ), rec( content := "\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Description", next := 395047, root := ~, start := 393366, stop := 395046 ) ], count := [ 9, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 395063, root := ~, start := 393210, stop := 395062 ), 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 := 395166, root := ~, start := 395088, stop := 395165 ), 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 := 395199, root := ~, start := 395171, stop := 395198 ), 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 := 395252, root := ~, start := 395224, stop := 395251 ), 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 := 395323, root := ~, start := 395309, stop := 395322 ), 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 := 395368, root := ~, start := 395356, stop := 395367 ), 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 := 395436, root := ~, start := 395422, stop := 395435 ), 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 := 395493, root := ~, start := 395479, stop := 395492 ), 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 := 395530, root := ~, start := 395516, stop := 395529 ), 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 := 395555, root := ~, start := 395541, stop := 395554 ), 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 := 395574, root := ~, start := 395560, stop := 395573 ), 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 := 395628, root := ~, start := 395600, stop := 395627 ), 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 := 395667, root := ~, start := 395663, stop := 395666 ), 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 := 395713, root := ~, start := 395685, stop := 395712 ), 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 := 395764, root := ~, start := 395750, stop := 395763 ), 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 := 395783, root := ~, start := 395769, stop := 395782 ), 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 := 395888, root := ~, start := 395864, stop := 395887 ), 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 := 395924, root := ~, start := 395920, stop := 395923 ), 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 := 396139, root := ~, start := 395932, stop := 396138 ), rec( content := "\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 396158, root := ~, start := 395204, stop := 396157 ) ], count := [ 9, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 396174, root := ~, start := 395071, stop := 396173 ), rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 396188, root := ~, start := 385352, stop := 396187 ) , 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 := 396258, root := ~, start := 396201, stop := 396257 ), 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 := 396407, root := ~, start := 396386, stop := 396406 ), 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 := 396437, root := ~, start := 396422, stop := 396436 ), 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 := 396558, root := ~, start := 396539, stop := 396557 ), 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 := 396891, root := ~, start := 396822, stop := 396890 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], name := "Item", next := 396906, root := ~, start := 396414, stop := 396905 ), 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 := 396936, root := ~, start := 396914, stop := 396935 ), 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 := 396967, root := ~, start := 396951, stop := 396966 ), 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 := 397123, root := ~, start := 397104, stop := 397122 ), 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 := 397326, root := ~, start := 397310, stop := 397325 ), 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 := 397386, root := ~, start := 397371, stop := 397385 ), 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 := 397608, root := ~, start := 397531, stop := 397607 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], name := "Item", next := 397623, root := ~, start := 396943, stop := 397622 ), 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 := 397654, root := ~, start := 397631, stop := 397653 ), 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 := 397684, root := ~, start := 397669, stop := 397683 ), 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 := 397725, root := ~, start := 397710, stop := 397724 ), 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 := 397927, root := ~, start := 397661, stop := 397926 ), 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 := 397959, root := ~, start := 397935, stop := 397958 ), 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 := 397990, root := ~, start := 397974, stop := 397989 ), 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 := 398032, root := ~, start := 398016, stop := 398031 ), 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 := 398280, root := ~, start := 398264, stop := 398279 ), 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 := 398339, root := ~, start := 397966, stop := 398338 ) ], count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "List", next := 398351, root := ~, start := 396373, stop := 398350 ), rec( content := "\n ", count := [ 9, 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromGrap\ h6String" ), content := 0, count := [ 9, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 398430, root := ~, start := 398371, stop := 398429 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDigr\ aph6String" ), content := 0, count := [ 9, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 398492, root := ~, start := 398433, stop := 398491 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromSpar\ se6String" ), content := 0, count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "Oper", next := 398554, root := ~, start := 398495, stop := 398553 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDiSp\ arse6String" ), content := 0, count := [ 9, 2, 1, 8 ], mathmode := "Text", name := "Oper", next := 398616, root := ~, start := 398557, stop := 398615 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "Returns", next := 398648, root := ~, start := 398619, stop := 398647 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "A", next := 398682, root := ~, start := 398672, stop := 398681 ), 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, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "P", next := 399025, root := ~, start := 399021, stop := 399024 ), rec( content := "\n\n \ Each of these functions takes an optional first argument ", count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "A", next := 399099, root := ~, start := 399088, stop := 399098 ), rec( content := ",\n w\ hich should be either ", count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "Ref", next := 399158, root := ~, start := 399128, stop := 399157 ), rec( content := "\n or\ ", count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 1, 10 ], mathmode := "Text", name := "Ref", next := 399198, root := ~, start := 399166, stop := 399197 ), 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, 1, 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> DigraphFromDigraph6String(IsMutableDigraph,\ \"&DOOOW?\");\n\ngap> DigraphFromDi\ graph6String(\"&CQFG\");\n\ngap> D\ igraphFromDigraph6String(\"&IM[SrKLc~lhesbU[F_\");\n\ngap> DigraphFromDiSparse6String(\".CaWBGA?b\");\n\n", count := [ 9, 2, 1, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 1, 11 ], mathmode := "Text", name := "Example", next := 400127, root := ~, start := 399382, stop := 400126 ), rec( content := "\n ", count := [ 9, 2, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "Description", next := 400146, root := ~, start := 398651, stop := 400145 ) ], count := [ 9, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 400160, root := ~, start := 398356, stop := 400159 ), rec( content := "\n\n ", count := [ 9, 2, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph6String" ) , content := 0, count := [ 9, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 400222, root := ~, start := 400181, stop := 400221 ), rec( attributes := rec( Arg := "digraph", Name := "Digraph6String"\ ), content := 0, count := [ 9, 2, 2, 4 ], mathmode := "Text", name := "Oper", next := 400268, root := ~, start := 400225, stop := 400267 ), rec( attributes := rec( Arg := "digraph", Name := "Sparse6String" ), content := 0, count := [ 9, 2, 2, 6 ], mathmode := "Text", name := "Oper", next := 400313, root := ~, start := 400271, stop := 400312 ), rec( attributes := rec( Arg := "digraph", Name := "DiSparse6String\ " ), content := 0, count := [ 9, 2, 2, 8 ], mathmode := "Text", name := "Oper", next := 400360, root := ~, start := 400316, stop := 400359 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Returns", next := 400391, root := ~, start := 400363, stop := 400390 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se four functions return a highly compressed string fully describing the\n \ digraph ", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "A", next := 400514, root := ~, start := 400500, stop := 400513 ), rec( content := ". ", 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 := 400520, root := ~, start := 400516, stop := 400519 ), 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, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ parse6", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "E", next := 400655, root := ~, start := 400641, stop := 400654 ), rec( content := " and\n \ ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ isparse6", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "E", next := 400680, root := ~, start := 400664, stop := 400679 ), rec( content := " (this l\ atter also preserves multiple edges). ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 2, 11 ], mathmode := "Text", name := "P", next := 400730, root := ~, start := 400726, stop := 400729 ), rec( content := "\n\n \ See ", count := [ 9, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 2, 11 ], mathmode := "Text", name := "Ref", next := 400767, root := ~, start := 400740, stop := 400766 ), rec( content := ".\n " , count := [ 9, 2, 2, 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, 2, 12 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "Example", next := 400954, root := ~, start := 400773, stop := 400953 ), rec( content := "\n ", count := [ 9, 2, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Description", next := 400971, root := ~, start := 400394, stop := 400970 ) ], count := [ 9, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 400985, root := ~, start := 400166, stop := 400984 ), rec( content := "\n\n ", count := [ 9, 2, 2, 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, 3, 2 ], mathmode := "Text", name := "Func", next := 401065, root := ~, start := 401006, stop := 401064 ), rec( attributes := rec( ), content := [ rec( content := "An IO fil\ e object.", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 401105, root := ~, start := 401068, stop := 401104 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "A", next := 401144, root := ~, start := 401129, stop := 401143 ), rec( content := " is a st\ ring representing the name of a file, then\n ", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "C", next := 401217, root := ~, start := 401199, stop := 401216 ), rec( content := " returns\ an ", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 3, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 3, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package\ file object for that file.\n ", count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "P", next := 401277, root := ~, start := 401273, stop := 401276 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "A", next := 401320, root := ~, start := 401308, stop := 401319 ), 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, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "A", next := 401566, root := ~, start := 401554, stop := 401565 ), 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, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401749, root := ~, start := 401739, stop := 401748 ), rec( content := ", ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ s6", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401761, root := ~, start := 401751, stop := 401760 ), rec( content := ", ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ d6", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401773, root := ~, start := 401763, stop := 401772 ), rec( content := ", ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ds6", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401786, root := ~, start := 401775, stop := 401785 ), rec( content := ", ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ txt", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401799, root := ~, start := 401788, stop := 401798 ), rec( content := ",\n " , count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401814, root := ~, start := 401805, stop := 401813 ), rec( content := ", or ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ pickle", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "C", next := 401833, root := ~, start := 401819, stop := 401832 ), rec( content := "; more d\ etails of these file formats is given\n below. ", count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "P", next := 401894, root := ~, start := 401890, stop := 401893 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "A", next := 401936, root := ~, start := 401925, stop := 401935 ), rec( content := " is spec\ ified, then it must be one of:\n ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "w\"", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "C", next := 401989, root := ~, start := 401979, stop := 401988 ), rec( content := " (for wr\ ite), ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "a\"", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "C", next := 402013, root := ~, start := 402003, stop := 402012 ), rec( content := " (for ap\ pend), or ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "C", next := 402041, root := ~, start := 402031, stop := 402040 ), rec( content := " (for re\ ad).\n If ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "A", next := 402072, root := ~, start := 402061, stop := 402071 ), rec( content := " is not \ specified, then ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "C", next := 402106, root := ~, start := 402096, stop := 402105 ), rec( content := " is used\ by default. ", count := [ 9, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "P", next := 402131, root := ~, start := 402127, stop := 402130 ), rec( content := "\n\n \ If ", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "A", next := 402155, root := ~, start := 402140, stop := 402154 ), rec( content := " ends in\ one of: ", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "C", next := 402182, root := ~, start := 402172, stop := 402181 ), rec( content := ", ", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "C", next := 402195, root := ~, start := 402184, stop := 402194 ), rec( content := ", or\n \ ", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "C", next := 402214, root := ~, start := 402204, stop := 402213 ), 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, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "P", next := 402355, root := ~, start := 402351, stop := 402354 ), rec( content := "\n\n \ The file object returned by ", count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "C", next := 402407, root := ~, start := 402389, stop := 402406 ), rec( content := " can be \ given as the first\n argument for either of the functions ", count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "Ref", next := 402501, root := ~, start := 402475, stop := 402500 ), rec( content := " or ", count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "Ref", next := 402538, root := ~, start := 402505, stop := 402537 ), 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, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "Ref", next := 402667, root := ~, start := 402641, stop := 402666 ), rec( content := "\n or\ ", count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "Ref", next := 402702, root := ~, start := 402675, stop := 402701 ), rec( content := " when, f\ or example, reading or writing many\n digraphs in a loop.\n ", count := [ 9, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 8 ], mathmode := "Text", name := "P", next := 402778, root := ~, start := 402774, stop := 402777 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 3, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], name := "Mar\ k", next := 402909, root := ~, start := 402884, stop := 402908 ), 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, \ 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], name := "Ite\ m", next := 403245, root := ~, start := 402916, stop := 403244 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 3, 1\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 11 ], name := "Mar\ k", next := 403278, root := ~, start := 403252, stop := 403277 ), 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, 3, 12 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 12 ], name := "Item", next := 4035\ 13, root := ~, start := 403285, stop := 403512 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 3, 13 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 13 ], name := "Mark", next := 4035\ 47, root := ~, start := 403520, stop := 403546 ), 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, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 14 ], name := "Ite\ m", next := 403777, root := ~, start := 403554, stop := 403776 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 3\ , 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 15 ], name := "Mar\ k", next := 403813, root := ~, start := 403784, stop := 403812 ), 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, 3, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 3, 16 ], name := "Item", next := 404025, root := ~, start := 403\ 820, stop := 404024 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 3, 17 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 3, 17 ], name := "Mark", next := 4040\ 62, root := ~, start := 404032, stop := 404061 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 3, 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, 3, 18 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 18 ], mathmode := "Text", name\ := "C", next := 404194, root := ~, start := 404149, stop := 404193 ), 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, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 3, 19 ], mathmode := "Text", name\ := "P", next := 404418, root := ~, start := 404414, stop := 404417 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 3, 19 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 3, 19 ], mathmode := "Text", name := "Ref\ ", next := 404466, root := ~, start := 404432, stop := 404465 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 3, 20 ], mathmode := "Text", name\ := "P", next := 404543, root := ~, start := 404539, stop := 404542 ), rec( cont\ ent := "\n ", count := [ 9, 2, 3, 20 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 3, 18 ], name := "Item", next := 404557, root := ~, start := 404\ 069, stop := 404556 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 3, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 3, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 3, 21 ], mathmode := "Text", name := "F", next := 404589, root := \ ~, start := 404580, stop := 404588 ), rec( content := " or ", count := [ 9, 2,\ 3, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 3, 21 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 21 ], math\ mode := "Text", name := "F", next := 404607, root := ~, start := 404593, stop \ := 404606 ), rec( content := ")", count := [ 9, 2, 3, 21 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 21 ], name := "Mark", next := 4046\ 15, root := ~, start := 404565, stop := 404614 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n http://gap-packages.github.io/io/\n", count := [ 9, 2, 3\ , 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, \ 3, 22 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17\ , stop := 77 ) ], count := [ 9, 2, 3, 22 ], name := "Alt", next := 84, root :=\ ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ), conten\ t := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :=\ [ 9, 2, 3, 22 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 22 ], n\ ame := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count\ := [ 9, 2, 3, 22 ], mathmode := "Text", name := "Alt", next := 128, root := ~ , star\ t := 84, stop := 127 ), rec( content := " package. This is particularly good\n\ when the ", count := [ 9, 2, 3, 22 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), content :\ = 0, count := [ 9, 2, 3, 22 ], mathmode := "Text", name := "Ref", next := 4047\ 51, root := ~, start := 404725, stop := 404750 ), rec( content := " is non-tri\ vial.\n ", count := [ 9, 2, 3, 22 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 9, 2, 3, 22 ], name := "Item", next := 404781, root := ~, start := 404\ 622, stop := 404780 ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "List", next := 404793, root := ~, start := 402871, stop := 404792 ), rec( content := "\n\n \ ", count := [ 9, 2, 3, 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", count := [ 9, 2, 3, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 3, 25 ], mathmode := "Text", name := "Example", next := 405168, root := ~, start := 404799, stop := 405167 ), rec( content := "\n ", count := [ 9, 2, 3, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 405187, root := ~, start := 401108, stop := 405186 ) ], count := [ 9, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 405201, root := ~, start := 400991, stop := 405200 ), rec( content := "\n\n ", count := [ 9, 2, 3, 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, 4, 2 ], mathmode := "Text", name := "Func", next := 405280, root := ~, start := 405222, stop := 405279 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or a list of digraphs.", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 405335, root := ~, start := 405283, stop := 405334 ), 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 := 405374, root := ~, start := 405359, stop := 405373 ), rec( content := " is a st\ ring containing the name of a file containing\n encoded digraphs or 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 http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 4, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), 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 := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 405517, root := ~, start := 405486, stop := 405516 ), rec( content := ", then " , count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 405543, root := ~, start := 405524, stop := 405542 ), rec( content := " returns\ the digraphs\n encoded in the file as a list. Note that 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 := 405629, root := ~, start := 405614, stop := 405628 ), rec( content := " is a\n \ compressed file, which has been compressed appropriately to give a filename\ \n extension of ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 405742, root := ~, start := 405732, stop := 405741 ), rec( content := ", ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 405755, root := ~, start := 405744, stop := 405754 ), rec( content := ", or ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 405770, root := ~, start := 405760, stop := 405769 ), rec( content := ", then\n\ ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 405800, root := ~, start := 405781, stop := 405799 ), rec( content := " can rea\ d ", 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 := 405825, root := ~, start := 405810, stop := 405824 ), rec( content := " without\ it first needing to be\n decompressed.\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 := 405883, root := ~, start := 405879, stop := 405882 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 405928, root := ~, start := 405914, stop := 405927 ), rec( content := " is spec\ ified\n and is a function which decodes a string into a digraph,\n then \ ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 406031, root := ~, start := 406012, stop := 406030 ), rec( content := " will us\ e ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 406055, root := ~, start := 406041, stop := 406054 ), rec( content := " to deco\ de the digraphs\n contained in ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 406111, root := ~, start := 406096, stop := 406110 ), rec( content := ".", 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 := 406116, root := ~, start := 406112, stop := 406115 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 406155, root := ~, start := 406147, stop := 406154 ), rec( content := " is spec\ ified, then ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 406194, root := ~, start := 406175, stop := 406193 ), rec( content := "\n re\ turns the ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 406219, root := ~, start := 406211, stop := 406218 ), rec( content := "th digra\ ph encoded in the file ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 406265, root := ~, start := 406250, stop := 406264 ), rec( content := ".\n " , 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 := 406275, root := ~, start := 406271, stop := 406274 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "A", next := 406320, root := ~, start := 406306, stop := 406319 ), rec( content := " is not \ specified, then\n ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 406367, root := ~, start := 406348, stop := 406366 ), rec( content := " will de\ duce which decoder to use based on the filename\n extension of ", 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 := 406455, root := ~, start := 406440, stop := 406454 ), rec( content := " (after \ removing the compression-related\n filename extensions ", 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 := 406530, root := ~, start := 406520, stop := 406529 ), 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 := 406543, root := ~, start := 406532, stop := 406542 ), rec( content := ", and ", 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 := 406559, root := ~, start := 406549, stop := 406558 ), rec( content := "). For \ example,\n if the filename extension is ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 406619, root := ~, start := 406609, stop := 406618 ), rec( content := ", then " , count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 406645, root := ~, start := 406626, stop := 406644 ), rec( content := " will us\ e\n the graph6 decoder ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ FromGraph6String" ), content := 0, count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "Ref", next := 406715, root := ~, start := 406678, stop := 406714 ), rec( content := ".\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 := 406725, root := ~, start := 406721, stop := 406724 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 4, 8 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 8 ], name := "Mar\ k", next := 406856, root := ~, start := 406831, stop := 406855 ), 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, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 9 ], name := "Ite\ m", next := 407192, root := ~, start := 406863, stop := 407191 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 4, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 10 ], name := "Mar\ k", next := 407225, root := ~, start := 407199, stop := 407224 ), 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, 11 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 11 ], name := "Item", next := 4074\ 60, root := ~, start := 407232, stop := 407459 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 4, 12 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 12 ], name := "Mark", next := 4074\ 94, root := ~, start := 407467, stop := 407493 ), 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, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 13 ], name := "Ite\ m", next := 407724, root := ~, start := 407501, stop := 407723 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 4\ , 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 14 ], name := "Mar\ k", next := 407760, root := ~, start := 407731, stop := 407759 ), 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, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 4, 15 ], name := "Item", next := 407972, root := ~, start := 407\ 767, stop := 407971 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 4, 16 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 4, 16 ], name := "Mark", next := 4080\ 09, root := ~, start := 407979, stop := 408008 ), 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, 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, 4, 17 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 17 ], mathmode := "Text", name\ := "C", next := 408141, root := ~, start := 408096, stop := 408140 ), 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, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 18 ], mathmode := "Text", name\ := "P", next := 408365, root := ~, start := 408361, stop := 408364 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 4, 18 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 4, 18 ], mathmode := "Text", name := "Ref\ ", next := 408413, root := ~, start := 408379, stop := 408412 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 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 := 408490, root := ~, start := 408486, stop := 408489 ), rec( cont\ ent := "\n", count := [ 9, 2, 4, 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, 4, 19 ], mathmode := "Text", name := "XMLCOMMENT", root \ := ~ ), rec( content := "\n ", count := [ 9, 2, 4, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 17 ], name := "Item", next := 4086\ 35, root := ~, start := 408016, stop := 408634 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "pickled (", count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "F", next := 408667 , root\ := ~, start := 408658, stop := 408666 ), rec( content := " or ", count := [ 9\ , 2, 4, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 4, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 20\ ], mathmode := "Text", name := "F", next := 408685, root := ~, start := 40867\ 1, stop := 408684 ), rec( content := ")", count := [ 9, 2, 4, 20 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 20 ], name := "Mar\ k", next := 408693, root := ~, start := 408643, stop := 408692 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n Digraphs are pickled using \ the ", count := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 4, 21 ], mathmode := "Text", name := "URL\ ", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 4, 21 ] , name\ := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes :=\ rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ \ rec( content := "IO", count := [ 9, 2, 4, 21 ], name := "PCDATA", root := ~ ) \ ], count := [ 9, 2, 4, 21 ], name := "Package", next := 122, root := ~, start \ := 101, stop := 121 ) ], count := [ 9, 2, 4, 21 ], mathmode := "Text", name :=\ "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " \ package. This is particularly good\n when the ", count := [ 9, 2, 4, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ Attr := "DigraphGroup" ), content := 0, count := [ 9, 2, 4, 21 ], mathmode :=\ "Text", name := "Ref", next := 408829, root := ~, start := 408803, stop := 40\ 8828 ), rec( content := " is non-trivial.\n ", count := [ 9, 2, 4, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], name := "Ite\ m", next := 408859, root := ~, start := 408700, stop := 408858 ) ], count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "List", next := 408871, root := ~, start := 406818, stop := 408870 ), rec( content := "\n\n \ ", count := [ 9, 2, 4, 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> ReadDigraphs(\n> Conc\ atenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \ \"/data/tree9.4.txt\"));\n[ , \n \ , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \ \n , \n ]", count := [ 9, 2, 4, 24 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 24 ], mathmode := "Text", name := "Example", next := 409917, root := ~, start := 408877, stop := 409916 ), rec( content := "\n ", count := [ 9, 2, 4, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 409936, root := ~, start := 405338, stop := 409935 ) ], count := [ 9, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 409950, root := ~, start := 405207, stop := 409949 ), rec( content := "\n\n ", count := [ 9, 2, 4, 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, 5, 2 ], mathmode := "Text", name := "Func", next := 410044, root := ~, start := 409971, stop := 410043 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 410083, root := ~, start := 410068, stop := 410082 ), rec( content := " is a li\ st of digraphs or a digraph and ", 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 := 410138, root := ~, start := 410123, stop := 410137 ), rec( content := "\n is\ a string 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 http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 5, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), 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 := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), 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 := 410217, root := ~, start := 410192, stop := 410216 ), rec( content := ",\n t\ hen ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 410248, root := ~, start := 410228, stop := 410247 ), rec( content := " writes \ the digraphs to the file represented by\n ", 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 := 410315, root := ~, start := 410300, stop := 410314 ), rec( content := ". If th\ e supplied filename ends in one of the extensions\n ", 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 := 410387, root := ~, start := 410377, stop := 410386 ), 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 := 410400, root := ~, start := 410389, stop := 410399 ), 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 := 410415, root := ~, start := 410405, stop := 410414 ), 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, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "P", next := 410775, root := ~, start := 410771, stop := 410774 ), rec( content := "\n\n \ ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO: e\ ncode and mode. ", count := [ 9, 2, 5, 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, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nfoDigraphs", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 410925, root := ~, start := 410906, stop := 410924 ), rec( content := " to 1 or\ higher, using ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etInfoLevel", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "C", next := 410967, root := ~, start := 410948, stop := 410966 ), 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 := 410972, root := ~, start := 410968, stop := 410971 ), rec( content := "\n\n \ The currently supported file formats are:\n ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 5, 6 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], name := "Mar\ k", next := 411062, root := ~, start := 411037, stop := 411061 ), 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, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 7 ], name := "Ite\ m", next := 411398, root := ~, start := 411069, stop := 411397 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 5, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 8 ], name := "Mar\ k", next := 411431, root := ~, start := 411405, stop := 411430 ), 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, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 9 ], name := "Item", next := 4116\ 66, root := ~, start := 411438, stop := 411665 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 5, 10 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 10 ], name := "Mark", next := 4117\ 00, root := ~, start := 411673, stop := 411699 ), 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, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 11 ], name := "Ite\ m", next := 411930, root := ~, start := 411707, stop := 411929 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 5\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 12 ], name := "Mar\ k", next := 411966, root := ~, start := 411937, stop := 411965 ), 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, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 5, 13 ], name := "Item", next := 412178, root := ~, start := 411\ 973, stop := 412177 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 5, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 5, 14 ], name := "Mark", next := 4122\ 16, root := ~, start := 412186, stop := 412215 ), 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, 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, 5, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 15 ], mathmode := "Text", name\ := "C", next := 412348, root := ~, start := 412303, stop := 412347 ), 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, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 16 ], mathmode := "Text", name\ := "P", next := 412572, root := ~, start := 412568, stop := 412571 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 5, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 5, 16 ], mathmode := "Text", name := "Ref\ ", next := 412620, root := ~, start := 412586, stop := 412619 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 5, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 17 ], mathmode := "Text", name\ := "P", next := 412697, root := ~, start := 412693, stop := 412696 ), rec( cont\ ent := "\n", count := [ 9, 2, 5, 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, 5, 17 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( cont\ ent := "\n ", count := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 5, 15 ], name := "Item", next := 412834, root := ~, start := 412\ 223, stop := 412833 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 5, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 5, 18 ], mathmode := "Text", name := "F", next := 412865, root := \ ~, start := 412856, stop := 412864 ), rec( content := " or ", count := [ 9, 2,\ 5, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 5, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 18 ], math\ mode := "Text", name := "F", next := 412883, root := ~, start := 412869, stop \ := 412882 ), rec( content := ")", count := [ 9, 2, 5, 18 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 18 ], name := "Mark", next := 4128\ 91, root := ~, start := 412841, stop := 412890 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 5, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n http://gap-packages.github.io/io/\n", count := [ 9, 2, 5\ , 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, \ 5, 19 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17\ , stop := 77 ) ], count := [ 9, 2, 5, 19 ], name := "Alt", next := 84, root :=\ ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ), conten\ t := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :=\ [ 9, 2, 5, 19 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 19 ], n\ ame := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count\ := [ 9, 2, 5, 19 ], mathmode := "Text", name := "Alt", next := 128, root := ~ , star\ t := 84, stop := 127 ), rec( content := " package. This is particularly good\n\ when the ", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), content :\ = 0, count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "Ref", next := 4130\ 27, root := ~, start := 413001, stop := 413026 ), rec( content := " is non-tri\ vial.\n ", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 9, 2, 5, 19 ], name := "Item", next := 413057, root := ~, start := 412\ 898, stop := 413056 ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "List", next := 413069, root := ~, start := 411024, stop := 413068 ), rec( content := "\n\n \ ", count := [ 9, 2, 5, 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, 5, 22 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 5, 22 ], mathmode := "Text", name := "Example", next := 413836, root := ~, start := 413075, stop := 413835 ), rec( content := "\n ", count := [ 9, 2, 5, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 413855, root := ~, start := 410047, stop := 413854 ) ], count := [ 9, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 413869, root := ~, start := 409956, stop := 413868 ), rec( content := "\n\n ", count := [ 9, 2, 5, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, deco\ der]", Name := "IteratorFromDigraphFile" ), content := 0, count := [ 9, 2, 6, 2 ], mathmode := "Text", name := "Func", next := 413955, root := ~, start := 413890, stop := 413954 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Returns", next := 413989, root := ~, start := 413958, stop := 413988 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", 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 := 414028, root := ~, start := 414013, stop := 414027 ), rec( content := " is a st\ ring representing the name of a file containing\n encoded digraphs, then ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 414141, root := ~, start := 414111, stop := 414140 ), rec( content := " returns\ an iterator\n for which the value of ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Oper := "NextIterator" ), content := 0, count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Ref", next := 414230, root := ~, start := 414189, stop := 414229 ), rec( content := " is the\\ n next digraph encoded in the file.\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 := 414284, root := ~, start := 414280, stop := 414283 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "A", next := 414329, root := ~, start := 414315, stop := 414328 ), rec( content := " is spec\ ified and is a function\n which decodes a string into a digraph, then ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "C", next := 414439, root := ~, start := 414409, stop := 414438 ), rec( content := "\n wi\ ll use ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "A", next := 414467, root := ~, start := 414453, stop := 414466 ), rec( content := " to deco\ de the digraphs contained in ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "A", next := 414519, root := ~, start := 414504, stop := 414518 ), rec( content := ".\n " , 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 := 414529, root := ~, start := 414525, stop := 414528 ), 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, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "P", next := 414707, root := ~, start := 414703, stop := 414706 ), rec( content := "\n\n \ To see what file types are available, see ", count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "Ref", next := 414782, root := ~, start := 414755, stop := 414781 ), rec( content := ".\n\n \ ", count := [ 9, 2, 6, 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, 6, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 7 ], mathmode := "Text", name := "Example", next := 415131, root := ~, start := 414789, stop := 415130 ), rec( content := "\n ", count := [ 9, 2, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 415148, root := ~, start := 413992, stop := 415147 ) ], count := [ 9, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 415162, root := ~, start := 413875, stop := 415161 ), rec( content := "\n\n ", count := [ 9, 2, 6, 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, 7, 2 ], mathmode := "Text", name := "Func", next := 415264, root := ~, start := 415183, stop := 415263 ), rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineDecoder" ), content := 0, count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "Oper", next := 415348, root := ~, start := 415267, stop := 415347 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "Returns", next := 415379, root := ~, start := 415351, stop := 415378 ), 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, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "P", next := 415504, root := ~, start := 415500, stop := 415503 ), rec( content := "\n\n \ ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineEncoder", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "A", next := 415544, root := ~, start := 415510, stop := 415543 ), 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, 7, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter2", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "A", next := 415774, root := ~, start := 415757, stop := 415773 ), rec( content := ", and th\ e edges themselves are separated by the\n string ", count := [ 9, 2, 7, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter1", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "A", next := 415850, root := ~, start := 415833, stop := 415849 ), rec( content := ". ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "A", next := 415887, root := ~, start := 415853, stop := 415886 ), 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, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "P", next := 416011, root := ~, start := 416007, stop := 416010 ), rec( content := "\n\n \ If only one delimiter is passed as an argument to\n ", count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "A", next := 416105, root := ~, start := 416071, stop := 416104 ), 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, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "P", next := 416215, root := ~, start := 416211, stop := 416214 ), rec( content := "\n\n \ The argument ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "A", next := 416247, root := ~, start := 416234, stop := 416246 ), 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, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 9 ], mathmode := "Text", name := "P", next := 416465, root := ~, start := 416461, stop := 416464 ), 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, 7, 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, 7, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 10 ], mathmode := "Text", name := "Example", next := 416932, root := ~, start := 416606, stop := 416931 ), rec( content := "\n ", count := [ 9, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "Description", next := 416949, root := ~, start := 415382, stop := 416948 ) ], count := [ 9, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 416963, root := ~, start := 415168, stop := 416962 ), rec( content := "\n\n ", count := [ 9, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TournamentLineD\ ecoder" ), content := 0, count := [ 9, 2, 8, 2 ], mathmode := "Text", name := "Oper", next := 417030, root := ~, start := 416984, stop := 417029 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "Returns", next := 417062, root := ~, start := 417033, stop := 417061 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "A", next := 417122, root := ~, start := 417112, stop := 417121 ), rec( content := ", decode\ s it, and then returns the\n tournament [see ", count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "Ref", next := 417203, root := ~, start := 417177, stop := 417202 ), rec( content := "] which \ it defines, according to\n the following rules. ", count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "P", next := 417265, root := ~, start := 417261, stop := 417264 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "A", next := 417310, root := ~, start := 417300, stop := 417309 ), 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, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "C", next := 417430, root := ~, start := 417422, stop := 417429 ), rec( content := " will be\ detected from the length of the string and will be as\n large as possible.\ ", count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "P", next := 417521, root := ~, start := 417517, stop := 417520 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "C", next := 417589, root := ~, start := 417576, stop := 417588 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "C", next := 417630, root := ~, start := 417617, stop := 417629 ), rec( content := " and so \ on until ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "C", next := 417660, root := ~, start := 417647, stop := 417659 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "C", next := 417719, root := ~, start := 417706, stop := 417718 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "C", next := 417796, root := ~, start := 417781, stop := 417795 ), rec( content := ". ", 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 := 417802, root := ~, start := 417798, stop := 417801 ), rec( content := "\n\n \ If a character of the string with corresponding edge ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "C", next := 417874, root := ~, start := 417861, stop := 417873 ), rec( content := " is equa\ l\n to ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "C", next := 417899, root := ~, start := 417891, stop := 417898 ), rec( content := ", then t\ he edge ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "C", next := 417928, root := ~, start := 417915, stop := 417927 ), rec( content := " is pres\ ent in the tournament.\n Otherwise, the edge ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "C", next := 417996, root := ~, start := 417983, stop := 417995 ), rec( content := " is pres\ ent instead. In this way, all the\n possible edges are encoded one-by-one.\ \n\n ", count := [ 9, 2, 8, 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, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "Example", next := 418271, root := ~, start := 418087, stop := 418270 ), rec( content := "\n ", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 3 ], mathmode := "Text", name := "Description", next := 418288, root := ~, start := 417065, stop := 418287 ) ], count := [ 9, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 418302, root := ~, start := 416969, stop := 418301 ), rec( content := "\n\n ", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "AdjacencyMatrix\ UpperTriangleLineDecoder" ), content := 0, count := [ 9, 2, 9, 2 ], mathmode := "Text", name := "Oper", next := 418387, root := ~, start := 418323, stop := 418386 ), 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 := 418419, root := ~, start := 418390, stop := 418418 ), 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 := 418479, root := ~, start := 418469, stop := 418478 ), rec( content := ", decode\ s it, and then returns the\n topologically sorted digraph [see ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Ref", next := 418588, root := ~, start := 418552, stop := 418587 ), rec( content := "]\n w\ hich it defines, according to 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 := 418650, root := ~, start := 418646, stop := 418649 ), 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 := 418695, root := ~, start := 418685, stop := 418694 ), rec( content := " represe\ nt the entries in the upper\n triangle of a digraph's adjacency matrix. The\ number of vertices ", 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 := 418808, root := ~, start := 418800, stop := 418807 ), 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, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "P", next := 418903, root := ~, start := 418899, stop := 418902 ), 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 := 418971, root := ~, start := 418958, stop := 418970 ), 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 := 419012, root := ~, start := 418999, stop := 419011 ), 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 := 419042, root := ~, start := 419029, stop := 419041 ), 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 := 419101, root := ~, start := 419088, stop := 419100 ), 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 := 419178, root := ~, start := 419163, stop := 419177 ), rec( content := ". If a \ character of the string with\n corresponding edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 419251, root := ~, start := 419238, stop := 419250 ), rec( content := " is equa\ l to ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 419272, root := ~, start := 419264, stop := 419271 ), 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, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "P", next := 419419, root := ~, start := 419415, stop := 419418 ), rec( content := "\n\n \ In particular, note that there exists no 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 := 419484, root := ~, start := 419471, stop := 419483 ), rec( content := " if ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ \\leq\n i", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "M", next := 419509, root := ~, start := 419488, stop := 419508 ), rec( content := ". In or\ der words, the digraph will be topologically sorted.\n\n ", count := [ 9, 2, 9, 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, 9, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 9, 7 ], mathmode := "Text", name := "Example", next := 419954, root := ~, start := 419575, stop := 419953 ), rec( content := "\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Description", next := 419971, root := ~, start := 418422, stop := 419970 ) ], count := [ 9, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 419985, root := ~, start := 418308, stop := 419984 ), rec( content := "\n\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TCodeDecoder" ) , content := 0, count := [ 9, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 420043, root := ~, start := 420006, stop := 420042 ), 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 := 420075, root := ~, start := 420046, stop := 420074 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", 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 := 420109, root := ~, start := 420099, stop := 420108 ), 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, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "P", next := 420294, root := ~, start := 420290, stop := 420293 ), rec( content := "\n\n \ The first integer of the string defines the number of vertices ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 420371, root := ~, start := 420363, stop := 420370 ), rec( content := " in\n \ the digraph, and the second defines the number of edges ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 420443, root := ~, start := 420435, stop := 420442 ), rec( content := ". The\n\ following ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 420473, root := ~, start := 420464, stop := 420472 ), rec( content := " integer\ s should be vertex numbers in the range ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 0 ..\n v-1]", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 420544, root := ~, start := 420521, stop := 420543 ), 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, 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 := 420664, root := ~, start := 420654, stop := 420663 ), rec( content := " has few\ er than\n ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e+2", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 420695, root := ~, start := 420684, stop := 420694 ), rec( content := " entries\ . ", 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 := 420709, root := ~, start := 420705, stop := 420708 ), 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, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ 6", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 420835, root := ~, start := 420825, stop := 420834 ), rec( content := " will de\ scribe the edge\n ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1,7]", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 420875, root := ~, start := 420863, stop := 420874 ), rec( content := ".\n " , count := [ 9, 2, 10, 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, 10, 6 ], name := "PCDATA" , root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "Example", next := 421243, root := ~, start := 420881, stop := 421242 ), rec( content := "\n ", count := [ 9, 2, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 421260, root := ~, start := 420078, stop := 421259 ) ], count := [ 9, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 421274, root := ~, start := 419991, stop := 421273 ), rec( content := "\n\n ", count := [ 9, 2, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlainTextString\ " ), content := 0, count := [ 9, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 421339, root := ~, start := 421295, stop := 421338 ), rec( attributes := rec( Arg := "s", Name := "DigraphFromPlai\ nTextString" ), content := 0, count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "Oper", next := 421391, root := ~, start := 421342, stop := 421390 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "Returns", next := 421422, root := ~, start := 421394, stop := 421421 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lainTextString", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "A", next := 421465, root := ~, start := 421443, stop := 421464 ), rec( content := " takes a\ single digraph, and returns a string\n describing the edges of that digrap\ h. ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "A", next := 421587, root := ~, start := 421554, stop := 421586 ), 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, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 6 ], mathmode := "Text", name := "P", next := 421918, root := ~, start := 421914, stop := 421917 ), 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, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "P", next := 422057, root := ~, start := 422053, stop := 422056 ), rec( content := "\n\n \ \n The operation ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "C", next := 422115, root := ~, start := 422082, stop := 422114 ), rec( content := "\n ta\ kes an optional first argument ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "Ref", next := 422183, root := ~, start := 422153, stop := 422182 ), rec( content := "\n or\ ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "Ref", next := 422223, root := ~, start := 422191, stop := 422222 ), 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, 11, 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, 11, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 11, 8 ], mathmode := "Text", name := "Example", next := 422644, root := ~, start := 422396, stop := 422643 ), rec( content := "\n ", count := [ 9, 2, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "Description", next := 422661, root := ~, start := 421425, stop := 422660 ) ], count := [ 9, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 422675, root := ~, start := 421280, stop := 422674 ), rec( content := "\n\n ", count := [ 9, 2, 11, 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, 12, 2 ], mathmode := "Text", name := "Func", next := 422775, root := ~, start := 422696, stop := 422774 ), rec( attributes := rec( Arg := "filename, delimi\ ter, offset, ignore", Name := "ReadPlainTextDigraph" ), content := 0, count := [ 9, 2, 12, 4 ], mathmode := "Text", name := "Oper", next := 422855, root := ~, start := 422778, stop := 422854 ), 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, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 423103, root := ~, start := 423087, stop := 423102 ), rec( content := ".", 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 := 423108, root := ~, start := 423104, stop := 423107 ), rec( content := "\n\n \ ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "A", next := 423129, root := ~, start := 423114, stop := 423128 ), rec( content := " should \ be the name of a file which will be written to or\n read from, and ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "A", next := 423219, root := ~, start := 423206, stop := 423218 ), 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, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ ritePlainTextDigraph", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "C", next := 423357, root := ~, start := 423329, stop := 423356 ), rec( content := " is call\ ed with ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "A", next := 423386, root := ~, start := 423373, stop := 423385 ), rec( content := " ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "C", next := 423396, root := ~, start := 423387, stop := 423395 ), rec( content := ", then\n\ the vertices will be numbered in the file starting from 0 instead of 1 -\n\ ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadPlainTextDigraph", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "C", next := 423511, root := ~, start := 423484, stop := 423510 ), rec( content := " would t\ hen need to be called with ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "A", next := 423559, root := ~, start := 423546, stop := 423558 ), rec( content := "\n ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "C", next := 423572, root := ~, start := 423564, stop := 423571 ), rec( content := " to conv\ ert back to the original graph.", 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 := 423615, root := ~, start := 423611, stop := 423614 ), rec( content := "\n\n \ ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ gnore", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "A", next := 423634, root := ~, start := 423621, stop := 423633 ), rec( content := " should \ be a list of characters which will be ignored when\n reading the graph.\n \ ", count := [ 9, 2, 12, 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, 12, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 8 ], mathmode := "Text", name := "Example", next := 424077, root := ~, start := 423720, stop := 424076 ), rec( content := "\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Description", next := 424094, root := ~, start := 422858, stop := 424093 ) ], count := [ 9, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 424108, root := ~, start := 422681, stop := 424107 ), 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", Name := "WriteDIMACSDigraph" ), content := 0, count := [ 9, 2, 13, 2 ], mathmode := "Text", name := "Oper", next := 424186, root := ~, start := 424129, stop := 424185 ), rec( attributes := rec( Arg := "filename", Name := "ReadDIMACSDigra\ ph" ), content := 0, count := [ 9, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 424236, root := ~, start := 424189, stop := 424235 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations write or read the single symmetric digraph ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "A", next := 424331, root := ~, start := 424317, stop := 424330 ), rec( content := "\n to\ or from a file in DIMACS format, as appropriate. The operation\n ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDIMACSDigraph", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "C", next := 424431, root := ~, start := 424406, stop := 424430 ), rec( content := " records\ the vertices and edges of ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "A", next := 424480, root := ~, start := 424466, stop := 424479 ), rec( content := ".\n T\ he vertex labels of ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "A", next := 424521, root := ~, start := 424507, stop := 424520 ), rec( content := " will be\ recorded only if they are\n integers. See ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Ref", next := 424606, root := ~, start := 424574, stop := 424605 ), rec( content := " and ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Ref", next := 424650, root := ~, start := 424611, stop := 424649 ), 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 := 424655, root := ~, start := 424651, stop := 424654 ), rec( content := "\n\n \ The first argument ", 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 := 424695, root := ~, start := 424680, stop := 424694 ), 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, 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 := 424852, root := ~, start := 424837, stop := 424851 ), rec( content := " ends in\ one of ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 424878, root := ~, start := 424868, stop := 424877 ), rec( content := ", ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 424891, root := ~, start := 424880, stop := 424890 ), rec( content := ",\n o\ r ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 424910, root := ~, start := 424900, stop := 424909 ), rec( content := ", then t\ he file is compressed, or decompressed, appropriately.\n ", 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 := 424981, root := ~, start := 424977, stop := 424980 ), 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, 13, 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, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "c", count := [ 9, 2, 13, 9 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 9 ] , math\ mode := "Text", name := "C", next := 425149, root := ~, start := 425141, stop \ := 425148 ), rec( content := " and followed by any number of characters\n \ is a comment line, and is ignored.\n ", count := [ 9, 2, 13, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 9 ], name := "Ite\ m", next := 425247, root := ~, start := 425104, stop := 425246 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "p", count := [ 9, 2, 13, 11 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 11\ ], mathmode := "Text", name := "C", next := 425299, root := ~, start := 42529\ 1, stop := 425298 ), rec( content := " defines the numbers of vertices and edg\ es\n the digraph. This line has the format ", count := [ 9, 2, 13, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "p edge \n ", count :\ = [ 9, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 13, 11 ], mathmode := "Text", name := "C", next := 425449, root :=\ ~, start := 425389, stop := 425448 ), rec( content := ", where ", count := [ \ 9, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "", count := [ 9, 2\ , 13, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 13, 11 ], mathmode := "Text", name := "C", next := 425483, root :\ = ~, start := 425457, stop := 425482 ), rec( content := " and\n ", coun\ t := [ 9, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2,\ 13, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 13, 11 ], mathmode := "Text", name := "C", next := 425519, root :\ = ~, start := 425496, stop := 425518 ), 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, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 13, 12 ], mathmode := "Text" , name\ := "P", next := 425688, root := ~, start := 425684, stop := 425687 ), rec( cont\ ent := "\n\n Although it is required to be present, the value of\n \ ", count := [ 9, 2, 13, 12 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 13, 12 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 13, 12 ], mathmode := "Text", name := "C", next :\ = 425781, root := ~, start := 425758, stop := 425780 ), 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, 13, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 11 ], name := "Ite\ m", next := 425910, root := ~, start := 425254, stop := 425909 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line of the form ", count\ := [ 9, 2, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "e ", count := [ 9, 2, \ 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2\ , 13, 14 ], mathmode := "Text", name := "C", next := 425979, root := ~, start \ := 425951, stop := 425978 ), rec( content := ", where ", count := [ 9, 2, 13, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "", count := [ 9, 2, 13, 14 ], mathmode\ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 13, 1\ 4 ], mathmode := "Text", name := "C", next := 426003, root := ~, start := 4259\ 87, stop := 426002 ), rec( content := "\n and ", count := [ 9, 2, 13, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "", count := [ 9, 2, 13, 14 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 13, 14\ ], mathmode := "Text", name := "C", next := 426032, root := ~, start := 42601\ 6, stop := 426031 ), rec( content := " are integers in the range ", count := [ \ 9, 2, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "[1 ..\n ]\ ", count := [ 9, 2, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 13, 14 ], mathmode := "Text", name := "C", next := 42610\ 2, root := ~, start := 426059, stop := 426101 ), rec( content := ", specifies \ that there is a (symmetric) edge\n in the digraph between the vertices \ ", count := [ 9, 2, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2, 13, 14\ ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ \ 9, 2, 13, 14 ], mathmode := "Text", name := "C", next := 426207, root := ~, star\ t := 426191, stop := 426206 ), rec( content := " and\n ", count := [ 9,\ 2, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "", count := [ 9, 2, 13, 14 ], \ mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, \ 2, 13, 14 ], mathmode := "Text", name := "C", next := 426236, root := ~, start\ := 426220, stop := 426235 ), rec( content := ". A symmetric edge only needs t\ o be defined once; an\n additional line ", count := [ 9, 2, 13, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "e ", count := [ 9, 2, 13, 14 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 14 ], mathmode :=\ "Text", name := "C", next := 426341, root := ~, start := 426313, stop := 4263\ 40 ), rec( content := ", or ", count := [ 9, 2, 13, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "e \n ", count := [ 9, 2, 13, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 14 ], mathmode := "Text", name\ := "C", next := 426384, root := ~, start := 426346, stop := 426383 ), rec( cont\ ent := ", will be interpreted as an additional, multiple, edge.\n Loops\ are permitted.\n ", count := [ 9, 2, 13, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 14 ], name := "Item", next := 4264\ 82, root := ~, start := 425917, stop := 426481 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n A line of the form ", count := [ 9, 2, 13, 16 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "n ", count := [ 9, 2, 13, 16 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 16 ], mathmode := "\ Text", name := "C", next := 426555, root := ~, start := 426523, stop := 426554\ ), rec( content := ", where\n ", count := [ 9, 2, 13, 16 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "", count := [ 9, 2, 13, 16 ], mathmode := "Text", name\ := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 13, 16 ], mathmode :=\ "Text", name := "C", next := 426587, root := ~, start := 426571, stop := 4265\ 86 ), rec( content := " is an integer in the range ", count := [ 9, 2, 13, 16 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 ..\n ]", count := [ 9, 2, 1\ 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2,\ 13, 16 ], mathmode := "Text", name := "C", next := 426658, root := ~, start :\ = 426615, stop := 426657 ), rec( content := " and ", count := [ 9, 2, 13, 16 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "", count := [ 9, 2, 13, 16 ], mathmode := "Te\ xt", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 13, 16 ], math\ mode := "Text", name := "C", next := 426683, root := ~, start := 426663, stop \ := 426682 ), rec( content := " is an integer,\n signifies that the vert\ ex ", count := [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", count :=\ [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ \ ) ], count := [ 9, 2, 13, 16 ], mathmode := "Text", name := "C", next := 42674\ 9, root := ~, start := 426733, stop := 426748 ), rec( content := " has the lab\ el\n ", count := [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 13, 16 ], mathmode := "Text", name := "C", next :\ = 426792, root := ~, start := 426772, stop := 426791 ), rec( content := " in t\ he digraph. If a label is not specified for a\n vertex, then ", count :\ = [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "ReadDIMACSDigraph", count := \ [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 9, 2, 13, 16 ], mathmode := "Text", name := "C", next := 426888, root := ~ , star\ t := 426864, stop := 426887 ), rec( content := " will assign the label\n \ ", count := [ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "1", count := [ \ 9, 2, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 9, 2, 13, 16 ], mathmode := "Text", name := "C", next := 426927, root := ~, star\ t := 426919, stop := 426926 ), rec( content := ", according to the DIMACS spec\ ification.\n ", count := [ 9, 2, 13, 16 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 2, 13, 16 ], name := "Item", next := 4269\ 81, root := ~, start := 426489, stop := 426980 ) ], count := [ 9, 2, 13, 8 ], mathmode := "Text", name := "List", next := 426993, root := ~, start := 425091, stop := 426992 ), rec( content := "\n\n \ A detailed definition of the DIMACS format can be found at\n ", count := [ 9, 2, 13, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://mat.gsia.cmu.edu/COLOR/general/ccformat.ps", count := [ 9, 2, 13, 18 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 18 ], mathmode := "Text", name := "URL", next := 427122, root := ~, start := 427062, stop := 427121 ), rec( content := ", in Sec\ tion\n 2.1. Note that optional descriptor lines, as described in Section 2.\ 1, will\n be ignored.\n ", count := [ 9, 2, 13, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 19 ], mathmode := "Text", name := "P", next := 427239, root := ~, start := 427235, stop := 427238 ), rec( content := "\n\n\n \ ", count := [ 9, 2, 13, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [1, 3, 4], [2, 4], [2, 3]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigrap\ h(filename, gr);;\ngap> ReadDIMACSDigraph(filename);\n", count := [ 9, 2, 13, 20 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 20 ], mathmode := "Text", name := "Example", next := 427601, root := ~, start := 427246, stop := 427600 ), rec( content := "\n ", count := [ 9, 2, 13, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 427618, root := ~, start := 424239, stop := 427617 ) ], count := [ 9, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 427632, root := ~, start := 424114, stop := 427631 ), rec( content := "\n\n ", count := [ 9, 2, 13, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 427646, root := ~, start := 396192, stop := 427645 ) , rec( content := "\n", count := [ 9, 2, 13, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 427657, root := ~, start := 385275, stop := 427656 ), rec( content := "\n ", count := [ 9, 2, 13, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " visualising, io ", count := [ 9, 2, 13, 21 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n", count := [ 9, 2, 13, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "Body", next := 427691, root := ~, start := 3662, stop := 427690 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Grape to Digraphs Command M\ ap\n ", count := [ "A", 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 427762, root := ~, start := 427706, stop := 427761 ), rec( content := "\n Below is a table of ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.ac.\ uk/~leonard/grape/", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 0, 0, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ "A", 0, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " commands with the ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " counterparts.\n The sections in this c\ hapter correspond to the chapters in the ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.ac.\ uk/~leonard/grape/", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 0, 0, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ "A", 0, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\n manual.\n\n ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to cons\ truct and modify graphs\n ", count := [ "A", 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 428007, root := ~, start := 427936, stop := 428006 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], name := "E", next := 428145, root := ~, start := 428035, stop := 428144 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "Alt", next := 428156, root := ~, start := 428012, stop := 428155 ), rec( content := "\n\n ", count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 4 ], name := "HorLine", next := 428202, root := ~, start := 428192, stop := 428201 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 1, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], name := "Item", next := 428271, root := ~, start := 428223, stop := 428270 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 7 ], name := "Item", next := 428331, root := ~, start := 428280, stop := 428330 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 1, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Alt", next := 428412, root := ~, start := 428357, stop := 428411 ), rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Item", next := 428428, root := ~, start := 428340, stop := 428427 ) ], count := [ "A", 1, 0, 4 ], name := "Row", next := 428441, root := ~, start := 428209, stop := 428440 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 10 ], name := "HorLine", next := 428458, root := ~, start := 428448, stop := 428457 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raph", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "C", next := 428509, root := ~, start := 428497, stop := 428508 ), rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], name := "Item", next := 428525, root := ~, start := 428480, stop := 428524 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ " ), content := 0, count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 428572, root := ~, start := 428551, stop := 428571 ), rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 13 ], name := "Item", next := 428588, root := ~, start := 428534, stop := 428587 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Used with the arguments of ", count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Graph", count := [ "A", 1, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "C", next := 4286\ 81, root := ~, start := 428669, stop := 428680 ), rec( HTML := true, content := "\ in ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Oper := "Digraph" ), content := 0, count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 428727, root := ~, start := 428694, stop := 4287\ 26 ), rec( HTML := true, content := " will return an equivalent digraph. ", count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0 , count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "P", next := 428767, root := ~, start := 428763, stop := 42876\ 6 ), rec( HTML := true, content := "\n ", count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Alt", next := 428784, root := ~, start := 428614, stop := 428783 ), rec( content := "\n \ ", count := [ "A", 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Item", next := 428800, root := ~, start := 428597, stop := 428799 ) ], count := [ "A", 1, 0, 10 ], name := "Row", next := 428813, root := ~, start := 428466, stop := 428812 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 17 ], name := "HorLine", next := 428830, root := ~, start := 428820, stop := 428829 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ dgeOrbitsGraph", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "C", next := 428891, root := ~, start := 428869, stop := 428890 ), rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 18 ], name := "Item", next := 428907, root := ~, start := 428852, stop := 428906 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeOrb\ itsDigraph" ), content := 0, count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "Ref", next := 428964, root := ~, start := 428933, stop := 428963 ), rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 20 ], name := "Item", next := 428980, root := ~, start := 428916, stop := 428979 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "EdgeOrbitsDigraph" ), content := 0 , count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "Ref", next := 429079, root := ~, start := 429048, stop := 4290\ 78 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "EdgeOrbitsDigraph", count := [ "A", 1, 0, 22 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "C", next := 429134, root := ~, start := 429110, stop := 42913\ 3 ), rec( HTML := true, content := " in ", count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 22 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 22 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 22 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Alt", next := 429163, root := ~, start := 429006, stop := 429162 ), rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Item", next := 429179, root := ~, start := 428989, stop := 429178 ) ], count := [ "A", 1, 0, 17 ], name := "Row", next := 429192, root := ~, start := 428838, stop := 429191 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 23 ], name := "HorLine", next := 429209, root := ~, start := 429199, stop := 429208 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ullGraph", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "C", next := 429264, root := ~, start := 429248, stop := 429263 ), rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], name := "Item", next := 429280, root := ~, start := 429231, stop := 429279 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "NullDig\ raph" ), content := 0, count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "Ref", next := 429331, root := ~, start := 429306, stop := 429330 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n or\n ", count := [ "A", 1, 0, 26 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EmptyDigrap\ h" ), content := 0, count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "Ref\ ", next := 429413, root := ~, start := 429387, stop := 429412 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Alt", next := 429430, root := ~, start := 429342, stop := 429429 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Item", next := 429446, root := ~, start := 429289, stop := 429445 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 28 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 28 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 28 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the arguments of ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "NullDigraph" ), content := 0 , count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "Ref", next := 429589, root := ~, start := 429550, stop := 4295\ 88 ), rec( HTML := true, content := " and ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EmptyD\ igraph" ), content := 0, count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Ref\ ", next := 429620, root := ~, start := 429594, stop := 429619 ), rec( HTML := true , content := "\ are just\n non-negative integers. Since digraphs in ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 1, 0, 28 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ are immutable,\n there is no reason (although it is possible) to c\ reate a null digraph\n and then add edges, as you might be used to \ doing in ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 28 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 28 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 28 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Alt", next := 429881, root := ~, start := 429472, stop := 429880 ), rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Item", next := 429897, root := ~, start := 429455, stop := 429896 ) ], count := [ "A", 1, 0, 23 ], name := "Row", next := 429910, root := ~, start := 429217, stop := 429909 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 29 ], name := "HorLine", next := 429927, root := ~, start := 429917, stop := 429926 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteGraph", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "C", next := 429986, root := ~, start := 429966, stop := 429985 ), rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 30 ], name := "Item", next := 430002, root := ~, start := 429949, stop := 430001 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eDigraph" ), content := 0, count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "Ref", next := 430057, root := ~, start := 430028, stop := 430056 ), rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 32 ], name := "Item", next := 430073, root := ~, start := 430011, stop := 430072 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 34 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 34 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 34 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 34 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the argument to ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "CompleteDigraph" ), content := 0 , count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Ref", next := 430218, root := ~, start := 430175, stop := 4302\ 17 ), rec( HTML := true, content := " is simply a non-negative integer\n \ indicating the number of vertices. By default a complete digraph in\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ does not have any loops. If you want a complete digraph\n with loo\ ps at every vertex, then use ", count := [ "A", 1, 0, 34 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "DigraphAddAllLoops" ), content := 0 , count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Ref", next := 430506, root := ~, start := 430460, stop := 4305\ 05 ), rec( HTML := true, content := ".\n ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Alt", next := 430524, root := ~, start := 430099, stop := 430523 ), rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Item", next := 430540, root := ~, start := 430082, stop := 430539 ) ], count := [ "A", 1, 0, 29 ], name := "Row", next := 430553, root := ~, start := 429935, stop := 430552 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 35 ], name := "HorLine", next := 430570, root := ~, start := 430560, stop := 430569 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnsonGraph", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "C", next := 430628, root := ~, start := 430609, stop := 430627 ), rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 36 ], name := "Item", next := 430644, root := ~, start := 430592, stop := 430643 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Johnson\ Digraph" ), content := 0, count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 430698, root := ~, start := 430670, stop := 430697 ), rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 38 ], name := "Item", next := 430714, root := ~, start := 430653, stop := 430713 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 40 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 40 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 40 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 40 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "JohnsonGraph", count :=\ [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], mathmode := \ "Text", name := "C", next := 430810, root := ~, start := 430791, stop := 43080\ 9 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 1, 0, 40 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Johnso\ nDigraph" ), content := 0, count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "Ref\ ", next := 430868, root := ~, start := 430840, stop := 430867 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Alt", next := 430886, root := ~, start := 430740, stop := 430885 ), rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Item", next := 430902, root := ~, start := 430723, stop := 430901 ) ], count := [ "A", 1, 0, 35 ], name := "Row", next := 430915, root := ~, start := 430578, stop := 430914 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 41 ], name := "HorLine", next := 430932, root := ~, start := 430922, stop := 430931 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ayleyGraph", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "C", next := 430989, root := ~, start := 430971, stop := 430988 ), rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], name := "Item", next := 431005, root := ~, start := 430954, stop := 431004 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "Ref", next := 431058, root := ~, start := 431031, stop := 431057 ), rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 44 ], name := "Item", next := 431074, root := ~, start := 431014, stop := 431073 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "CayleyDigraph" ), content := 0 , count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "Ref", next := 431171, root := ~, start := 431144, stop := 4311\ 70 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CayleyGraph", count := [ "A", 1, 0, 46 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "C", next := 431222, root := ~, start := 431204, stop := 43122\ 1 ), rec( HTML := true, content := " in ", count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 46 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 46 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 46 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 46 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Alt", next := 431251, root := ~, start := 431100, stop := 431250 ), rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Item", next := 431267, root := ~, start := 431083, stop := 431266 ) ], count := [ "A", 1, 0, 41 ], name := "Row", next := 431280, root := ~, start := 430940, stop := 431279 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 47 ], name := "HorLine", next := 431297, root := ~, start := 431287, stop := 431296 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ddEdgeOrbit", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "C", next := 431355, root := ~, start := 431336, stop := 431354 ), rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 48 ], name := "Item", next := 431371, root := ~, start := 431319, stop := 431370 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ AddEdgeOrbit" ), content := 0, count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "Ref", next := 431430, root := ~, start := 431397, stop := 431429 ), rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 50 ], name := "Item", next := 431446, root := ~, start := 431380, stop := 431445 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 52 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 52 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 52 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 52 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 52 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , digraphs in ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ are immutable,\n and this operation returns a new digraph, i.e. it\ does not modify the\n digraph given as an argument.\n ", count := [ "A", 1, 0, 52 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Alt", next := 431710, root := ~, start := 431472, stop := 431709 ), rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Item", next := 431726, root := ~, start := 431455, stop := 431725 ) ], count := [ "A", 1, 0, 47 ], name := "Row", next := 431739, root := ~, start := 431305, stop := 431738 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 53 ], name := "HorLine", next := 431756, root := ~, start := 431746, stop := 431755 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ emoveEdgeOrbit", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "C", next := 431817, root := ~, start := 431795, stop := 431816 ), rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 54 ], name := "Item", next := 431833, root := ~, start := 431778, stop := 431832 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ RemoveEdgeOrbit" ), content := 0, count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "Ref", next := 431895, root := ~, start := 431859, stop := 431894 ), rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 56 ], name := "Item", next := 431911, root := ~, start := 431842, stop := 431910 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 58 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 58 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 58 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 58 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 58 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , digraphs in ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ are immutable,\n and this operation returns a new digraph, i.e. it\ does not modify the\n digraph given as an argument.\n ", count := [ "A", 1, 0, 58 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Alt", next := 432175, root := ~, start := 431937, stop := 432174 ), rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Item", next := 432191, root := ~, start := 431920, stop := 432190 ) ], count := [ "A", 1, 0, 53 ], name := "Row", next := 432204, root := ~, start := 431764, stop := 432203 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 59 ], name := "HorLine", next := 432221, root := ~, start := 432211, stop := 432220 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ssignVertexNames", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "C", next := 432284, root := ~, start := 432260, stop := 432283 ), rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], name := "Item", next := 432300, root := ~, start := 432243, stop := 432299 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "SetDigr\ aphVertexLabels" ), content := 0, count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "Ref", next := 432362, root := ~, start := 432326, stop := 432361 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n and\n ", count := [ "A", 1, 0, 62 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "SetDigraph\ VertexLabel" ), content := 0, count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "Ref\ ", next := 432454, root := ~, start := 432419, stop := 432453 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Alt", next := 432471, root := ~, start := 432373, stop := 432470 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Item", next := 432487, root := ~, start := 432309, stop := 432486 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Use ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "SetDigraphVertexLabel" ), content := 0 , count := [ "A", 1, 0, 64 ], mathmode := \ "Text", name := "Ref", next := 432583, root := ~, start := 432548, stop := 4325\ 82 ), rec( HTML := true, content := " to set an individual\n vertex\ label and ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( Oper := "SetDigraphVertexLabels" ), content := 0, count := [ "A", 1, 0, 64 ], mathmode := \ "Text", name := "Ref", next := 432670, root := ~, start := 432634, stop := 4326\ 69 ), rec( HTML := true, content := " to set all of\n the vertex la\ bels.\n ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Alt", next := 432732, root := ~, start := 432513, stop := 432731 ), rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Item", next := 432748, root := ~, start := 432496, stop := 432747 ) ], count := [ "A", 1, 0, 59 ], name := "Row", next := 432761, root := ~, start := 432229, stop := 432760 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 65 ], name := "HorLine", next := 432778, root := ~, start := 432768, stop := 432777 ) ], count := [ "A", 1, 0, 4 ], mathmode := "Text", name := "Table", next := 432791, root := ~, start := 428162, stop := 432790 ), rec( content := "\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "Section", next := 432804, root := ~, start := 427922, stop := 432803 ), rec( content := "\n\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to insp\ ect graphs, vertices and edges\n ", count := [ "A", 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 432900, root := ~, start := 432822, stop := 432899 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], name := "E", next := 433038, root := ~, start := 432928, stop := 433037 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 433049, root := ~, start := 432905, stop := 433048 ), rec( content := "\n\n ", count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 4 ], name := "HorLine", next := 433095, root := ~, start := 433085, stop := 433094 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 2, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], name := "Item", next := 433165, root := ~, start := 433117, stop := 433164 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 7 ], name := "Item", next := 433225, root := ~, start := 433174, stop := 433224 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 2, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Alt", next := 433306, root := ~, start := 433251, stop := 433305 ), rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Item", next := 433322, root := ~, start := 433234, stop := 433321 ) ], count := [ "A", 2, 0, 4 ], name := "Row", next := 433335, root := ~, start := 433103, stop := 433334 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 10 ], name := "HorLine", next := 433352, root := ~, start := 433342, stop := 433351 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sGraph", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "C", next := 433405, root := ~, start := 433391, stop := 433404 ), rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], name := "Item", next := 433421, root := ~, start := 433374, stop := 433420 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "Ref", next := 433470, root := ~, start := 433447, stop := 433469 ), rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 13 ], name := "Item", next := 433486, root := ~, start := 433430, stop := 433485 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Filt := "IsDigraph" ), content := 0, count := [ "A", 2, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 433565, root := ~, start := 433542, stop := 4335\ 64 ), rec( HTML := true, content := " is a category and not a function like\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsGraph", count := [ "A\ ", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], mathmode := \ "Text", name := "C", next := 433630, root := ~, start := 433616, stop := 43362\ 9 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Alt", next := 433648, root := ~, start := 433512, stop := 433647 ), rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Item", next := 433664, root := ~, start := 433495, stop := 433663 ) ], count := [ "A", 2, 0, 10 ], name := "Row", next := 433677, root := ~, start := 433360, stop := 433676 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 16 ], name := "HorLine", next := 433694, root := ~, start := 433684, stop := 433693 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ rderGraph", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "C", next := 433750, root := ~, start := 433733, stop := 433749 ), rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], name := "Item", next := 433766, root := ~, start := 433716, stop := 433765 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ NrVertices" ), content := 0, count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "Ref", next := 433823, root := ~, start := 433792, stop := 433822 ), rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 19 ], name := "Item", next := 433839, root := ~, start := 433775, stop := 433838 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphNrVertices" ), content := 0 , count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 433940, root := ~, start := 433909, stop := 4339\ 39 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "OrderGraph", count := [ "A", 2, 0, 21 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "C", next := 433990, root := ~, start := 433973, stop := 43398\ 9 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Alt", next := 434019, root := ~, start := 433865, stop := 434018 ), rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Item", next := 434035, root := ~, start := 433848, stop := 434034 ) ], count := [ "A", 2, 0, 16 ], name := "Row", next := 434048, root := ~, start := 433702, stop := 434047 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 22 ], name := "HorLine", next := 434065, root := ~, start := 434055, stop := 434064 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sVertex(", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "graph", count := [ "A", 2, 0, 23 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "A", next := 434128, root := ~, start := 434116, stop := 43412\ 7 ), rec( content := ", ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "A", next := 4341\ 38, root := ~, start := 434130, stop := 434137 ), rec( content := ")", count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "C", next := 434143, root := ~, start := 434104, stop := 434142 ), rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], name := "Item", next := 434159, root := ~, start := 434087, stop := 434158 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "v", count := [ "A", 2, 0, 25 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "A", next := 434196, root := ~, start := 434188, stop := 43419\ 5 ), rec( content := " in DigraphVertices(", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "A", next := 4342\ 30, root := ~, start := 434216, stop := 434229 ), rec( content := ")", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "C", next := 434235, root := ~, start := 434185, stop := 434234 ), rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], name := "Item", next := 434251, root := ~, start := 434168, stop := 434250 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphVertices" ), content := 0 , count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 434350, root := ~, start := 434321, stop := 4343\ 49 ), rec( HTML := true, content := " can be used to return\n the l\ ist of vertices of a digraph, or if ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "v", count := [ "A", 2, 0, 27 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "A", next := 4344\ 34, root := ~, start := 434426, stop := 434433 ), rec( HTML := true, content := "\ is known to be\n a positive integer, then you can simply do\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( HTML := \ true, content := "v", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := "Text", name \ := "A", next := 434528, root := ~, start := 434520, stop := 434527 ), rec( HTML\ := true, content := " <= DigraphNrVertices(", count := [ "A", 2, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( HTML := true, content := "digraph", count := [ "A", 2, 0, 27 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ] , math\ mode := "Text", name := "A", next := 434567, root := ~, start := 434553, stop \ := 434566 ), rec( HTML := true, content := ")", count := [ "A", 2, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "C", next := 434572, root := ~, start := 434517, stop := 43457\ 1 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Alt", next := 434589, root := ~, start := 434277, stop := 434588 ), rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Item", next := 434605, root := ~, start := 434260, stop := 434604 ) ], count := [ "A", 2, 0, 22 ], name := "Row", next := 434618, root := ~, start := 434073, stop := 434617 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 28 ], name := "HorLine", next := 434635, root := ~, start := 434625, stop := 434634 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexName", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "C", next := 434691, root := ~, start := 434674, stop := 434690 ), rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], name := "Item", next := 434707, root := ~, start := 434657, stop := 434706 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabel" ), content := 0, count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "Ref", next := 434765, root := ~, start := 434733, stop := 434764 ), rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 31 ], name := "Item", next := 434781, root := ~, start := 434716, stop := 434780 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Unlike ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexName", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "C", next := 4348\ 61, root := ~, start := 434844, stop := 434860 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the return value of\n ", count := [ "A", 2, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hVertexLabel" ), content := 0, count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "Ref\ ", next := 434938, root := ~, start := 434906, stop := 434937 ), rec( HTML := true , content := "\ is mutable.\n ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Alt", next := 434967, root := ~, start := 434807, stop := 434966 ), rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Item", next := 434983, root := ~, start := 434790, stop := 434982 ) ], count := [ "A", 2, 0, 28 ], name := "Row", next := 434996, root := ~, start := 434643, stop := 434995 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 34 ], name := "HorLine", next := 435013, root := ~, start := 435003, stop := 435012 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexNames", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "C", next := 435070, root := ~, start := 435052, stop := 435069 ), rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], name := "Item", next := 435086, root := ~, start := 435035, stop := 435085 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "Ref", next := 435145, root := ~, start := 435112, stop := 435144 ), rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 37 ], name := "Item", next := 435161, root := ~, start := 435095, stop := 435160 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Unlike ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexNames", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "C", next := 4352\ 42, root := ~, start := 435224, stop := 435241 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the return value of\n ", count := [ "A", 2, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hVertexLabels" ), content := 0, count := [ "A", 2, 0, 39 ], mathmode := "Text" , name := "Ref\ ", next := 435320, root := ~, start := 435287, stop := 435319 ), rec( HTML := true , content := "\ is mutable.\n ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Alt", next := 435349, root := ~, start := 435187, stop := 435348 ), rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Item", next := 435365, root := ~, start := 435170, stop := 435364 ) ], count := [ "A", 2, 0, 34 ], name := "Row", next := 435378, root := ~, start := 435021, stop := 435377 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 40 ], name := "HorLine", next := 435395, root := ~, start := 435385, stop := 435394 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertices", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "C", next := 435449, root := ~, start := 435434, stop := 435448 ), rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], name := "Item", next := 435465, root := ~, start := 435417, stop := 435464 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "Ref", next := 435520, root := ~, start := 435491, stop := 435519 ), rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 43 ], name := "Item", next := 435536, root := ~, start := 435474, stop := 435535 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The vertices of a digraph in ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 45 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ is always the range ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "[1\n .. n]", count := [ "A", 2, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "\ Text", name := "C", next := 435681, root := ~, start := 435652, stop := 435680\ ), rec( HTML := true, content := " where ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "n", count := [ "A", 2, 0, 45 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "C", next := 4356\ 96, root := ~, start := 435688, stop := 435695 ), rec( HTML := true, content := "\ is the number of vertices of the\n digraph. The operation ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digrap\ hVertices" ), content := 0, count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 435795, root := ~, start := 435766, stop := 435794 ), rec( HTML := true , content := "\ behaves the\n same as ", count := [ "A", 2, 0, 45 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Vertices", count := [ "A", 2, 0, 45 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "T\ ext", name := "C", next := 435843, root := ~, start := 435828, stop := 435842 ), rec( HTML := true , content := "\ in ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Alt", next := 435872, root := ~, start := 435562, stop := 435871 ), rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Item", next := 435888, root := ~, start := 435545, stop := 435887 ) ], count := [ "A", 2, 0, 40 ], name := "Row", next := 435901, root := ~, start := 435403, stop := 435900 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 46 ], name := "HorLine", next := 435918, root := ~, start := 435908, stop := 435917 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexDegree", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "C", next := 435976, root := ~, start := 435957, stop := 435975 ), rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 47 ], name := "Item", next := 435992, root := ~, start := 435940, stop := 435991 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "Ref", next := 436049, root := ~, start := 436018, stop := 436048 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also: ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "InDegreeOfVertex" ), content := 0 , count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "Ref", next := 436131, root := ~, start := 436101, stop := 4361\ 30 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Alt", next := 436149, root := ~, start := 436060, stop := 436148 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Item", next := 436165, root := ~, start := 436001, stop := 436164 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "OutDegreeOfVertex" ), content := 0 , count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "Ref", next := 436266, root := ~, start := 436235, stop := 4362\ 65 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "VertexDegree", count := [ "A", 2, 0, 51 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "C", next := 436318, root := ~, start := 436299, stop := 43631\ 7 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 51 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 51 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 51 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 51 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Alt", next := 436347, root := ~, start := 436191, stop := 436346 ), rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Item", next := 436363, root := ~, start := 436174, stop := 436362 ) ], count := [ "A", 2, 0, 46 ], name := "Row", next := 436376, root := ~, start := 435926, stop := 436375 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 52 ], name := "HorLine", next := 436393, root := ~, start := 436383, stop := 436392 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexDegrees", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "C", next := 436452, root := ~, start := 436432, stop := 436451 ), rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 53 ], name := "Item", next := 436468, root := ~, start := 436415, stop := 436467 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ eeSet" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref", next := 436520, root := ~, start := 436494, stop := 436519 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also:\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDeg\ reeSequence" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 436615, root := ~, start := 436584, stop := 436614 ), rec( HTML := true , content := "\ ,\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "OutDegrees" ), content := 0 , count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "Ref", next := 436653, root := ~, start := 436629, stop := 4366\ 52 ), rec( HTML := true, content := ",\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InDegr\ ees" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 436690, root := ~, start := 436667, stop := 436689 ), rec( HTML := true , content := "\ ,\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "InDegreeSequence" ), content := 0 , count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "Ref", next := 436734, root := ~, start := 436704, stop := 4367\ 33 ), rec( HTML := true, content := ",\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InDegr\ eeSet" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 436773, root := ~, start := 436748, stop := 436772 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Alt", next := 436791, root := ~, start := 436531, stop := 436790 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Item", next := 436807, root := ~, start := 436477, stop := 436806 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "OutDegreeSet" ), content := 0 , count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "Ref", next := 436903, root := ~, start := 436877, stop := 4369\ 02 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "VertexDegrees", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "C", next := 436956, root := ~, start := 436936, stop := 43695\ 5 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 57 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 57 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 57 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 57 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Alt", next := 436985, root := ~, start := 436833, stop := 436984 ), rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Item", next := 437001, root := ~, start := 436816, stop := 437000 ) ], count := [ "A", 2, 0, 52 ], name := "Row", next := 437014, root := ~, start := 436401, stop := 437013 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 58 ], name := "HorLine", next := 437031, root := ~, start := 437021, stop := 437030 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sLoopy", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "C", next := 437084, root := ~, start := 437070, stop := 437083 ), rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], name := "Item", next := 437100, root := ~, start := 437053, stop := 437099 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "Ref", next := 437155, root := ~, start := 437126, stop := 437154 ), rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 61 ], name := "Item", next := 437171, root := ~, start := 437109, stop := 437170 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0 , count := [ "A", 2, 0, 63 ], mathmode := \ "Text", name := "Ref", next := 437270, root := ~, start := 437241, stop := 4372\ 69 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsLoopy", count := [ "A", 2, 0, 63 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "C", next := 4373\ 17, root := ~, start := 437303, stop := 437316 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 63 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 63 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 63 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Alt", next := 437346, root := ~, start := 437197, stop := 437345 ), rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Item", next := 437362, root := ~, start := 437180, stop := 437361 ) ], count := [ "A", 2, 0, 58 ], name := "Row", next := 437375, root := ~, start := 437039, stop := 437374 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 64 ], name := "HorLine", next := 437392, root := ~, start := 437382, stop := 437391 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sSimpleGraph", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "C", next := 437451, root := ~, start := 437431, stop := 437450 ), rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 65 ], name := "Item", next := 437467, root := ~, start := 437414, stop := 437466 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Prop := "IsSymmetricDigraph" ), content := 0 , count := [ "A", 2, 0, 67 ], mathmode := \ "Text", name := "Ref", next := 437555, root := ~, start := 437523, stop := 4375\ 54 ), rec( HTML := true, content := "\n and not ", count := [ "A", 2, 0, 67 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digrap\ hHasLoops" ), content := 0, count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Ref\ ", next := 437606, root := ~, start := 437577, stop := 437605 ), rec( HTML := true , content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Alt", next := 437623, root := ~, start := 437493, stop := 437622 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\\ n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Prop := "IsSymmetricDigraph" ), content := 0, count := [ "A", 2, 0, 67 ], name := "Ref\ ", next := 437695, root := ~, start := 437663, stop := 437694 ), rec( content := "\ \n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Alt", next := 437712, root := ~, start := 437634, stop := 437711 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Item", next := 437728, root := ~, start := 437476, stop := 437727 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 69 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 69 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 69 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 69 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsSimpleGraph", count :\ = [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "C", next := 437825, root := ~, start := 437805, stop := 43782\ 4 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymm\ etricDigraph" ), content := 0, count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "Ref\ ", next := 437887, root := ~, start := 437855, stop := 437886 ), rec( HTML := true , content := "\ \n and not ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0 , count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "Ref", next := 437938, root := ~, start := 437909, stop := 4379\ 37 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Alt", next := 437956, root := ~, start := 437754, stop := 437955 ), rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Item", next := 437972, root := ~, start := 437737, stop := 437971 ) ], count := [ "A", 2, 0, 64 ], name := "Row", next := 437985, root := ~, start := 437400, stop := 437984 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 70 ], name := "HorLine", next := 438002, root := ~, start := 437992, stop := 438001 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacency", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "C", next := 438057, root := ~, start := 438041, stop := 438056 ), rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], name := "Item", next := 438073, root := ~, start := 438024, stop := 438072 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutNeig\ hboursOfVertex" ), content := 0, count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "Ref", next := 438134, root := ~, start := 438099, stop := 438133 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "InNeighboursOfVertex" ), content := 0 , count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "Ref", next := 438219, root := ~, start := 438185, stop := 4382\ 18 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Alt", next := 438237, root := ~, start := 438145, stop := 438236 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Item", next := 438253, root := ~, start := 438082, stop := 438252 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "OutNeighboursOfVertex" ), content := 0 , count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "Ref", next := 438358, root := ~, start := 438323, stop := 4383\ 57 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Adjacency", count := [ "A", 2, 0, 75 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], mathmode := "\ Text", name := "C", next := 438407, root := ~, start := 438391, stop := 438406\ ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 75 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 75 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 75 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 75 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Alt", next := 438436, root := ~, start := 438279, stop := 438435 ), rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Item", next := 438452, root := ~, start := 438262, stop := 438451 ) ], count := [ "A", 2, 0, 70 ], name := "Row", next := 438465, root := ~, start := 438010, stop := 438464 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 76 ], name := "HorLine", next := 438482, root := ~, start := 438472, stop := 438481 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEdge", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "C", next := 438534, root := ~, start := 438521, stop := 438533 ), rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], name := "Item", next := 438550, root := ~, start := 438504, stop := 438549 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graph and list", Oper := "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "Ref", next := 438632, root := ~, start := 438576, stop := 438631 ), rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 79 ], name := "Item", next := 438648, root := ~, start := 438559, stop := 438647 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The arguments of\n ", count := [ "A", 2, 0, 81 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for d\ igraph and list", Oper := "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 81 ], mathmode := \ "Text", name := "Ref", next := 438789, root := ~, start := 438733, stop := 4387\ 88 ), rec( HTML := true, content := "\n can be a digraph and an edg\ e, or a digraph and two vertices. In the\n first form the operation\ \n ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Label := "for digraph and list", Oper := \ "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 81 ], mathmode := "Text\ ", name := "Ref", next := 438975, root := ~, start := 438919, stop := 438974 ) , rec( HTML := true , content := "\ \n behaves the same as ", count := [ "A", 2, 0, 81 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsEdge", count := [ "A", 2, 0, 81 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "C", next := 4390\ 21, root := ~, start := 439008, stop := 439020 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 81 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 81 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 81 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Alt", next := 439050, root := ~, start := 438674, stop := 439049 ), rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Item", next := 439066, root := ~, start := 438657, stop := 439065 ) ], count := [ "A", 2, 0, 76 ], name := "Row", next := 439079, root := ~, start := 438490, stop := 439078 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 82 ], name := "HorLine", next := 439096, root := ~, start := 439086, stop := 439095 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ irectedEdges", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "C", next := 439155, root := ~, start := 439135, stop := 439154 ), rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 83 ], name := "Item", next := 439171, root := ~, start := 439118, stop := 439170 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "Ref", next := 439223, root := ~, start := 439197, stop := 439222 ), rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 85 ], name := "Item", next := 439239, root := ~, start := 439180, stop := 439238 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphEdges" ), content := 0 , count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "Ref", next := 439335, root := ~, start := 439309, stop := 4393\ 34 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "DirectedEdges", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "C", next := 439388, root := ~, start := 439368, stop := 43938\ 7 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 87 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 87 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 87 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 87 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Alt", next := 439417, root := ~, start := 439265, stop := 439416 ), rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Item", next := 439433, root := ~, start := 439248, stop := 439432 ) ], count := [ "A", 2, 0, 82 ], name := "Row", next := 439446, root := ~, start := 439104, stop := 439445 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 88 ], name := "HorLine", next := 439463, root := ~, start := 439453, stop := 439462 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedEdges", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "C", next := 439524, root := ~, start := 439502, stop := 439523 ), rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 89 ], name := "Item", next := 439540, root := ~, start := 439485, stop := 439539 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 2, 0, 91 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 91 ], name := "Item", next := 439586, root := ~, start := 439549, stop := 439585 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 2, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 93 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 2, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Alt", next := 439745, root := ~, start := 439612, stop := 439744 ), rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Item", next := 439761, root := ~, start := 439595, stop := 439760 ) ], count := [ "A", 2, 0, 88 ], name := "Row", next := 439774, root := ~, start := 439471, stop := 439773 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 94 ], name := "HorLine", next := 439791, root := ~, start := 439781, stop := 439790 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istance", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "C", next := 439845, root := ~, start := 439830, stop := 439844 ), rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], name := "Item", next := 439861, root := ~, start := 439813, stop := 439860 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and two vertices", Oper := "DigraphShortestDistance" ), content := 0, count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "Ref", next := 439978, root := ~, start := 439887, stop := 439977 ), rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 97 ], name := "Item", next := 439994, root := ~, start := 439870, stop := 439993 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for a digraph and two ver\ tices", Oper := "DigraphShortestDistance" ), content := 0, count := [ "A", 2, 0, 99 ], mathmode := \ "Text", name := "Ref", next := 440152, root := ~, start := 440064, stop := 4401\ 51 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 99 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Distance", count := [ "A", 2, 0, 99 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], mathmode := "T\ ext", name := "C", next := 440200, root := ~, start := 440185, stop := 440199 ), rec( HTML := true , content := "\ in ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 99 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 99 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 99 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Alt", next := 440229, root := ~, start := 440020, stop := 440228 ), rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Item", next := 440245, root := ~, start := 440003, stop := 440244 ) ], count := [ "A", 2, 0, 94 ], name := "Row", next := 440258, root := ~, start := 439799, stop := 440257 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 100 ], name := "HorLine", next := 440275, root := ~, start := 440265, stop := 440274 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ iameter", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ] , mathmode := "Text", name := "C", next := 440329, root := ~, start := 440314, stop := 440328 ), rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ], name := "Item", next := 440345, root := ~, start := 440297, stop := 440344 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Diameter" ), content := 0, count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "Ref", next := 440400, root := ~, start := 440371, stop := 440399 ), rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 103 ], name := "Item", next := 440416, root := ~, start := 440354, stop := 440415 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 105 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphDiameter" ), content := 0 , count := [ "A", 2, 0, 105 ] , mathmode := \ "Text", name := "Ref", next := 440515, root := ~, start := 440486, stop := 4405\ 14 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 105 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Diameter", count := [ "A", 2, 0, 105 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], mathmode := \ "Text", name := "C", next := 440563, root := ~, start := 440548, stop := 44056\ 2 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 105 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 105 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 105 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 105 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 105 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 105 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Alt", next := 440592, root := ~, start := 440442, stop := 440591 ), rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Item", next := 440608, root := ~, start := 440425, stop := 440607 ) ], count := [ "A", 2, 0, 100 ], name := "Row", next := 440621, root := ~, start := 440283, stop := 440620 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 106 ], name := "HorLine", next := 440638, root := ~, start := 440628, stop := 440637 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ irth", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ] , mathmode := "Text", name := "C", next := 440689, root := ~, start := 440677, stop := 440688 ), rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ], name := "Item", next := 440705, root := ~, start := 440660, stop := 440704 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ UndirectedGirth" ), content := 0, count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "Ref", next := 440767, root := ~, start := 440731, stop := 440766 ), rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 109 ], name := "Item", next := 440783, root := ~, start := 440714, stop := 440782 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 111 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphUndirectedGirth" ), content := 0 , count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "Ref", next := 440889, root := ~, start := 440853, stop := 4408\ 88 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Girth", count := [ "A", 2, 0, 111 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "C", next := 4409\ 34, root := ~, start := 440922, stop := 440933 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 111 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 111 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 111 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 111 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ except that in the case that the\n argument is a forest, ", count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "infinity", count := [ "A", 2, 0, 111 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "K", next := 441028, root := ~, start := 441013, stop := 44102\ 7 ), rec( HTML := true, content := " is returned by ", count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 111 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ \n whereas ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "-1", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "C", next := 4410\ 84, root := ~, start := 441075, stop := 441083 ), rec( HTML := true, content := "\ is returned by ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 111 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 111 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 111 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 111 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 111 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Alt", next := 441125, root := ~, start := 440809, stop := 441124 ), rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Item", next := 441141, root := ~, start := 440792, stop := 441140 ) ], count := [ "A", 2, 0, 106 ], name := "Row", next := 441154, root := ~, start := 440646, stop := 441153 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 112 ], name := "HorLine", next := 441171, root := ~, start := 441161, stop := 441170 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sConnectedGraph", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "C", next := 441233, root := ~, start := 441210, stop := 441232 ), rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], name := "Item", next := 441249, root := ~, start := 441193, stop := 441248 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsStron\ glyConnectedDigraph" ), content := 0, count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "Ref", next := 441315, root := ~, start := 441275, stop := 441314 ), rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 115 ], name := "Item", next := 441331, root := ~, start := 441258, stop := 441330 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 117 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsStronglyConnectedDigraph\ " ), content := 0, count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "Ref\ ", next := 441441, root := ~, start := 441401, stop := 441440 ), rec( HTML := true , content := "\ behaves the\n same as ", count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsConnectedGraph", count := [ "A", 2, 0, 117 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "C", next := 441497, root := ~, start := 441474, stop := 44149\ 6 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 117 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 117 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 117 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 117 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 117 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Alt", next := 441526, root := ~, start := 441357, stop := 441525 ), rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Item", next := 441542, root := ~, start := 441340, stop := 441541 ) ], count := [ "A", 2, 0, 112 ], name := "Row", next := 441555, root := ~, start := 441179, stop := 441554 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 118 ], name := "HorLine", next := 441572, root := ~, start := 441562, stop := 441571 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBipartite", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 119 ] , mathmode := "Text", name := "C", next := 441629, root := ~, start := 441611, stop := 441628 ), rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 119 ], name := "Item", next := 441645, root := ~, start := 441594, stop := 441644 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "Ref", next := 441703, root := ~, start := 441671, stop := 441702 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n and ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsSymmetricDigraph" ), content := 0 , count := [ "A", 2, 0, 121 ] , mathmode := \ "Text", name := "Ref", next := 441780, root := ~, start := 441748, stop := 4417\ 79 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 121 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Alt", next := 441797, root := ~, start := 441714, stop := 441796 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Item", next := 441813, root := ~, start := 441654, stop := 441812 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 123 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 123 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 123 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 123 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 123 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsBipartite", count := [ \ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ] , mathmode := \ "Text", name := "C", next := 441908, root := ~, start := 441890, stop := 44190\ 7 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 123 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipa\ rtiteDigraph" ), content := 0, count := [ "A", 2, 0, 123 ], mathmode := "Text" , name := "Ref\ ", next := 441970, root := ~, start := 441938, stop := 441969 ), rec( HTML := true , content := "\ and ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Prop := "IsSymmetricDigraph" ), content := 0, count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "Ref", next := 442021, root := ~, start := 441975, stop := 4420\ 20 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Alt", next := 442039, root := ~, start := 441839, stop := 442038 ), rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Item", next := 442055, root := ~, start := 441822, stop := 442054 ) ], count := [ "A", 2, 0, 118 ], name := "Row", next := 442068, root := ~, start := 441580, stop := 442067 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 124 ], name := "HorLine", next := 442085, root := ~, start := 442075, stop := 442084 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sNullGraph", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 125 ] , mathmode := "Text", name := "C", next := 442142, root := ~, start := 442124, stop := 442141 ), rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 125 ], name := "Item", next := 442158, root := ~, start := 442107, stop := 442157 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsNullD\ igraph" ), content := 0, count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "Ref", next := 442211, root := ~, start := 442184, stop := 442210 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n or ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsEmptyDigraph" ), content := 0 , count := [ "A", 2, 0, 127 ] , mathmode := \ "Text", name := "Ref", next := 442283, root := ~, start := 442255, stop := 4422\ 82 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 127 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Alt", next := 442300, root := ~, start := 442222, stop := 442299 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Item", next := 442316, root := ~, start := 442167, stop := 442315 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 129 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 129 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 129 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 129 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 129 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsNullGraph", count := [ \ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "C", next := 442411, root := ~, start := 442393, stop := 44241\ 0 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsEmpt\ yDigraph" ), content := 0, count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "Ref\ ", next := 442469, root := ~, start := 442441, stop := 442468 ), rec( HTML := true , content := "\ and its synonym ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsNullDigraph" ), content := 0 , count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "Ref", next := 442527, root := ~, start := 442486, stop := 4425\ 26 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 129 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Alt", next := 442545, root := ~, start := 442342, stop := 442544 ), rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Item", next := 442561, root := ~, start := 442325, stop := 442560 ) ], count := [ "A", 2, 0, 124 ], name := "Row", next := 442574, root := ~, start := 442093, stop := 442573 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 130 ], name := "HorLine", next := 442591, root := ~, start := 442581, stop := 442590 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCompleteGraph", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "C", next := 442652, root := ~, start := 442630, stop := 442651 ), rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], name := "Item", next := 442668, root := ~, start := 442613, stop := 442667 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsCompl\ eteDigraph" ), content := 0, count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "Ref", next := 442725, root := ~, start := 442694, stop := 442724 ), rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 133 ], name := "Item", next := 442741, root := ~, start := 442677, stop := 442740 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 135 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 135 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 135 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 135 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 135 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsCompleteGraph", count\ := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ] , mathmode := \ "Text", name := "C", next := 442840, root := ~, start := 442818, stop := 44283\ 9 ), rec( HTML := true, content := " is roughly\n equivalent to\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( Prop := "IsCompleteDigraph" ), content := 0, count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "Ref", next := 442921, root := ~, start := 442890, stop := 4429\ 20 ), rec( HTML := true, content := " except that in ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 135 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ \n a complete digraph has ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "no loops", count := [ "A", 2, 0, 135 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "E", next := 442998, root := ~, start := 442983, stop := 44299\ 7 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Alt", next := 443016, root := ~, start := 442767, stop := 443015 ), rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Item", next := 443032, root := ~, start := 442750, stop := 443031 ) ], count := [ "A", 2, 0, 130 ], name := "Row", next := 443045, root := ~, start := 442599, stop := 443044 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 136 ], name := "HorLine", next := 443062, root := ~, start := 443052, stop := 443061 ) ], count := [ "A", 2, 0, 4 ], mathmode := "Text", name := "Table", next := 443075, root := ~, start := 433055, stop := 443074 ), rec( content := "\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "Section", next := 443088, root := ~, start := 432808, stop := 443087 ), rec( content := "\n\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to dete\ rmine regularity properties of graphs\n ", count := [ "A", 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 443191, root := ~, start := 443106, stop := 443190 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], name := "E", next := 443329, root := ~, start := 443219, stop := 443328 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "Alt", next := 443340, root := ~, start := 443196, stop := 443339 ), rec( content := "\n\n ", count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 4 ], name := "HorLine", next := 443386, root := ~, start := 443376, stop := 443385 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 3, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], name := "Item", next := 443456, root := ~, start := 443408, stop := 443455 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 7 ], name := "Item", next := 443516, root := ~, start := 443465, stop := 443515 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 3, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Alt", next := 443597, root := ~, start := 443542, stop := 443596 ), rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Item", next := 443613, root := ~, start := 443525, stop := 443612 ) ], count := [ "A", 3, 0, 4 ], name := "Row", next := 443626, root := ~, start := 443394, stop := 443625 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 10 ], name := "HorLine", next := 443643, root := ~, start := 443633, stop := 443642 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sRegularGraph", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "C", next := 443703, root := ~, start := 443682, stop := 443702 ), rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 11 ], name := "Item", next := 443719, root := ~, start := 443665, stop := 443718 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref", next := 443778, root := ~, start := 443745, stop := 443777 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also\n ", count := [ "A", 3, 0, 13 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInRe\ gularDigraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref\ ", next := 443873, root := ~, start := 443841, stop := 443872 ), rec( HTML := true , content := "\ \n and\n ", count := [ "A", 3, 0, 13 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegularD\ igraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref\ ", next := 443932, root := ~, start := 443902, stop := 443931 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Alt", next := 443950, root := ~, start := 443789, stop := 443949 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Item", next := 443966, root := ~, start := 443728, stop := 443965 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsRegularGraph", count \ := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], mathmode := \ "Text", name := "C", next := 444064, root := ~, start := 444043, stop := 44406\ 3 ), rec( HTML := true, content := " is\n equivalent to ", count := [ "A", 3, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutR\ egularDigraph" ), content := 0, count := [ "A", 3, 0, 15 ], mathmode := "Text" , name := "Ref\ ", next := 444127, root := ~, start := 444094, stop := 444126 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Alt", next := 444145, root := ~, start := 443992, stop := 444144 ), rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Item", next := 444161, root := ~, start := 443975, stop := 444160 ) ], count := [ "A", 3, 0, 10 ], name := "Row", next := 444174, root := ~, start := 443651, stop := 444173 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 16 ], name := "HorLine", next := 444191, root := ~, start := 444181, stop := 444190 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ocalParameters", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "C", next := 444252, root := ~, start := 444230, stop := 444251 ), rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 17 ], name := "Item", next := 444268, root := ~, start := 444213, stop := 444267 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 19 ], name := "Item", next := 444314, root := ~, start := 444277, stop := 444313 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "LocalParameters", count\ := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "C", next := 444412, root := ~, start := 444390, stop := 44441\ 1 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Alt", next := 444525, root := ~, start := 444339, stop := 444524 ), rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Item", next := 444541, root := ~, start := 444323, stop := 444540 ) ], count := [ "A", 3, 0, 16 ], name := "Row", next := 444554, root := ~, start := 444199, stop := 444553 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 22 ], name := "HorLine", next := 444571, root := ~, start := 444561, stop := 444570 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ lobalParameters", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "C", next := 444633, root := ~, start := 444610, stop := 444632 ), rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], name := "Item", next := 444649, root := ~, start := 444593, stop := 444648 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 25 ], name := "Item", next := 444695, root := ~, start := 444658, stop := 444694 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 27 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "GlobalParameters", coun\ t := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "C", next := 444794, root := ~, start := 444771, stop := 44479\ 3 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Alt", next := 444907, root := ~, start := 444720, stop := 444906 ), rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Item", next := 444923, root := ~, start := 444704, stop := 444922 ) ], count := [ "A", 3, 0, 22 ], name := "Row", next := 444936, root := ~, start := 444579, stop := 444935 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 28 ], name := "HorLine", next := 444953, root := ~, start := 444943, stop := 444952 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDistanceRegular", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "C", next := 445016, root := ~, start := 444992, stop := 445015 ), rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], name := "Item", next := 445032, root := ~, start := 444975, stop := 445031 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDista\ nceRegularDigraph" ), content := 0, count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "Ref", next := 445096, root := ~, start := 445058, stop := 445095 ), rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 31 ], name := "Item", next := 445112, root := ~, start := 445041, stop := 445111 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 33 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsDistanceRegular", coun\ t := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "C", next := 445213, root := ~, start := 445189, stop := 44521\ 2 ), rec( HTML := true, content := " is\n equivalent to ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDist\ anceRegularDigraph" ), content := 0, count := [ "A", 3, 0, 33 ], mathmode := "\ Text", name := "Ref", next := 445281, root := ~, start := 445243, stop := 4452\ 80 ), rec( HTML := true, content := ".\n ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Alt", next := 445299, root := ~, start := 445138, stop := 445298 ), rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Item", next := 445315, root := ~, start := 445121, stop := 445314 ) ], count := [ "A", 3, 0, 28 ], name := "Row", next := 445328, root := ~, start := 444961, stop := 445327 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 34 ], name := "HorLine", next := 445345, root := ~, start := 445335, stop := 445344 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedAdjacencyMat", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "C", next := 445412, root := ~, start := 445384, stop := 445411 ), rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], name := "Item", next := 445428, root := ~, start := 445367, stop := 445427 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 37 ], name := "Item", next := 445474, root := ~, start := 445437, stop := 445473 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 39 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "CollapsedAdjacencyMat", coun\ t := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "C", next := 445578, root := ~, start := 445550, stop := 44557\ 7 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Alt", next := 445691, root := ~, start := 445499, stop := 445690 ), rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Item", next := 445707, root := ~, start := 445483, stop := 445706 ) ], count := [ "A", 3, 0, 34 ], name := "Row", next := 445720, root := ~, start := 445353, stop := 445719 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 40 ], name := "HorLine", next := 445737, root := ~, start := 445727, stop := 445736 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ rbitalGraphColadjMats", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "C", next := 445805, root := ~, start := 445776, stop := 445804 ), rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], name := "Item", next := 445821, root := ~, start := 445759, stop := 445820 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 43 ], name := "Item", next := 445867, root := ~, start := 445830, stop := 445866 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 45 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "OrbitalGraphColadjMats" , coun\ t := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "C", next := 445972, root := ~, start := 445943, stop := 44597\ 1 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Alt", next := 446085, root := ~, start := 445892, stop := 446084 ), rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Item", next := 446101, root := ~, start := 445876, stop := 446100 ) ], count := [ "A", 3, 0, 40 ], name := "Row", next := 446114, root := ~, start := 445745, stop := 446113 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 46 ], name := "HorLine", next := 446131, root := ~, start := 446121, stop := 446130 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexTransitiveDRGs", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "C", next := 446197, root := ~, start := 446170, stop := 446196 ), rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], name := "Item", next := 446213, root := ~, start := 446153, stop := 446212 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 49 ], name := "Item", next := 446259, root := ~, start := 446222, stop := 446258 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 51 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 51 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 51 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 51 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "VertexTransitiveDRGs", coun\ t := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "C", next := 446362, root := ~, start := 446335, stop := 44636\ 1 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Alt", next := 446475, root := ~, start := 446284, stop := 446474 ), rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Item", next := 446491, root := ~, start := 446268, stop := 446490 ) ], count := [ "A", 3, 0, 46 ], name := "Row", next := 446504, root := ~, start := 446139, stop := 446503 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 52 ], name := "HorLine", next := 446521, root := ~, start := 446511, stop := 446520 ) ], count := [ "A", 3, 0, 4 ], mathmode := "Text", name := "Table", next := 446535, root := ~, start := 443346, stop := 446534 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "Section", next := 446549, root := ~, start := 443092, stop := 446548 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Some special vert\ ex subsets of a graph\n ", count := [ "A", 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 446636, root := ~, start := 446567, stop := 446635 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], name := "E", next := 446774, root := ~, start := 446664, stop := 446773 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "Alt", next := 446785, root := ~, start := 446641, stop := 446784 ), rec( content := "\n\n ", count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 4 ], name := "HorLine", next := 446831, root := ~, start := 446821, stop := 446830 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 4, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], name := "Item", next := 446901, root := ~, start := 446853, stop := 446900 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 7 ], name := "Item", next := 446961, root := ~, start := 446910, stop := 446960 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 4, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Alt", next := 447042, root := ~, start := 446987, stop := 447041 ), rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Item", next := 447058, root := ~, start := 446970, stop := 447057 ) ], count := [ "A", 4, 0, 4 ], name := "Row", next := 447071, root := ~, start := 446839, stop := 447070 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 10 ], name := "HorLine", next := 447088, root := ~, start := 447078, stop := 447087 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ onnectedComponent", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "C", next := 447152, root := ~, start := 447127, stop := 447151 ), rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], name := "Item", next := 447168, root := ~, start := 447110, stop := 447167 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ConnectedComponent" ), content := 0, count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "Ref", next := 447233, root := ~, start := 447194, stop := 447232 ), rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 13 ], name := "Item", next := 447249, root := ~, start := 447177, stop := 447248 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple di/graph and a vertex,\n ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "ConnectedComponent", count := [ "A", 4, 0, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "C", next := 447390, root := ~, start := 447365, stop := 44738\ 9 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphCon\ nectedComponent" ), content := 0, count := [ "A", 4, 0, 15 ], mathmode := "Tex\ t", name := "Ref", next := 447470, root := ~, start := 447431, stop := 447469 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "DigraphConnectedComponent"\ ), content := 0, count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Ref\ ", next := 447523, root := ~, start := 447484, stop := 447522 ), rec( HTML := true , content := "\ can also be applied to\n non-simple graphs (in the language of ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ).\n ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Alt", next := 447623, root := ~, start := 447275, stop := 447622 ), rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Item", next := 447639, root := ~, start := 447258, stop := 447638 ) ], count := [ "A", 4, 0, 10 ], name := "Row", next := 447652, root := ~, start := 447096, stop := 447651 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 16 ], name := "HorLine", next := 447669, root := ~, start := 447659, stop := 447668 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ onnectedComponents", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "C", next := 447734, root := ~, start := 447708, stop := 447733 ), rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], name := "Item", next := 447750, root := ~, start := 447691, stop := 447749 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "Ref", next := 447816, root := ~, start := 447776, stop := 447815 ), rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 19 ], name := "Item", next := 447832, root := ~, start := 447759, stop := 447831 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple di/graph,\n ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "ConnectedComponents", count := [ "A", 4, 0, 21 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "C", next := 447961, root := ~, start := 447935, stop := 44796\ 0 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 21 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphCon\ nectedComponents" ), content := 0, count := [ "A", 4, 0, 21 ], mathmode := "Te\ xt", name := "Ref", next := 448042, root := ~, start := 448002, stop := 448041\ ), rec( HTML := true, content := ".\n ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digrap\ hConnectedComponents" ), content := 0, count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 448096, root := ~, start := 448056, stop := 4480\ 95 ), rec( HTML := true, content := " can also be applied to\n non-\ simple graphs (in the language of ", count := [ "A", 4, 0, 21 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ).\n ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Alt", next := 448196, root := ~, start := 447858, stop := 448195 ), rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Item", next := 448212, root := ~, start := 447841, stop := 448211 ) ], count := [ "A", 4, 0, 16 ], name := "Row", next := 448225, root := ~, start := 447677, stop := 448224 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 22 ], name := "HorLine", next := 448242, root := ~, start := 448232, stop := 448241 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ icomponents", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "C", next := 448300, root := ~, start := 448281, stop := 448299 ), rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 23 ], name := "Item", next := 448316, root := ~, start := 448264, stop := 448315 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "Ref", next := 448375, root := ~, start := 448342, stop := 448374 ), rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 25 ], name := "Item", next := 448391, root := ~, start := 448325, stop := 448390 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple bipartite di/graph,\n ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Bicomponents", count := [ "A", 4, 0, 27 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "C", next := 448523, root := ~, start := 448504, stop := 44852\ 2 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 27 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphBic\ omponents" ), content := 0, count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Ref\ ", next := 448597, root := ~, start := 448564, stop := 448596 ), rec( HTML := true , content := "\ .\n The definition of ", count := [ "A", 4, 0, 27 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipart\ iteDigraph" ), content := 0, count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Ref\ ", next := 448661, root := ~, start := 448629, stop := 448660 ), rec( HTML := true , content := "\ does not require\n its argument to be simple (in the language of " , count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ), i.e. the\n bipartite graphs in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ are a proper subset of the bipartite\n digraphs in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 4, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Alt", next := 448885, root := ~, start := 448417, stop := 448884 ), rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Item", next := 448901, root := ~, start := 448400, stop := 448900 ) ], count := [ "A", 4, 0, 22 ], name := "Row", next := 448914, root := ~, start := 448250, stop := 448913 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 28 ], name := "HorLine", next := 448931, root := ~, start := 448921, stop := 448930 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceSet", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "C", next := 448988, root := ~, start := 448970, stop := 448987 ), rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], name := "Item", next := 449004, root := ~, start := 448953, stop := 449003 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph, a pos int, and an int", Oper := "DigraphDistanceSet" ), content := 0, count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "Ref", next := 449123, root := ~, start := 449032, stop := 449122 ), rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 31 ], name := "Item", next := 449139, root := ~, start := 449013, stop := 449138 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "DistanceSet", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "C", next := 4492\ 13, root := ~, start := 449195, stop := 449212 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a dig\ raph, a pos int, and an int", Oper := "DigraphDistanceSet" ), content := 0, count := [ "A", 4, 0, 33 ], mathmode := \ "Text", name := "Ref", next := 449345, root := ~, start := 449254, stop := 4493\ 44 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 4, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Alt", next := 449377, root := ~, start := 449165, stop := 449376 ), rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Item", next := 449393, root := ~, start := 449148, stop := 449392 ) ], count := [ "A", 4, 0, 28 ], name := "Row", next := 449406, root := ~, start := 448939, stop := 449405 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 34 ], name := "HorLine", next := 449423, root := ~, start := 449413, stop := 449422 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ayers", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "C", next := 449475, root := ~, start := 449462, stop := 449474 ), rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], name := "Item", next := 449491, root := ~, start := 449445, stop := 449490 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Layers" ), content := 0, count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "Ref", next := 449544, root := ~, start := 449517, stop := 449543 ), rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 37 ], name := "Item", next := 449560, root := ~, start := 449500, stop := 449559 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Layers", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "C", next := 4496\ 29, root := ~, start := 449616, stop := 449628 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 39 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphLay\ ers" ), content := 0, count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "Ref\ ", next := 449697, root := ~, start := 449670, stop := 449696 ), rec( HTML := true , content := "\ in ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Alt", next := 449729, root := ~, start := 449586, stop := 449728 ), rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Item", next := 449745, root := ~, start := 449569, stop := 449744 ) ], count := [ "A", 4, 0, 34 ], name := "Row", next := 449758, root := ~, start := 449431, stop := 449757 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 40 ], name := "HorLine", next := 449775, root := ~, start := 449765, stop := 449774 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ ndependentSet", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "C", next := 449835, root := ~, start := 449814, stop := 449834 ), rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 41 ], name := "Item", next := 449851, root := ~, start := 449797, stop := 449850 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ IndependentSet" ), content := 0, count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "Ref", next := 449912, root := ~, start := 449877, stop := 449911 ), rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 43 ], name := "Item", next := 449928, root := ~, start := 449860, stop := 449927 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "IndependentSet", count := [ "A", 4, 0, 45 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "C", next := 4500\ 05, root := ~, start := 449984, stop := 450004 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphInd\ ependentSet" ), content := 0, count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 450081, root := ~, start := 450046, stop := 450080 ), rec( HTML := true , content := "\ in ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Alt", next := 450113, root := ~, start := 449954, stop := 450112 ), rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Item", next := 450129, root := ~, start := 449937, stop := 450128 ) ], count := [ "A", 4, 0, 40 ], name := "Row", next := 450142, root := ~, start := 449783, stop := 450141 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 46 ], name := "HorLine", next := 450159, root := ~, start := 450149, stop := 450158 ) ], count := [ "A", 4, 0, 4 ], mathmode := "Text", name := "Table", next := 450173, root := ~, start := 446791, stop := 450172 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "Section", next := 450187, root := ~, start := 446553, stop := 450186 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to cons\ truct new graphs from old\n ", count := [ "A", 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 450278, root := ~, start := 450205, stop := 450277 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], name := "E", next := 450416, root := ~, start := 450306, stop := 450415 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "Alt", next := 450427, root := ~, start := 450283, stop := 450426 ), rec( content := "\n\n ", count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 4 ], name := "HorLine", next := 450473, root := ~, start := 450463, stop := 450472 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 5, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], name := "Item", next := 450543, root := ~, start := 450495, stop := 450542 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 7 ], name := "Item", next := 450603, root := ~, start := 450552, stop := 450602 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 5, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Alt", next := 450684, root := ~, start := 450629, stop := 450683 ), rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Item", next := 450700, root := ~, start := 450612, stop := 450699 ) ], count := [ "A", 5, 0, 4 ], name := "Row", next := 450713, root := ~, start := 450481, stop := 450712 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 10 ], name := "HorLine", next := 450730, root := ~, start := 450720, stop := 450729 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nducedSubgraph", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "C", next := 450791, root := ~, start := 450769, stop := 450790 ), rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 11 ], name := "Item", next := 450807, root := ~, start := 450752, stop := 450806 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "Ref", next := 450864, root := ~, start := 450833, stop := 450863 ), rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 13 ], name := "Item", next := 450880, root := ~, start := 450816, stop := 450879 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "InducedSubdigraph" ), content := 0 , count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 450967, root := ~, start := 450936, stop := 4509\ 66 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ does not allow a third\n argument ", count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "G", count := [ "A", 5, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "C", next := 4510\ 34, root := ~, start := 451026, stop := 451033 ), rec( HTML := true, content := "\ specifying a group of automorphisms on the\n resultant graph. This\ may be included in a future release.\n ", count := [ "A", 5, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Alt", next := 451165, root := ~, start := 450906, stop := 451164 ), rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Item", next := 451181, root := ~, start := 450889, stop := 451180 ) ], count := [ "A", 5, 0, 10 ], name := "Row", next := 451194, root := ~, start := 450738, stop := 451193 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 16 ], name := "HorLine", next := 451211, root := ~, start := 451201, stop := 451210 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceSetInduced", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "C", next := 451275, root := ~, start := 451250, stop := 451274 ), rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], name := "Item", next := 451291, root := ~, start := 451233, stop := 451290 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 19 ], name := "Item", next := 451337, root := ~, start := 451300, stop := 451336 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Alt", next := 451496, root := ~, start := 451363, stop := 451495 ), rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Item", next := 451512, root := ~, start := 451346, stop := 451511 ) ], count := [ "A", 5, 0, 16 ], name := "Row", next := 451525, root := ~, start := 451219, stop := 451524 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 22 ], name := "HorLine", next := 451542, root := ~, start := 451532, stop := 451541 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceGraph", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "C", next := 451601, root := ~, start := 451581, stop := 451600 ), rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 23 ], name := "Item", next := 451617, root := ~, start := 451564, stop := 451616 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graph and list", Oper := "DistanceDigraph" ), content := 0, count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "Ref", next := 451701, root := ~, start := 451643, stop := 451700 ), rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 25 ], name := "Item", next := 451717, root := ~, start := 451626, stop := 451716 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "DistanceGraph", count := [ "A", 5, 0, 27 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "C", next := 4517\ 93, root := ~, start := 451773, stop := 451792 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 27 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for digra\ ph and list", Oper := "DistanceDigraph" ), content := 0, count := [ "A", 5, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 451892, root := ~, start := 451834, stop := 4518\ 91 ), rec( HTML := true, content := " in\n ", count := [ "A", 5, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Alt", next := 451936, root := ~, start := 451743, stop := 451935 ), rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Item", next := 451952, root := ~, start := 451726, stop := 451951 ) ], count := [ "A", 5, 0, 22 ], name := "Row", next := 451965, root := ~, start := 451550, stop := 451964 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 28 ], name := "HorLine", next := 451982, root := ~, start := 451972, stop := 451981 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ omplementGraph", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "C", next := 452043, root := ~, start := 452021, stop := 452042 ), rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 29 ], name := "Item", next := 452059, root := ~, start := 452004, stop := 452058 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Dual" ), content := 0, count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "Ref", next := 452110, root := ~, start := 452085, stop := 452109 ), rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 31 ], name := "Item", next := 452126, root := ~, start := 452068, stop := 452125 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "DigraphDual" ), content := 0 , count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "Ref", next := 452207, root := ~, start := 452182, stop := 4522\ 06 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ is equivalent to\n ", count := [ "A", 5, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "ComplementGraph", count := [ "A", 5, 0, 33 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "C", next := 452273, root := ~, start := 452251, stop := 45227\ 2 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 5, 0, 33 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ with the optional argument\n ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "comploops", count := [ "A", 5, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "\ Text", name := "A", next := 452340, root := ~, start := 452324, stop := 452339\ ), rec( HTML := true, content := " set to ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "true", count := [ "A", 5, 0, 33 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "K", next := 4523\ 59, root := ~, start := 452348, stop := 452358 ), rec( HTML := true, content := "\ (loops/nonloops are\n complemented).\n ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Alt", next := 452423, root := ~, start := 452152, stop := 452422 ), rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Item", next := 452439, root := ~, start := 452135, stop := 452438 ) ], count := [ "A", 5, 0, 28 ], name := "Row", next := 452452, root := ~, start := 451990, stop := 452451 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 34 ], name := "HorLine", next := 452469, root := ~, start := 452459, stop := 452468 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ ointGraph", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "C", next := 452525, root := ~, start := 452508, stop := 452524 ), rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], name := "Item", next := 452541, root := ~, start := 452491, stop := 452540 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 37 ], name := "Item", next := 452587, root := ~, start := 452550, stop := 452586 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 39 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Alt", next := 452746, root := ~, start := 452613, stop := 452745 ), rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Item", next := 452762, root := ~, start := 452596, stop := 452761 ) ], count := [ "A", 5, 0, 34 ], name := "Row", next := 452775, root := ~, start := 452477, stop := 452774 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 40 ], name := "HorLine", next := 452792, root := ~, start := 452782, stop := 452791 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ dgeGraph", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "C", next := 452847, root := ~, start := 452831, stop := 452846 ), rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], name := "Item", next := 452863, root := ~, start := 452814, stop := 452862 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeUnd\ irectedDigraph" ), content := 0, count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "Ref", next := 452924, root := ~, start := 452889, stop := 452923 ), rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 43 ], name := "Item", next := 452940, root := ~, start := 452872, stop := 452939 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "EdgeGraph", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "C", next := 4530\ 12, root := ~, start := 452996, stop := 453011 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeUndire\ ctedDigraph" ), content := 0, count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 453088, root := ~, start := 453053, stop := 453087 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Alt", next := 453120, root := ~, start := 452966, stop := 453119 ), rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Item", next := 453136, root := ~, start := 452949, stop := 453135 ) ], count := [ "A", 5, 0, 40 ], name := "Row", next := 453149, root := ~, start := 452800, stop := 453148 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 46 ], name := "HorLine", next := 453166, root := ~, start := 453156, stop := 453165 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ witchedGraph", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "C", next := 453225, root := ~, start := 453205, stop := 453224 ), rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 47 ], name := "Item", next := 453241, root := ~, start := 453188, stop := 453240 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 49 ], name := "Item", next := 453287, root := ~, start := 453250, stop := 453286 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 51 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Alt", next := 453446, root := ~, start := 453313, stop := 453445 ), rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Item", next := 453462, root := ~, start := 453296, stop := 453461 ) ], count := [ "A", 5, 0, 46 ], name := "Row", next := 453475, root := ~, start := 453174, stop := 453474 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 52 ], name := "HorLine", next := 453492, root := ~, start := 453482, stop := 453491 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ nderlyingGraph", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "C", next := 453553, root := ~, start := 453531, stop := 453552 ), rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 53 ], name := "Item", next := 453569, root := ~, start := 453514, stop := 453568 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "Ref", next := 453632, root := ~, start := 453595, stop := 453631 ), rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 55 ], name := "Item", next := 453648, root := ~, start := 453578, stop := 453647 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "UnderlyingGraph", count := [ "A", 5, 0, 57 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "C", next := 4537\ 26, root := ~, start := 453704, stop := 453725 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 57 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 57 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 57 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 57 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphSym\ metricClosure" ), content := 0, count := [ "A", 5, 0, 57 ], mathmode := "Text" , name := "Ref\ ", next := 453804, root := ~, start := 453767, stop := 453803 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Alt", next := 453836, root := ~, start := 453674, stop := 453835 ), rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Item", next := 453852, root := ~, start := 453657, stop := 453851 ) ], count := [ "A", 5, 0, 52 ], name := "Row", next := 453865, root := ~, start := 453500, stop := 453864 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 58 ], name := "HorLine", next := 453882, root := ~, start := 453872, stop := 453881 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ uotientGraph", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "C", next := 453941, root := ~, start := 453921, stop := 453940 ), rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 59 ], name := "Item", next := 453957, root := ~, start := 453904, stop := 453956 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Quotien\ tDigraph" ), content := 0, count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "Ref", next := 454012, root := ~, start := 453983, stop := 454011 ), rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 61 ], name := "Item", next := 454028, root := ~, start := 453966, stop := 454027 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "QuotientDigraph" ), content := 0 , count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "Ref", next := 454113, root := ~, start := 454084, stop := 4541\ 12 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 63 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ takes a partition\n ", count := [ "A", 5, 0, 63 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "p", count := [ "A", 5, 0, 63 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "C", next := 4541\ 66, root := ~, start := 454158, stop := 454165 ), rec( HTML := true, content := "\ of the vertices of the digraph, rather than a set of\n generating \ pairs. Hence, the equivalence might not be\n group-invariant. Als\ o note that in ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ multiple edges may be\n created, whereas in ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 5, 0, 63 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 63 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 63 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 63 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ they are effectively combined into a\n single edge.\n ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Alt", next := 454486, root := ~, start := 454054, stop := 454485 ), rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Item", next := 454502, root := ~, start := 454037, stop := 454501 ) ], count := [ "A", 5, 0, 58 ], name := "Row", next := 454515, root := ~, start := 453890, stop := 454514 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 64 ], name := "HorLine", next := 454532, root := ~, start := 454522, stop := 454531 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ipartiteDouble", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "C", next := 454593, root := ~, start := 454571, stop := 454592 ), rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 65 ], name := "Item", next := 454609, root := ~, start := 454554, stop := 454608 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Biparti\ teDoubleDigraph" ), content := 0, count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "Ref", next := 454671, root := ~, start := 454635, stop := 454670 ), rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 67 ], name := "Item", next := 454687, root := ~, start := 454618, stop := 454686 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "BipartiteDouble", count := [ "A", 5, 0, 69 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "C", next := 4547\ 65, root := ~, start := 454743, stop := 454764 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 69 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 69 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 69 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 69 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "BipartiteD\ oubleDigraph" ), content := 0, count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "Ref\ ", next := 454842, root := ~, start := 454806, stop := 454841 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Alt", next := 454874, root := ~, start := 454713, stop := 454873 ), rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Item", next := 454890, root := ~, start := 454696, stop := 454889 ) ], count := [ "A", 5, 0, 64 ], name := "Row", next := 454903, root := ~, start := 454540, stop := 454902 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 70 ], name := "HorLine", next := 454920, root := ~, start := 454910, stop := 454919 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eodesicsGraph", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "C", next := 454980, root := ~, start := 454959, stop := 454979 ), rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 71 ], name := "Item", next := 454996, root := ~, start := 454942, stop := 454995 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 73 ], name := "Item", next := 455042, root := ~, start := 455005, stop := 455041 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 75 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Alt", next := 455201, root := ~, start := 455068, stop := 455200 ), rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Item", next := 455217, root := ~, start := 455051, stop := 455216 ) ], count := [ "A", 5, 0, 70 ], name := "Row", next := 455230, root := ~, start := 454928, stop := 455229 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 76 ], name := "HorLine", next := 455247, root := ~, start := 455237, stop := 455246 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedIndependentOrbitsGraph", count := [ "A", 5, 0, 77 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "C", next := 455324, root := ~, start := 455286, stop := 455323 ), rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], name := "Item", next := 455340, root := ~, start := 455269, stop := 455339 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 79 ], name := "Item", next := 455386, root := ~, start := 455349, stop := 455385 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 81 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 81 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Alt", next := 455574, root := ~, start := 455412, stop := 455573 ), rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Item", next := 455590, root := ~, start := 455395, stop := 455589 ) ], count := [ "A", 5, 0, 76 ], name := "Row", next := 455603, root := ~, start := 455255, stop := 455602 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 82 ], name := "HorLine", next := 455620, root := ~, start := 455610, stop := 455619 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedCompleteOrbitsGraph", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "C", next := 455694, root := ~, start := 455659, stop := 455693 ), rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], name := "Item", next := 455710, root := ~, start := 455642, stop := 455709 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 85 ], name := "Item", next := 455756, root := ~, start := 455719, stop := 455755 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 87 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Alt", next := 455944, root := ~, start := 455782, stop := 455943 ), rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Item", next := 455960, root := ~, start := 455765, stop := 455959 ) ], count := [ "A", 5, 0, 82 ], name := "Row", next := 455973, root := ~, start := 455628, stop := 455972 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 88 ], name := "HorLine", next := 455990, root := ~, start := 455980, stop := 455989 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ewGroupGraph", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "C", next := 456049, root := ~, start := 456029, stop := 456048 ), rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 89 ], name := "Item", next := 456065, root := ~, start := 456012, stop := 456064 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 91 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 91 ], name := "Item", next := 456111, root := ~, start := 456074, stop := 456110 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 93 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ . However, the same\n results would be possible by calling ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hCopy" ), content := 0, count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "Ref\ ", next := 456306, root := ~, start := 456281, stop := 456305 ), rec( HTML := true , content := "\ on\n the digraph to produce a copy ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "gr", count := [ "A", 5, 0, 93 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "C", next := 4563\ 61, root := ~, start := 456352, stop := 456360 ), rec( HTML := true, content := "\ , and then applying\n ", count := [ "A", 5, 0, 93 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "SetDigraphGroup(gr, G)", count := [ "A", 5, 0, 93 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "C", next := 456422, root := ~, start := 456393, stop := 45642\ 1 ), rec( HTML := true, content := ".\n ", count := [ "A", 5, 0, 93 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Alt", next := 456440, root := ~, start := 456137, stop := 456439 ), rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Item", next := 456456, root := ~, start := 456120, stop := 456455 ) ], count := [ "A", 5, 0, 88 ], name := "Row", next := 456469, root := ~, start := 455998, stop := 456468 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 94 ], name := "HorLine", next := 456486, root := ~, start := 456476, stop := 456485 ) ], count := [ "A", 5, 0, 4 ], mathmode := "Text", name := "Table", next := 456500, root := ~, start := 450433, stop := 456499 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "Section", next := 456514, root := ~, start := 450191, stop := 456513 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Vertex-Colouring \ and Complete Subgraphs\n ", count := [ "A", 6, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 2 ], mathmode := "Text", name := "Heading", next := 456602, root := ~, start := 456532, stop := 456601 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], name := "E", next := 456740, root := ~, start := 456630, stop := 456739 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "Alt", next := 456751, root := ~, start := 456607, stop := 456750 ), rec( content := "\n\n ", count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 4 ], name := "HorLine", next := 456797, root := ~, start := 456787, stop := 456796 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 6, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], name := "Item", next := 456867, root := ~, start := 456819, stop := 456866 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 7 ], name := "Item", next := 456927, root := ~, start := 456876, stop := 456926 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 6, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Alt", next := 457008, root := ~, start := 456953, stop := 457007 ), rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Item", next := 457024, root := ~, start := 456936, stop := 457023 ) ], count := [ "A", 6, 0, 4 ], name := "Row", next := 457037, root := ~, start := 456805, stop := 457036 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 10 ], name := "HorLine", next := 457054, root := ~, start := 457044, stop := 457053 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexColouring", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "C", next := 457115, root := ~, start := 457093, stop := 457114 ), rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 11 ], name := "Item", next := 457131, root := ~, start := 457076, stop := 457130 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "Ref", next := 457215, root := ~, start := 457157, stop := 457214 ), rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 13 ], name := "Item", next := 457231, root := ~, start := 457140, stop := 457230 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexColouring", count := [ "A", 6, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "C", next := 4573\ 09, root := ~, start := 457287, stop := 457308 ), rec( HTML := true, content := "\ in ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGreedyColouring", Label :=\ "for a digraph" ), content := 0, count := [ "A", 6, 0, 15 ], mathmode := "Tex\ t", name := "Ref", next := 457408, root := ~, start := 457338, stop := 457407 ), rec( HTML := true , content := "\ in ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ ,\n except it returns a transformation rather than a list of vertex\ \n colors.\n ", count := [ "A", 6, 0, 15 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Alt", next := 457536, root := ~, start := 457257, stop := 457535 ), rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Item", next := 457552, root := ~, start := 457240, stop := 457551 ) ], count := [ "A", 6, 0, 10 ], name := "Row", next := 457565, root := ~, start := 457062, stop := 457564 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 16 ], name := "HorLine", next := 457582, root := ~, start := 457572, stop := 457581 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteSubgraphs", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "C", next := 457645, root := ~, start := 457621, stop := 457644 ), rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], name := "Item", next := 457661, root := ~, start := 457604, stop := 457660 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ Cliques" ), content := 0, count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "Ref", next := 457715, root := ~, start := 457687, stop := 457714 ), rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 19 ], name := "Item", next := 457731, root := ~, start := 457670, stop := 457730 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The functionality of ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CompleteSubgraphs", count := [ "A", 6, 0, 21 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "C", next := 457832, root := ~, start := 457808, stop := 45783\ 1 ), rec( HTML := true, content := " in ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 6, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is provided\n by ", count := [ "A", 6, 0, 21 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphCliques" ), content := 0 , count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 457899, root := ~, start := 457871, stop := 4578\ 98 ), rec( HTML := true, content := " and ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digrap\ hMaximalCliques" ), content := 0, count := [ "A", 6, 0, 21 ], mathmode := "Tex\ t", name := "Ref", next := 457939, root := ~, start := 457904, stop := 457938 ), rec( HTML := true , content := "\ \n in ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Alt", next := 457983, root := ~, start := 457757, stop := 457982 ), rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Item", next := 457999, root := ~, start := 457740, stop := 457998 ) ], count := [ "A", 6, 0, 16 ], name := "Row", next := 458012, root := ~, start := 457590, stop := 458011 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 22 ], name := "HorLine", next := 458029, root := ~, start := 458019, stop := 458028 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteSubgraphsOfGivenSize", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "C", next := 458103, root := ~, start := 458068, stop := 458102 ), rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], name := "Item", next := 458119, root := ~, start := 458051, stop := 458118 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ Cliques" ), content := 0, count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "Ref", next := 458173, root := ~, start := 458145, stop := 458172 ), rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 25 ], name := "Item", next := 458189, root := ~, start := 458128, stop := 458188 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Much of the functionality of ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CompleteSubgraphsOfGivenSize", count := [ "A", 6, 0, 27 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "C", next := 458309, root := ~, start := 458274, stop := 45830\ 8 ), rec( HTML := true, content := " in\n ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 6, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is provided by ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphCliques" ), content := 0 , count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 458376, root := ~, start := 458348, stop := 4583\ 75 ), rec( HTML := true, content := " and ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digrap\ hMaximalCliques" ), content := 0, count := [ "A", 6, 0, 27 ], mathmode := "Tex\ t", name := "Ref", next := 458430, root := ~, start := 458381, stop := 458429 ), rec( HTML := true , content := "\ in ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , except that the\n functionality relating to vertex weights and pa\ rtial colourings is\n not implemented.\n ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Alt", next := 458586, root := ~, start := 458215, stop := 458585 ), rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Item", next := 458602, root := ~, start := 458198, stop := 458601 ) ], count := [ "A", 6, 0, 22 ], name := "Row", next := 458615, root := ~, start := 458037, stop := 458614 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 28 ], name := "HorLine", next := 458632, root := ~, start := 458622, stop := 458631 ) ], count := [ "A", 6, 0, 4 ], mathmode := "Text", name := "Table", next := 458646, root := ~, start := 456757, stop := 458645 ), rec( content := "\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "Section", next := 458659, root := ~, start := 456518, stop := 458658 ), rec( content := "\n\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Automorphism grou\ ps and isomorphism testing for graphs\n ", count := [ "A", 7, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 2 ], mathmode := "Text", name := "Heading", next := 458762, root := ~, start := 458677, stop := 458761 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], name := "E", next := 458900, root := ~, start := 458790, stop := 458899 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "Alt", next := 458911, root := ~, start := 458767, stop := 458910 ), rec( content := "\n\n ", count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 4 ], name := "HorLine", next := 458957, root := ~, start := 458947, stop := 458956 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 7, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], name := "Item", next := 459027, root := ~, start := 458979, stop := 459026 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 7 ], name := "Item", next := 459087, root := ~, start := 459036, stop := 459086 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 7, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Alt", next := 459168, root := ~, start := 459113, stop := 459167 ), rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Item", next := 459184, root := ~, start := 459096, stop := 459183 ) ], count := [ "A", 7, 0, 4 ], name := "Row", next := 459197, root := ~, start := 458965, stop := 459196 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 10 ], name := "HorLine", next := 459214, root := ~, start := 459204, stop := 459213 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utGroupGraph", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "C", next := 459273, root := ~, start := 459253, stop := 459272 ), rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 11 ], name := "Item", next := 459289, root := ~, start := 459236, stop := 459288 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "Ref", next := 459368, root := ~, start := 459315, stop := 459367 ), rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 13 ], name := "Item", next := 459384, root := ~, start := 459298, stop := 459383 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "AutGroupGraph", count := [ "A", 7, 0, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "C", next := 4594\ 60, root := ~, start := 459440, stop := 459459 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 7, 0, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automorphi\ smGroup", Label := "for a digraph" ), content := 0, count := [ "A", 7, 0, 15 ] , mathmode := \ "Text", name := "Ref", next := 459554, root := ~, start := 459501, stop := 4595\ 53 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Alt", next := 459586, root := ~, start := 459410, stop := 459585 ), rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Item", next := 459602, root := ~, start := 459393, stop := 459601 ) ], count := [ "A", 7, 0, 10 ], name := "Row", next := 459615, root := ~, start := 459222, stop := 459614 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 16 ], name := "HorLine", next := 459632, root := ~, start := 459622, stop := 459631 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphIsomorphism", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "C", next := 459694, root := ~, start := 459671, stop := 459693 ), rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], name := "Item", next := 459710, root := ~, start := 459654, stop := 459709 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "Ref", next := 459790, root := ~, start := 459736, stop := 459789 ), rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 19 ], name := "Item", next := 459806, root := ~, start := 459719, stop := 459805 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "GraphIsomorphism", count := [ "A", 7, 0, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "C", next := 4598\ 85, root := ~, start := 459862, stop := 459884 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for digraphs", Oper := "I\ somorphismDigraphs" ), content := 0, count := [ "A", 7, 0, 21 ], mathmode := "\ Text", name := "Ref", next := 459980, root := ~, start := 459914, stop := 4599\ 79 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n Note however that ", count := [ "A", 7, 0, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "GraphIsomorphism", count := [ "A", 7, 0, 21 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "C", next := 460049, root := ~, start := 460026, stop := 46004\ 8 ), rec( HTML := true, content := "'s optional argument\n ", count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "firstunbindcanon", count := [ "A", 7, 0, 21 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "A", next := 460105, root := ~, start := 460082, stop := 46010\ 4 ), rec( HTML := true, content := " is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Alt", next := 460232, root := ~, start := 459832, stop := 460231 ), rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Item", next := 460248, root := ~, start := 459815, stop := 460247 ) ], count := [ "A", 7, 0, 16 ], name := "Row", next := 460261, root := ~, start := 459640, stop := 460260 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 22 ], name := "HorLine", next := 460278, root := ~, start := 460268, stop := 460277 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sIsomorphicGraph", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "C", next := 460341, root := ~, start := 460317, stop := 460340 ), rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], name := "Item", next := 460357, root := ~, start := 460300, stop := 460356 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "Ref", next := 460437, root := ~, start := 460383, stop := 460436 ), rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 25 ], name := "Item", next := 460453, root := ~, start := 460366, stop := 460452 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "IsIsomorphicGraph", count := [ "A", 7, 0, 27 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "C", next := 4605\ 33, root := ~, start := 460509, stop := 460532 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for digraphs", Oper := "I\ sIsomorphicDigraph" ), content := 0, count := [ "A", 7, 0, 27 ], mathmode := "\ Text", name := "Ref", next := 460628, root := ~, start := 460562, stop := 4606\ 27 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n Note however that ", count := [ "A", 7, 0, 27 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "IsIsomorphicGraph", count := [ "A", 7, 0, 27 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "C", next := 460698, root := ~, start := 460674, stop := 46069\ 7 ), rec( HTML := true, content := "'s optional argument\n ", count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "firstunbindcanon", count := [ "A", 7, 0, 27 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "A", next := 460754, root := ~, start := 460731, stop := 46075\ 3 ), rec( HTML := true, content := " is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Alt", next := 460881, root := ~, start := 460479, stop := 460880 ), rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Item", next := 460897, root := ~, start := 460462, stop := 460896 ) ], count := [ "A", 7, 0, 22 ], name := "Row", next := 460910, root := ~, start := 460286, stop := 460909 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 28 ], name := "HorLine", next := 460927, root := ~, start := 460917, stop := 460926 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphIsomorphismClassRepresentatives", count := [ "A", 7, 0, 29 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "C", next := 461009, root := ~, start := 460966, stop := 461008 ), rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], name := "Item", next := 461025, root := ~, start := 460949, stop := 461024 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 7, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 31 ], name := "Item", next := 461071, root := ~, start := 461034, stop := 461070 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 7, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Alt", next := 461259, root := ~, start := 461097, stop := 461258 ), rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Item", next := 461275, root := ~, start := 461080, stop := 461274 ) ], count := [ "A", 7, 0, 28 ], name := "Row", next := 461288, root := ~, start := 460935, stop := 461287 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 34 ], name := "HorLine", next := 461305, root := ~, start := 461295, stop := 461304 ) ], count := [ "A", 7, 0, 4 ], mathmode := "Text", name := "Table", next := 461319, root := ~, start := 458917, stop := 461318 ), rec( content := "\n ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "Section", next := 461332, root := ~, start := 458663, stop := 461331 ), rec( content := "\n", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "Appendix", next := 461344, root := ~, start := 427693, stop := 461343 ), rec( content := " Grape to Digraphs command map ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( attributes := rec( Databases := "digraphs" ), content := 0, count := [ "Bib", 0, 0, 1 ], mathmode := "Text", name := "Bibliography", next := 461424, root := ~, start := 461387, stop := 461423 ), rec( attributes := rec( ), content := 0, count := [ "Ind", 0, 0, 1 ], mathmode := "Text", name := "TheIndex", next := 461436, root := ~, start := 461425, stop := 461435 ) ], count := [ 0, 0, 0, 1 ], name := "Book", next := 461445, root := ~, start := 1624, stop := 461444 ) ], count := [ 0, 0, 0, 1 ], indent := "", indextext := "< (for digraphs) 4.1
\n= \ (for digraphs) 4.1
\n<\ code class=\"func\">AdjacencyMatrix 5.2-1
\nAdjacencyMatrixMutableCopy 5.2-1 \nAdjacencyMatrixUpperTriangleLineDecoder 9.2-9 \nArticulatio\ nPoints 5.3-14 \ \nAsBinaryRelation 3.2-1 \nAsDigraph 3.2-2 \n\ AsGraph 3.2-4 \ \nAsMonoid 5.5-1 \nAsSemigroup, for a filter\ and a digraph 5.5-1 \ \n    for a filter, semilattice digraph, and two lists 5.5-2 \nAsTransformation 3.2-5\ \nAutomorphismGroup, for a digraph 7.2-2 \n   \ ; for a digraph and a homogeneous list 7.2-5 \n    for a digraph, homogeneo\ us list, and list 7.2-6 \nBipartiteDoubleDigraph 3.3-37 \nBlissAutomorphi\ smGroup, for a digraph 7.2\ -3 \n    for a digraph and homogenous list 7.2-3 \n   \ for a digraph, homogenous list, and list 7.2-3 \nBlissCanonicalDigraph 7.2-9 \nBlissCanonicalLabelling, for a digraph 7.2-7 \n    for a digr\ aph and a list 7.2-8 \ \nBooleanAdjacencyMatrix 5.2-3 \nBooleanAdjacencyMa\ trixMutableCopy 5.2-3 \ \nCayleyDigraph 3.1-12 \nChainDigraph\ 3.5-1 \nCharacteristicPolynomial 5.2-2 \nChromaticNumber 7.3-17 \nCliqueNumber 8.1-5 <\ br />\nCliquesFinder 8.1-2 \nCompleteBipartiteDigra\ ph 3.5-3 \nCompleteDigraph 3.5-2 \nCompleteMultipartiteDigraph 3.5-4 \nCycleDigraph 3.\ 5-5 \nDegreeMatrix 5.2-16 \nDigraph 3.1-7 \n  \ ;  for a group, list, function, and function 3.1-7 \n    for a list an\ d function 3.1-7 \nDigraph6String 9.2-2 \nDigraphAddAllLoops 3.3-38 \nDigraphAddAllLoopsAttr \ 3.3-38 \nDigraphAddEdge, for a digraph \ and an edge 3.3-16 \n&\ nbsp;   for a digraph, source, and range 3.3-16 \nDigraphAddEdge\ Orbit 3.3-17 \n\ DigraphAddEdges 3.3-18 \nDigraphAddVertex <\ a href=\"chap3_mj.html#X83B2506D79453208\">3.3-14 \nDigraphAddVertices, for a digraph and a list of labels 3.3-15 \n   &nbs\ p;for a digraph and an integer 3.\ 3-15 \nDigraphAdjacencyFunction 5.2-4 \n\ DigraphAllSimpleCircuits 5\ .3-21 \nDigraphBicomponents 5.3-13 \nDig\ raphByAdjacencyMatrix 3.1-\ 8 \nDigraphByEdges 3.1-9 \nDigraphByInNe\ ighbors 3.1-11 \ \nDigraphByInNeighbours 3.1-11 \nDigraphByOutNeighb\ oursType 3.1-6 \ \nDigraphCartesianProduct, for a list of digraphs \ 3.3-30 \n  &\ nbsp; for a positive number of digraphs 3.3-30 \nDigraphCartesianProductPr\ ojections 3.3-32 \nDigraphClique 8.1-3 \nDigraphCliques 8.1-4 \nDigraphCliquesReps 8.1\ -4 \nDigraphClosure 3.3-40 \nDigraphColo\ uring, for a digraph and a number of colours 7.3-14 \nDigraphConnectedCo\ mponent 5.3-10 \ \nDigraphConnectedComponents 5.3-9 \nDigraphCopy 3.3-1 \nDigraphCopySameMutability 3.3-1 \nDigraphCore 7.3-18 \nDigraphDegeneracy 5.3-24<\ /a> \nDigraphDegeneracyOrdering 5.3-25 \nDig\ raphDiameter 5.3-1 \nDigraphDirectProduct, for a list of digraphs\ 3.3-31 \n  \   for a positive number of digraphs 3.3-31 \nDigraphDirectProductProj\ ections 3.3-33 \ \nDigraphDisjointUnion, for a list of digraphs 3.3-27 \n  &nbs\ p; for an\n arbitrary number of digraphs 3.3-27 \nDigraphDistanceSet, for a digraph, a pos int, and a list 5.3-5 \n    for a digraph, a pos int,\ and an int 5.3-5 \nDigraphDual 3.3-10 \nDigraphDualAttr 3.3-10 \nD\ igraphEdgeLabel 5.1-10\ \nDigraphEdgeLabels 5.1-11 \nDigraphEdges 5.1-3 \nDigraphEdgeUnion, for a list of digraphs 3.3-28 \n    for a \ positive number of digraphs 3.3-2\ 8 \nDigraphEmbedding 7.3-8 \nDigraphEpim\ orphism 7.3-6 \ \nDigraphFamily 3.1-6 \nDigraphFile 9.2-3 \nD\ igraphFloydWarshall 5.3-16\ \nDigraphFromDigraph6String 9.2-1 \nDi\ graphFromDiSparse6String 9\ .2-1 \nDigraphFromGraph6String 9.2-1 \nD\ igraphFromPlainTextString \ 9.2-11 \nDigraphFromSparse6String 9.2-1 \nDigraphGirth 5.3-6 \ \nDigraphGreedyColouring, for a digraph 7.3-15 \n  &nbs\ p; for a digraph and vertex order 7.3-15 \n    for a digraph and vertex ord\ er function 7.3-15 \n<\ code class=\"func\">DigraphGroup 7.2-10 \nDigraphHasLoops 6.1-1 \n\ DigraphHomomorphism 7.3-2<\ /a> \nDigraphImmutableCopy 3.3-1 \nDigraphIm\ mutableCopyIfImmutable 3.3\ -2 \nDigraphImmutableCopyIfMutable 3.3-2 \nDigraphIndependentSet 8\ .2-2 \nDigraphIndependentSets 8.2-3 \nDi\ graphIndependentSetsReps 8\ .2-3 \nDigraphInEdges 5.1-12 \nDigraphJo\ in, for a list of digraphs 3.3-29 \n    for a positive number of digraphs\ 3.3-29 \nDigraphLayers 5.3\ -23 \nDigraphLongestDistanceFromVertex \ 5.3-4 \nDigraphLongestSimpleCircuit 5.3-22 \nDigraphLoops 5.2-14 \n\ DigraphMaximalClique 8.1-3\ \nDigraphMaximalCliques 8.1-4 \nDigraph\ MaximalCliquesAttr 8.1-4 \nDigraphMaximalCliquesReps 8.1-4 \nDigra\ phMaximalCliquesRepsAttr 8\ .1-4 \nDigraphMaximalIndependentSet 8.2-2 \nDigraphMaximalIndependentSets 8.2-3 \nDigraphMaximalIndependentSetsA\ ttr 8.2-3 \nDigraphMaximalIndependentSetsReps 8.2-3 \nDigraphMaxim\ alIndependentSetsRepsAttr \ 8.2-3 \nDigraphMonomorphism 7.3-4 \nDigr\ aphMutableCopy 3.3-1 \ \nDigraphMutableCopyIfImmutable 3.3-2 \nDigra\ phMutableCopyIfMutable 3.3\ -2 \nDigraphMycielskian 3.3-41 \nDigraph\ MycielskianAttr 3.3-41\ \nDigraphNrConnectedComponents 5.3-9 \nDigr\ aphNrEdges 5.1-4 \nDigraphNrStronglyConnectedComponents 5.3-11 \nD\ igraphNrVertices 5.1-2\ \nDigraphOddGirth 5.3-7 \nDigraphOrbitReps<\ /code> 7.2-12 \nDigraphOrbits 7.2-11 \nDigraphOutEdges 5.1-13 \nDig\ raphPath 5.3-18 \nDigraphPeriod 5.3-15 \nDigraphPlainTextLineDecod\ er 9.2-7 \nDigraphPlainTextLineEncoder 9.2-7 \nDigraphRange\ 5.2-5 \nDigraphReflexiveTransitiveClosure 3.3-12 \nDigraphReflexiveTran\ sitiveClosureAttr 3.3-12 \nDigraphReflexiveTransitiveReduction <\ a href=\"chap3_mj.html#X82AD17517E273600\">3.3-13 \nDigraphReflexiveTransitiveReductionAttr 3.3-13 \nDigraphRemoveAllM\ ultipleEdges 3.3-25 <\ br />\nDigraphRemoveAllMultipleEdgesAttr 3.3-25 \nD\ igraphRemoveEdge, for a digraph and an edge 3.3-21 \n    for a digraph, s\ ource, and range 3.3-21 \nDigraphRemoveEdgeOrbit 3.3-22 \nDigraphRemoveEd\ ges 3.3-23 \nDigraphRemoveLoops 3.3-24 \nDigraphRemoveLoopsAttr 3.3-24 \nDigraphRemoveVertex 3.3-19 \nDigraphRemoveVertices \ 3.3-20 \nDigraphReverse 3.3-\ 9 \nDigraphReverseAttr 3.3-9 \nDigraphRe\ verseEdge, for a digraph and an edge 3.3-26 \n    for a digraph, source, \ and range 3.3-26 \nDigraphReverseEdges, for a digraph and a list of edge\ s 3.3-26 \nDigraphs package overview 1. \nDigraphSchreierVector\ 7.2-13 \nDigraphShortestDistance, for a digraph and a list 5.3-2 \n    fo\ r a digraph and two vertices 5.3-\ 2 \n    for a digraph, a list, and a list 5.3-2 \nDigraphShortestDistances 5.3-3 \nDigraphShortestPath 5.3-19 \n\ DigraphSinks 5.1-5 \nDigraphsMakeDoc 2.4-1 \nDigraphSource \ 5.2-5 \nDigraphSources 5.1-\ 6 \nDigraphsRespectsColouring 7.3-12 \nD\ igraphStabilizer 7.2-14 \nDigraphsTestExtreme 2.5-3 \nDigraphsTest\ Install 2.5-1 \ \nDigraphsTestStandard 2.5-2 \nDigraphStronglyConne\ ctedComponent 5.3-12 \ \nDigraphStronglyConnectedComponents 5.3-11 \nDigraphsUseBliss 7.2-1 \nDigraphsUseNauty 7.2-1 \nDigraphSymmetri\ cClosure 3.3-11 \nDigraphSymmetricClosureAttr 3.3-11 \nDigraphTopo\ logicalSort 5.1-7 \nDigraphTransitiveClosure 3.3-12 \nDigraphTrans\ itiveClosureAttr 3.3-12 \nDigraphTransitiveReduction 3.3-13 \nDigr\ aphTransitiveReductionAttr 3.3-13 \nDigraphUndirectedGirth 5.3-8 \nDigraphVertexLabel 5.1-8<\ /a> \nDigraphVertexLabels 5.1-9 \nDigraphVer\ tices 5.1-1 \n<\ code class=\"func\">DigraphWelshPowellOrder 7.3-16
\nDiSparse6String 9.2-2 \nDistanceDigraph, for digraph and int 3.3-39 \n    for digraph \ and list 3.3-39 \nDotDigraph 9.1-2
\nDotHighlightedDigraph 9.1-6
\nDotPartialOrderDigraph 9.\ 1-4
\nDotPreorderDigraph 9.1-5
\nDotQuas\ iorderDigraph 9.1-5 <\ br />\nDotSymmetricDigraph 9.1-3
\nDotVertexLabelle\ dDigraph 9.1-2
\ \nDoubleDigraph 3.3-36
\nEdgeDigraph 3.3-34
\nEdgeOrbitsDigraph 3.1-10<\ /a>
\nEdgeUndirectedDigraph 3.3-35
\nEmbeddi\ ngsDigraphs 7.3-9 \nEmbeddingsDigraphsRepresentatives 7.3-9
\nEmpt\ yDigraph 3.5-6
\ \nEpimorphismsDigraphs 7.3-7
\nEpimorphismsDigraphs\ Representatives 7.3-7 \
\nGeneralisedPetersenGraph 3.5-9
\nGenerator\ sOfCayleyDigraph 5.4-2\
\nGeneratorsOfEndomorphismMonoid 7.3-13
\nG\ eneratorsOfEndomorphismMonoidAttr 7.3-13
\nGraph 3.2-3
\nGraph6Str\ ing 9.2-2
\nGroupOfCayleyDigraph 5.4-1
\nHamiltonianPath <\ a href=\"chap5_mj.html#X863FDFC4839A3B82\">5.3-26
\nHomomorphismDigraphsFinder 7.3-1
\nHomomorphismsDigraphs <\ a href=\"chap7_mj.html#X8653EDA183E06D05\">7.3-3
\nHomomorphismsDigraphsRepresentatives 7.3-3
\nInDegreeOfVertex 5.2-12 \nInDegrees 5.2-9<\ /a> \nInDegreeSequence 5.2-9 \nInDegreeSet 5.2-9 \nInducedSubdigraph 3.3-3 \nInNeighbors 5.2-7 \nInNeigh\ borsMutableCopy 5.2-7 \ \nInNeighborsOfVertex 5.2-13 \nInNeighbours<\ /code> 5.2-7 \nInNeighboursMutableCopy 5.2-7 \nInNeighboursOfVertex 5.2-13 \nIsAcyclicDigraph 6.3-1 \nIsAntisymmetricDigraph 6.1-2 \n\ IsAntiSymmetricDigraph 6.1\ -2 \nIsAperiodicDigraph 6.3-6 \nIsBiconn\ ectedDigraph 6.3-4 \nIsBipartiteDigraph 6.1-3 \nIsCayleyDigraph 3.1-4 \nIsChainDigraph 6.3-2 \nIsClique 8.1-1 \nIsCompleteBipa\ rtiteDigraph 6.1-4 \nIsCompleteDigraph 6.1-5 \nIsCompleteMultipart\ iteDigraph 6.1-6 \nIsConnectedDigraph 6.3-3 \nIsCycleDigraph 6.3-11 \nIsDigraph 3.1-1<\ /a> \nIsDigraphAutomorphism 7.2-20 \n    for \ a digraph and a transformation or permutation 7.2-20 \nIsDigraphColouring 7.2-21 \n  \ ;  for a transformation 7.2-21 \nIsDigraphCore 6.5-1 \nIsDigrap\ hEdge, for digraph and list 5.1-14 \n    for digraph and two pos ints <\ a href=\"chap5_mj.html#X7BB8ED88835F07B4\">5.1-14 \nIsDigraphEmbedding 7\ .3-11 \n    for digraphs and a permutation or t\ ransformation 7.3-11 \ \nIsDigraphEndomorphism 7.3-10 \n    for digraphs an\ d a permutation or transformation 7.3-10 \nIsDigraphEpimorphism 7.3-10 \n   &nb\ sp;for digraphs and a permutation or transformation 7.3-10 \nIsDigraphHomomorph\ ism 7.3-10 \n&n\ bsp;   for digraphs and a permutation or transformation 7.3-10 \nIsDigraphIsomorphism 7.2-\ 20 \n    for digraphs and transformation or per\ mutation 7.2-20 \nIsDigraphMonomorphism 7.3-10 \n    for digraphs and a pe\ rmutation or transformation 7.3-1\ 0 \nIsDigraphWithAdjacencyFunction 3.1-5 \nIsDirectedTree 6.3-7 \nIsDistanceRegularDigraph 6.2-4 \nIsEdgeT\ ransitive 6.5-2 \nIsEmptyDigraph 6.1-7 \nIsEquivalenceDigraph 6.1-8 \nIsEulerianDigraph 6.3-9 \nIsFunctionalDigraph 6.1-9 \nI\ sHamiltonianDigraph 6.3-10\ \nIsImmutableDigraph 3.1-3 \nIsIndepend\ entSet 8.2-1 \n\ IsInRegularDigraph 6.2-1 \nIsIsomorphicDigraph, for digraphs 7.2-15 \n    for digraphs and homogeneous lists 7.2-16 \nIsJoinS\ emilatticeDigraph 6.1-17 \nIsLatticeDigraph 6.1-17 \nIsMatching 5.1-15 \nIsMaximalClique 8.1-1 \nIsMaximalIndependentSet 8.2-1 \nIsMaximalMatching 5.1-1\ 5 \nIsMeetSemilatticeDigraph 6.1-17 \nI\ sMultiDigraph 6.1-10 \ \nIsMutableDigraph 3.1-2 \nIsNullDigraph 6.1-7 \nIsomorphismDigraphs, for digraphs 7.2-17 \n    for digraphs and\ homogeneous lists 7.2-18 \nIsOuterPlanarDigraph 6.4-2 \nIsOutRegularDigr\ aph 6.2-2 \nIsPartialOrderDigraph 6.1-16 \nIsPerfectMatching 5.1-15 \nIsPlanarDigraph \ 6.4-1 \nIsPreorderDigraph 6.1-15 \nIsQua\ siorderDigraph 6.1-15 \ \nIsReachable 5.3-17 \nIsReflexiveDigraph 6.1-11 \nIsRegularDigraph 6.2-3 \nIsStronglyConnectedDigraph \ 6.3-5 \nIsSubdigraph 4.1-1<\ /a> \nIsSymmetricDigraph 6.1-12 \nIsTourname\ nt 6.1-13 \nIsTransitiveDigraph 6.1-14 \nIsUndirectedForest\ 6.3-8 \nIsUndirectedSpanningForest 4.1-2 \nIsUndirectedSpanningTree4.1-2 \nIsUndirectedTree 6.3-8 \nIsVertexTransitive 6.5-3 \nIt\ eratorFromDigraphFile 9.2-\ 6 \nIteratorOfPaths 5.3-20 \nJohnsonDigr\ aph 3.5-7 \nKuratowskiOuterPlanarSubdigraph 5.6-2 \nKuratowskiPlan\ arSubdigraph 5.6-1 \nLaplacianMatrix 5.2-17 \nLineDigraph \ 3.3-34 \nLineUndirectedDigraph 3.3-35 \nMaximalAntiSymmetricSubdigraph 3.3-6 \nMaximalAntiSymmetricSubdigraphAttr 3.3-6 \nMaximalSymmetricSu\ bdigraph 3.3-5 \ \nMaximalSymmetricSubdigraphAttr 3.3-5 \nMaximalSym\ metricSubdigraphWithoutLoops 3.3-5 \nMaximalSymmetricSubdigraphWithoutLo\ opsAttr 3.3-5 \ \nMonomorphismsDigraphs 7.3-5 \nMonomorphismsDigrap\ hsRepresentatives 7.3-5 \nNautyAutomorphismGroup 7.2-4 \nNautyCano\ nicalDigraph 7.2-9 \nNautyCanonicalLabelling, for a digraph 7.2-7 \n   &\ nbsp;for a digraph and a list 7.2\ -8 \nNrSpanningTrees 5.3-27 \nNullDigrap\ h 3.5-6 \nOnDigraphs, for a digraph and a perm 7.1-1 \n    for a di\ graph and a transformation 7.1-1<\ /a> \nOnMultiDigraphs 7.1-2 \n    for a digra\ ph, perm, and perm 7.1-2 \nOutDegreeOfVertex 5.2-10 \nOutDegrees \ 5.2-8 \nOutDegreeSequence 5.\ 2-8 \nOutDegreeSet 5.2-8 \nOuterPlanarEm\ bedding 5.6-4 \ \nOutNeighbors 5.2-6 \nOutNeighborsMutableCopy 5.2-6 \nOutNeighborsOfVertex 5.2-11 \nOutNeighbours 5.2-6 \nOutN\ eighboursMutableCopy 5.2-6\ \nOutNeighboursOfVertex 5.2-11 \nPartia\ lOrderDigraphJoinOfVertices 5.2-15 \nPartialOrderDigraphMeetOfVertices<\ /code> 5.2-15 \nPetersenGraph 3.5-8 \nPlainTextString 9.2-11 \nPlan\ arEmbedding 5.6-3 \nQuotientDigraph 3.3-8 \nRandomDigraph \ 3.4-1 \nRandomLattice 3.4-4<\ /a> \nRandomMultiDigraph 3.4-2 \nRandomTourn\ ament 3.4-3 \n<\ code class=\"func\">ReadDigraphs 9.2-4 \nReadDIMACSDigraph 9.2-13 \nReadPlainTextDigraph 9.\ 2-12 \nReducedDigraph 3.3-4 \nReducedDig\ raphAttr 3.3-4 \ \nRepresentativeOutNeighbours 7.2-19 \nSemigroupOfC\ ayleyDigraph 5.4-1 \nSetDigraphEdgeLabel 5.1-10 \nSetDigraphEdgeLa\ bels, for a digraph and a function 5.1-11 \n    for a digraph and a list \ of lists 5.1-11 \nSetDigraphVertexLabel 5.1-8 \nSetDigraphVertexLabels 5.1-9 \nSparse6String \ 9.2-2 \nSplash 9.1-1 \nSubdigraphHomeomo\ rphicToK23 5.6-5 \nSubdigraphHomeomorphicToK33 5.6-5 \nSubdigraphH\ omeomorphicToK4 5.6-5 \ \nTCodeDecoder 9.2-10 \nTournamentLineDecode\ r 9.2-8 \nUndirectedSpanningForest 3.3-7 \nUndirectedSpanningFores\ tAttr 3.3-7 \n<\ code class=\"func\">UndirectedSpanningTree 3.3-7
\nUndirectedSpanningTre\ eAttr 3.3-7
\n<\ code class=\"func\">WriteDigraphs 9.2-5
\nWriteDIMACSDigraph 9.2-13
\nWritePlainTextDigraph \ 9.2-12
\n", input := "Digraphs\">\n http://www.maths.qmul.ac.uk/~leonard/grape/Grape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suite<\ /URL>edge-addition-planarity-suite\">\n h\ ttp://pallini.di.uniroma1.it/nauty\">\n https://github.com/sebasguts/NautyTracesInterfa\ ceNautyTracesInterface\">\n \n http://gap-pac\ kages.github.io/io/\nIO\ \">\n \n http://http://\ gap-packages.github.io/orb/\n Orb\">\n \n\n\n\n\n\n\n\n ]>\n\n\n\n &Digraphs;\n Version &VERS\ ION;\n\n Jan De Beule\n jdebeule@cage.ugent.be\n http://homepages.vub.ac.be/~jdbeule\n \n\n Julius Jonušas\n julius.jonusas@tuwien.ac.at\n http://julius.jonusas.work\n \n\n <\ Author>James D. Mitchell\n jdm3@st-and.ac.uk\n \ http://www-groups.mcs.st-andrews.ac.uk/~jamesm\n \n\n Michael Torpey\n mct25@st-andrews.ac.uk\n https://mtorpey.github.io\n \n\n Wilf A. Wilson\ \n gap@wilf-wilson.net\n http://wilf.me\n \n\n Stuart Burrell\n\n Reinis Cirp\ ons\n\n Luke Elliott\n\n Max Horn\ \n\n Christopher Jefferson\n\n Markus Pfeiffer\n\n Chris Russell\n\n Finn Smith\n\n \ Murray White\n\n ©right; ©RIGHTYEARS; by\ \n Jan De Beule, Julius Jonušas, James D. Mitchell, Michael\n Torpe\ y, Wilf A. Wilson et al.
\n\n &Digraphs; is free software; you can redis\ tribute it and/or modify it\n under the terms of the \n http://www.fsf.org/licenses/gpl.html as publish\ ed by the Free Software\n Foundation; either version 3 of the License, or (\ at your option) any later\n version.\n \n\n \n T\ he &Digraphs; package is a &GAP; package containing methods for graphs,\n d\ igraphs, and multidigraphs.\n \n\n \n We wou\ ld like to thank Christopher Jefferson for his help in including \n &bliss;\ in &Digraphs;.\n\n This package's methods for computing digraph homomorphi\ sms are based on work\n by Max Neunhöffer, and independently Artur Sch\ äfer.\n \n\n\n\n\n\n\n<\ Body>\n \n \n The &Digra\ phs; package\n \n &Digraphs; package overv\ iew\n\n
Introduction\n\n This is the manua\ l for version &VERSION; of the &Digraphs; package.\n This package was develop\ ed at the University of St Andrews by:\n \n Jan De Beule,\n Julius Jonušas,\n James D. Mitchell,\ \n Michael C. Torpey, and\n Wilf A. Wilson.\n \ \n Additional contributions were made by:\n \n Stuart\ Burrell,\n Reinis Cirpons,\n Luke Elliott,\n Max Horn,\n Christopher Jefferson,\n \ Markus Pfeiffer,\n Chris Russell,\n Fin\ n Smith, and\n Murray Whyte.\n \n\n The &Digra\ phs; package contains a variety of methods for efficiently creating\n and sto\ ring mutable and immutable digraphs and computing information about\n them.\ Full explanations of all the functions contained in the package are\n pro\ vided below.
\n\n If the &Grape; package is available, it will be loaded \ automatically.\n Digraphs created with the &Digraphs; package can be converte\ d to &Grape;\n graphs with , and conversely &Grape; grap\ hs can be\n converted to &Digraphs; objects with . &G\ rape; is not\n required for &Digraphs; to run.
\n\n The &bliss; tool is included in this package. It\n is an open-source tool f\ or computing automorphism groups and canonical forms\n of graphs, written by \ Tommi Junttila and Petteri Kaski. Several of the methods\n in the &Digraphs; \ package rely on &bliss;. If the &NautyTracesInterface;\n package for GAP is a\ vailable then it is also possible to use &nauty; for\ computing automorphism groups and canonical forms\n in &Digraphs;. See Secti\ on \n for more details.\ \n
\n\n From version 1.0.0 of this package, digraphs can be either mut\ able or immutable.\n Mutable digraphs can be changed in-place by many of the \ methods in the\n package, which avoids unnecessary copying. Immutable digraph\ s cannot be\n changed in-place, but their advantage is that the value of an a\ ttribute of\n an immutable digraph is only ever computed once. Mutable digrap\ hs can be\n converted into immutable digraphs in-place using . One of the motivations for introducing m\ utable\n digraphs in version 1.0.0 was that in practice the authors often wan\ ted\n to create a digraph 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, with each copy being discarded almost immediately.\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 desirable).\n\n Definitions\n\n For the purpose\ s of this package and its documentation, the following\n definitions apply:\
\n\n A digraph E=(E^0,E^1,r,s), also known as a direc\ ted\n graph, consists of a set of vertices E^0 and a set of ed\ ges\n E^1 together with functions s, r: E^1 \\to E^0, called \ the\n source and range, respectively.\n\n The source and r\ ange of an edge is respectively the values of s, r at\n that edge.\n\ \n An edge is called a loop if its source and range are the same.\n\ \n A digraph is called a multidigraph if there exist two or more edg\ es\n with the same source and the same range.
\n\n A directed wal\ k on a digraph is a sequence of alternating vertices\n and edges (v_\ 1, e_1, v_2, e_2, ..., e_{n-1}, v_n) such that each edge\n e_i h\ as source v_i and range v_{i+1}. A directed\n path i\ s a directed walk where no vertex (and hence no edge) is\n repeated. A \ directed circuit is a directed walk where v_1 =\n v_n, and a <\ E>directed cycle is a directed circuit where where no\n vertex is repea\ ted, except for v_1 = v_n.
\n\n The length of a directed\ walk (v_1, e_1, v_2, e_2, ..., e_{n-1},\n v_n) is equal to n-1\ , the number of edges it contains. A\n directed walk (or path) (v_1\ , e_1, v_2, e_2, ..., e_{n-1}, v_n) is\n sometimes called a directed wa\ lk (or path) from vertex v_1 to\n vertex v_n. A dir\ ected walk of zero length, i.e. a sequence\n (v) for some vertex \ v, is called trivial. A trivial\n directed walk is considered to\ be both a circuit and a cycle, as is the\n empty directed walk (). \ A simple circuit is another name for a\n non-trivial and non-empty d\ irected cycle.
\n\n \n
\n\n \n \n\n
\n Optional package dependencies\n\n \n\n The &Digraphs; package is written in &GAP; and C code and \ requires the\n &IO; package. The &IO; package is used to read and write tr\ ansformations,\n partial permutations, and bipartitions to a file.
\n\n\ \n \n\ \n
\n Compiling th\ e kernel module\n The &Digraphs; package has a &GAP; kernel compo\ nent in\n C which should be compiled. This component contains certain low-l\ evel\n functions required by &Digraphs;.\n
\n\n It is not possibl\ e to use the &Digraphs; package without compiling it.
\n\n To compile th\ e kernel component inside the pkg/&ARCHIVENAME;\n directory, type\n<\ Listing>\n\n
\n\n If you in\ stalled the package in another 'pkg' directory than the standard 'pkg'\n di\ rectory in your &GAP; installation, then you have to do two things. Firstly\n \ during compilation you have to use the option '--with-gaproot=PATH' of the\ \n 'configure' script where 'PATH' is a path to the main GAP root directory\ (if\n not given the default '../..' is assumed).
\n\n If you install\ ed &GAP; on several architectures, you must execute the\n configure/make st\ ep for each of the architectures. You can either do this\n immediately afte\ r configuring and compiling GAP itself on this architecture, or\n alternati\ vely set the environment\n variable 'CONFIGNAME' to the name of the configu\ ration you used when compiling\n GAP before running './configure'. Note ho\ wever 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 compi\ lation. For example you have to specify 32-bit\n or 64-bit mode correctly!\ \n
\n\n \n \n\n
\n\ Rebuilding the documentation\n The &Digraphs; packag\ e comes complete with pdf, html, and text versions of\n the documentation. \ However, you might find it necessary, at some point, to\n rebuild the docum\ entation. To rebuild the documentation, please use the\n function .\n\n \n \n Nothing\n \n This function s\ hould be called with no argument to compile the &Digraphs;\n documentation.\ \n \n\n\n
\n\n \n \n\n
\n Testing your installation\n\n In this section we describe how to test that &Digraphs; is working a\ s\n intended. To test that &Digraphs; is installed correctly use\n or for more extensive tests use\n .
\n\n If something goes wrong, then please \ review the instructions in Section \ and ensure that &Digraphs; has been\n properly installed. If you continue\ having problems, please use the https://gi\ thub.com/gap-packages/Digraphs/issues\n to report the issues you are \ having.\n\n \n \n\ true or false.\n \n This f\ unction can be called without arguments to test your installation of\n &Dig\ raphs; is working correctly. These tests should take no more than a\n few s\ econds to complete. To test more comprehensively that\n &Digraphs; is work\ ing correctly, use .\n \n\n\n \n \n true or false.\n \n \ This function can be called without arguments to test all of the methods\n \ included in &Digraphs;.\n These tests should take less than a minute to com\ plete.
\n\n To quickly test that &Digraphs; is installed correctly use \ . For a more thorough test, use .\n \n\n\n \ \n \n Noth\ ing.\n \n This function should be called with no ar\ gument. It executes a series of\n very demanding tests, which measure the p\ erformance of a variety of\n functions on large examples. These tests take\ a long time to\n complete, at least several minutes.
\n\n For these\ tests to complete, the digraphs library digraphs-lib must\n be down\ loaded and placed in the digraphs directory in a subfolder\n named <\ F>digraphs-lib.\n This library can be found on the &Digraphs; website.\ \n \n\n\n\n
\n\n \n \n\n\n\n \ \n Creating \ digraphs\n\n In this chapter we describe how to create digraphs.
\n\n
Creating digraphs\n \n \n \n Every digraph in\ &Digraphs; belongs to the category\n IsDigraph. Some basic at\ tributes and operations for digraphs\n are ,\ , and\n .\n \n\n\n \n \n \n IsMutableDigraph is a syno\ nym for and . A mutable digraph may be changed\n in-place by methods in the \ &Digraphs; package, and is not attribute-storing\n – see .\n
\n\n A mutable digraph ma\ y be converted into an immutable attribute-storing\n digraph by calling on the\n digraph.\n \n\n\n \n \n \n IsImmutableDigraph is a subcat\ egory of .\n Digraphs that lie in IsImmutableD\ igraph are immutable and\n attribute-storing. In particular, they lie i\ n\n .
\n\n A \ mutable digraph may be converted to an immutable\n digraph that lies in the\ category IsImmutableDigraph by calling\n on the digraph.
\n\n The operation can be used\n to construct a mutable copy of an immutab\ le digraph. It is however not\n possible to convert an immutable digraph in\ to a mutable digraph in-place.\n \n\n\n \n \n \n \ IsCayleyDigraph is a subcategory of IsDigraph. Digraphs that\n\ are Cayley digraphs of a group and that are constructed by the operation\n\ are constructed in this category, and are\n \ always immutable.\n \n\n\n \n \ \n \n IsDigraphWithAdjacencyFunction is a subcategory of IsDigrap\ h.\n Digraphs that are created using an adjacency function are c\ onstructed\n in this category.\n \n\n\n \n \n \n \n The type of all digraphs is DigraphByOutNei\ ghboursType.\n The family of all digraphs is DigraphFamily.\n <\ /Description>\n\n\n \n \n \n \n A digraph.\n \n If the optional first argument filt is present, then this shou\ ld\n specify the category or representation the digraph being created will\ \n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n \ If the optional first argument filt is not present, then is used by default.\n \n for a\ list (i.e. an adjacency list)\n \n if obj is \ a list of lists of positive integers in\n the range from 1 to Length(obj), then this function\n returns the digraph with \ vertices\n E ^ 0 = [1 .. Length(obj)],\n an\ d edges corresponding to the entries of obj.
\n\n More preci\ sely, there is an edge from vertex i to j if and\n only i\ f j is in obj[i]; the source of this edge is\n \ i and the range is j. If j occurs in\n obj[\ i] with multiplicity k, then there are k\n edges from\ i to j.\n \n\n for three lists\n \ \n if obj is a duplicate-free list, and source\ and\n range are lists of equal length consisting of positive in\ tegers\n in the list [1 .. Length(obj)], then this functi\ on\n returns a digraph with vertices\n E ^ 0 = [1 .. L\ ength(obj)], and\n Length(source) edges. For \ each i in\n [1 .. Length(source)] there exists an \ edge with source\n vertex source[i] and range vertex range[i]\ .\n See and .\n
\n\n The vertices of the digraph will be labelled by\ the elements of\n obj.\n \n\n for an int\ eger, and two lists\n \n if obj is an integer,\ and source and\n range are lists of equal length consist\ ing of positive integers\n in the list [1 .. obj], then t\ his function\n returns a digraph with vertices\n E ^ 0 = \ [1 .. obj], and\n Length(source) edges. Fo\ r each i in\n [1 .. Length(source)] there exists a\ n edge with source\n vertex source[i] and range vertex range[\ i].\n See and .\n \n\n for a list and a function\n <\ Item>\n if list is a list and func is a function taking 2\ \n arguments that are elements of list, and func returns\ \n true or false, then this operation creates a digraph w\ ith\n vertices [1 .. Length(list)] and an edge from verte\ x\n i to vertex j if and only if\n func(\ list[i], list[j]) returns true.\n \n\n \ for a group, a list, and two functions\n \n \ The arguments will be G, list, act, adj.
\n\n Let G b\ e a group acting on the objects in list via\n the action act<\ /A>, and let adj be a function taking\n two objects from list\ as arguments and returning true or\n false. The func\ tion adj will describe the adjacency\n between objects from l\ ist, which is invariant under the action of\n G. This varian\ t of the constructor returns a digraph with\n vertices the objects of <\ A>list and directed edges [x, y]\n when f(x, y) is true.
\n\n The action of the group G on the objects in \ list is stored\n in the attribute , \ and is used to speed up\n operations like .\n \n\n for a Grape package graph\n \n if obj is a &Grape; package graph (i.e. a record for which \ the\n function IsGraph returns true), then this function \ returns\n a digraph isomorphic to obj.\n \n\n <\ Mark>for a binary relation\n \n if obj is a bi\ nary relation on the points [1 .. n] for\n some posititve intege\ r n, then this function returns the digraph\n defined by obj<\ /A>. Specifically, this function returns a digraph\n which has n\ vertices, and which has an edge with source i\n and range j<\ /C> if and only if [i,j] is a pair in\n the binary relation o\ bj.\n \n \n\n gr := Digr\ aph([\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 := Graph(SymmetricGroup\ (5), [[1, 2]], OnSets,\n> function(x, y) return Intersection(x, y) = []; end);\ ;\ngap> Digraph(Petersen);\n\nga\ p> gr := Digraph([1 .. 10], ReturnTrue);\n]]>\n\n The next example illustrates the uses of the\ fourth and fifth variants of\n this constructor. The resulting digraph i\ s a strongly regular graph, and\n it is actually the point graph of the v\ an Lint-Schrijver partial\n geometry, . The algebra\ ic description\n is taken from the seminal paper of Calderbank and Kantor\ .\n 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 -> 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)^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)^47, Z(3^4)^48, \n Z(3^4)^5\ 3, 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> retur\ n x - y in omega;\n> end;\nfunction( x, y ) ... end\ngap> digraph := Digraph(A\ sList(f), adj);\n\ngap> group \ := Group(Z(3));\n\ngap> act := \\*;\n\ngap> digraph := Digraph(group, List(f), act, adj);\n\n]]>\n \n\n\n \ \n \n A digraph.\n \n If the optional fi\ rst argument filt is present, then this should\n specify the categor\ y or representation the digraph being created will\n belong to. For example\ , if filt is ,\n then the digraph be\ ing created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first argu\ ment filt is not present, then \ is used by default.
\n\n If list is the adjacency matrix of a dig\ raph in the sense of\n , then this operation\ returns the digraph\n which is defined by list.
\n\n Alterna\ tively, if list is a square boolean matrix, then this\n operation re\ turns the digraph with Length(list)\n vertices which h\ as the edge [i,j] if and only if\n list[i][j] is tr\ ue.
\n\n 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 := DigraphByAdjacencyM\ atrix(IsMutableDigraph, \n> [[true, false, true],\n> [false, false, true],\n>\ [false, true, false]]);\n\n]]>\n \n\n\n \n \n A digraph.\n \ \n If the optional first argument filt is present, the\ n this should\n specify the category or representation the digraph being cr\ eated will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt<\ /A> is , then the digraph will be immut\ able. \n If the optional first argument filt is not present, then is used by default.
\n\n If lis\ t is list of pairs of positive integers, then this function\n returns t\ he digraph with the minimum number of vertices m such that\n its lis\ t equal list.
\n\n If the optional second argument n is a \ positive integer with\n n >= m (with m defined as abov\ e),\n then this function returns the digraph with n vertices and lis\ t\n list.
\n\n See .\n\n 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]]>\n \n\n\n \n \n \n An immutable digraph.\n \n \ \n If G is a permutation group, edges is an edge\ or list of\n edges, and n is a non-negative integer such that G<\ /A> fixes\n [1 .. n] setwise, then\n this operation return\ s an immutable digraph with n vertices and the union\n of the orbits\ of the edges in edges under the action of the\n permutation group\ G. An edge in this context is simply a pair of\n positive integers\ .\n
\n\n If the optional third argument n is not present, the\ n the largest\n moved point of the permutation group G is used by de\ fault.
\n\n 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> Represent\ ativeOutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ ] ]\n]]>\n \n\n\n \n \n \n A digraph.\n \n If the optiona\ l first argument filt is present, then this should\n specify the categ\ ory or representation the digraph being created will\n belong to. For example\ , if filt is ,\n then the digraph bein\ g created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first argument <\ A>filt is not present, then is used \ by default.
\n\n If list is a list of lists of positive integers li\ st the range\n [1 .. Length(list)], then this function returns \ the\n digraph with vertices E^0=[1 .. Length(list)], and\ \n edges corresponding to the entries of list. More precisely, there i\ s an\n edge with source vertex i and range vertex j if i\ is list\n list[j].
\n\n If i occurs list lis\ t[j] with multiplicity k,\n then there are k multiple e\ dges from i to j.
\n\n See .\n\ \n 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, 1\ 0], [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 := DigraphByIn\ Neighbours(IsMutableDigraph, \n> [[2, 3, 2], [1]\ , [1, 2, 3]]);\n\n]]>\ \n \n\n\n \n \n An immutable digraph.\n \n Let G be any group and let gens be a list of ele\ ments of\n G. This operation returns an immutable digraph that corre\ sponds to\n the Cayley graph of G with respect\n gens. The\ vertices are the elements of G. There exists an edge\n from the ver\ tex u to the vertex v if and only if there exists\n a generat\ or g in gens such that x * g = y.
\n\n \n\n If the op\ tional second argument gens is not present, then the\n generators of\ G are used by default.
\n\n The digraph created by this operatio\ n belongs to the category , the group G can be recovered from the\n digraph using , and the generators\n gens can be obtained using .
\n\n Note that this function can only\ return an immutable digraph.\n\n G := DihedralGro\ up(8);\n\ngap> CayleyDigraph(G);\n\ngap> G := DihedralGroup(IsPermGroup, \ 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\n\ngap> digraph := CayleyDigraph(G, [()]);\n\ngap> GroupOfCayleyDigraph(digraph) = G;\nt\ rue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]]]>\n \n\n\n
\n\n
Changing represen\ tations\n \n \n A binary relation.\n \n If <\ A>digraph is a digraph with a positive number of vertices\n n, a\ nd no multiple edges, then this operation returns a binary\n relation on th\ e points [1..n].\n The pair [i,j] is in the binary relation i\ f and only if\n [i,j] is an edge in digraph.\n
\n D := Digraph([[3, 2], [1, 2], [2], [3, 4]]);\n\ngap> AsBinaryRelation(D);\nBinary Relation o\ n 4 points\n]]>\n \n\n\n \ \n \n A digrap\ h, or fail.\n \n If the optional first argum\ ent filt is present, then this should\n specify the category or repr\ esentation the digraph being created will\n belong to. For example, if f\ ilt is ,\n then the digraph being creat\ ed will be mutable, if filt is ,\ then the digraph will be immutable. \n If the optional first argument f\ ilt is not present, then is used b\ y default.
\n\n If trans is a transformation, and n is a n\ on-negative integer\n such that the restriction of trans to [1 ..\ n] defines\n a transformation of [1 .. n], then <\ C>AsDigraph\n returns the functional digraph with n vertices def\ ined by\n trans. See .\n
\ \n\n Specifically, the digraph returned by AsDigraph has n e\ dges:\n for each vertex x in [1 .. n], there is a uniq\ ue edge\n with source x; this edge has range x^trans.\ \n
\n\n If the optional second argument n is not supplied, th\ en the degree of\n the transformation trans is used by default. If \ the restriction of\n trans to [1 .. n] does not define\ a transformation of\n [1 .. n], then AsDigraph(trans, n)\n returns fail.\n\n 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 := BinaryRelationOnPoints(\n> \ [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation on 5 points\nga\ p> D := AsDigraph(b);\n\n]]>\n \n\n\n \n \n A &Grape; package graph.\n \n If digraph is a mutable or immutable digraph without mult\ iple edges, then this operation\n returns a &Grape; package graph that is i\ somorphic to digraph.
\n\n If digraph is a multidigraph, \ then since &Grape; does not support\n multiple edges, the multiple edges wi\ ll be reduced to a single edge in the\n result. In order words, for a multi\ digraph this operation will return the\n same as Graph(DigraphRemoveAllM\ ultipleEdges(digraph)).\n\n Pete\ rsen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y) return In\ tersection(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 schreier\ Vector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digraph(Petersen);\n\ngap> Graph(last) = Petersen;\ntrue]]\ >\n \n\n\n \n \n A &Grape; package graph.\ \n \n If digraph is a digraph, then this method return\ s the same as\n , except that if digraph is imm\ utable, then the\n result will be stored as a mutable attribute of digra\ ph.\n In this latter case, when AsGraph(digraph) i\ s called\n subsequently, the same &GAP; object will be returned as before.\ \n\n 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]]>\n \n\n\n \n \n A trans\ formation, or fail\n \n If digraph is\ a functional digraph, then AsTransformation\n returns the transform\ ation which is defined by digraph. See . Otherwise,\n AsTransformation(digraph) re\ turns fail.
\n\n If digraph is a functional digraph with\ n vertices, then\n AsTransformation(digraph) w\ ill return the\n transformation f of degree at most n where f\ or each 1\n \\leq i \\leq n, i ^ f is equal to the unique o\ ut-neighbour of\n vertex i in digraph.\n\n D := Digraph([[1], [3], [2]]);\n\ngap> AsTransformation(D);\nTransformation( [ 1, 3, 2 ] )\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] )\ngap> AsPermutation\ (last);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> AsTransformation(D);\nfail]]>\n \n\n\n
\n\n
New digraphs fr\ om old\n \n \n \n \n \n A digraph.\n \n \ Each of these operations returns a new copy of digraph,\n of the app\ ropriate mutability, retaining none of the attributes or\n properties of digraph.
\n\n DigraphCopy is a synonym for DigraphCopySa\ meMutability.\n D := CycleDigraph(10);\n\ngap> DigraphCopy(D) = D;\ntrue\ngap> IsIdent\ icalObj(DigraphCopy(D), D);\nfalse\ngap> DigraphMutableCopy(D);\n\n]]>\n \n\n\n \n \n \ \n \n \n A digraph.\n \n Each of these operations returns either the orig\ inal argument\n digraph, or a new copy of digraph of the appr\ opriate\n mutability, according to the mutability of digraph.\n C := CycleDigraph(10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntru\ e\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D := D\ igraphMutableCopyIfMutable(C);\n\nga\ p> IsMutableDigraph(D);\nfalse\ngap> D := DigraphMutableCopyIfImmutable(C);\n<\ mutable digraph with 10 vertices, 10 edges>\ngap> IsMutableDigraph(D);\ntrue\n\ gap> C := CycleDigraph(IsMutableDigraph, 10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\nt\ rue\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap>\ D := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> D := DigraphMutableCopyIfImmutabl\ e(C);\n\ngap> IsIdenticalObj(C, D)\ ;\ntrue\ngap> IsMutableDigraph(D);\ntrue]]>\n \n\n\n \n \n A digraph.\n \n If digra\ ph is a digraph, and verts is a subset of the\n vertices of d\ igraph, then this operation returns a digraph\n constructed from dig\ raph by retaining precisely those vertices in\n verts, and those\ edges whose source and range vertices are both\n contained in verts\ .
\n\n The vertices of the induced subdigraph are\n [1..Length(ve\ rts)] but the original vertex labels can be\n accessed via .
\n\n If digraph belongs to , then\n digraph is modified in place. If dig\ raph belongs to , a new immutable\ digraph containing the\n appropriate\n vertices and edges is returned.<\ P/>\n\n 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(last);\n[ 1 .. 3 ]\ngap> D := DigraphMutableCopy(\ D);\n\ngap> new := InducedSubd\ igraph(D, [1, 3, 4]);\n\ngap> D\ = new;\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\n \n This function returns a digraph isomorphic to the subdigraph of\n \ digraph induced by the set of non-isolated vertices,\n i.e. the set \ of those vertices of digraph which are the source or\n range of some\ edge in digraph.\n See .\n
\ \n\n The ordering of the remaining vertices of digraph is preserved,\ as are\n the labels of the remaining vertices and edges;\n see and .\n This c\ an allow one to match a vertex in the reduced digraph to the\n correspondin\ g vertex in digraph.\n
\n\n If digraph is immutable, t\ hen a new immutable digraph is returned.\n Otherwise, the isolated vertices\ of the mutable digraph digraph are\n removed in-place.\n D := Digraph([[1, 2], [], [], [1, 4], []]);\n\ngap> R := ReducedDigraph(D);\n\ngap> OutNeighbours(R);\n[ [ 1, 2 ], [ ], [ 1, 3 \ ] ]\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ]\ngap> D\ igraphEdges(R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]\ngap> DigraphVerte\ xLabel(R, 3);\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap> D := Digraph(IsMutab\ leDigraph, [[], [3], [2]]);\n\ngap> \ ReducedDigraph(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \ \n \n A digraph.\n \n If digraph is \ a digraph, then MaximalSymmetricSubdigraph\n returns a symmetric dig\ raph without multiple edges which has the same\n vertex set as digraph, and whose edge list is formed from\n digraph by ignoring the mul\ tiplicity of edges, and by ignoring\n edges [u,v] for which there do\ es not exist an edge [v,u].\n
\n\n The digraph returned by MaximalSymmetricSubdigraphWithoutLoops is\n the same, except that loop\ s are removed.
\n \n If digraph is immutable, then a new immut\ able digraph is returned.\n Otherwise, the mutable digraph digraph i\ s changed in-place into such\n a digraph described above.
\n\n See , , and\n for more information.\n D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and IsMultiDigraph(D);\nt\ rue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap> S :\ = MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigraph(S);\ntrue\ngap> \ OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMu\ tableDigraph, 3);\n\ngap> MaximalSym\ metricSubdigraph(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n A digraph.\n \n If digraph is a digraph, then MaximalAntiSymmetricSubdigraph\n returns an ant\ i-symmetric subdigraph of digraph formed by\n retaining the vertices\ of digraph, discarding any duplicate edges,\n and discarding any ed\ ge [i,j] of digraph where i > j\n and the reverse edge\ [j,i] is an edge of digraph.\n In other words, for every\n \ symmetric pair of edges [i,j] and [j,i] in digraph, whe\ re \n i and j are distinct, it discards the the edge\n \ [\\max(i,j),\\min(i,j)].\n
\n\n If digraph is immutable, \ then a new immutable digraph is returned.\n Otherwise, the mutable digraph \ digraph is changed in-place.
\n\n See for more information.\n D := Digraph(\ [[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph(D);\ntrue\ngap> Ou\ tNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap> D := MaximalAnt\ iSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigraph(D);\ntrue\ngap> \ OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D := Digraph(IsMutable\ Digraph, [[2], [1]]);\n\ngap> Maxima\ lAntiSymmetricSubdigraph(D);\n\ngap> \ D;\n\n]]>\n \ \n\n\n \n \n \n \n \n A digraph, or fail.\n \n If digraph is a digraph with\ at least one vertex, then\n UndirectedSpanningForest returns an und\ irected spanning forest of\n digraph, otherwise this attribute retur\ ns fail. See for the\ definition of an undirected\n spanning forest.
\n\n If digraph is a digraph with at least one vertex and whose is connected (see ), then UndirectedSpanningTree returns\n an undirected spanning\ tree of digraph, otherwise this attribute\n returns fail. S\ ee for the\n definition of an und\ irected spanning tree.
\n\n If digraph is immutable, then an immu\ table digraph is returned.\n Otherwise, the mutable digraph digraph \ is changed in-place into an\n undirected spanning tree of digraph.\n\n Note that for an immutable digraph that has known undirected spannin\ g tree,\n the attribute UndirectedSpanningTree returns the same digr\ aph as\n the attribute UndirectedSpanningForest.\n\n D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);\n\ngap> UndirectedSpanningTree(D);\nfail\ng\ ap> forest := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 \ ] ]\ngap> IsUndirectedSpanningForest(D, forest);\ntrue\ngap> DigraphConnectedC\ omponents(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> DigraphConnectedCompone\ nts(MaximalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> Undire\ ctedSpanningForest(MaximalSymmetricSubdigraph(D))\n> = forest;\ntrue\ngap> D :\ = CompleteDigraph(4);\n\ngap> tree\ := UndirectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> tree = Undirecte\ dSpanningForest(D);\ntrue\ngap> UndirectedSpanningForest(EmptyDigraph(0));\nfa\ il\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n A digraph.\n \n If digraph is a digraph, and p is a partition of \ the vertices\n of digraph, then this operation returns a digraph con\ structed by\n amalgamating all vertices of digraph which lie in the \ same part of\n p.\n
\n\n A partition of the vertices of digraph is a list of non-empty\n disjoint lists, such that the union o\ f all the sub-lists is equal\n to vertex set of digraph. In particul\ ar, each vertex must appear\n in precisely one sub-list.\n
\n\n T\ he vertices of digraph in part i of p will become\n ve\ rtex i in the quotient, and there exists some edge in digraph\n \ with source in part i and range in part j if and only if ther\ e\n is an edge from i to j in the quotient.\n In particula\ r, 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 &Digraphs; package.\n
\ \n\n If digraph belongs to , then\n \ digraph is modified in place. If digraph belongs to , a new immutable digraph with the above\n \ properties is returned.
\n\n 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 := QuotientDigraph(D, p);\n\ngap> DigraphVertices(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]]>\n \n\n\n \ \n \n \n A digraph.\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 [i, j] o\ f a digraph, the\n reverse contains the corresponding edge [j, i].\n\n DigraphReverse returns the reverse of the digraph digraph<\ /A>. If\n digraph is immutable, then a new immutable digraph is retu\ rned. Otherwise,\n the mutable digraph digraph is changed in-place i\ nto its reverse.\n D := Digraph([[3], [1, 3, 5], [\ 1], [1, 2, 4], [2, 3, 5]]);\n\nga\ p> DigraphReverse(D);\n\ngap> Out\ Neighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ]\nga\ p> 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(IsMutab\ leDigraph, 3);\n\ngap> OutNeighbours\ (D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n]]>\n \n\n\n \n \n \n A digraph.\n \n The dual of\ digraph has the same vertices as digraph,\n and there is an \ edge in the dual from i to j\n whenever there is no edge from\ i to j in digraph.\n The dual is sometimes cal\ led the complement.
\n\n DigraphDual returns the dual of t\ he digraph digraph. If\n digraph is an immutable digraph, the\ n a new immutable digraph is\n returned. Otherwise, the mutable digraph \ digraph is changed in-place\n into its dual.\n 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]]>\n \n\n\n \n\ \n \n A digraph.\ \n \n If digraph is a digraph, then this attribute giv\ es the minimal\n symmetric digraph which has the same vertices and contains\ all the edges of\n digraph.
\n \n A digraph is symmetri\ c if its adjacency matrix \n is symmetr\ ic. For a digraph with multiple\n edges this means that there are the same \ number of edges from a vertex\n u to a vertex v as there are \ from v to u; \n see .
\n\ \n If digraph is immutable, then a new immutable digraph is returned\ .\n Otherwise, the mutable digraph digraph is changed in-place into \ its\n symmetric closure.\n D := Digraph([[1, 2, 3],\ [2, 4], [1], [3, 4]]);\n\ngap> D \ := DigraphSymmetricClosure(D);\n\ngap> IsSymmetricDigraph(D);\ntrue\ngap> List(OutNeighbours(D), AsSe\ t);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ]\ngap> D := Digraph([[\ 2, 2], [1]]);\n\ngap> D := Di\ graphSymmetricClosure(D);\n\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 1 ] ]\ngap> D := CycleDigra\ ph(IsMutableDigraph, 3);\n\ngap> Dig\ raphSymmetricClosure(D);\n\ngap> D;\ \n\n]]>\n \ \n\n\n \n \n \n \n \ \n A digraph.\n \n If digraph is a digrap\ h with no multiple edges, then these\n attributes return the (reflexive) tr\ ansitive closure of digraph.
\n\n A digraph is reflexive i\ f it has a loop at every vertex, and it is\n transitive if whenever \ [i,j] and [j,k] are edges of\n digraph, [i,k] i\ s also an edge. The (reflexive)\n transitive closure of a digraph\ digraph is the least\n (reflexive and) transitive digraph containin\ g digraph.
\n\n If digraph is immutable, then a new immuta\ ble digraph is returned.\n Otherwise, the mutable digraph digraph is\ changed in-place into its\n (reflexive) transitive closure.
\n\n Let\ n be the number of vertices of digraph, and let\n m b\ e the number of edges. For an arbitrary digraph, these\n attributes will u\ se a version of the Floyd-Warshall algorithm, with\n complexity O(n^3).\n\n However, for a topologically sortable digraph [see ], these attributes will use methods\n with co\ mplexity O(m + n + m \\cdot n) when this is faster.
\n\n D := DigraphFromDiSparse6String(\".H`eOWR`Ul^\");\n\ngap> IsReflexiveDigraph(D) or IsTransitive\ Digraph(D);\nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1, 3 ], [ ], [ 5 ],\ [ ], [ 7, 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := DigraphTransitiveClosure\ (D);\n\ngap> OutNeighb\ ours(T);\n[ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [ ], [ 5 ], \n \ [ ], [ 7, 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphReflexiveTransitiveC\ losure(D);\n\ngap> OutNe\ ighbours(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 := CycleD\ igraph(IsMutableDigraph, 3);\n\ngap>\ DigraphReflexiveTransitiveClosure(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \n \n A digraph.\n \n If dig\ raph is a topologically sortable digraph\n [see ]\n with no multiple edges, then these operations return th\ e\n (reflexive) transitive reduction of digraph.
\n\n The (ref\ lexive) transitive reduction of such a digraph is the unique\n least subgra\ ph such that the (reflexive) transitive closure of the\n subgraph is equal \ to the (reflexive) transitive closure of digraph\n [see ].\n In order words, it is the least s\ ubgraph of digraph which\n retains the same reachability as digra\ ph.
\n\n If digraph is immutable, then a new immutable digrap\ h is returned.\n Otherwise, the mutable digraph digraph is changed i\ n-place into its\n (reflexive) transitive reduction.
\n\n Let n be the number of vertices of an arbitrary digraph, and let\n m\ be the number of edges.\n Then these operations\n use methods with comp\ lexity O(m + n + m \\cdot n).\n
\n D\ := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(D);\ntrue\ngap> D1 \ := DigraphReflexiveTransitiveReduction(D);\n\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeighbours(D1);\n[ [ 2 ]\ , [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\n\ngap> DigraphHasLoops(D2);\ntrue\ngap> OutNeighbour\ s(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveTransitiveClosure(D)\ \n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> DigraphTransitiveClo\ sure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D := Digraph(IsMutable\ Digraph, [[1], [1], [1, 2, 3]]);\n\n\ gap> DigraphReflexiveTransitiveReduction(D);\n\ngap> D;\n\n]]>\ \n \n\n\n \n \n A digraph.\n \n The operation returns a digraph constructed from digraph by adding a\n single new vertex, and no new edges.
\n\n If the opt\ ional second argument label is a &GAP; object,\n then the new vertex\ will be labelled label.\n
\n\n If digraph belongs to \ , then the\n vertex is added directly to digraph. If digraph belongs to\n , an immutable copy of digraph with\n the additional vertex is r\ eturned.
\n\n D := CompleteDigraph(3);\n\ngap> new := DigraphAddVertex(D);\n\ngap> D = new;\nfalse\ngap> DigraphVer\ tices(new);\n[ 1 .. 4 ]\ngap> new := DigraphAddVertex(D, Group([(1, 2)]));\n\ngap> DigraphVertexLabels(new);\n[ \ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := CompleteBipartiteDigraph(IsMutableDigra\ ph, 2, 3);\n\ngap> new := DigraphAd\ dVertex(D);\n\ngap> D = new;\ntrue\ \n]]>\n \n\n\n \n \n \n \n A digraph.\n \n For a non-negative integer m, this operation returns a digraph\ \n constructed from digraph by adding m new vertices.\n \n\n Otherwise, if labels is a list\n consisting of k &G\ AP; objects, then this operation returns a\n digraph constructed from di\ graph by adding k new vertices,\n which are labelled according t\ o this list.\n
\n\n If digraph belongs to , then the\n vertices are added directly to digraph, wh\ ich is changed in-place.\n If digraph belongs to , then the\n result is returned as an immutable digraph.\n \
\n\n D := CompleteDigraph(3);\n\ngap> new := DigraphAddVertices(D, 3);\n\ngap> DigraphVertices(new);\n[ 1 .. 6 ]\ \ngap> new := DigraphAddVertices(D, [Group([(1, 2)]), \"d\"]);\n\ngap> DigraphVertexLabels(new);\n[ 1, 2, 3, Gro\ up([ (1,2) ]), \"d\" ]\ngap> DigraphAddVertices(D, 0) = D;\ntrue\ngap> D := Co\ mpleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> new := DigraphAddVertices(D, 4);\n\ngap> D = new;\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\n \n If edge is \ a pair of vertices of digraph, or src and\n ran are ve\ rtices of digraph,\n then this operation returns a digraph construct\ ed from digraph\n by adding a new edge with source edge[1]\ [src] and range\n edge[2] [ran].\n
\ \n \n If digraph belongs to , the\ n the edge\n is added directly to digraph. If digraph belongs\ to , then an immutable copy of di\ graph with the\n additional edge is returned.
\n\n D1 := Digraph([[2], [3], []]);\n\ngap> DigraphEdges(D1);\n[ [ 1, 2 ], [ 2, 3 ] ]\ngap> D2 := Digrap\ hAddEdge(D1, [3, 1]);\n\ngap> Digr\ aphEdges(D2);\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D3 := DigraphAddEdge(D2,\ [2, 3]);\n\ngap> DigraphEdge\ s(D3);\n[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D := CycleDigraph(IsM\ utableDigraph, 4);\n\ngap> new := Di\ graphAddEdge(D, [1, 3]);\n\ngap> Dig\ raphEdges(new);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ]\ngap> D \ = new;\ntrue\n]]>\n \n\n\n \n \n \ \n A new digraph.\n \n \n This operation return\ s a new digraph with the same vertices and edges as\n digraph and wi\ th additional edges consisting of the orbit of the\n edge edge under\ the action of the of\n digraph. If edge is already an edge in digraph, then\n digraph is re\ turned unchanged. The argument digraph must be\n an immutable digrap\ h.\n
\n\n An edge is simply a pair of vertices of digraph.\n\ \n 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, 8 ], [ 2, 6 ], \n [ 3, 5 ]\ , [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ \ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7\ ], [ 8, 6 ], [ 8, 5 ] ]\ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, \ 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ],\ [ 3, 7 ], [ 3, 2 ], \n [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5\ , 2 ], \n [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7\ , 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \n [ 8, 5 ], [ 8, 1 \ ] ]\ngap> gr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);\n\ngap> gr3 = gr1;\ntrue\n]]>\n \ \n\n\n \n \n A digraph.\n \n If \ edges is a (possibly empty) list of pairs of vertices of\n digraph, then this operation returns a digraph constructed from\n digraph\ by adding the edges specified by edges. More\n precisely, for ever\ y edge in edges, a new edge\n will be added with source ed\ ge[1] and range edges[2].\n
\n\n If an edge is included i\ n edges with multiplicity k,\n then it will be added k\ times.\n\n If digraph belongs to , \ then the edges\n are added directly to digraph. If digraph be\ longs to , then the result is returne\ d as an immutable\n digraph.\n
\n\n 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\ngap> new := DigraphAddEd\ ges(D,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n\ngap> D = new;\nfalse\ngap> D2 := Di\ graphMutableCopy(func(1024));\n\ngap> new := DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1024], [1023, 1024], \ [1024, 1]]);\n\ngap> D2 =\ new;\ntrue\n]]>\n \n\n\n \n \n A dig\ raph.\n \n If v is a vertex of digraph, then this operation returns a\n digraph constructed from digraph \ by removing vertex v,\n along with any edge whose source or range ve\ rtex is v.
\n\n If digraph has n vertices, then the\ vertices of the returned\n digraph are [1..n-1], but the original l\ abels can be\n accessed via .
\n\n \ If digraph belongs to , then the\n \ vertex is removed directly from digraph. If digraph belongs\n \ to , an immutable copy of digraph\n\ without the vertex is returned.
\n\n D := D\ igraph([\"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]\ ]>\n \n\n\n \n \n A digraph.\n \n If verts is a (possibly empty) duplicate-\ free list of vertices of\n digraph, then this operation returns a di\ graph constructed from\n digraph by removing every vertex in vert\ s, along with any edge\n whose source or range vertex is in verts.
\n\n If digraph has n vertices, then the vertices of th\ e new\n digraph are [1 .. n-Length(verts)], but the\n orig\ inal labels can be accessed via .
\n\n \ If digraph belongs to , then the\n \ vertices are removed directly from digraph. If digraph belongs\ \n to , an immutable copy of digraph\n without the vertices is returned.
\n\n \ 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]]>\n \n\n\n \ \n \n \n A digraph.\n \n If digraph is a digrap\ h with no multiple edges and edge is a\n pair of vertices of digr\ aph, or src and ran are\n vertices of digraph, the\ n this operation returns a digraph\n constructed from digraph by rem\ oving the edge specified by\n edge or [src, ran].
\n\n \ If digraph belongs to , then \ the edge is removed directly from\n digraph. If digraph belon\ gs to , an immutable copy of digra\ ph without\n the edge is returned.\n
\n\n Note that if dig\ raph belongs to ,\n then a new copy o\ f digraph will be returned even if edge or \n [src, ran] does not define an edge of digraph.
\n\n D := CycleDigraph(250000);\n\ \ngap> D := DigraphRemoveEdge(D, [250000, 1]);\n\ngap> new := DigraphRemoveEdge(D, [25000, 2]);;\ngap>\ new = D;\ntrue\ngap> IsIdenticalObj(new, D);\nfalse\ngap> D := DigraphMutable\ Copy(D);;\ngap> new := DigraphRemoveEdge(D, 2500, 2);;\ngap> IsIdenticalObj(ne\ w, D);\ntrue]]>\n\n \n\n\n \n \n \n A new digraph.\n \n \n This operation retu\ rns a new digraph with the same vertices as\n digraph and with the o\ rbit of the edge edge (under the action\n of the of digraph) removed.\n If edge is not an edge in digraph, then digraph is\n returned unchanged. The argument \ digraph must be an immutable\n digraph.\n
\n\n An edge is sim\ ply a pair of vertices of digraph.\n 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, 8 ], [ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, \ 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ],\ \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ]\ngap> Digrap\ hEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], \n\ [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], \n [ 4, 6 ], [ \ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5, 8 ], [ 5, 4 ], [ 6, 4\ ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], \ [ 8, 7 ], [ 8, 6 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := DigraphRemoveEdgeOrb\ it(gr2, [1, 8]);\n\ngap> gr3 = gr\ 1;\ntrue\n]]>\n \n\n\n \n\ \n A dig\ raph.\n \n If one of the following holds:\n \n \n digraph is a digraph with no multiple edges, \ and\n edges is a list of pairs of vertices of digraph, or\ \n \n digraph is a digraph and edges is \ an empty list \n \n\n then this operation returns a digrap\ h constructed from digraph\n by removing all of the edges specified \ by edges\n (see ).\n
\n\n If digraph belongs to , then the edge\n is removed directly from digraph. If digraph belongs to , th\ e edge is removed from an immutable copy of\n digraph and this new d\ igraph is returned.
\n\n Note that if edges is empty, then this o\ peration\n will always return digraph rather than a copy. Also, if a\ ny element\n of edges is invalid (i.e. does not define an edge of digraph)\n then that element will simply be ignored.\n\n <\ ![CDATA[\ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> D := DigraphMutableCopy(D);\n\ngap> new := DigraphRemoveEdg\ es(D, [[1, 2], [2, 3], [3, 100]]);\n\ngap> new = D;\ntrue\n]]>\n \n\n\n \n \n\ \n A digrap\ h.\n \n If digraph is a digraph, then this o\ peration returns a\n digraph constructed from digraph by removing e\ very loop. A loop\n is an edge with equal source and range.
\n\n If <\ A>digraph is immutable, then a new immutable digraph is returned.\n Oth\ erwise, the loops are removed from the mutable digraph digraph\n in-\ place.\n D := Digraph([[1, 2, 4], [1, 4], [3, 4], \ [1, 4, 5], [1, 5]]);\n\ngap> Digr\ aphRemoveLoops(D);\n\ngap> D := Di\ graph(IsMutableDigraph, [[1, 2], [1]]);\n\ngap> DigraphRemoveLoops(D);\n\ \ngap> D;\n\n]]>\n \n\n\n \n \n \n A digraph.\n \n \ If digraph is a digraph, then this operation returns a\n digraph c\ onstructed from digraph by removing all multiple edges.\n The result\ is the largest subdigraph of digraph which does not\n contain multi\ ple edges.
\n\n If digraph is immutable, then a new immutable dig\ raph is returned.\n Otherwise, the multiple edges of the mutable digraph digraph are\n removed in-place.\n D1 := Di\ graph([[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> DigraphRemoveAllMultipleEdges(D);\ \n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \n A digraph.\n \n \ If digraph is a digraph without multiple edges, and edges is a\ \n list of pairs of vertices of digraph (the entries of each pair\n \ corresponding to the source and the range of an edge, respectively),\n t\ hen DigraphReverseEdges returns a digraph constructed from\n digr\ aph by reversing the orientation of every edge specified by\n edges<\ /A>. If only one edge is to be reversed, then\n DigraphReverseEdge c\ an be used instead. In this case, the second\n argument should just be a si\ ngle vertex-pair, or the second and third\n arguments should be the source \ and range of an edge respectively.
\n\n Note that even though digrap\ h cannot have multiple edges, the\n output may have multiple edges.
\n\n If digraph belongs to , then t\ he edges\n are reversed in digraph. If digraph belongs to , an immutable copy of digraph wi\ th the\n specified edges reversed is returned.
\n 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\ng\ ap> D = new;\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\n \n In the first form, if D1, D2, etc. \ are digraphs, then\n DigraphDisjointUnion returns their disjoint uni\ on.\n\n In the second form, if list is a non-empty list of digraphs,\ then\n DigraphDisjointUnion returns the disjoint union of the digra\ phs\n contained in the list.
\n\n For a disjoint union of digraphs,\ the vertex set is the disjoint union of\n the vertex sets, and the edge li\ st is the disjoint union of the edge lists.\n
\n\n More specifically\ , for a collection of digraphs D1, D2,\n ..., the disj\ oint union with have\n DigraphNrVertices(D1) +\ \n DigraphNrVertices(D2) + ... vertices.\ \n The edges of D1 will remain unchanged, whilst the edges of the\n \ ith digraph, D[i], will be changed so that they\n b\ elong to the vertices of the disjoint union corresponding to\n D[\ i]. In particular, the edges of D[i] will\n have their s\ ource and range increased by\n DigraphNrVertices(D1) <\ C>+ ... +\n DigraphNrVertices(D[i-1]).\n\n
\n\n Note that previously set will be lost.
\n\n If the first digraph D1 [list[1]] \ belongs to , then D1 [list[1]] is modified in\n place to contain the appropriate vertices and edges. I\ f digraph belongs\n to , a new imm\ utable digraph containing the\n appropriate vertices and edges is returned.\ \n D1 := CycleDigraph(3);\n\ngap> OutNeighbours(D1);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> D2 := C\ ompleteDigraph(3);\n\ngap> OutNeig\ hbours(D2);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> union := DigraphDisjointUn\ ion(D1, D2);\n\ngap> OutNeighbours\ (union);\n[ [ 2 ], [ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ]\n]]>\ \n \n\n\n \n \n \n A digraph.\n \n In the first \ form, if D1, D2, etc. are digraphs, then\n DigraphEdgeUnio\ n returns their edge union.\n\n In the second form, if list is a\ non-empty list of digraphs, then\n DigraphEdgeUnion returns the edg\ e union of the digraphs contained\n in the list.
\n\n The vertex se\ t of the edge union of a collection of digraphs is the\n union of th\ e vertex sets, whilst the edge list of the edge union is\n the concatena\ tion of the edge lists. The number of vertices of the\n edge union is \ equal to the maximum number of vertices of one of the\n digraphs, wh\ ilst the number of edges of the edge union will equal the\n sum of t\ he number of edges of each digraph.
\n\n Note that previously set will be lost.\n
\n\n If the first dig\ raph D1 [list[1]] belongs to , then D1 [list[1]] is modified in\n place to contain the a\ ppropriate vertices and edges. If digraph belongs\n to , a new immutable digraph containing the\n appropriate\ vertices and edges is returned.\n D := CycleDigraph(1\ 0);\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 = D\ igraphByEdges(\n> Concatenation(DigraphEdges(D1), DigraphEdges(D2)));\ntrue\n]\ ]>\n \n\n\n \n \n \n A digraph.\n \n In the first\ form, if D1, D2, etc. are digraphs, then\n DigraphJoin returns their join.\n\n In the second form, if list is a non-empty\ list of digraphs,\n then DigraphJoin returns the join of the digrap\ hs contained in the\n list.
\n\n The join of a collection of digraph\ s D1, D2, ... is\n formed by first taking the of the co\ llection.\n\n In the disjoint union, if i \\neq j then there are no \ edges between\n vertices corresponding to digraphs D[i] and\n\ D[j] in the collection; the join is created by including\n \ all such edges.
\n\n For example, the join of two empty digraphs is a \ complete bipartite\n digraph.
\n\n Note that previously set will be lost.\n
\n\n If the first digraph\ D1 [list[1]] belongs to , \ then D1 [list[1]] is modified in\n place to contain the appro\ priate vertices and edges. If digraph belongs\n to , a new immutable digraph containing the\n appropriate ver\ tices and edges is returned.\n D := CompleteDigraph(3)\ ;\n\ngap> IsCompleteDigraph(Digrap\ hJoin(D, D));\ntrue\ngap> D2 := CycleDigraph(3);\n\ngap> DigraphJoin(D, D2);\n\n]]>\n \n\n\n \n\ \n \n A digraph.\n \n In the first form, if gr1, gr2, et\ c. are digraphs, then\n DigraphCartesianProduct returns a digraph is\ omorphic to their\n Cartesian product.\n
\n\n In the second form,\ if list is a non-empty list of digraphs,\n then DigraphCartesian\ Product returns a digraph isomorphic to the \n Cartesian product of the\ digraphs contained in the list.\n
\n \n Mathematically, the Cartesi\ an product of two digraphs G, H\n is a digraph with vertex se\ t Cartesian(DigraphVertices(G),\n DigraphVertices(H)) such that ther\ e is an edge from [u, u'] to\n [v, v'] iff u = v and\ there is an edge from u' to\n v' in H or u' = v'<\ /C> and there is an edge from\n u to v in G.\n
\ \n\n Due to technical reasons, the digraph D returned by\n Dig\ raphCartesianProduct has vertex set\n [1 .. DigraphNrVertices(G)*Di\ graphNrVertices(H)] instead, and the\n exact\n method of encoding p\ airs of vertices into integers is implementation\n specific. The original \ vertex pair can be somewhat regained by using\n . In addition,\n \ are preserved: if vertex pair\n [u,u'] was encoded as i then\ the vertex label of i\n will be the pair of vertex labels of u and u' i.e.\n DigraphVertexLabel(D,i) = [DigraphVertexLabel(G,\ u),\n DigraphVertexLabel(H,u')].\n
\n\n As the Cartesian pr\ oduct is associative, the Cartesian product of a collection \n of digraphs \ gr1, gr2, ... is computed in the obvious \n fashion. \ \n
\n\n gr := ChainDigraph(4);\n\ngap> gr2 := CycleDigraph(3);\n\ngap> gr3 := DigraphCartesianProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digr\ aph([[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]]>\n \n\n\n \n \ \n \n A digraph.\n \n In the f\ irst form, if gr1, gr2, etc. are digraphs, then\n DigraphD\ irectProduct returns a digraph isomorphic to their direct\n product.\n \
\n\n In the second form, if list is a non-empty list of digra\ phs, then\n DigraphDirectProduct returns a digraph isomorphic to the\ direct\n product of the digraphs contained in the list.\n
\n\n M\ athematically, the direct product of two digraphs G, H is a\n \ digraph with vertex set Cartesian(DigraphVertices(G),\n DigraphVertices\ (H)) such that there is an edge from [u, u']\n to [v, v']\ iff there is an edge from u to v in\n G and an edge f\ rom u' to v' in H. \n
\n\n Due to technical re\ asons, the digraph D returned by\n DigraphDirectProduct has v\ ertex set\n [1 .. DigraphNrVertices(G)*DigraphNrVertices(H)] instea\ d, and the\n exact method of encoding pairs of vertices into integers is im\ plementation\n specific. The original vertex pair can be somewhat regained\ by using\n . In addition\n \ are preserved: if vertex pair\n [\ u,u'] was encoded as i then the vertex label of i\n will \ be the pair of vertex labels of u and u' i.e.\n DigraphVer\ texLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u')].\ \n
\n\n As the direct product is associative, the direct product of \ a collection \n of digraphs gr1, gr2, ... is computed \ in the obvious \n fashion. \n
\n\n gr := Cha\ inDigraph(4);\n\ngap> gr2 := CycleDig\ raph(3);\n\ngap> gr3 := DigraphDirect\ Product(gr, gr2);\n\ngap> IsIsomo\ rphicDigraph(gr3, \n> Digraph([[6], [7], [8], [], \n> [10], [11], [12\ ], [],\n> [2], [3], [4], []]));\ntrue\n]]>\n \n\n\n \n \n A list of transformations.\ \n \n If digraph is a Cartesian product digraph, \n \ digraph = DigraphCartesianProduct(gr_1, gr_2, ... ), then\n Digr\ aphCartesianProductProjections returns a list proj\n such that <\ C>proj[i] is the projection onto the i-th\n coordinate of the pr\ oduct.\n
\n\n A projection is an idempotent endomorphism of digra\ ph. If\n gr1, gr2, ... are all loopless digraphs, then the induc\ ed \n subdigraph of digraph on the image of proj[i] is \n \ isomorphic to gr_i.\n
\n\n Currently this attribute is only s\ et upon creating an immutable \n digraph via DigraphCartesianProduct\ and there is no way \n of calculating it for other digraphs.\n
\n\n\ For more information see\n \n\n D := DigraphCart\ esianProduct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap>\ HasDigraphCartesianProductProjections(D);\ntrue\ngap> proj := DigraphCartesia\ nProductProjections(D);; Length(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\n\ gap> RankOfTransformation(proj[3]);\n2\ngap> S := ImageSetOfTransformation(pro\ j[2]);;\ngap> IsIsomorphicDigraph(CycleDigraph(4), InducedSubdigraph(D, S));\n\ true\n]]>\n \n\n\n \n \n A l\ ist of transformations.\n \n If digraph is a\ direct product digraph, \n digraph = DigraphDirectProduct(gr_1, gr_2, .\ .. ), then\n DigraphDirectProductProjections returns a list p\ roj\n such that proj[i] is the projection onto the i-th\n\ coordinate of the product.\n
\n\n A projection is an idempotent \ endomorphism of digraph. If\n gr1, gr2, ... are all loopless \ digraphs, then the image \n of digraph under proj[i] is isomo\ rphic to gr_i.\n
\n\n Currently this attribute is only set up\ on creating an immutable\n digraph via DigraphDirectProduct and ther\ e is no way of \n calculating it for other digraphs.\n
\n\n For m\ ore information, see\n \n\n D := DigraphDirectProduct(C\ hainDigraph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigraphDir\ ectProductProjections(D);\ntrue\ngap> proj := DigraphDirectProductProjections(\ D);; Length(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\ngap> RankOfTransform\ ation(proj[3]);\n2\ngap> P := DigraphRemoveAllMultipleEdges(\n> ReducedDigraph\ (OnDigraphs(D, proj[2])));; \ngap> IsIsomorphicDigraph(CycleDigraph(4), P);\nt\ rue\n]]>\n \n\n\n \n \n \n A digraph.\n \n Given a dig\ raph digraph, the operation returns the digraph\n obtained by associ\ ating a vertex with each edge of digraph, and\n creating an edge fro\ m a vertex v to a vertex u if and only\n if the terminal vert\ ex of the edge associated with v is the start\n vertex of the edge a\ ssociated with u.
\n\n Note that the returned digraph is always a\ new immutable digraph, and the\n argument digraph is never modified\ .\n LineDigraph(CompleteDigraph(3));\n\ngap> LineDigraph(ChainDigraph(3));\n\n]]>\n \n\n\n \n \n \n A\ digraph.\n \n Given a symmetric digraph digraph\ , the operation returns the\n symmetric digraph obtained by associating\ a vertex with each edge of\n digraph, ignoring directions and multi\ plicites, and adding an edge\n between two vertices if and only if the corr\ esponding edges have a vertex in\n common.
\n\n Note that the returne\ d digraph is always a new immutable digraph, and the\n argument digraph<\ /A> is never modified.\n LineUndirectedDigraph(Com\ pleteDigraph(3));\n\ngap> LineUndi\ rectedDigraph(DigraphSymmetricClosure(ChainDigraph(3)));\n\n]]>\n \n\n\n \ \n \n A\ digraph.\n \n Let digraph be a digraph with\ vertex set V. This function\n returns the double digraph of digr\ aph. The vertex set of the double\n digraph is the orginal vertex set t\ ogether with a duplicate. The edges are\n [u_1, v_2] and [u_2, v_\ 1] if and only if [u, v] is an\n edge in digraph, togethe\ r with the original edges and their\n duplicates.
\n\n If digraph<\ /A> is mutable, then digraph is modified in-place. If\n digraph is immutable, then a new immutable digraph constructed as\n described ab\ ove is returned.\n gamma := Digraph([[2], [3], [1]\ ]);\n\ngap> DoubleDigraph(gamma);\ \n\n]]>\n \n\n\n \n \n A digraph.\n \n Let \ digraph be a digraph with vertex set V. This function\n retur\ ns the bipartite double digraph of digraph. The vertex set of\n the \ double digraph is the original vertex set together with a duplicate. The\n \ edges are [u_1, v_2] and [u_2, v_1] if and only if [u,\n \ v] is an edge in digraph. The resulting graph is bipartite,\n s\ ince the orignal edges are not included in the resulting digraph.
\n\n I\ f digraph is mutable, then digraph is modified in-place. If\n \ digraph is immutable, then a new immutable digraph constructed as\n \ described above is returned.\n gamma := Digraph([\ [2], [3], [1]]);\n\ngap> Bipartite\ DoubleDigraph(gamma);\n\n]]>\n \n\n\n \n \n \n A digraph.\n \n For a d\ igraph digraph this operation returns a new digraph constructed\n fr\ om digraph, such that a loop is added for every vertex which did not\n \ have a loop in digraph.
\n\n If digraph is immutable, t\ hen a new immutable digraph is returned.\n Otherwise, the loops are added t\ o the loopless vertices of the mutable digraph\n digraph in-place.\n\ 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\ngap> D;\n\n]]\ >\n \n\n\n \n \n \n A digraph.\n \n \ The first argument is a digraph, the second argument is a non-negative\n in\ teger or a list of positive integers. This operation returns a digraph on\n \ the same set of vertices as digraph, with two vertices being adjacent\ \n if and only if the distance between them in digraph equals i\n or is a number in list. See .
\n\n If digraph\ is mutable, then digraph is modified in-place. If\n digraph<\ /A> is immutable, then a new immutable digraph constructed as\n described a\ bove is returned.\n digraph := DigraphFromSparse6S\ tring(\n> \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfHMjKM\");\n\ngap> DistanceDigraph(digraph, 1);\n\ngap> DistanceDigraph(digraph, [1, 2])\ ;\n\n]]>\n \n\n\n \n \n A digraph.\n \n Given \ a symmetric loopless digraph with no multiple edges digraph,\n the <\ E>k-closure of digraph is defined to be the unique\n smal\ lest symmetric loopless digraph C with no multiple edges on the\n ve\ rtices of digraph that contains all the edges of digraph\n an\ d satsifies the property that the sum of the degrees of every two\n non-adj\ acenct vertices in C is less than k. See , , , and .
\n\n The operation \ DigraphClosure returns the k-closure of\n digraph.\n <\ Example> D := CompleteDigraph(6);\n\ngap> D := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);\n\ngap> closure := DigraphClosure(D, 6);\n\ \ngap> IsCompleteDigraph(closure)\ ;\ntrue\n]]>\n \n\n\n \n \ \n \n A digraph.\n \n If digraph is a symmetric digraph, then\n DigraphMycie\ lskian returns the Mycielskian of digraph.
\n \n The Mycie\ lskian of a symmetric digraph is a larger symmetric digraph\n constructed f\ rom it, which has a larger chromatic number. For further\n information, see\ https://en.wikipedia.org/wiki/Mycielskian.
\n\n If digrap\ h is immutable, then a new immutable digraph is returned.\n Otherwise, \ the mutable digraph digraph is changed in-place into\n its Mycielski\ an.
\n\n D := CycleDigraph(2);\n\ngap> ChromaticNumber(D);\n2\ngap> D := DigraphMyci\ elskian(D);\n\ngap> ChromaticNumb\ er(D);\n3\ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n4\ngap> D := CompleteBipartiteDigraph\ (IsMutable, 2, 3);\n\ngap> DigraphM\ ycielskian(D);\n\ngap> D;\n\n]]>\n \n\n\n
\n\n
Random digraphs\n \ \n \n A digraph.\n \n If the optional first argument \ filt is present, then this should\n specify the category or represen\ tation the digraph being created will\n belong to. For example, if filt<\ /A> is ,\n then the digraph being created w\ ill be mutable, if filt is , the\ n the digraph will be immutable. \n If the optional first argument filt<\ /A> is not present, then is used by \ default.
\n\n If n is a positive integer, then this function retu\ rns a random\n digraph with n vertices and without multiple edges. T\ he result\n may or may not have loops.
\n\n If the optional second ar\ gument p is a float with value\n 0 \\leq p \\le\ q 1, then an edge will exist between each\n pair of vertices with proba\ bility approximately p.\n If p is not specified, then a rando\ m probability will be assumed\n (chosen with uniform probability).\n RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2)\ ;\n\n]]>\n \n\n\n \n \n A digraph.\n \n If n\ is a positive integer, then this function returns a random\n digraph w\ ith n vertices. If the optional second argument\n m is a posi\ tive integer, then the digraph will have m\n edges. If m is n\ ot specified, then the number of edges will be\n chosen randomly (with unif\ orm probability) from the range\n [1 .. {n \\choose 2}].\n
\n\n The method used by this function chooses each edge from th\ e set of\n all possible edges with uniform probability. No effort is made t\ o avoid\n creating multiple edges, so it is possible (but not guaranteed) t\ hat the\n result will have multiple edges. The result may or may not have l\ oops.
\n RandomMultiDigraph(1000);\n\ngap> RandomMultiDigraph(1000, 950)\ ;\n\n]]>\n \n\n\n \n \n A digraph.\n \n If\ the optional first argument filt is present, then this should\n spe\ cify the category or representation the digraph being created will\n belong\ to. For example, if filt is ,\n the\ n the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the opt\ ional first argument filt is not present, then is used by default.
\n\n If n is a non-negative\ integer, this function returns a random\n tournament with n vertice\ s. See .
\n RandomTour\ nament(10);\n\ngap> RandomTournament(Is\ MutableDigraph, 10);\n\n]]><\ /Log>\n \n\n\n \n \n A digraph.\n \n \ If the optional first argument filt is present, then this should\n \ specify the category or representation the digraph being created will\n be\ long to. For example, if filt is ,\n \ then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the\ optional first argument filt is not present, then is used by default.
\n\n If n is a positive\ integer, this function return a random lattice\n with m vertices, w\ here it is guaranteed that m is between\n n and 2 * n. See .
\n R\ andomLattice(10);\n\ngap\ > RandomLattice(IsMutableDigraph, 10);\n\n]]>\n \n\n\n
\n\n Standard examples\n \n \n A digraph.\n \n If n is a positive integer, this function returns a chain\n \ with n vertices and n - 1 edges.\n Specifically,\n \ for each vertex i (with i < n), there is a directed\n \ edge with source i and range i + 1.
\n\n If the option\ al first argument filt is present, then this should\n specify the ca\ tegory or representation the digraph being created will\n belong to. For ex\ ample, if filt is ,\n then the digra\ ph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first\ argument filt is not present, then is used by default.
\n\n The of a chain represents a\n total order.\n ChainDigraph(42);\n\ngap> Ch\ ainDigraph(IsMutableDigraph, 10);\n\ \n]]>\n \n\n\n \n \n A digraph.\ \n \n If n is a non-negative integer, this function re\ turns the complete\n digraph with n vertices. See .
\n\n If the optional first argument filt is pre\ sent, then this should\n specify the category or representation the digraph\ being created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, i\ f filt is , then the digraph wil\ l be immutable. \n If the optional first argument filt is not presen\ t, then is used by default.\n CompleteDigraph(20);\n\ngap> CompleteDigraph(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \ \n \n A digraph.\n \n A complete bipartite\ digraph is a digraph whose vertices\n can be partitioned into two non-empt\ y vertex sets, such there exists\n a unique edge with source i and r\ ange j if and only if\n i and j lie in different verte\ x sets.
\n\n If m and n are positive integers, this funct\ ion\n returns the complete bipartite digraph with vertex sets of sizes\n \ m (containing the vertices [1 .. m]) and\n n (contain\ ing the vertices [m + 1 .. m + n]).\n
\n\n If the optional fi\ rst argument filt is present, then this should\n specify the categor\ y or representation the digraph being created will\n belong to. For example\ , if filt is ,\n then the digraph be\ ing created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first argu\ ment filt is not present, then is used by default.\n\n CompleteBipartiteDigraph\ (2, 3);\n\ \ngap> CompleteBipartiteDigraph(IsMutableDigraph, 3, 2);\n\n]]>\n \n\n\n \ \n \n A digraph.\n \n For \ a list orders of n positive integers,\n this function retur\ ns the digraph containing n independent sets\n of vertices of orde\ rs [l[1] .. l[n]]. Moreover,\n each vertex is adjace\ nt to every other not contained in the\n same independent set.\n
\n\n If the optional first argument filt is present, then this sh\ ould\n specify the category or representation the digraph being created w\ ill\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt\ is , then the digraph will be immuta\ ble. \n If the optional first argument filt is not present, then <\ Ref\n Filt=\"IsImmutableDigraph\"/> is used by default.\n CompleteMultipartiteDigraph([5, 4, 2]);\n\ngap> CompleteMultipartiteDig\ raph(IsMutableDigraph, [5, 4, 2]);\n\n]]>\n \n\n\n \n \n A digraph.\n\ \n If n is a positive integer, this function returns \ a cycle\n digraph with n vertices and n edges. Specifi\ cally,\n for each vertex i (with i < n), there is a\ directed\n edge with source i and range i + 1. In addition, \ there is\n an edge with source n and range 1.
\n\n If \ the optional first argument filt is present, then this should\n spec\ ify the category or representation the digraph being created will\n belong \ to. For example, if filt is ,\n then\ the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the opt\ ional first argument filt is not present, then is used by default.\n CycleDigraph(1\ );\n\ngap> CycleDigraph(123);\n\ngap> CycleDigraph(IsMutableDigraph, 10)\ ;\n\n]]>\n \n\n\n \n \n \n A dig\ raph.\n \n If n is a non-negative integer, t\ his function returns the\n empty or null digraph with n vertices. An empty\n digraph is one with no edges.\n
\n \n \ If the optional first argument filt is present, then this should\n \ specify the category or representation the digraph being created will\n bel\ ong to. For example, if filt is ,\n \ then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the \ optional first argument filt is not present, then is used by default.
\n\n NullDigraph is a sy\ nonym for EmptyDigraph.\n\n EmptyDigraph(20\ );\n\ngap> NullDigraph(10);\n\ngap> EmptyDigraph(IsMutableDigraph, 10);\ \n]]>\n \n\n\n \n \n A digraph.\n \n If n an\ d k are non-negative integers, then this operation\n returns a symme\ tric digraph which corresponds to the undirected Johnson\n graph \ J(n, k).
\n\n If the optional first argument filt is present\ , then this should\n specify the category or representation the digraph bei\ ng created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if \ filt is , then the digraph will be \ immutable. \n If the optional first argument filt is not present, th\ en is used by default.
\n\n Th\ e Johnson graph J(n, k) has vertices given by all the\n k<\ /A>-subsets of the range [1 .. n], and two vertices are\n c\ onnected by an edge iff their intersection has size k - 1.\n\n \ gr := JohnsonDigraph(3, 1);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3 ], [ 1, 3 \ ], [ 1, 2 ] ]\ngap> gr := 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> Jo\ hnsonDigraph(IsMutableDigraph, 1, 0);\n\n\ ]]>\n \n\n\n \n \n A digraph.\n \n\n From https://en.wikipedia.org/wiki/Petersen_graph:\
\n\n 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 counter\ example for\n many problems in graph theory. The Petersen graph is named af\ ter Julius\n Petersen, who in 1898 constructed it to be the smallest bridge\ less cubic\n graph with no three-edge-coloring.
\n \n If the o\ ptional first argument filt is present, then this should\n specify t\ he category or representation the digraph being created will\n belong to. F\ or example, if filt is ,\n then the \ digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional \ first argument filt is not present, then is used by default.
\n\n See also .\n\n ChromaticNumber(PetersenGraph());\ \n3\ngap> PetersenGraph(IsMutableDigraph);\n\n]]>\n \n\n\n \ \n \n A digraph.\n \n If n is a positive integer \ and k is a non-negative\n integer less than n / 2, the\ n this operation returns\n the generalised Petersen graph GPG(\ n, k).
\n\n From\n https://en.wikipedia.org/wiki/Genera\ lized_Petersen_graph:\n
\n\n The generalized Petersen graph\ s are a family of cubic\n graphs formed by connecting the vertices of a reg\ ular polygon to the\n corresponding vertices of a star polygon. They includ\ e the Petersen graph\n and generalize one of the ways of constructing the P\ etersen graph. The\n generalized Petersen graph family was introduced in 19\ 50 by H. S. M.\n Coxeter and was given its name in 1969 by Mark Watkins.\n
\n\n If the optional first argument filt is present, then\ this should\n specify the category or representation the digraph being cre\ ated will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is , then the digraph will be immuta\ ble. \n If the optional first argument filt is not present, then is used by default.
\n\n See also\ .\n\n GeneralisedPeterse\ nGraph(7, 2);\n\ngap> \ GeneralisedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, 2);\n\ngap> IsIsomorphicDigraph(D, PetersenGrap\ h());\ntrue\ngap> GeneralisedPetersenGraph(IsMutableDigraph, 9, 4);\n\n]]>\n \n\n\n
\n\n\n \n Operators\n\n
Operators for di\ graphs\n \n digraph1 = digraph2<\ /C>\n \n = (for digraphs)\n returns true if digraph1 and digraph2 have the\ same\n vertices, and DigraphEdges(digraph1) =\n Digra\ phEdges(digraph2), up to some re-ordering of the edge\n lists\ .
\n\n Note that this operator does not compare the vertex labels\n \ of digraph1 and digraph2.\n \n\n <\ A>digraph1 < digraph2\n \n < (for digraphs)\n\n This operator returns <\ K>true if one of the following holds:\n\n \n \n \ The number n_1 of vertices in digraph1 is less than\n \ the number n_2 of vertices in digraph2;\n \n \n n_1 = n_2, and the number m_1 o\ f edges in\n digraph1 is less than the number m_2 of e\ dges in\n digraph2;\n \n \n \ n_1 = n_2, m_1 = m_2, and\n DigraphEdges(d\ igraph1) is less than\n DigraphEdges(digraph2)\ after having both of these sets\n have been sorted with respect to \ the lexicographical order.\n \n \n \n \ \n\n \n \n true or false.\n \n If \ super and sub are digraphs, then this operation returns\n \ true if sub is a subdigraph of super, and false\n \ if it is not.
\n\n A digraph sub is a subdigraph of a dig\ raph super if\n sub and super share the same number of\ vertices, and the\n collection of edges of super (including repeats\ ) contains the\n collection of edges of sub (including repeats). \n\n In other words, sub is a subdigraph of super if and on\ ly if\n DigraphNrVertices(sub) = DigraphNrVertices(super)<\ /C>, and\n for each pair of vertices i and j, there are at le\ ast as many\n edges of the form [i, j] in super as there are \ in sub.\n
\n\n g := Digraph([[2, 3],\ [1], [2, 3]]);\n\ngap> h := Digra\ ph([[2, 3], [], [2]]);\n\ngap> IsS\ ubdigraph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse\ngap> IsSubdigraph(Com\ pleteDigraph(4), CycleDigraph(4));\ntrue\ngap> IsSubdigraph(CycleDigraph(4), C\ hainDigraph(4));\ntrue\ngap> g := Digraph([[2, 2], [1]]);\n\ngap> h := Digraph([[2], [1]]);\n\ngap> IsSubdigraph(g, h);\ntrue\ngap> IsSubdigr\ aph(h, g);\nfalse]]>\n \n\n\n \n \n \n true or false.\n \n The operation IsUndirec\ tedSpanningTree returns true if the\n digraph sub is an u\ ndirected spanning tree of the digraph\n super, and the operation IsUndirectedSpanningForest returns\n true if the digraph sub\ is an undirected spanning forest of\n the digraph super.
\n\ \n An undirected spanning tree of a digraph super is a subdig\ raph\n of super that is an undirected tree (see \n and ). Note that a digraph who\ se is not connected has no u\ ndirected\n spanning trees (see ).
\n\ \n An undirected spanning forest of a digraph super is a\n \ subdigraph of super that is an undirected forest (see and ), and is not\n c\ ontained in any larger such subdigraph of super. Equivalently, an\n \ undirected spanning forest is a subdigraph of super whose connected\n \ components coincide with those of the of super (see ).
\n\n Note that an undirected spanning tree is an undirected spa\ nning forest that\n is connected.\n\n D := Comp\ leteDigraph(4);\n\ngap> tree := Di\ graph([[3], [4], [1, 4], [2, 3]]);\n\ngap> IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntrue\ngap> IsUndir\ ectedSpanningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(4);\n\ngap> IsSubdigraph(D, forest) and IsUndirected\ Forest(forest);\ntrue\ngap> IsUndirectedSpanningForest(D, forest);\nfalse\ngap\ > IsSubdigraph(tree, forest);\ntrue\ngap> D := DigraphDisjointUnion(CycleDigra\ ph(2), CycleDigraph(2));\n\ngap> I\ sUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUndirectedSpannin\ gForest(D, D);\ntrue]]>\n \n\n\n \n\n \n Attributes and operations\n
Vertices and edges\n \n \n A list of integers.\n \n Returns the vertices of the digraph digraph.
\n\n Note \ that the vertices of a digraph are always a range of\n positive integers fr\ om 1 to the number of vertices of the graph.\n 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> DigraphVer\ tices(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 100 \ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\n]]>\n \n\n\n \n \n An integer.\n \n Retu\ rns the number of vertices of the digraph digraph.
\n\n gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\ \", \"b\", \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Dig\ raph([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> \ [2, 7, 5, 3, 7]);\n\ngap> Digr\ aphNrVertices(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(100));\n100\ngap> \ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrVertices(D);\n3\n]]>\n \n\n\n \n \n \ A list of lists.\n \n DigraphEdges\ returns a list of edges of the digraph digraph,\n where each edge i\ s a pair of elements of of\n the form [s\ ource,range].\n
\n\n The entries of DigraphEdges(digra\ ph) are in one-to-one\n corresponence with the edges of digra\ ph. Hence\n DigraphEdges(digraph) is duplicate-fr\ ee if and only if\n digraph contains no multiple edges.
\n\n \ The entries of DigraphEdges are guaranteed to be sorted by their\n f\ irst component (i.e. by the source of each edge), but they are not\n necess\ arily then sorted by the second component.\n gr :=\ DigraphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n\ngap> edges := ShallowCopy(DigraphEdges(gr));; Sort(edg\ es);\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\ngap> DigraphEdges(D);\n[ [ 1, 2\ ], [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n \ \n \n \ An integer.\n \n This function returns the number o\ f edges of the digraph digraph.\n gr := Dig\ raph([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;\ngap> Digr\ aphNrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ngap> DigraphNrEdges(gr);\n3\ngap\ > D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrEdges(D);\n3\n]]>\n \n\n\n \n \n \ A list of vertices.\n \n This function ret\ urns a list of the sinks of the digraph\n digraph.\n A sink of a \ digraph is a vertex with out-degree zero.\n See .\n gr := Digraph([[3, 5, 2, 2], [3], [], [5\ , 2, 5, 3], []]);\n\ngap> Dig\ raphSinks(gr);\n[ 3, 5 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSinks(D);\n[ ]\n]]>\n \n\n\n \n \n A list of vertices.\n \n This function returns an immutable list of the sources of the d\ igraph\n digraph.\n A source of a digraph is a vertex with in-deg\ ree zero.\n See .\n 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]]>\n \n\n\n \n \n A list of positive integers, or fail.\n \ \n If digraph is a digraph whose only directed cycles a\ re loops, then\n DigraphTopologicalSort returns the vertices of d\ igraph ordered\n so that every edge's source appears no earlier in the \ list than its range.\n If the digraph digraph contains directed cycl\ es of length greater\n than 1, then this operation returns fail.\n
\n\n See section for the definition\n of a directed cycle, and the definition of a loop.\ \n\n
\n\n The method used for this attribute has complexity O(m+n\ ) where\n m is the number of edges (counting multiple edges as o\ ne) and\n n is the number of vertices in the digraph.
\n D := Digraph([\n> [2, 3], [], [4, 6], [5], [], [7, 8, 9],\ [], [], []]);\n\ngap> DigraphTopo\ logicalSort(D);\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ngap> D := Digraph(IsMutableDig\ raph, [[2, 3], [3], [4], []]);\n\nga\ p> DigraphTopologicalSort(D);\n[ 4, 3, 2, 1 ]\n]]>\n \ \n\n\n \n \n \n \n If digraph is a digraph, then the first op\ eration returns the label\n of the vertex i. The second operation ca\ n be used to set the label\n of the vertex i in digraph to th\ e arbitrary &GAP; object\n obj.
\n\n The label of a vertex c\ an be changed an arbitrary number of times. If no\n label has been set for \ the vertex i, then the default value is\n i.
\n\n If <\ A>digraph is a digraph created from a record with a component\n vert\ ices, then the labels of the vertices are set to the value of\n this co\ mponent.
\n\n Induced subdigraphs, and some other operations which creat\ e new digraphs from\n old ones, inherit their labels from their parents.\n\ \n D := DigraphFromDigraph6String(\"&DHUEe_\");\n<\ immutable digraph with 5 vertices, 11 edges>\ngap> DigraphVertexLabel(D, 3);\n\ 3\ngap> D := Digraph([\"a\", \"b\", \"c\"], [], []);\n\ngap> DigraphVertexLabel(D, 2);\n\"b\"\ngap> SetDigraphVerte\ xLabel(D, 2, \"d\");\ngap> DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := Induced\ Subdigraph(D, [1, 2]);\n\ngap> Digrap\ hVertexLabel(D, 2);\n\"d\"\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\",\ \"g\"], [], []);\n\ngap> DigraphVertex\ Label(D, 1);\n\"e\"\ngap> SetDigraphVertexLabel(D, 1, \"h\");\ngap> DigraphVer\ texLabel(D, 1);\n\"h\"\ngap> InducedSubdigraph(D, [1, 2]);\n\ngap> DigraphVertexLabel(D, 1);\n\"h\"\n]]>\n \ \n\n\n \n \n \n \n If digraph is a digraph, then \ DigraphVertexLabels returns a\n copy of the labels of the vertices in <\ A>digraph.\n SetDigraphVertexLabels can be used to set the label\ s of the vertices\n in digraph to the list of\n arbitrary &GAP; o\ bjects list.
\n\n The label of a vertex can be changed an arbitr\ ary number of times. If no\n label has been set for the vertex i, th\ en the default value is\n i.
\n\n If digraph is a digr\ aph created from a record with a component\n vertices, then the labe\ ls of the vertices are set to the value of\n this component.
\n\n Ind\ uced subdigraphs, and other operations which create new digraphs from\n old\ ones, inherit their labels from their parents.\n\n D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabels(D);\n[ 1 .. 5 ]\ngap> D := Digraph([\ \"a\", \"b\", \"c\"], [], []);\n\ngap\ > DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" ]\ngap> SetDigraphVertexLabel\ (D, 2, \"d\");\ngap> DigraphVertexLabels(D);\n[ \"a\", \"d\", \"c\" ]\ngap> D \ := InducedSubdigraph(D, [1, 3]);\n\ng\ ap> DigraphVertexLabels(D);\n[ \"a\", \"c\" ]\ngap> D := Digraph(IsMutableDigr\ aph, [\"e\", \"f\", \"g\"], [], []);\n\ \ngap> SetDigraphVertexLabels(D, [\"h\", \"i\", \"j\"]);\ngap> DigraphVertexLa\ bels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"h\", \"j\"\ ]\n]]>\n \n\n\n \n \n \n \n If digra\ ph is a digraph without multiple edges, then the first\n operation retu\ rns the label of the edge from vertex i to vertex\n j. The s\ econd operation can be used to set the label of the edge\n between vertex <\ A>i and vertex j to the arbitrary &GAP; object\n obj.
\n\n The label of an edge can be changed an arbitrary number of times. If \ no\n label has been set for the edge, then the default value is 1. <\ P/>\n\n Induced subdigraphs, and some other operations which create new dig\ raphs from\n old ones, inherit their edge labels from their parents.\n\n \ See also .\n\n D\ := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabel(D, 3, 1);\n1\ngap> SetDigraphEdgeLabel(D, \ 2, 5, [42]);\ngap> DigraphEdgeLabel(D, 2, 5);\n[ 42 ]\ngap> D := InducedSubdig\ raph(D, [2, 5]);\n\ngap> DigraphEd\ geLabel(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[ 17\ 29 ]\ngap> InducedSubdigraph(D, [4, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 1729 ]\n]]>\n \n\n\n \n \n \n \n \n If digraph is a digraph without multiple edges\ , then\n DigraphEdgeLabels returns a copy of the labels of the edges\ \n in digraph as a list of lists labels such that\n lab\ els[i][j] is the label on the edge from vertex i\n to vertex \ OutNeighbours(digraph)[i][j].\n\n SetDigraphEdgeLabels can be us\ ed to set the labels of the edges in\n digraph without multiple edge\ s to the list labels of lists of\n arbitrary &GAP; objects such that\ list[i][j] is the label on the edge\n from vertex i to the v\ ertex OutNeighbours(digraph>[i][j].\n\n Alternatively SetDigraphE\ dgeLabels can be called with binary function\n func that as its \ second argument that when passed two vertices i\n and j retur\ ns the label for the edge between vertex i and vertex\n j. \n\n The label of an edge can be changed an arbitrary number of times. If\ no\n label has been set for an edge, then the default value is 1. <\ P/>\n\n Induced subdigraphs, and some other operations which create new dig\ raphs\n from old ones, inherit their labels from their parents.\n\n 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> DigraphEdgeLabels(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 ] ]\ngap> OutNeighbours(D);\n[ [ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := Co\ mpleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1\ ], [ 1, 1 ], [ 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 4, \"a\");\ngap> Digr\ aphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, \"a\", 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 \ ] ]\ngap> InducedSubdigraph(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\ ]]>\n \n\n\n \n \n A list of edges.\n \n DigraphInEdges returns the list of all e\ dges of digraph\n which have vertex as their range.\n D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphInEdges(D, 2);\n[ [ 1, \ 2 ], [ 1, 2 ] ]\n]]>\n \n\n\n \n \n \ A list of edges.\n \n DigraphOutEdges return\ s the list of all edges of digraph\n which have vertex as the\ ir source.\n D := Digraph([[2, 2], [3, 3], [4, 4],\ [1, 1]]);\n\ngap> DigraphOut\ Edges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] ]\n]]>\n \n\n\n \n \n \n true or false.\n \n In the first\ form, this function returns true if and only if the list\n list<\ /A> specifies an edge in the digraph digraph. Specifically,\n this \ operation returns true if list is a pair of positive\n intege\ rs where list[1] is the source and list[2]\n is\ the range of an edge in digraph, and false otherwise.
\n\n \ The second form simply returns true if [u, v] i\ s\n an edge in digraph, and false otherwise.\n\n D := Digraph([[2, 2], [6], [], [3], [], [1]]);\n\ngap> IsDigraphEdge(D, [1, 1]);\nfalse\ngap> I\ sDigraphEdge(D, [1, 2]);\ntrue\ngap> IsDigraphEdge(D, [1, 8]);\nfalse\n]]>\n \n\n\n \n \n \n \n\ true or false.\n \n If \ digraph is a digraph and list is a list of pairs of\n vertices o\ f digraph, then IsMatching returns true if\n list is a matching of digraph. The operations\n IsMaximalMatching<\ /C> and IsPerfectMatching return true if\n list is a m\ aximal, or perfect, matching of digraph,\n respectively. Otherwise,\ these operations return false.
\n\n A matching M \ of a digraph digraph is a subset of the\n edges of digraph, i\ .e. DigraphEdges(digraph), such\n that no pair of distinct ed\ ges in M are incident to the same vertex\n of digraph. Note \ that this definition allows a matching to contain\n loops. See . The matching M is\n maximal if it is\ contained in no larger matching of the digraph, and\n is perfect if\ every vertex of the digraph is incident to an edge in\n the matching. Eve\ ry perfect matching is maximal.\n\n D := Digraph([\ [2], [1], [2, 3, 4], [3, 5], [1]]);\n\ngap> IsMatching(D, [[2, 1], [3, 2]]);\nfalse\ngap> edges := [[3, 2]];;\ng\ ap> IsMatching(D, edges);\ntrue\ngap> IsMaximalMatching(D, edges);\nfalse\ngap\ > edges := [[5, 1], [3, 3]];;\ngap> IsMaximalMatching(D, edges);\ntrue\ngap> I\ sPerfectMatching(D, edges);\nfalse\ngap> edges := [[1, 2], [3, 3], [4, 5]];;\n\ gap> IsPerfectMatching(D, edges);\ntrue\n]]>\n \n\n\n
\n\n
Neighbours and degree\n \n \n \n A square\ matrix of non-negative integers.\n \n This functio\ n returns the adjacency matrix mat of the digraph\n digraph.\ \n The value of the matrix entry mat[i][j] is the number of edges\n \ in digraph with source i and range j. If digraph\n has no vertices, then the empty list is returned.
\n\n The funct\ ion AdjacencyMatrix returns an immutable list of\n lists, whereas th\ e function AdjacencyMatrixMutableCopy returns a copy\n of Adjacen\ cyMatrix that is a mutable list of mutable lists.
\n\n 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]]>\n \n\n\ \n \n \n A polynomial with integer coefficients.\n \n This function returns the characteristic polynomial of the digraph\n \ digraph. That is it returns the characteristic polynomial\n of \ the adjacency matrix of the digraph digraph\n 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+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\ngap> CharacteristicPolynom\ ial(D);\nx_1^5-10*x_1^3-20*x_1^2-15*x_1-4\ngap> D := CycleDigraph(IsMutableDig\ raph, 3);\n\ngap> CharacteristicPoly\ nomial(D);\nx_1^3-1\n]]>\n \n\n\n \n \n \n A squ\ are matrix of booleans.\n \n If digraph is a\ digraph with a positive number of vertices\n n, then BooleanAdja\ cencyMatrix(digraph)\n returns the boolean adjacency matr\ ix mat of digraph. The\n value of the matrix entry mat[j]\ [i] is true if and only if\n there exists an edge in digraph<\ /A> with source j and range\n i. If digraph has no ve\ rtices, then the empty list is\n returned.
\n\n Note that the boolea\ n adjacency matrix loses information about multiple\n edges.
\n\n T\ he attribute BooleanAdjacencyMatrix returns an immutable list of\n i\ mmutable lists, whereas the function\n BooleanAdjacencyMatrixMutableCopy\ returns a copy of the\n BooleanAdjacencyMatrix that is a mutabl\ e list of mutable lists.
\n gr := Digraph([[3,\ 4], [2, 3], [1, 2, 4], [4]]);\n\n\ gap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ false, false, true, tr\ ue ],\n [ false, true, true, false ],\n [ true, true, false, tr\ ue ],\n [ false, false, false, true ] ]\ngap> gr := CycleDigraph(4);;\ng\ ap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ false, true, false, fals\ e ],\n [ false, false, true, false ],\n [ false, false, false, tru\ e ],\n [ true, false, false, false ] ]\ngap> BooleanAdjacencyMatrix(Empt\ yDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> PrintArray(BooleanAdjacencyMatrix(D));\n\ [ [ false, true, false ],\n [ false, false, true ],\n [ true, fal\ se, false ] ]\n]]>\n \n\n\n \n \n \ A function.\n \n If digraph is a digraph, th\ en DigraphAdjacencyFunction returns\n a function which takes two int\ eger parameters x, y and returns\n true if there exists an ed\ ge from vertex x to vertex y\n in digraph and false\ if not.\n\n digraph := Digraph([[1, 2], [3], \ []]);\n\ngap> foo := DigraphAdjace\ ncyFunction(digraph);\nfunction( u, v ) ... end\ngap> foo(1, 1);\ntrue\ngap> f\ oo(1, 2);\ntrue\ngap> foo(1, 3);\nfalse\ngap> foo(3, 1);\nfalse\ngap> gr := Di\ graph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \"b\"],\n> \ [\"b\", \"a\", \"a\"]);\n\ngap> foo := DigraphAdjacencyFunction(gr);\nfunction( u, v ) ... en\ d\ngap> foo(1, 2);\ntrue\ngap> foo(3, 2);\nfalse\ngap> foo(3, 1);\nfalse\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> foo := DigraphAdjacencyFunction(D);\nfunction( u, v ) ... end\nga\ p> foo(1, 2);\ntrue\ngap> foo(2, 1);\nfalse\n]]>\n \n\ \n\n \n \n \n A list of p\ ositive integers.\n \n DigraphRange and D\ igraphSource return the range and source of\n the digraph digraph. More precisely, position i in\n DigraphRange(digraph) is the range of the ith edge of\n digraph.\n\n 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 := Cyc\ leDigraph(IsMutableDigraph, 3);\n\ng\ ap> DigraphRange(D);\n[ 2, 3, 1 ]\ngap> DigraphSource(D);\n[ 1, 2, 3 ]\n]]>\n \n\n\n \n \n \n \n \n The adjacencies of\ a digraph.\n \n This function returns the list \ out of out-neighbours of each vertex\n of the digraph digraph.\n\ More specifically, a vertex j appears in out[i] each time\n \ there exists an edge with source i and range j in\n dig\ raph.
\n\n The function OutNeighbours returns an immutable l\ ist of\n lists, whereas the function OutNeighboursMutableCopy return\ s a copy\n of OutNeighbours which is a mutable list of mutable lists\ .
\n\n 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\ngap> OutNeighbours(gr);\n[ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutN\ eighboursMutableCopy(gr);\n[ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> D := Cycl\ eDigraph(IsMutableDigraph, 3);\n\nga\ p> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\n]]>\n \ \n\n\n \n \n \n \n \n A list of lists of vertices.\n \n This function returns the list inn of in-neighbours of eac\ h vertex\n of the digraph digraph.\n More specifically, a vertex \ j appears in inn[i] each time\n there exists an edge with sou\ rce j and range i in\n digraph.
\n\n The functi\ on InNeighbours returns an immutable list of\n lists, whereas the fu\ nction InNeighboursMutableCopy returns a copy\n of InNeighbours which is a mutable list of mutable lists.
\n\n Note that each entry \ of inn is sorted into ascending order.
\n\n 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]]>\n \n\n\n \n \n \n \n \ A list of non-negative integers.\n \n\n Given a dig\ raph digraph with n vertices, the function\n OutDegrees returns an immutable list out of length n, such that\n for \ a vertex i in digraph, the value of out[i] is the\n ou\ t-degree of vertex i.\n See .
\ \n\n The function OutDegreeSequence returns the same list,\n afte\ r it has been sorted into non-increasing order.
\n\n The function Ou\ tDegreeSet returns the same list, sorted into\n increasing order with d\ uplicate entries removed.
\n\n D := Digraph([[\ 1, 3, 2, 2], [], [2, 1], []]);\n\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(IsMutableDi\ graph, 3);\n\ngap> OutDegrees(D);\n[\ 1, 1, 1 ]\n]]>\n \n\n\n \ \n \n \n \n A list of non-negative integers.\n \n\n Given a\ digraph digraph with n vertices, the function\n InDegrees\ returns an immutable list inn of length n, such that\n f\ or a vertex i in digraph, the value of inn[i] is\n the\ in-degree of vertex i.\n See .
\ \n\n The function InDegreeSequence returns the same list,\n after\ it has been sorted into non-increasing order.
\n\n The function InD\ egreeSet returns the same list, sorted into\n increasing order with dup\ licate entries removed.
\n\n D := Digraph([[1,\ 3, 2, 2, 4], [], [2, 1, 4], []]);\n\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(IsMutableD\ igraph, 3);\n\ngap> InDegrees(D);\n[\ 1, 1, 1 ]\n]]>\n \n\n\n \ \n \n The\ non-negative integer.\n \n This operation returns \ the out-degree of the vertex vertex in the\n digraph digraph.\ \n The out-degree of vertex is the number of edges in digraph\ \n whose source is vertex.\n
\n D\ := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> OutDegreeOfVertex(D, 1)\ ;\n3\ngap> OutDegreeOfVertex(D, 2);\n2\ngap> OutDegreeOfVertex(D, 3);\n4\ngap>\ OutDegreeOfVertex(D, 4);\n7\n]]>\n \n\n\ \n \n \n \n A list of vertices.\n \n\n This operation ret\ urns the list out of vertices of the digraph\n digraph.\n \ A vertex i appears in the list out each time there exists an\n \ edge with source vertex and range i in digraph; in\n \ particular, this means that out may contain duplicates.
\n D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1,\ 2, 2, 1, 2, 2]]);\n\ngap> O\ utNeighboursOfVertex(D, 1);\n[ 2, 2, 3 ]\ngap> OutNeighboursOfVertex(D, 3);\n[\ 2, 2, 3 ]\n]]>\n \n\n\n \ \n \n A no\ n-negative integer.\n \n This operation returns the\ in-degree of the vertex vertex in the\n digraph digraph.\n \ The in-degree of vertex is the number of edges in digraph\n \ whose range is vertex.\n 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> InDegr\ eeOfVertex(D, 2);\n9\ngap> InDegreeOfVertex(D, 3);\n0\ngap> InDegreeOfVertex(D\ , 4);\n2\n]]>\n \n\n\n \n\ \n \n A list of postit\ ive vertices.\n \n\n This operation returns the lis\ t inn of vertices of the digraph\n digraph.\n A vertex \ i appears in the list inn each time there exists an\n edge with\ source i and range vertex in digraph; in\n particular\ , this means that inn may contain duplicates.
\n\n D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1,\ 2, 2]]);\n\ngap> InNeighbou\ rsOfVertex(D, 1);\n[ 2, 4, 4, 4 ]\ngap> InNeighboursOfVertex(D, 2);\n[ 1, 1, 3\ , 3, 4, 4, 4, 4 ]\n]]>\n \n\n\n \n \n A list of\ vertices.\n \n If digraph is a digraph, the\ n DigraphLoops returns the list\n consisting of the of digraph\n at which there is a loop. See .
\n\n D := Digraph([[2]\ , [3], []]);\n\ngap> DigraphHasLoo\ ps(D);\nfalse\ngap> DigraphLoops(D);\n[ ]\ngap> D := Digraph([[3, 5], [1], [2\ , 4, 3], [4], [2, 1]]);\n\ngap> Di\ graphLoops(D);\n[ 3, 4 ]\ngap> D := Digraph(IsMutableDigraph, [[1], [1]]);\n\ngap> DigraphLoops(D);\n[ 1 ]\n]]>\n \n\n\n \n \n \n A positive integer or fail\n \n If the fi\ rst argument is a partial order digraph then these operations return the meet, or\n the join, of the two input\ vertices. If the meet (or join) is does not\n exist then fail is re\ turned. The meet (or join) is guaranteed to\n exist when the first argument\ satisfies\n (or\n ) - see the documentation for these\n propert\ ies for the definition of the meet (or the join).\n\n D := Digraph([[1], [1, 2], [1, 3], [1, 2, 3, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 2, 3);\n4\ng\ ap> PartialOrderDigraphJoinOfVertices(D, 2, 3);\n1\ngap> PartialOrderDigraphMe\ etOfVertices(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);\nfail\ \ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\nfail\ngap> PartialOrderDig\ raphMeetOfVertices(D, 1, 2);\nfail\ngap> PartialOrderDigraphJoinOfVertices(D, \ 1, 2);\nfail\n]]>\n \n\n\n \n \n A square matri\ x of non-negative integers.\n \n This function retu\ rns the outdegree matrix mat of the digraph\n digraph. The va\ lue of the ith diagonal matrix entry is the\n outdegree of the verte\ x i in digraph. All off-diagonal entries \n are 0.\n \ If digraph has no vertices, then the empty list is returned.
\n\n\ See for more information.\n 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]]>\n \n\n\ \n \n \n A square matrix of integers.\n \n This functio\ n returns the outdegree Laplacian matrix mat of the\n digraph dig\ raph. The outdegree Laplacian matrix is defined as\n DegreeMatrix(di\ graph) - AdjacencyMatrix(digraph). If\n digraph has no vertices\ , then the empty list is returned.
\n\n See and for more\n information.\n 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\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(LaplacianMatrix(D));\n[ [ 1, -1, 0 ],\n [ 0, 1, \ -1 ],\n [ -1, 0, 1 ] ]\n]]>\n \n\n\ \n
\n\n
Reachability and connectivity\ \n \n \n An integer or fail.\n \n This function \ returns the diameter of the digraph digraph.\n
\n\n If a digr\ aph digraph is strongly connected and has at least 1\n vertex, then \ the diameter is the maximum shortest distance between\n any pair of \ distinct vertices. Otherwise then the diameter of\n digraph is undef\ ined, and this function returns the value\n fail.
\n\n See .
\n D :\ = Digraph([[2], [3], [4, 5], [5], [1, 2, 3, 4, 5]]);\n\ngap> DigraphDiameter(D);\n3\ngap> D := ChainDigraph(2);\ \n\ngap> DigraphDiameter(D);\nfail\ng\ ap> IsStronglyConnectedDigraph(D);\nfalse\ngap> D := GeneralisedPetersenGraph(\ IsMutableDigraph, 6, 2);\n\ngap> D\ igraphDiameter(D);\n4\n]]>\n \n\n\n <\ ManSection>\n \n \n \n An integer o\ r fail\n \n If there is a directed path in t\ he digraph digraph between vertex\n u and vertex v, th\ en this operation returns the length of the\n shortest such directed path. \ If no such directed path exists, then this\n operation returns fail\ . See section for the \ definition of a directed path.
\n\n If the second form is used, then list should be a list of length two,\n containing two positive integer\ s which correspond to the vertices u\n and v.
\n\n Not\ e that as usual, a vertex is considered to be at distance 0 from\n itself .\
\n\n If the third form is used, then list1 and list2 are \ both lists\n of vertices. The shortest directed path between list1 a\ nd\n list2 is then the length of the shortest directed path which st\ arts\n with a vertex in list1 and terminates at a vertex in list2\ , if\n such directed path exists. If list1 and list2 have\ non-empty\n intersection, the operation returns 0.
\n D := Digraph([[2], [3], [1, 4], [1, 3], [5]]);\n\ngap> DigraphShortestDistance(D, 1, 3);\n2\ng\ ap> DigraphShortestDistance(D, [3, 3]);\n0\ngap> DigraphShortestDistance(D, 5,\ 2);\nfail\ngap> DigraphShortestDistance(D, [1, 2], [4, 5]);\n2\ngap> DigraphS\ hortestDistance(D, [1, 3], [3, 5]);\n0\n]]>\n \n\n\n \n \n A square matrix.\n \n If digraph is a digraph with n vertices, then this\n function retu\ rns an n \\times n matrix mat, where each entry is\n either a\ non-negative integer, or fail. If n = 0, then an\n empty li\ st is returned.
\n\n If there is a directed path from vertex i t\ o vertex j, then\n the value of mat[i][j] is the length of th\ e shortest such directed\n path. If no such directed path exists, then the \ value of mat[i][j] is\n fail. We use the convention that the\ distance from every vertex to\n itself is 0, i.e. mat[i][i] = 0<\ /C> for all vertices i.\n
\n\n The method used in this functi\ on is a version of the Floyd-Warshall\n algorithm, and has complexity O(\ n^3).\n\n D := Digraph([[1, 2], [3], [1, 2], [\ 4]]);\n\ngap> mat := DigraphShorte\ stDistances(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\ngap> DigraphShortestDistances(D);\n[ [ 0\ , 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]\n]]>\n \n\n\n \n \n An integer, or infinity.\n \ \n If digraph is a digraph and v is a vertex in \ digraph,\n then this operation returns the length of the longest dir\ ected walk in\n digraph which begins at vertex v. See sectio\ n for the definitions \ of directed\n walk, directed cycle, and loop.\n
\n\n \n \ \n If there exists a directed walk starting at vertex v\ \n which traverses a loop or a directed cycle,\n then we conside\ r there to be a walk of infinite length from v\n (realised by re\ peatedly traversing the loop/directed cycle),\n and so the result is infinity.\n To disallow walks using loops, try using\n :
\n\n DigraphLongestDistanceFromVe\ rtex(DigraphRemoveLoops(digraph,v)).\n \n\n \ \n Otherwise, if all directed walks\n starting at vertex \ v have finite length,\n then the length of the longest such walk\ is returned.\n \n \n\n Note that the result is 0<\ /C> if and only if v is a sink of\n digraph. See .\n\n D := Digraph([[2], [3, 4], [],\ [5], [], [6]]);\n\ngap> DigraphLo\ ngestDistanceFromVertex(D, 1);\n3\ngap> DigraphLongestDistanceFromVertex(D, 3)\ ;\n0\ngap> 3 in DigraphSinks(D);\ntrue\ngap> DigraphLongestDistanceFromVertex(\ D, 6);\ninfinity\n]]>\n \n\n\n \n \n \n A list of vertices\n \n T\ his operation returns the list of all vertices in digraph digraph\n \ such that the shortest distance to a vertex vertex is\n distance<\ /A> or is in the list distances.
\n\n digraph should be a\ digraph, vertex should be a positive\n integer, distance sho\ uld be a non-negative integer, and\n distances should be a list of n\ on-negative integers.\n\n D := Digraph([[2], [3], \ [1, 4], [1, 3]]);\n\ngap> DigraphD\ istanceSet(D, 2, [1, 2]);\n[ 3, 1, 4 ]\ngap> DigraphDistanceSet(D, 3, 1);\n[ 1\ , 4 ]\ngap> DigraphDistanceSet(D, 2, 0);\n[ 2 ]\n]]>\n \n\n\n \n \n An integer, or infinity.\n \n This attribute returns the girth of the digraph digraph.\ \n The girth of a digraph is the length of its shortest simple circu\ it.\n See section for the \ definitions\n of simple circuit, directed cycle, and loop.\n
\n\n \ If digraph has no directed cycles, then this function will return\n \ infinity. If digraph contains a loop, then this function will\ \n return 1.\n
\n\n In the worst case, the method used in \ this function is a version of the\n Floyd-Warshall algorithm, and has compl\ exity O(n ^ 3), where\n n is the number of vertices in\ digraph. If the digraph has\n known automorphisms [see ], then the performance\n is likely to be better.\n \n\n For symmetric digraphs, see also .
\n D := Digraph([[1], [1]]);\n\ngap> DigraphGirth(D);\n1\ngap> D := Digrap\ h([[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(IsMutableDigraph, 6, 2);\n\ngap> DigraphGirth(D);\n2\n]]>\n \n\ \n\n \n \n An integer, or infinity.\n \ \n This attribute returns the odd girth of the digraph digraph.\n The odd girth of a digraph is the length of its shortest simple\ \n circuit of odd length. See Section for the definitions of simple circuit, directed\n cycle, \ and loop.\n
\n\n If digraph has no directed cycles of odd len\ gth, then this function\n will return infinity, even if digraph has a directed cycle\n of even length. If digraph contains a loo\ p, then this function\n will return 1.\n
\n\n See also .
\n D := Digraph([[2\ ], [3, 1], [1]]);\n\ngap> DigraphO\ ddGirth(D);\n3\ngap> D := CycleDigraph(4);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := Digraph([[2], [3], [], \ [3], [4]]);\n\ngap> DigraphOddGirt\ h(D);\ninfinity\ngap> D := CycleDigraph(IsMutableDigraph, 4);\n\ngap> DigraphDisjointUnion(D, CycleDigraph(5));\n<\ mutable digraph with 9 vertices, 9 edges>\ngap> DigraphOddGirth(D);\n5\n]]>\n \n\n\n \n \n An integer or infinit\ y.\n \n If digraph is a symmetric digrap\ h, then this function returns the\n girth of digraph when treated as\ an undirected graph (i.e. each pair\n of edges [i, j] and [j, i]\ is treated as a single edge between\n i and j).
\n\n\ The girth of an undirected graph is the length of its shortest simp\ le\n cycle, i.e. the shortest non-trivial path starting and ending at the s\ ame\n vertex and passing through no vertex or edge more than once.
\n\n\ If digraph has no cycles, then this function will return\n in\ finity.
\n D := Digraph([[2, 4], [1, 3], [\ 2, 4], [1, 3]]);\n\ngap> DigraphUn\ directedGirth(D);\n4\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> DigraphUndirectedGirth(D);\ninfinity\ngap> \ D := Digraph([[1], [], [4], [3]]);\n\ngap> DigraphUndirectedGirth(D);\n1\ngap> D := GeneralisedPetersenGraph(IsM\ utableDigraph, 9, 2);\n\ngap> Digr\ aphUndirectedGirth(D);\n5\n]]>\n \n\n\n \ \n \ \n \n \ A record.\n \n This function returns the record \ wcc corresponding to the weakly\n connected components of the digraph <\ A>digraph. Two vertices of\n digraph are in the same weakly con\ nected component whenever they are\n equal, or there exists a directed path\ (ignoring the orientation of edges)\n between them. More formally, two ve\ rtices are in the same weakly connected\n component of digraph if an\ d only if they are in the same strongly\n connected component (see )\n of the of digraph.
\n\n The set of weakly connected \ components is a partition of\n the vertex set of digraph.
\n\n \ The record wcc has 2 components: comps and id.\n The\ component comps is a list of the weakly connected components\n of \ digraph (each of which is a list of vertices).\n The component id\ is a list such that the vertex i is an\n element of the weakly \ connected component comps[id[i]].
\n\n The method used in this f\ unction has complexity O(m+n), where\n m is the number of ed\ ges and\n n is the number of vertices in the digraph.
\n\n DigraphNrConnectedComponents(digraph) is simply a shortcut\n fo\ r Length(DigraphConnectedComponents(digraph).comps),\n and is\ no more efficient.\n\n gr := Digraph([[2], [3, 1]\ , []]);\n\ngap> DigraphConnectedCo\ mponents(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 := EmptyDigrap\ h(0);\n\ngap> DigraphConnectedCompone\ nts(gr);\nrec( comps := [ ], id := [ ] )\ngap> D := CycleDigraph(IsMutableDi\ graph, 2);\n\ngap> G := CycleDigraph\ (3);\n\ngap> DigraphDisjointUnion(D, \ G);\n\ngap> DigraphConnectedComponen\ ts(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n]]\ >\n \n\n\n \n \n A list of\ vertices.\n \n If vertex is a vertex in the\ digraph digraph, then\n this operation returns the connected compon\ ent of vertex\n in digraph.\n See for more information.\n D := Di\ graph([[3], [2], [1, 2], [4]]);\n\ \ngap> DigraphConnectedComponent(D, 3);\n[ 1, 2, 3 ]\ngap> DigraphConnectedCom\ ponent(D, 2);\n[ 1, 2, 3 ]\ngap> DigraphConnectedComponent(D, 4);\n[ 4 ]\n]]><\ /Example>\n \n\n\n \n \n \n A record.\n \n This function returns the record scc corres\ ponding to the strongly\n connected components of the digraph digraph. Two vertices of\n digraph are in the same strongly connected com\ ponent whenever they\n are equal, or there is a directed path from each ver\ tex to the other. The\n set of strongly connected components is a partitio\ n of the vertex set of\n digraph.\n
\n\n The record scc\ has 2 components: comps and id.\n The component comps\ is a list of the strongly connected components\n of digraph (ea\ ch of which is a list of vertices).\n The component id is a list suc\ h that the vertex i is an\n element of the strongly connected compon\ ent comps[id[i]].
\n\n The method used in this function is a non\ -recursive version of Gabow's\n Algorithm and has com\ plexity O(m+n) where\n m is the number of edges (counting mul\ tiple edges as one) and\n n is the number of vertices in the digraph\ .
\n\n DigraphNrStronglyConnectedComponents(digraph) is s\ imply a\n shortcut for\n Length(DigraphStronglyConnectedComponents(digraph).comps),\n and is no more efficient.\n\n gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphStronglyConnectedComponents(gr);\nrec( comps := [ [ \ 3 ], [ 1, 2 ] ], id := [ 2, 2, 1 ] )\ngap> DigraphNrStronglyConnectedComponent\ s(gr);\n2\ngap> D := DigraphDisjointUnion(CycleDigraph(4), CycleDigraph(5));\n\ \ngap> DigraphStronglyConnectedCom\ ponents(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);\n\ 2\ngap> 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]]>\n \n\n\n \n \n A list of vertices.\n <\ Description>\n If vertex is a vertex in the digraph digraph, \ then\n this operation returns the strongly connected component of vertex\ \n in digraph.\n See for more information.\n D := Digraph([[\ 3], [2], [1, 2], [3]]);\n\ngap> Di\ graphStronglyConnectedComponent(D, 3);\n[ 1, 3 ]\ngap> DigraphStronglyConnecte\ dComponent(D, 2);\n[ 2 ]\ngap> DigraphStronglyConnectedComponent(D, 4);\n[ 4 ]\ \n]]>\n \n\n\n \n \n A pair of lists of \ vertices, or fail.\n \n If digraph is\ a bipartite digraph, i.e. if it satisfies , then DigraphBicomponents returns a\n pair of bicomponents of\ digraph. Otherwise,\n DigraphBicomponents returns fail.
\n\n For a bipartite digraph, the vertices can be partitioned into tw\ o non-empty\n sets such that the source and range of any edge are in distin\ ct sets. The\n parts of this partition are called bicomponents of digraph.\n Equivalently, a pair of bicomponents of digraph cons\ ists of the\n color-classes of a 2-coloring of digraph.
\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 for more information.\n D := CycleDigraph(3);\n\n\ gap> DigraphBicomponents(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 ]\ ]\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 2 ], [ 3,\ 4, 5 ] ]\n]]>\n \n\n\n \ \n \n A list of \ vertices.\n \n\n A connected digraph is biconnec\ ted if it is still connected (in the\n sense of ) when any vertex is removed. \n If the digraph digraph i\ s not biconnected but is connected, then any\n vertex v of digrap\ h whose removal makes the resulting digraph\n disconnected is called an\ articulation point.
\n\n ArticulationPoints returns a lis\ t of the articulation points of\n digraph, if any, and, in particula\ r, returns the empty list if\n digraph is not connected.
\n\n \ Multiple edges and loops are ignored by this method.
\n\n The method u\ sed in this operation has complexity O(m+n) where\n m is the \ number of edges (counting multiple edges as one, and not\n counting loops) \ and n is the number of vertices in the digraph.\n\n See also .\n ArticulationPoints(Cy\ cleDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [3, 5], [4], [2], [6], [1], \ []]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap> ArticulationPoints(ChainDi\ graph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(NullDigraph(5));\n[ ]\ngap> \ D := ChainDigraph(IsMutableDigraph, 4);\n\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n]]>\n \ \n\n\n \n \n An integer.\n \n This function \ returns the period of the digraph digraph.
\n\n If a digraph d\ igraph has at least one directed cycle, then the period\n is the greate\ st positive integer which divides the lengths of all directed\n cycles of <\ A>digraph. If digraph has no directed cycles, then\n this funct\ ion returns 0. See section for the definition of a directed cycle.
\n\n A digraph wi\ th a period of 1 is said to be aperiodic. See\n .
\n 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\ngap> DigraphPeriod(D);\n1\ \ngap> D := ChainDigraph(2);\n\ngap> \ DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\ngap> D := GeneralisedPe\ tersenGraph(IsMutableDigraph, 5, 2);\n\ngap> DigraphPeriod(D);\n1\n]]>\n \n\n\n \n \n A matrix.\n \n If \ digraph is a digraph with n vertices, then\n this operation r\ eturns an n \\times n matrix mat containing\n the output of a\ generalised version of the Floyd-Warshall algorithm,\n applied to digra\ ph.
\n\n The operation DigraphFloydWarshall is customised by\ the arguments\n func, nopath, and edge.\n The argu\ ments nopath and edge can be arbitrary &GAP; objects.\n The a\ rgument func must be a function which accepts 4 arguments:\n the mat\ rix mat, followed by 3 postive integers. The function\n func \ is where the work to calculate the desired outcome must be\n performed.
\n\n This method initialises the matrix mat by setting entry\n <\ C>mat[i][j] to equal edge if there is an edge with source\n i\ and range j, and by setting entry mat[i][j] to equal\n <\ A>nopath otherwise.\n The final part of DigraphFloydWarshall the\ n calls the function\n func inside three nested for loops, over the \ vertices of\n digraph:
\n\n
\n\n The matrix mat is then returned as the result. An ex\ ample of using\n DigraphFloydWarshall to calculate the shortest (non\ -zero) distances\n between the vertices of a digraph is shown below:
\n\ \n 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\ng\ ap> 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]]>\n \n\n\n \ \n \n true\ or false.\n \n This operation returns tr\ ue if there exists a non-trivial directed\n walk from vertex u t\ o vertex v in the digraph digraph,\n and false if ther\ e does not exist such a directed walk. See section\n for the definition of a\n non-trivial directed \ walk.\n
\n\n The method for IsReachable has worst case comple\ xity of O(m +\n n) where m is the number of edges and n<\ /M> the number of\n vertices in digraph.\n\n\ D := Digraph([[2], [3], [2, 3]]);\n\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]]>\ \n \n\n\n \n \n A pair of lists, or fail.\n \n If there exists a non-trivial directed path (or a\ non-trivial cycle, in the\n case that u = v) from ver\ tex u to vertex\n v in the digraph digraph, then this \ operation returns such a\n directed path (or directed cycle). Otherwise, th\ is operation returns\n fail. See Section for the\n definition of a directed path and a directed \ cycle.\n
\n\n A directed path (or directed cycle) of non-zero length\ n-1,\n (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n),\n is repr\ esented by a pair of lists [v,a] as follows:\n\n \n \n v is the list [v_1, v_2, ..., v_n].\n \n \ \n a is the list of positive integers [a_1, a_2, .\ .., a_{n-1}]\n where for each each i < n, a_i is t\ he position of\n v_{i+1} in\n OutNeighboursOfVertex(digraph,v_i)\n corresponding to the edge \ e_i. This is can be useful if the\n position of a vertex in a l\ ist of out-neighours is significant, for\n example in orbit digraphs.\n\ \n \n\n The method for DigraphPath has worst \ case complexity of O(m +\n n) where m is the number of edge\ s and n the number of\n vertices in digraph.\n\n D := Digraph([[2], [3], [2, 3]]);\n\ngap> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ngap> Dig\ raphPath(D, 2, 1);\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ 2 ] ]\ngap\ > DigraphPath(D, 1, 1);\nfail\n]]>\n \n\ \n\n \n \n A pair of lists, or fail.\n \n\ Returns the shortest directed path in the digraph digraph from the vertex\ \n u to the vertex v, if such a path exists. If u =\ \n v, then the shortest non-trivial cycle is returned, again, if\ it\n exists. Otherwise, this operation returns fail. See Section\n\ for the definition of a dire\ cted\n path and a directed cycle.\n
\n\n See for details on the output.\n\n The method for DigraphShortestPat\ h has worst case complexity of\n O(m + n) where m is the \ number of edges and n the\n number of vertices in digraph.\n\ \n 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> DigraphShortest\ Path(D, 1, 1);\n[ [ 1, 1 ], [ 1 ] ]\n]]>\n \n\n\n \n \n An iterator.\n \n If digraph is a digraph or a list of adjacencies which defines a\n digraph - see - then this operation returns an\n iterator of \ the non-trivial directed paths (or directed cycles, in the case\n that u\ = v) in digraph from the vertex u\n to the\ vertex v.\n
\n\n See for more in\ formation about the repesentation\n of a directed path or directed cycle wh\ ich is used, and see for mor\ e information about iterators.\n See Section for the definition of\n a directed path and a directed cycl\ e.\n
\n\n 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> NextIter\ ator(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]]>\n \n\n\n \n \n A list of lists of vertices.\n \n \ If digraph is a digraph, then DigraphAllSimpleCircuits\n re\ turns a list of the simple circuits in digraph.
\n\n See \ section for the definition\n \ of a simple circuit, and related notions. Note that a loop is a simple\n \ circuit.
\n\n For a digraph without multiple edges, a simple circuit i\ s uniquely\n determined by its subsequence of vertices. However this is not\ the case for\n a multidigraph. The attribute DigraphAllSimpleCircuits<\ /C> ignores\n multiple edges, and identifies a simple circuit using only it\ s subsequence\n of vertices. For example, although the simple circuits (\ v, e, v) and\n (v, e', v) (for distinct edges e and e'\ ) are\n mathematically distinct, DigraphAllSimpleCircuits consid\ ers them to\n be the same.
\n\n With this approach, a directed circu\ it of length n can be determined\n by a list of its first n v\ ertices. Thus a simple circuit (v_1,\n e_1, v_2, e_2, ..., e_{n-1}, v_\ n, e_{n+1}, v_1) can be represented as\n the list [v_1, \\ldots, v_n\ ], or any cyclic permutation thereof. For\n each simple circuit of \ digraph,\n DigraphAllSimpleCircuits(digraph) includes pre\ cisely one such\n list to represent the circuit.
\n\n 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 := Digrap\ h([[1, 1]]);\n\ngap> DigraphAll\ SimpleCircuits(D);\n[ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n<\ mutable digraph with 3 vertices, 3 edges>\ngap> DigraphAllSimpleCircuits(D);\n\ [ [ 1, 2, 3 ] ]\n]]>\n \n\n\n \n \n A list of vertices, or fail.\n \n If \ digraph is a digraph, then DigraphLongestSimpleCircuit\n returns\ the longest simple circuit in digraph. See section\n for the definition of simple\n ci\ rcuit, and the definition of length for a simple circuit.
\n\n This attr\ ibute computes\n DigraphAllSimpleCircuits(digraph) to \ find all the\n simple circuits of digraph, and returns one of maxima\ l length. A\n simple circuit is represented as a list of vertices, in the \ same way as\n described in .
\n\ \n If digraph has no simple circuits, then this attribute returns\n \ fail. If digraph has multiple simple circuits of maximal\n \ length, then this attribute returns one of them.
\n\n D := Digraph([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSi\ mpleCircuit(D);\n[ 4, 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSi\ mpleCircuit(D);\nfail\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> D\ igraphLongestSimpleCircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGrap\ h(IsMutableDigraph, 4, 1);\n\ngap> \ DigraphLongestSimpleCircuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n]]>\n \ \n\n\n \n \n A list.\n \n\ This operation returns a list list such that list[i] is the\ \n list of vertices whose minimum distance from the vertex vertex in\ \n digraph is i - 1. Vertex vertex is\n assumed to \ be at distance 0 from itself.\n D := Comple\ teDigraph(4);;\ngap> DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4 ] ]\n]]>\n \n\n\n \n \n A non-negative integer, or fail\ .\n \n If digraph is a symmetric digraph\ without multiple edges - see\n and - then\n this attribute returns the degeneracy o\ f digraph.
\n\n The degeneracy of a digraph is the least integer\ k such\n that every induced of digraph contains a vertex who\ se number of\n neighbours (excluding itself) is at most k. Note that\ this means\n that loops are ignored.
\n\n If digraph is not s\ ymmetric or has multiple edges then this\n attribute returns fail.\n\ D := DigraphSymmetricClosure(ChainDigraph(5));;\n\ gap> DigraphDegeneracy(D);\n1\ngap> D := CompleteDigraph(5);;\ngap> DigraphDeg\ eneracy(D);\n4\ngap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]\ );\n\ngap> DigraphDegeneracy(D);\ \n1\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 10, 3);\n\ngap> DigraphDegeneracy(D);\n3\n]]>\n \n\n\n \n \n A list of integers, or f\ ail.\n \n If digraph is a digraph for wh\ ich\n DigraphDegeneracy(digraph) is a non-negative int\ eger\n k - see - then this attribu\ te\n returns a degeneracy ordering of the vertices of the vertices of\n \ digraph.
\n\n A degeneracy ordering of digraph is a list <\ C>ordering of the\n vertices of digraph ordered such that for an\ y\n position i of the list, the vertex ordering[i] has at mos\ t\n k neighbours in later position of the list.
\n\n If Dig\ raphDegeneracy(digraph) returns fail,\n then this \ attribute returns fail.\n D := DigraphSymme\ tricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracyOrdering(D);\n[ 5, 4, 3,\ 2, 1 ]\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracyOrdering(D);\n[\ 5, 4, 3, 2, 1 ]\ngap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], [\ ]]);\n\ngap> DigraphDegeneracyOrd\ ering(D);\n[ 1, 6, 5, 2, 4, 3 ]\ngap> D := GeneralisedPetersenGraph(IsMutableD\ igraph, 3, 1);\n\ngap> DigraphDegen\ eracyOrdering(D);\n[ 6, 5, 4, 1, 3, 2 ]\n]]>\n \n\n\n \n \n A list or fail.\n \n Return\ s a Hamiltonian path if one exists, fail if not.
\n\n A Hamilt\ onian path of a digraph with n vertices is directed\n cycle of l\ ength n. If digraph is a digraph that contains a\n Hamiltonia\ n path, then this function returns one, described in the form\n used by . Note if digraph has\n 0 or 1 \ vertices, then HamiltonianPath returns [] or\n [1], re\ spectively.
\n\n The method used in this attribute has the same worst ca\ se complexity as\n .\n\n D := Digraph([[]]);\n\ngap\ > HamiltonianPath(D);\n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D := Di\ graph([[3], [], [2]]);\n\ngap> Ham\ iltonianPath(D);\nfail\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := \ GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> HamiltonianPath(D);\nfail\n]]>\n \n\n\n \n \n An integer.\n \n Returns \ the number of spanning trees of the symmetric digraph\n digraph.\n \ NrSpanningTrees will return an error if digraph is not a\n \ symmetric digraph.
\n\n See and\n \ for more information. \n\n D := CompleteDigraph(5);\n\ngap> NrSpanningTrees(D);\n125\ngap> D := DigraphSymmetricClosur\ e(CycleDigraph(24));;\ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(Empt\ yDigraph(0));\n0\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\ \n\ngap> NrSpanningTrees(D);\n1134\ 225\n]]>\n \n\n\n
\n\n Cayley graphs of groups\n \n \n \n A group or semigroup.\n \n If digraph is an immutable Cayley graph of a group G<\ /C> and\n digraph belongs to the category , then\n GroupOfCayleyDigraph returns G.\n
\n\n \ If digraph is a Cayley graph of a semigroup S and\n digr\ aph belongs to the category , then\n \ SemigroupOfCayleyDigraph returns S.\n
\n\n See also .\n G := Dihed\ ralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph := Cayley\ Digraph(G);\n\ngap> GroupOfCayley\ Digraph(digraph) = G;\ntrue\n]]>\n \n\n\ \n \n \n A list of generators.\n \n If dig\ raph is an immutable Cayley graph of a group or semigroup with\n respec\ t to a set of generators gens and digraph belongs to\n the ca\ tegory , then\n GeneratorsOfCayleyDigraph\ return the list of generators gens\n over which digraph \ is defined.\n
\n\n See also \n \ or .\n \ G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph\ := CayleyDigraph(G);\n\ngap> Gen\ eratorsOfCayleyDigraph(digraph) = GeneratorsOfGroup(G);\ntrue\ngap> digraph :=\ CayleyDigraph(G, [()]);\n\ngap> G\ eneratorsOfCayleyDigraph(digraph) = [()];\ntrue]]>\n \ \n\n\n
\n\n
Associated semigroups\n \n \n \n A semilattice of partial perms.\n \n The operation AsSemigroup requires that \ filt be equal to\n .\n If digraph is a\n or \n then AsSemigroup \ returns a semigroup of partial perms which is\n isomorphic to the semigro\ up whose elements are the vertices of\n digraph with the binary op\ eration\n .\n If digraph\n satisfies but no\ t\n then\n AsSemigroup returns a semigroup of partial perms which is\n isomorphic to the semi\ group whose elements are the vertices of\n digraph with the binary\ operation\n .\n \n\n The operation AsMonoid behaves similarly to AsSemigroup<\ /C>\n except that filt may also be equal to\n ,\n digraph must satisfy , and the\n output satisfies .
\n\n The output of both of these operation\ s is guaranteed to be of minimal\n degree (see ).\n Furthermore the of\n the output is guaranteed to be\ the unique generating set of minimal size.\n di\ := Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 2, 3, 5]]);\n\ngap> S := AsSemigroup(IsPartialPermSemigroup, di);\ \n\ngap> ForAll(Elements(S\ ), IsIdempotent);\ntrue\ngap> IsInverseSemigroup(S);\ntrue\ngap> Size(S);\n5\n\ gap> di := Digraph([[1], [1, 2], [1, 2, 3]]);\n\ngap> M := AsMonoid(IsPartialPermMonoid, di);\n\ngap> Size(M);\n3]]>\n \n \n\n \n \n A Clifford semigroup of partial perms. \n \n The operation AsSemigroup requires that filt be equal to\n .\n If Y is a or\n \ ,\n gps is a list of \ groups corresponding to each vertex, and\n homs is a list containi\ ng for each edge (i, j) in\n the transitive reduction of digrap\ h a triple [i, j, hom]\n where hom is a group homomorph\ ism from gps[i] to\n gps[j], and the diagram of homomorphis\ ms commutes, then\n AsSemigroup returns a semigroup of partial per\ ms which is\n isomorphic to the strong semilattice of groups S[Y; gps;\ homs].\n 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 := GroupHomomorphis\ mByFunction(G1, G3, sgn);;\ngap> hom23 := GroupHomomorphismByFunction(G2, G3, \ sgn);;\ngap> T := AsSemigroup(IsPartialPermSemigroup,\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]]>\n \n \n\n
\n\n
Planarity\n \n \n A list or fail.\n \n K\ uratowskiPlanarSubdigraph returns the immutable list of\n lists of out-\ neighbours of a (not necessarily induced) subdigraph of the\n digraph di\ graph that witnesses the fact that digraph is not\n planar, or <\ K>fail if digraph is planar. In other words,\n KuratowskiPla\ narSubdigraph returns the out-neighbours of a\n subdigraph of digrap\ h that is homeomorphic to the complete graph\n with 5 vertices, \ or to the complete bipartite graph with vertex sets\n of sizes 3 and\ 3.
\n \n The directions and multiplicities of any edges in <\ A>digraph are\n ignored when considering whether or not digraph \ is planar.
\n\n See also \n and .\n
\n \n This method uses \ the reference implementation in\n &edge-addition-planarity-suite; by John B\ oyer of the algorithms described\n in .\n\n 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 :\ = 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> 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> KuratowskiPlanarSubdigr\ aph(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);\n\ false\ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, \ 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\n]]>\n \n\n\n \n \n A list or fail.\n \n KuratowskiOuterPlanarSubdigraph returns the imm\ utable list of\n immutable lists of out-neighbours of a (not necessarily in\ duced)\n subdigraph of the digraph digraph that witnesses the fact\n\ that digraph is not outer planar, or fail if\n digraph\ is outer planar. In other words,\n KuratowskiOuterPlanarSubdigraph\ returns the out-neighbours of a\n subdigraph of digraph that is\ homeomorphic to the complete graph\n with 4 vertices, or to the com\ plete bipartite graph with vertex sets\n of sizes 2 and 3. \n \n The directions and multiplicities of any edges in digraph\ are\n ignored when considering whether or not digraph is outer plan\ ar. \n
\n\n See also \n ,\n \ , and \n .\n
\n\n This method uses the reference impleme\ ntation in\n &edge-addition-planarity-suite; by John Boyer of the algorithm\ s described\n in .\n\n D := Dig\ raph([[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> KuratowskiOuterPlanarSubd\ igraph(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> KuratowskiOuterPlanarSubd\ igraph(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\ngap> IsOu\ terPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ],\ [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]]]>\n \n\n\n \n \n A list or fail.\ \n \n If digraph is a planar digraph, then \n Pl\ anarEmbedding returns the immutable list of lists of\n out-neighbours o\ f a subdigraph of digraph such that each vertex's\n neighbours are g\ iven in clockwise order. If digraph is not planar,\n then fail is returned.
\n \n The directions and multiplicities of any edges \ in digraph are\n ignored by PlanarEmbedding.\n
\n\n \ See also \n .\n
\n\n This method \ uses the reference implementation in\n &edge-addition-planarity-suite; by J\ ohn Boyer of the algorithms described\n in .\n\n 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(IsMut\ ableDigraph, [[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> PlanarEmbedding(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\ngap> PlanarEmbedding(D);\nfail\n]]>\n \n\n\n \n \n A list or fail.\n \n If digraph is an outer planar digraph, then\n OuterPlanar\ Embedding returns the immutable list of lists\n of out-neighbours of a \ subdigraph of digraph such that each\n vertex's neighbours are given\ in clockwise order. If digraph is not\n outer planar, then fail<\ /K> is returned.
\n \n The directions and multiplicities of any edge\ s in digraph are\n ignored by OuterPlanarEmbedding.\n
\ \n\n See also .\n
\n\n This m\ ethod uses the reference implementation in\n &edge-addition-planarity-suite\ ; by John Boyer of the algorithms described\n in .\n\n<\ Example> 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\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbeddin\ g(CompleteBipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 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> OuterPlanarEmbedding(D);\nfail\ngap> D := Dig\ raph(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> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarE\ mbedding(CompleteBipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ \n]]>\n \n\n\n \n \n \n \n A list or fail.\n \n These attributes return the immutable list of lists of\n out\ -neighbours of a subdigraph of the digraph digraph which is\n homeom\ orphic to one of the following:\n the complete bipartite graph with vertex \ sets of sizes 2 and\n 3; the complete bipartite graph with ve\ rtex sets of sizes 3\n and 3; or the complete graph with 4\ vertices. If\n digraph has no such subdigraphs, then fail is returned. \n
\n\n See also and \ for more details.
\n\n This \ method uses the reference implementation in\n &edge-addition-planarity-suit\ e; by John Boyer of the algorithms described\n in .\n\n\ 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\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphH\ omeomorphicToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 \ ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\ \nfail\ngap> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> Subdigra\ phHomeomorphicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK\ 33(CompleteBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], \ [ ], [ ], [ ] ]\ngap> 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, 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 ], [ ], [ ] ]\nga\ p> D := Digraph(IsMutableDigraph, [[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> S\ ubdigraphHomeomorphicToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ \ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphi\ cToK33(D);\nfail\ngap> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap\ > SubdigraphHomeomorphicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeo\ morphicToK33(CompleteBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6\ , 5, 4 ], [ ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(\ 3));\nfail\n]]>\n \n\n\n
\n\n\ \n \n Properties of digraphs\n\n
E\ dge properties\n \n \n true or false.\n \n Returns true if the digraph digraph has loops, and\n\ false if it does not. A loop is an edge with equal source and range\ .\n
\n \n If the argument digraph is mutable, then the r\ eturn value of\n this property is recomputed every time it is called. \n\n<\ Example> D := Digraph([[1, 2], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ \ngap> DigraphHasLoops(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 := Comp\ leteDigraph(IsMutableDigraph, 4);\n\ \ngap> DigraphHasLoops(D);\nfalse]]>\n \n\n\n \n \n \n t\ rue or false.\n \n This property is t\ rue if the digraph digraph\n is antisymmetric, and false \ if it is not.\n
\n\n A digraph is antisymmetric if whenever t\ here is an edge with source\n u and range v, and an edge with\ source v and range\n u, then the vertices u and v<\ /C> are equal.\n
\n \n If the argument digraph is mutabl\ e, then the return value of\n this property is recomputed every time it is \ called. \n\n gr1 := Digraph([[2], [1, 3], [2, 3]])\ ;\n\ngap> IsAntisymmetricDigraph(g\ r1);\nfalse\ngap> DigraphEdges(gr1){[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2\ := Digraph([[1, 2], [3, 3], [1]]);\n\ngap> IsAntisymmetricDigraph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ \ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph is bipar\ tite, and\n false if it is not. A digraph is bipartite if and only if\ the\n vertices of digraph can be partitioned into two non-empty sets \ such\n that the source and range of any edge of digraph lie in distinc\ t sets.\n Equivalently, a digraph is bipartite if and only if it is 2-colorab\ le; see\n .
\n\n See also .\n
\n \n If\ the argument digraph is mutable, then the return value of\n this pr\ operty is recomputed every time it is called. \n D\ := ChainDigraph(4);\n\ngap> IsBipart\ iteDigraph(D);\ntrue\ngap> D := CycleDigraph(3);\n\ngap> IsBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartite\ Digraph(IsMutableDigraph, 5, 4);\n\ \ngap> IsBipartiteDigraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n \n\ Returns true if the digraph digraph is a complete bipartite\ \n digraph, and false if it is not.
\n\n A digraph is a co\ mplete bipartite digraph if it is bipartite, see\n , and there exists a unique edge with\n source i and r\ ange j if and only if i and j lie\n in different bicom\ ponents of digraph, see .
\n\n Equivalently, a bipartite digraph with bicomponents of size m \ and\n n is complete precisely when it has 2mn edges, none of \ which\n are multiple edges.
\n\n See also .\n
\n \n If the argument digraph is mutab\ le, then the return value of\n this property is recomputed every time it is\ called. \n\n D := CycleDigraph(2);\n\ngap> IsCompleteBipartiteDigraph(D);\ntrue\ngap> \ D := CycleDigraph(4);\n\ngap> IsBipar\ titeDigraph(D);\ntrue\ngap> IsCompleteBipartiteDigraph(D);\nfalse\ngap> D := C\ ompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteBipartiteDigraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n <\ Description>\n Returns true if the digraph digraph is complet\ e, and\n false if it is not.
\n\n A digraph is complete if it has no loops, and for all\n distinct vertices i and <\ C>j,\n there is exactly one edge with source i and range j.\n\n Equivalently, a digraph with n vertices is complete precisely\ when\n it has n(n - 1) edges, no loops, and no multiple edges.\n \
\n \n If the argument digraph is mutable, then the return \ value of\n this property is recomputed every time it is called. \n\n D := Digraph([[2, 3], [1, 3], [1, 2]]);\n\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Dig\ raph([[2, 2], [1]]);\n\ngap> \ IsCompleteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigr\ aph, 5, 4);\n\ngap> IsCompleteDigra\ ph(D);\nfalse]]>\n \n\n\n \n \n true or false.\n \n This property r\ eturns true if digraph is a complete\n multipartite digraph, \ and false if not.
\n\n A digraph is a complete multipartite d\ igraph if and only if\n its vertices can be partitioned into at least t\ wo maximal independent sets,\n where every possible edge between these inde\ pendent sets occurs in the\n digraph exactly once.\n
\n \n I\ f the argument digraph is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n D := CompleteMultipartiteDigraph([2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph(D);\ \ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteMultipartiteDigraph(D);\ntr\ ue]]>\n \n\n\n \n \n \n true or false.\n \n Returns true if the digraph digraph is empty, and\n \ false if it is not. A digraph is empty if it has no\n edges.<\ P/>\n\n IsNullDigraph is a synonym for IsEmptyDigraph.\n <\ P/>\n \n If the argument digraph is mutable, then the return va\ lue of\n this property is recomputed every time it is called. \n\n D := Digraph([[], []]);\n\ngap> IsEmptyDigraph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D \ := Digraph([[], [1]]);\n\ngap> IsEm\ ptyDigraph(D);\nfalse\ngap> IsNullDigraph(D);\nfalse]]>\n \n\n\n \n \n true or false.\n \n A digraph is an equivalence digraph if and only if the digraph\ satisifies\n all of ,\n and .\n A partial \ order digraph corresponds to an equivalence relation.\n
\n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n D := Digraph([[1, 3], [2], [1, 3]]);\n\ngap> IsEquivalenceDigraph(D);\ntrue\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph is\n f\ unctional.
\n\n A digraph is functional if every vertex is the s\ ource of a\n unique edge.\n
\n \n If the argument digraph\ is mutable, then the return value of\n this property is recomputed eve\ ry time it is called. \n gr1 := Digraph([[3], [2],\ [2], [1], [6], [5]]);\n\ngap> IsF\ unctionalDigraph(gr1);\ntrue\ngap> gr2 := Digraph([[1, 2], [1]]);\n\ngap> IsFunctionalDigraph(gr2);\nfalse\ngap>\ gr3 := Digraph(3, [1, 2, 3], [2, 3, 1]);\n\ngap> IsFunctionalDigraph(gr3);\ntrue\n]]>\n \n\n\n \n \n true or false.\n \ \n A multidigraph is one that has at least two\n edges with equal\ source and range.
\n\n If the argument digraph is mutable, then \ the return value of\n this property is recomputed every time it is called. \ \n\n D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \ \"b\", \"b\"], [\"b\", \"c\", \"a\"]);\n\ngap> IsMultiDigraph(D);\nfalse\ngap> D := DigraphFromDigraph6String(\"\ &Bug\");\n\ngap> IsDuplicateFree(D\ igraphEdges(D));\ntrue\ngap> IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1,\ 2, 3, 2], [2, 1], [3]]);\n\n\ gap> IsDuplicateFree(DigraphEdges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\n\ gap> D := DigraphMutableCopy(D); \n\ngap> IsMultiDigraph(D);\ntrue]]>\n \n\n\n \n \ \n true or false.\n \n This\ property is true if the digraph digraph is\n reflexive, and \ false if it is not.\n A digraph is reflexive if it has a loop\ at every vertex.
\n
\n \n If the argument digraph i\ s mutable, then the return value of\n this property is recomputed every tim\ e it is called. \n 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\ngap> IsReflexiveDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n <\ Description>\n This property is true if the digraph digraph\n\ is symmetric, and false if it is not.
\n\n A symmetric dig\ raph is one where for each non-loop edge, having\n source u and \ range v, there is a corresponding edge with\n source v and ra\ nge u. If there are n edges with\n source u and range\ v, then there must be precisely n\n edges with source v and range u. In other words, a symmetric\n digraph has a symmetr\ ic adjacency matrix .\n
\n \n If\ the argument digraph is mutable, then the return value of\n this pr\ operty is recomputed every time it is called. \n g\ r1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr1);\ntrue\ngap> adj1 := AdjacencyMatrix(gr1);\ ;\ngap> Display(adj1);\n[ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ]\ ]\ngap> adj1 = TransposedMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(gr1);\nt\ rue\ngap> gr2 := Digraph([[2, 3], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 := Adjace\ ncyMatrix(gr2);;\ngap> Display(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, 1 ],\n \ [ 0, 1, 1 ] ]\ngap> adj2 = TransposedMat(adj2);\nfalse\n]]>\n <\ /Description>\n\n\n \n \n true or false.\n \n This property is true if the digraph digraph is \ a tournament,\n and false if it is not.
\n\n A tournament is \ an orientation of a complete (undirected) graph.\n Specifically, a tourname\ nt is a digraph which has a unique directed edge\n (of some orientation) be\ tween any pair of distinct vertices, and no loops.\n
\n \n If t\ he argument digraph is mutable, then the return value of\n this prop\ erty is recomputed every time it is called. \n D :\ = Digraph([[2, 3, 4], [3, 4], [4], []]);\n\ngap> IsTournament(D);\ntrue\ngap> D := Digraph([[2], [1], [3]]);\n\ngap> IsTournament(D);\nfalse\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> IsTournament(D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\n\ngap> IsTournament(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph di\ graph\n is transitive, and false if it is not.\n\n A digraph \ is transitive if whenever [ i, j ] and\n [ j, k ] are \ edges of the digraph, then [ i, k ] is also an\n edge of the digraph\ .
\n\n Let n be the number of vertices of an arbitrary digraph, \ and let\n m be the number of edges.\n For general digraphs, t\ he methods used for this property use a version\n of the Floyd-Warshall alg\ orithm, and have complexity O(n^3).\n\n However for digraphs which a\ re topologically sortable\n [], then \ methods with\n complexity O(m + n + m \\cdot n) will be used when ap\ propriate.\n
\n \n If the argument digraph is mutable, t\ hen the return value of\n this property is recomputed every time it is call\ ed. \n
\n D := Digraph([[1, 2], [3], [3]]);\ \n\ngap> IsTransitiveDigraph(D);\n\ false\ngap> gr2 := Digraph([[1, 2, 3], [3], [3]]);\n\ngap> IsTransitiveDigraph(gr2);\ntrue\ngap> gr2 = DigraphTr\ ansitiveClosure(D);\ntrue\ngap> gr3 := Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ \ngap> IsTransitiveDigraph(gr\ 3);\ntrue\n]]>\n \n\n\n \ \n \n \n true or false.\n \n A digraph is a preorder digraph if and only if the\ digraph satisifies both\n and .\n A preorder digraph (or quasiorder digraph) digraph corresponds to\n the preorder relation \\leq defined by\ x \\leq y if and only\n if [x, y] is an edge of digraph.\n
\n \n If the argument digraph is mutable, then the\ return value of\n this property is recomputed every time it is called. \n\ \n D := Digraph([[1], [2, 3], [2, 3]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := D\ igraph([[1 .. 4], [1 .. 4], [1 .. 4], [1 .. 4]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], \ [3], [4], [5], [1]]);\n\ngap> IsPr\ eorderDigraph(D);\nfalse\ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n\ngap> IsQuasiorderDigraph(D);\nfalse\n]]><\ /Example>\n \n\n\n \n \n true or fals\ e.\n \n A digraph is a partial order digraph if\ and only if the digraph satisifies\n all of ,\n and .\n A partial order digraph corresponds\n to the parti\ al order relation \\leq defined by x \\leq y if and\n only if\ [x, y] is an edge of digraph.\n
\n \n If the arg\ ument digraph is mutable, then the return value of\n this property i\ s recomputed every time it is called. \n\n D := Digrap\ h([[1, 3], [2, 3], [3]]);\n\ngap> \ IsPartialOrderDigraph(D);\ntrue\ngap> D := CycleDigraph(5);\n\ngap> IsPartialOrderDigraph(D);\nfalse\ngap> D := Dig\ raph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\n]]>\n \n\n\n \n \n \n \n true or false.\n \n IsMeetSemilat\ ticeDigraph returns true if the digraph\n digraph is a me\ et semilattice; IsJoinSemilatticeDigraph\n returns true if th\ e digraph digraph is a join semilattice;\n and IsLatticeDigraph<\ /C> returns true if the digraph\n digraph is both a meet and \ a join semilattice.\n
\n\n For a partial order digraph the\n corresponding partial order is the relation \ \\leq, defined by\n x \\leq y if and only if [x, y] is\ an edge.\n A digraph is a meet semilattice if it is a partial order\ and every\n pair of vertices has a greatest lower bound (meet) with respec\ t to the\n aforementioned relation. A join semilattice is a partial \ order where\n every pair of vertices has a least upper bound (join) with re\ spect to\n the relation.\n
\n \n If the argument digraph<\ /A> is mutable, then the return value of\n this property is recomputed ever\ y time it is called. \n\n D := Digraph([[1, 3], [2\ , 3], [3]]);\n\ngap> IsMeetSemilat\ ticeDigraph(D);\nfalse\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLattic\ eDigraph(D);\nfalse\ngap> D := Digraph([[1], [2], [1 .. 3]]);\n\ngap> IsJoinSemilatticeDigraph(D);\nfalse\ngap> \ IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D :\ = Digraph([[1 .. 4], [2, 4], [3, 4], [4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDi\ graph(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> IsMeetSemilatticeDigraph(D);\nt\ rue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\ \n]]>\n \n\n\n
\n\n Regularity\n \n \n true or false.\n\ \n This property is true if there is an integer n<\ /C> such that\n for every vertex v of digraph digraph there a\ re exactly\n n edges terminating in v.\n\n See also and\n .\n \
\n \n If the argument digraph is mutable, then the return v\ alue of\n this property is recomputed every time it is called. \n IsInRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInReg\ ularDigraph(ChainDigraph(4));\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n \ This property is true if there is an integer n such that\n fo\ r every vertex v of digraph digraph there are exactly\n n<\ /C> edges starting at v.\n
\n\n See also and\n .\n
\n \n \ If the argument digraph is mutable, then the return value of\n th\ is property is recomputed every time it is called. \n IsOutRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph(C\ hainDigraph(4));\nfalse\n]]>\n \n\n\n \ \n \n true or false.\n \n This property is\ true if there is an integer n such that\n for every vertex <\ C>v of digraph digraph there are exactly\n n edges starti\ ng and terminating at v. In other words,\n the property is true if digraph is both in-regular and\n and out-regular.\n\n See a\ lso and\n .\n
\n \n If the argument digraph is mutable, then \ the return value of\n this property is recomputed every time it is called. \ \n IsRegularDigraph(CompleteDigraph(4));\ntrue\nga\ p> IsRegularDigraph(ChainDigraph(4));\nfalse\n]]>\n \ \n\n\n \n \n true or false.\n \n If digraph is a connected symmetric graph, this property \ returns\n true if for any two vertices u and v of d\ igraph\n and any two integers i and j between 0 an\ d the\n diameter of digraph, the number of vertices at distance i\ \n from u and distance j from v depends only on\n \ i, j, and the distance between vertices u and\n \ v.
\n\n Alternatively, a distance regular graph is a graph for which\ there exist\n integers b_i, c_i, and i such that for \ any two\n vertices u, v in digraph which are distance \ i\n apart, there are exactly b_i neighbors of v which \ are at\n distance i - 1 away from u, and c_i neighbors\ of\n v which are at distance i + 1 away from u. This\ \n definition is used to check whether digraph is distance regular.<\ P/>\n\n In the case where digraph is not symmetric or not connected,\ the\n property is false.\n
\n \n If the argument \ digraph is mutable, then the return value of\n this property is recompu\ ted every time it is called. \n\n D := DigraphSymm\ etricClosure(ChainDigraph(5));;\ngap> IsDistanceRegularDigraph(D);\nfalse\ngap\ > D := Digraph([[2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]]);\n\ngap> IsDistanceRegularDigraph(D);\ntrue\n]]>\n \n\n\n \n\n
Connectivity and cycles\n \n \n true or false.\n \n This property is true if the digraph digrap\ h is acyclic, and\n false if it is not. A digraph is acyclic<\ /E> if every directed\n cycle on the digraph is trivial. See section for the definition of a dire\ cted cycle, and of a trivial\n directed cycle.
\n\n The method used i\ n this operation has complexity O(m+n) where\n m is the numbe\ r of edges (counting multiple edges as one) and\n n is the number of\ vertices in the digraph.
\n\n If the argument digraph is mutabl\ e, then the return value of\n this property is recomputed every time it is \ called. \n\n Petersen := Graph(SymmetricGroup(5), \ [[1, 2]], OnSets,\n> function(x, y)\n> return IsEmpty(Intersection(x, y));\n\ > end);;\ngap> D := Digraph(Petersen);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> D := DigraphFromDiSparse6Stri\ ng(\n> \".b_OGCIDBaPGkULEbQHCeRIdrHcuZMfRyDAbPhTi|zF\");\n\ngap> IsAcyclicDigraph(D);\ntrue\ngap> IsAcyclicDig\ raph(ChainDigraph(10));\ntrue\ngap> D := CompleteDigraph(IsMutableDigraph, 4);\ \n\ngap> IsAcyclicDigraph(D);\nfals\ e\ngap> IsAcyclicDigraph(CycleDigraph(10));\nfalse]]>\n \n\n\n \n \n true or false.\n \n IsChainDigraph returns true if the digraph digraph\ is\n isomorphic to the chain digraph with the same number of vertices as\n\ digraph, and false if it is not; see .
\n\n A digraph is a chain if and only if it is a di\ rected tree, in which\n every vertex has out degree at most one; see \n and .\n
\n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n 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> IsChai\ nDigraph(D);\ntrue\ngap> D := ChainDigraph(IsMutableDigraph, 5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphRev\ erse(D);\n\ngap> IsChainDigraph(D);\ \ntrue]]>\n \n\n\n \n \n true or <\ K>false.\n \n This property is true if t\ he digraph digraph\n is weakly connected and false if it is n\ ot. A digraph\n digraph is weakly connected if it is possibl\ e to travel\n from any vertex to any other vertex by traversing edges in ei\ ther\n direction (possibly against the orientation of some of them).
\n\ \n The method used in this function has complexity O(m) if the\n \ digraph's attribute is set, otherwise it has\n \ complexity O(m+n) (where\n m is the number of edges and\n \ n is the number of vertices of the digraph).\n
\n \n I\ f the argument digraph is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n D := Digraph([[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D :=\ Digraph([[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\ngap> D\ := Digraph(IsMutableDigraph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D);\n\ true\ngap> D := Digraph(IsMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> IsCo\ nnectedDigraph(D);\nfalse]]>\n \n\n\n \ \n \n true or false.\n \n A connected \ digraph is biconnected if it is still connected (in the\n sense of <\ Ref Prop=\"IsConnectedDigraph\"/>) when any vertex is removed. \n IsBico\ nnectedDigraph returns true if the digraph\n digraph is b\ iconnected, and false if it is not. In\n particular, IsBiconnecte\ dDigraph returns false if\n digraph is not connected.
\n\n Multiple edges and loops are ignored by this method.
\n\n The \ method used in this operation has complexity O(m+n) where\n m\ is the number of edges (counting multiple edges as one, and not\n counting\ loops) and n is the number of vertices in the digraph.\n
\n\n \ See also .\n
\n \n If the a\ rgument digraph is mutable, then the return value of\n this property\ is recomputed every time it is called. \n IsBiconnect\ edDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBiconnectedDigraph(C\ ycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4\ ]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigrap\ h(IsMutableDigraph, 5, 4);\n\ngap> \ IsBiconnectedDigraph(D);\ntrue]]>\n \n\n\ \n \n \n true or false.\n \n \ This property is true if the digraph digraph is strongly\n co\ nnected and false if it is not.
\n\n A digraph digraph is\ strongly connected if there is a directed\n path from every vertex \ to every other vertex. See section for the definition of a directed\n path.
\n\n The m\ ethod used in this operation is based on Gabow's Algorithm and has complexity O(m+n), where m is\n the number \ of edges (counting multiple edges as one) and n is the\n number of v\ ertices in the digraph.\n
\n \n If the argument digraph \ is mutable, then the return value of\n this property is recomputed every ti\ me it is called. \n\n D := CycleDigraph(250000);\n\ \ngap> IsStronglyConnectedDigrap\ h(D);\ntrue\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> IsStronglyConnectedDigraph(D);\nf\ alse\ngap> D := CycleDigraph(IsMutableDigraph, 250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\nga\ p> DigraphRemoveEdge(D, [250000, 1]);\n\ngap> IsStronglyConnectedDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \ \n This property is true if the digraph digraph\ \n is aperiodic, i.e. if its is equal to 1.\ \n Otherwise, the property is false.\n
\n \n If the a\ rgument digraph is mutable, then the return value of\n this property\ is recomputed every time it is called. \n D := Di\ graph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Digraph([[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> IsAper\ iodicDigraph(D);\ntrue]]>\n \n\n\n \n \n t\ rue or false.\n \n Returns true i\ f the digraph digraph is a directed tree, and\n false if it i\ s not.
\n\n A directed tree is an acyclic digraph with precisely\ 1 source,\n such that no two vertices share an out-neighbour.\n Note th\ e empty digraph is not considered a directed\n tree as it has no source. \n\n See also .\n
\n \n If t\ he argument digraph is mutable, then the return value of\n this prop\ erty is recomputed every time it is called. \n\n D\ := Digraph([[], [2]]);\n\ngap> IsD\ irectedTree(D);\nfalse\ngap> D := Digraph([[3], [3], []]);\n\ngap> IsDirectedTree(D);\nfalse\ngap> D := Digraph(\ [[2], [3], []]);\n\ngap> IsDirecte\ dTree(D);\ntrue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [], []]);\n\ngap> IsDirectedTree(D);\ntrue\n]]>\n \n\n\n \n \n \n true or false.\n \n The property IsUndirectedTree returns true if the digraph\ \n digraph is an undirected tree, and the property\n IsUndirec\ tedForest returns true if digraph is an\n undirected fore\ st; otherwise, these properties return false.
\n\n An undirec\ ted tree is a symmetric digraph without loops, in which for\n any pair \ of distinct vertices u and v, there is exactly one\n directed\ path from u to v. See \ and , and see\n section for the definition of\n directed path. This d\ efinition implies that an undirected tree has\n no multiple edges.
\n\n\ An undirected forest is a digraph, each of whose connected componen\ ts\n is an undirected tree. In other words, an undirected forest is isomorp\ hic to\n a disjoint union of undirected trees. See and . In particular, every\n undirected tree is a\ n undirected forest.
\n\n Please note that the digraph with zero vertic\ es is considered to be neither\n an undirected tree nor an undirected fores\ t.\n
\n \n If the argument digraph is mutable, then the \ return value of\n this property is recomputed every time it is called. \n\n\ D := Digraph([[3], [3], [1, 2]]);\n\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmet\ ricDigraph(D) and not DigraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], \ [1, 4], [3], [2]]);\n\ngap> IsConn\ ectedDigraph(D);\nfalse\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedFo\ rest(D);\ntrue\ngap> D := Digraph([[1, 2], [1], [2]]);\n\ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfa\ lse\ngap> IsSymmetricDigraph(D) or not DigraphHasLoops(D);\nfalse]]>\ \n \n\n\n \n \n true or false.\n \n This property returns true if the digraph digraph<\ /A> is Eulerian.\n
\n\n A connected digraph is called Eulerian if there exists a directed\n circuit on the digraph which includes every \ edge exactly once. See\n section for the definition of\n a directed circuit. Note that the empty dig\ raph with at most one vertex is\n considered to be Eulerian.\n
\n \ \n If the argument digraph is mutable, then the return value of\n\ this property is recomputed every time it is called. \n\n D := Digraph([[]]);\n\ngap>\ IsEulerianDigraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digra\ ph([[3], [], [2]]);\n\ngap> IsEule\ rianDigraph(D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> IsEulerianDigraph(D);\ntrue\n]]>\n\ \n\n\n \n \n true or false.\n \n If digraph is Hamiltonian, then this property\ returns\n true, and false if it is not.
\n\n A digrap\ h with n vertices is Hamiltonian if it has a\n directed cycle\ of length n. See Section for the definition of a directed cycle.\n Note the empty digraphs\ on 0 and 1 vertices are considered to be\n Hamiltonian.
\n\n The met\ hod used in this operation has the worst case complexity as\n .\n
\n \n If the argument digraph \ is mutable, then the return value of\n this property is recomputed every ti\ me it is called. \n\n g := Digraph([[]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g \ := Digraph([[2], [1]]);\n\ngap> Is\ HamiltonianDigraph(g);\ntrue\ngap> g := Digraph([[3], [], [2]]);\n\ngap> IsHamiltonianDigraph(g);\nfalse\ngap> g\ := Digraph([[2], [3], [1]]);\n\ng\ ap> IsHamiltonianDigraph(g);\ntrue\n]]>\n \n\n\n \n \n \ true or false.\n \n IsCy\ cleDigraph returns true if the digraph digraph is\n isomo\ rphic to the cycle digraph with the same number of vertices as\n digraph\ , and false if it is not; see .<\ P/>\n\n A digraph is a cycle if and only if it is strongly connected\ and has\n the same number of edges as vertices. \n
\n \n If\ the argument digraph is mutable, then the return value of\n this pr\ operty is recomputed every time it is called. \n\n D :\ = Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsCycleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\n\ngap> IsCycleDigraph(D);\ntrue\ngap> D := OnDigraph\ s(D, (1, 2, 3));\n\ngap> D = Cycle\ Digraph(5);\nfalse\ngap> IsCycleDigraph(D);\ntrue]]>\n \n\n\n
\n \n
Planarity\ \n \n \n true or false.\n \n A planar<\ /E> digraph is a digraph that can be embedded in the plane in\n such a way \ that its edges do not intersect. A digraph is planar if and only\n if it do\ es not have a subdigraph that is homeomorphic to either the\n complete grap\ h on 5 vertices or the complete bipartite graph with\n vertex sets o\ f sizes 3 and 3.\n
\n \n IsPlanarDigraph ret\ urns true if the digraph digraph is\n planar and false\ if it is not. The directions and multiplicities of\n any edges in digra\ ph are ignored by IsPlanarDigraph.\n
\n\n See also .\n
\n\n This method uses the referen\ ce implementation in\n &edge-addition-planarity-suite; by John Boyer of the\ algorithms described\n in .\n\n IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigraph(Comple\ teDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));\n\ true\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsPla\ narDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanarDigraph(\ CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(CompleteBi\ partiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteB\ ipartiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n An outer planar digraph is a digraph that can be embedded \ in the\n plane in such a way that its edges do not intersect, and all verti\ ces\n belong to the unbounded face of the embedding. A digraph is outer pl\ anar\n if and only if it does not have a subdigraph that is homeomorphic to\ either\n the complete graph on 4 vertices or the complete bipartite\ graph\n with vertex sets of sizes 2 and 3.\n
\n \n\ IsOuterPlanarDigraph returns true if the digraph\n dig\ raph is outer planar and false if it is not. The\n directions an\ d multiplicities of any edges in digraph are ignored by\n IsPlana\ rDigraph.
\n\n See also .\n\n Th\ is method uses the reference implementation in\n &edge-addition-planarity-s\ uite; by John Boyer of the algorithms described\n in .\ \n\n IsOuterPlanarDigraph(CompleteDigraph(4));\nfalse\ \ngap> IsOuterPlanarDigraph(CompleteDigraph(5));\nfalse\ngap> IsOuterPlanarDig\ raph(CompleteBipartiteDigraph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph(Comple\ teBipartiteDigraph(3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(10))\ ;\ntrue\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));\nfal\ se\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ng\ ap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterPlanarDi\ graph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(\ IsMutableDigraph, 10));\ntrue\n\n]]>\n \n\n\n
\n\n
Homomorphisms and transformations\n \n \n \ true or false.\n \n This pro\ perty returns true if digraph is a core,\n and false i\ f it is not.
\n\n A digraph D is a core if and only if it \ has no proper\n subdigraphs A such that there exists a homomorphism \ from D\n to A. In other words, a digraph D is a core i\ f and only if\n every endomorphism on D is an automorphism on D.\n
\n \n If the argument digraph is mutable, then the\ return value of\n this property is recomputed every time it is called. \n\ \n D := CompleteDigraph(6);\n\ngap> IsDigraphCore(D);\ntrue\ngap> D := DigraphSymmet\ ricClosure(CycleDigraph(6));\n\ngap> DigraphHomomorphism(D, CompleteDigraph(2));\nTransformation( [ 1\ , 2, 1, 2, 1, 2 ] )\ngap> IsDigraphCore(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n If digraph is a digraph without multiple edges, then\n IsEdgeTransitive returns true if digraph\n is edge trans\ itive, and false otherwise. A digraph is\n edge transitive if\ its automorphism group acts\n transitively on its edges (via the action\n \ ).\n
\n \n If the ar\ gument digraph is mutable, then the return value of\n this property \ is recomputed every time it is called. \n\n IsEdge\ Transitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(ChainDigraph(3));\ \nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\nError, the arg\ ument must be a digraph with no multiple edges,\n]]>\n \n\n\n \n \n true or false.\n \n If digraph is a digraph, then IsVertexTransitive r\ eturns\n true if digraph is vertex transitive, and false\n otherwise. A digraph is vertex transitive if its automorphism g\ roup\n acts transitively on its vertices.\n
\n \n If the arg\ ument digraph is mutable, then the return value of\n this property i\ s recomputed every time it is called. \n\n IsVerte\ xTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitive(ChainDigraph(3\ ));\nfalse\n]]>\n \n\n\n
\n\n\ \n\n \n Homomorphisms\n \n\n
Acting on digraphs<\ /Heading>\n \n \n \n A digraph.\n \n If digraph is a digraph, \ and the second argument perm is a\n permutation of the vertic\ es of digraph, then this operation\n returns a digraph constructed b\ y relabelling the vertices of\n digraph according to perm. N\ ote that for an automorphism\n f of a digraph, we have OnDigraphs\ (digraph, f) =\n digraph.\n
\n\n If the second \ argument is a transformation trans of the\n vertices of di\ graph, then this operation returns a digraph\n constructed by transform\ ing the source and range of each edge according to\n trans. Thus a v\ ertex which does not appear in the image of\n trans will be isolated\ in the returned digraph, and the returned\n digraph may contain multiple e\ dges, even if digraph does not.\n If trans is mathematically \ a permutation, then the result coincides\n with OnDigraphs(digraph, AsPermutation(trans)).\n
\n\n The of digraph will not be retained\n in the returned di\ graph.
\n\n If digraph belongs to , then\n relabelling of the vertices is performed directly on digraph<\ /A>.\n If digraph belongs to , an \ immutable\n copy of digraph with the vertices relabelled is returned\ .
\n\n D := Digraph([[3], [1, 3, 5], [1], [1, 2\ , 4], [2, 3, 5]]);\n\ngap> new :=\ OnDigraphs(D, (1, 2));\n\ngap> O\ utNeighbours(new);\n[ [ 2, 3, 5 ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ]\ng\ ap> 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]]>\n \n\n\ \n \n \n\ \n A digraph.\n \n If digraph is a digraph, and pair is a pair consisting \ of a\n permutation of the vertices and a permutation of the edges of\n <\ A>digraph, then this operation returns a digraph\n constructed by relab\ elling the vertices and edges of digraph\n according to perm[1] and perm[2], respectively.
\n\n In its second form, OnMult\ iDigraphs returns a digraph with vertices\n and edges permuted by pe\ rm1 and perm2, respectively.
\n\n Note that OnDigraphs(digraph,\n perm)=OnMultiDigraphs(digraph, [perm, ()]) whe\ re perm is\n a permutation of the vertices of digraph. If you\ are only interested\n in the action of a permutation on the vertices of a \ digraph, then you can\n use OnDigraphs instead of OnMultiDigraphs\ .\n\n If digraph belongs to , th\ en\n relabelling of the vertices is performed directly on digraph.\n\ If digraph belongs to , an immuta\ ble\n copy of digraph with the vertices relabelled is returned.
\ \n\n D1 := Digraph([\n> [3, 6, 3], [], [3], [9, 10\ ], [9], [], [], [10, 4, 10], [], []]);\n\ngap> p := BlissCanonicalLabelling(D1);\n[ (1,7)(3,6)(4,10)(5,9\ ), () ]\ngap> D2 := OnMultiDigraphs(D1, p);\n\ngap> OutNeighbours(D2);\n[ [ ], [ ], [ ], [ ], [ ], [\ 6 ], [ 6, 3, 6 ], [ 4, 10, 4 ], \n [ 5 ], [ 5, 4 ] ]]]>\n \n\n\n
\n\n
\n Isomorphisms and canonical labellings\n From version 0.11.0 of &Digraphs; it is possible to use either &b\ liss; or\n &nauty; (via &NautyTracesInterface;) to calculate canonical labe\ llings and\n automorphism groups of digraphs; see and \ for more details about &bliss; and &nauty;,\n r\ espectively. \n\n \n \n \n Nothing.\ \n \n These functions can be used to specify whether &nauty; \ or &bliss; should be\n used by default by &Digraphs;. If &NautyTracesInterf\ ace; is not available,\n then these functions do nothing. Otherwise, by cal\ ling\n DigraphsUseNauty subsequent computations will default to usin\ g\n &nauty; rather than &bliss;, where possible.\n
\n\n You can c\ all these functions at any point in a &GAP; session, as many\n times as you\ like, it is guaranteed that existing digraphs remain\n valid, and that com\ parison of existing digraphs and newly created digraphs\n via\n ,\n ,\n , and\n are \ also valid.
\n\n It is also possible to compute the automorphism group o\ f a specific digraph\n using both &nauty; and &bliss; using \n and , respectively.\n \n\n\n \n \n A permutation group.\n \n If digraph is a digraph, then this attribute contains the group of\n auto\ morphisms of digraph. An automorphism of digraph\n is\ an isomorphism from digraph to itself. See for more information\n about isomor\ phisms of digraphs.
\n\n If digraph is not a multidigraph then t\ he automorphism group is\n returned as a group of permutations on the set o\ f vertices of\n digraph.
\n\n If digraph is a multidig\ raph then the automorphism group is returned\n as the direct product of a g\ roup of permutations on the set of vertices of\n digraph with a grou\ p of permutations on the set of edges of\n digraph. These groups can\ be accessed using on the\n returned group.
\n\n B\ y 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 , \n , ,\n and .\n
\n\n If the argument digraph is mutabl\ e, then the return value of\n this attribute is recomputed every time it is\ called. \n\n johnson := DigraphFromGraph6String(\ \"E}lw\");\n\ngap> G := Automorph\ ismGroup(johnson);\nGroup([ (3,4), (2,3)(4,5), (1,2)(5,6) ])\ngap> cycle := Cy\ cleDigraph(9);\n\ngap> G := Automorph\ ismGroup(cycle);\nGroup([ (1,2,3,4,5,6,7,8,9) ])\ngap> IsCyclic(G) and Size(G)\ = 9;\ntrue]]>\n \n \n\n \n \n \n \n A permutation group\ .\n \n If digraph is a digraph, then this at\ tribute contains the group of\n automorphisms of digraph as calculat\ ed using &bliss; by Tommi Junttila\n and Petteri Kaski.
\n\n The att\ ribute and\n oper\ ation returns\n the value of either BlissAutomorphismGroup or . These groups are, of cours\ e, equal but\n their generating sets may differ.
\n The attribute returns the value of\n BlissAutomorphismGroup as it is\ not implemented for &nauty; The \n requirements for the optional arguments\ vertex_colours and \n edge_colours are documented in .\n\n See also , and .\n
\n\n If the argument digraph is mutable\ , then the return value of\n this attribute is recomputed every time it is \ called. \n\n G := BlissAutomorphismGroup(JohnsonDi\ graph(5, 2));;\ngap> IsSymmetricGroup(G);\ntrue\ngap> Size(G);\n120]]>\n \n \n\n \n \n A permuta\ tion group.\n \n If digraph is a digraph, th\ en this attribute contains the group of\n automorphisms of digraph a\ s calculated using &nauty; by\n Brendan Mckay and Adolfo Piperno via &Nauty\ TracesInterface;.\n\n \n\n The attribute and\n operation returns\n the value of either Nau\ tyAutomorphismGroup or .\n These groups are, of course, equal but their generat\ ing sets may differ.
\n\n See also , and\ .\n
\n\n If the argument digraph\ is mutable, then the return value of\n this attribute is recomputed ev\ ery time it is called. \n\n NautyAutomorphismGroup(Joh\ nsonDigraph(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) ])]]>\n \n \n\n <\ ManSection>\n \n A permutatio\ n group.\n \n\n This operation computes the automor\ phism group of a vertex-coloured digraph.\n A vertex-coloured digraph can b\ e specified by its underlying digraph\n digraph and its colouring vert_colours. Let n be the\n number of vertices of digraph. The colouring vert_colours may\n have one of the following two f\ orms:\n\n \n \n a list of n integers, where \ vert_colours[i] is the\n colour of vertex i, using\ the colours [1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint lists whose union is\ \n DigraphVertices(digraph), such that\n vert_c\ olours[i] is the list of all vertices with colour\n i\ .\n \n \n\n The automorphism group of a colour\ ed digraph digraph with\n colouring vert_colours is the group\ consisting of its automorphisms; an\n automorphism of digraph is an isomorphism of coloured\n digraphs from digraph to itself. \ This group is equal to the\n subgroup of AutomorphismGroup(digraph) consisting of those\n automorphisms that preserve the colouring spec\ ified by vert_colours. See\n , and see \n for more information about isom\ orphisms of coloured digraphs.
\n\n If digraph is not a multidig\ raph then the automorphism group is\n returned as a group of permutations o\ n the set of vertices of\n digraph.
\n\n If digraph is\ a multidigraph then the automorphism group is returned\n as the direct pro\ duct of a group of permutations on the set of vertices of\n digraph \ with a group of permutations on the set of edges of\n digraph. These\ groups can be accessed using on the\n returned group.\n\n By default, the automorphism group is found using &bliss; by Tommi J\ unttila\n and Petteri Kaski. If &NautyTracesInterface; is available, then &\ nauty; by\n Brendan Mckay and Adolfo Piperno can be used instead; see\n \ ,\n , ,\n and .\n\n cycle := CycleDigraph(9);\n\ \ngap> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and Size(G) = 9;\ntru\ e\ngap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := Automorphism\ Group(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 -> OnTupl\ esSets(colours, p) = colours);\ntrue\ngap> IsTrivial(AutomorphismGroup(cycle, \ [1, 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue]]>\n \n\n\n \n \n A permutation group.\n \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 digraph and colourings vert_colours, edge_colour\ s.\n Let n be the number of vertices of digraph. The colo\ urings\n must have the following forms:\n\n \n \n \ vert_colours must be fail or a list of n integers,\n \ where vert_colours[i] is the colour of vertex i,\n \ using the colours [1 .. m] for some m <= n;\n \n \n edge_colours must be fail or a list o\ f n lists of\n integers of the same shape as OutNeighbours(di\ graph), where\n edge_colours[i][j] is the colour of t\ he edge\n OutNeighbours(digraph)[i][j], using the colours [1 \ .. k]\n for some k <= n;\n \n \n\n \ Giving vert_colours [edge_colours] as fail is\n equ\ ivalent to setting all vertices [edges] to be the same colour.
\n\n Unl\ ike , it is\n poss\ ible to obtain the automorphism group of an edge-coloured multidigraph\n (s\ ee ) when no two edges share the same\n sour\ ce, range, and colour.\n\n The automorphism group of a vertex/edge-c\ oloured digraph\n digraph with colouring c is the group consi\ sting of its\n vertex/edge-colour preserving automorphisms; an\n auto\ morphism of digraph is an isomorphism of vertex/edge-coloured\n \ digraphs from digraph to itself. This group is equal to the\n subgr\ oup of AutomorphismGroup(digraph) consisting of those\n autom\ orphisms that preserve the colouring specified by colours. See\n , and see \n \ for more information about isomorphisms of coloured digraphs.
\n\n I\ f digraph is not a multidigraph then the automorphism group is\n ret\ urned as a group of permutations on the set of vertices of\n digraph\ .
\n\n If digraph is a multidigraph then the automorphism group \ is returned\n as the direct product of a group of permutations on the set o\ f vertices of\n digraph with a group of permutations on the set of e\ dges of\n digraph. These groups can be accessed using on the\n returned group.
\n\n By default, the automorphism group\ is found using &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTra\ cesInterface; is available, then &nauty; by\n Brendan Mckay and Adolfo Pipe\ rno can be used instead; see\n ,\n , , and\n .\n\n cycle := CycleDigraph(12\ );\n\ngap> 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));\n12\ngap> Size(AutomorphismGrou\ p(cycle, vert_colours));\n4\ngap> Size(AutomorphismGroup(cycle, fail, edge_col\ ours));\n6\ngap> Size(AutomorphismGroup(cycle, vert_colours, edge_colours));\n\ 2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> vert_colours, List([1 .. 12], x \ -> [x mod 4 + 1])));\ntrue\n]]>\n \n\n\n\ \n \n \n A permutation, or a list of two permu\ tations.\n \n A function \\rho that maps a d\ igraph to a digraph is a canonical\n representative map if the fol\ lowing two conditions hold for all\n digraphs G and H:
\n\ \n \n \n \\rho(G) and G are isomorphi\ c as digraphs; and\n \n \n \\rho(G)=\\rho(H)<\ /M> if and only if G and H are\n isomorphic as digraphs.\ \n \n \n\n A canonical labelling of a digraph \ G (under \\rho) is an\n isomorphism of G onto its c\ anonical representative,\n \\rho(G). See \n for more information about isomorph\ isms of digraphs.
\n\n BlissCanonicalLabelling returns a canonic\ al labelling of the digraph\n digraph found using &bliss; by Tommi J\ unttila and Petteri Kaski.\n NautyCanonicalLabelling returns a canon\ ical labelling of the digraph\n digraph found using &nauty; by Brend\ an McKay and Adolfo Piperno.\n Note that the canonical labellings returned \ by &bliss; and &nauty; are not\n usually the same (and may depend of the ve\ rsion used).
\n\n BlissCanonicalLabelling can only be computed if\ digraph has\n no multiple edges; see .
\n\n digraph1 := DigraphFromDiSparse6String\ (\".ImNS_AiB?qRN\");\n\ngap> Blis\ sCanonicalLabelling(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, Bli\ ssCanonicalLabelling(digraph2));\ntrue]]>\n \n\n\n \n \n \n A permutation.\n \n A functi\ on \\rho that maps a coloured digraph to a coloured digraph is\n a <\ E>canonical representative map if the following two conditions hold\n f\ or all coloured digraphs G and H:\n\n \n \n \ \\rho(G) and G are isomorphic as coloured digraphs; and\n\ \n \n \\rho(G)=\\rho(H) if and only if <\ M>G and H are\n isomorphic as coloured digraphs.\n \n \n\n A canonical labelling of a coloured digraph \ G (under\n \\rho) is an isomorphism of G onto its cano\ nical\n representative, \\rho(G). See for more inf\ ormation about\n isomorphisms of coloured digraphs.
\n\n A coloured \ digraph can be specified by its underlying digraph digraph\n and its\ colouring colours. Let n be the number of vertices of\n \ digraph. The colouring colours may have one of the following\n t\ wo forms:
\n\n \n \n a list of n integer\ s, where colours[i] is the\n colour of vertex i, u\ sing the colours [1 .. m] for some\n m <= n; or\n \ \n \n a list of non-empty disjoint lists whose union\ is\n DigraphVertices(digraph), such that\n col\ ours[i] is the list of all vertices with colour\n i.\ \n \n \n\n If digraph and colours toget\ her form a coloured digraph,\n BlissCanonicalLabelling returns a can\ onical labelling of the digraph\n digraph found using &bliss; by Tom\ mi Junttila and Petteri Kaski.\n Similarly, NautyCanonicalLabelling \ returns a canonical labelling of\n the digraph digraph found using &\ nauty; by Brendan McKay and Adolfo\n Piperno. Note that the canonical labe\ llings returned by &bliss; and\n &nauty; are not usually the same (and may \ depend of the version used).
\n\n BlissCanonicalLabelling can onl\ y be computed if digraph has\n no multiple edges; see . The canonical\n labelling of digraph is given as \ a permutation of its vertices.\n The canonical representative of digraph\ can be created from\n digraph and its canonical labelling p<\ /C> by using the\n operation :\n gap> OnDigraphs(digraph, p);\n\n The colo\ uring of the canonical representative can easily be\n constructed. A vertex\ v (in digraph) has colour i if\n and only if the vert\ ex v ^ p (in the canonical representative) has\n colour i, wh\ ere p is the permutation of the canonical\n labelling that acts on t\ he vertices of digraph. In particular, if\n colours has the f\ irst form that is described above, then the\n colouring of the canonical re\ presentative is given by:\n\n gap> List(DigraphVertices(digraph), i ->\ colours[i / p]);\n\n On the other hand, if colours has the se\ cond form above, then the\n canonical representative has colouring:\n\n \ gap> OnTuplesSets(colours, p);\n\n
\n\n 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 := BlissCanon\ icalLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnDigraphs(digra\ ph, p);\n\ngap> List(DigraphVerti\ ces(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ]]]>\n \n\n\n \n \n \n A digraph.\n \n The a\ ttribute BlissCanonicalLabelling returns the canonical\n representat\ ive found by applying . The digraph returned is canonical in the sense\n that\n\n \ \n \n BlissCanonicalDigraph(digraph) a\ nd digraph are\n isomorphic as digraphs; and\n \n \ \n If gr is any digraph then BlissCanonicalDigraph(g\ r)\n and BlissCanonicalDigraph(digraph) are equal if \ and only\n if gr and digraph are isomorphic as digraphs.\ \n \n \n\n Analogously, the attribute NautyCanonic\ alLabelling returns the\n canonical representative found by applying .\n
\n\n If the argument digraph is mutable, then the return value of\n\ this attribute is recomputed every time it is called. \n\n 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]]>\n \n\n\n \n \n\ A permutation group.\n \n If digraph<\ /A> is immutable and was created knowing a subgroup of\n its automorphism g\ roup, then this group is stored in the attribute\n DigraphGroup. If \ digraph is mutable, or was not created\n knowing a subgroup of its a\ utomorphism group, then DigraphGroup\n returns the entire automorphi\ sm group of digraph. \n Note that if digraph is mutable, t\ hen the automorphism group is\n recomputed every time this function is call\ ed.\n
\n\n Note that certain other constructor operations such as <\ Ref\n Oper=\"CayleyDigraph\"/>, , a\ nd , may not require a group as one of the\ arguments,\n but use the standard constructor method using a group, and he\ nce set the\n DigraphGroup attribute for the resulting digraph.\n\n \ n := 4;;\ngap> adj := function(x, y)\n> retur\ n (((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(IsMut\ ableDigraph, group, [1 .. n], \\^, adj);\n\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\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2\ ,3,4) ])\ngap> D := DoubleDigraph(D);\n\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> AutomorphismGroup(D) =\n> Group([(6\ , 8), (5, 7), (4, 6), (3, 5), (2, 4),\n> (1, 2)(3, 4)(5, 6)(7, 8)]);\nt\ rue\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> HasDigraphGroup(D);\nfalse\ngap> HasAutomorphismGroup(D);\nfal\ se\ngap> DigraphGroup(D);\nGroup([ (2,3) ])\ngap> HasAutomorphismGroup(D);\ntr\ ue\ngap> group := DihedralGroup(8);\n\ng\ ap> D := CayleyDigraph(group);\n\ \ngap> HasDigraphGroup(D);\ntrue\ngap> GeneratorsOfGroup(DigraphGroup(D));\n[ \ (1,2)(3,8)(4,6)(5,7), (1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]]]>\ \n \n\n\n \n \n \n An immutable list of lists of inte\ gers.\n \n \n DigraphOrbits returns the orb\ its of the action of the\n on the set of verti\ ces of digraph.\n\n G := Group([(2, 3)(7, 8\ , 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := EdgeOrbitsDigraph(G, [1, 2]);\n<\ immutable digraph with 9 vertices, 6 edges>\ngap> DigraphOrbits(D);\n[ [ 1, 2,\ 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphOrbits(D);\n[ [ 1, 2, 3 ], [ 4, \ 5, 6, 7, 8, 9 ] ]\n]]>\n \n\n\n \n \n \n \ An immutable list of integers.\n \n \n Digraph\ OrbitReps returns a list of orbit representatives\n of the action of th\ e on the set of vertices of\n digraph.\ \n\n D := CayleyDigraph(AlternatingGroup(4));\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ \ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphFromDigraph6String(\"&I\ GO??S?`?_@?a?CK?O\");\n\ngap> Di\ graphOrbitReps(D);\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> DigraphOrbitReps(D\ igraphMutableCopy(D));\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n]]>\n \n\n\n \n \n \n An immutable list of integers.\n\ \n \n DigraphSchreierVector returns the so\ -called Schreier vector\n of the action of the on the set of vertices of\n digraph. The Schreier vector is \ a list sch of integers with\n length DigraphNrVertices(digraph\ ) where:\n \n sch[i] < 0:\n \ \n implies that i is an orbit representative and\n \ DigraphOrbitReps(digraph)[-sch[i]] = i.\n \n\n \ sch[i] > 0:\n \n implies that i / \ gens[sch[i]]\n is one step closer to the root (or representative) o\ f the tree, where\n gens is the generators of DigraphGroup(digraph).\n \n \n\n \ 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\ngap> sch := Digraph\ SchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(AlternatingGroup(\ 4));\n\ngap> sch := DigraphSchre\ ierVector(D);\n[ -1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1 ]\ngap> DigraphOrbitReps(\ D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,5,7)(2,4,8)\ (3,6,9)(10,11,12), (1,2,3)(4,7,10)(5,9,11)(6,8,12) ]\ngap> 10 / gens[sch[10]];\ \n7\ngap> 7 / gens[sch[7]];\n5\ngap> 5 / gens[sch[5]];\n1]]>\n \n\n\n \n \n \n A permutation group.\n \ \n \n DigraphStabilizer returns the stabilizer of the \ vertex v\n under of the action of the o\ n the set of\n vertices of digraph.\n\n \ D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> DigraphSta\ bilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> Dig\ raphStabilizer(D, 2);\nGroup(())\n]]>\n \n\n\n \n \n true or false.\n \n This operation returns true if there exis\ ts an isomorphism from the\n digraph digraph1 to the digraph digr\ aph2. See for more information\n about isomorphisms of digraphs.
\n\n By d\ efault, an isomorphism is found using the canonical labellings of the\n dig\ raphs obtained from &bliss; by Tommi Junttila\n and Petteri Kaski. If &Naut\ yTracesInterface; is available, then &nauty; by\n Brendan Mckay and Adolfo \ Piperno can be used instead; see\n , and .\n\n digraph1 := Cy\ cleDigraph(4);\n\ngap> digraph2 := Cy\ cleDigraph(5);\n\ngap> IsIsomorphicDi\ graph(digraph1, digraph2);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\ \n\ngap> IsIsomorphicDigraph(digra\ ph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], []]);\n\ngap> digraph2 := Digraph([[], [], [2]]);\n\ngap> IsIsomorphicDigraph(digraph1, di\ graph2);\ntrue]]>\n \n\n\n \n \n t\ rue or false.\n \n This operation tests \ for isomorphism of coloured digraphs. A coloured\n digraph can be specifie\ d by its underlying digraph digraph1 and its\n colouring colours1\ . Let n be the number of vertices of\n digraph1. The col\ ouring colours1 may have one of the following\n two forms:\n\n \n \n a list of n integers, where colours<\ C>[i] is the\n colour of vertex i, using the colours [1 .\ . m] for some\n m <= n; or\n \n \n\ a list of non-empty disjoint lists whose union is\n DigraphV\ ertices(digraph), such that\n colours[i] is th\ e list of all vertices with colour\n i.\n \n \n\n If digraph1 and digraph2 are digraphs without multipl\ e edges,\n and colours1 and colours2 are colourings of dig\ raph1\n and digraph2, respectively, then this operation returns \ true\n if there exists an isomorphism between these two coloured dig\ raphs. See\n for more information about isomorphisms of coloure\ d digraphs.
\n\n By default, an isomorphism is found using the canonica\ l 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 Adolfo Piperno can be used instead; see\n , and .\n\n 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, di\ graph2,\n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsomorphicDigraph(digra\ ph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse]]>\n \n\n\n \n \n A permutat\ ion, or a pair of permutations, or fail.\n \n \ This operation returns an isomorphism between the digraphs digraph1\n\ and digraph2 if one exists, else this operation returns fail\ .\n
\n\n An isomorphism from a digraph digraph1 to a d\ igraph\n digraph2 is a bijection p from the vertices of\n \ digraph1 to the vertices of digraph2 with the following\n pro\ perty: for all vertices i and j of digraph1,\n [i, \ j] is an edge of digraph1 if and only if [i ^ p, j\n ^ p] is an edge of digraph2.
\n\n If there exists such an isomorph\ ism, then this operation returns one.\n The form of this isomorphism is a p\ ermutation p of the vertices of\n digraph1 such that
\n\n\ OnDigraphs(digraph1, p) = digraph2.\n\n By default, an is\ omorphism is found using the canonical labellings of the\n digraphs obtaine\ d from &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterf\ ace; is available, then &nauty; by\n Brendan Mckay and Adolfo Piperno can b\ e used instead; see\n , and .\n\n digraph1 := CycleDigraph(4)\ ;\n\ngap> digraph2 := CycleDigraph(5)\ ;\n\ngap> IsomorphismDigraphs(digraph\ 1, digraph2);\nfail\ngap> digraph1 := CompleteBipartiteDigraph(10, 5);\n\ngap> digraph\ 2 := CompleteBipartiteDigraph(5, 10);\n\ngap> p := IsomorphismDigraphs(digraph1, digra\ ph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)\ngap> OnDigraphs(digraph1, p)\ = digraph2;\ntrue\n]]>\n \n\n\n \n \n A permutation, or fail.\n \n This operati\ on searches for an isomorphism between coloured digraphs. A\n coloured dig\ raph can be specified by its underlying digraph digraph1\n and its c\ olouring colours1. Let n be the number of vertices\n of d\ igraph1. The colouring colours1 may have one of the\n following \ two forms:\n\n \n \n a list of n integers, w\ here colours[i] is the\n colour of vertex i, using\ the colours [1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint lists whose union is\ \n DigraphVertices(digraph), such that\n colour\ s[i] is the list of all vertices with colour\n i.\n \ \n \n\n An isomorphism between coloured digraph\ s is an isomorphism between\n the underlying digraphs that preserves the co\ lourings. See for more information\n about isomorphisms of digraphs. More precisely, l\ et f be an\n isomorphism of digraphs from the digraph digraph1 (with colouring\n colours1) to the digraph digraph2 (with c\ olouring\n colours2), and let p be the permutation of the ver\ tices of\n digraph1 that corresponds to f. Then f pre\ serves the\n colourings of digraph1 and digraph2 – and henc\ e is an\n isomorphism of coloured digraphs – if colours1[i] =\n\ colours2[i ^ p] for all vertices i in digraph1.\ \n
\n\n This operation returns such an isomorphism if one exists, el\ se it returns\n fail.
\n\n By default, an isomorphism is foun\ d 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 Adolfo Piperno can be used instead; s\ ee\n , and \ .\n\n 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(digra\ ph1);\n\ngap> colours1 := [1, 1, 1\ , 1];;\ngap> colours2 := [1, 1, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph\ 1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = \ digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> colours1[i ^ p]) = \ colours2;\ntrue\ngap> colours1 := [1, 1, 2, 2];;\ngap> colours2 := [2, 2, 1, 1\ ];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colours1, colours2);\n(\ 1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVe\ rtices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> IsomorphismDig\ raphs(digraph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfail]]>\ \n \n\n\n \n \n An immutable list of lists\ .\n \n This function returns the list out of\ out-neighbours of each\n representative of the orbits of the action\ of on\n the vertex set of the digraph digr\ aph.
\n\n More specifically, if reps is the list of orbit re\ presentatives,\n then a vertex j appears in out[i] each time\ \n there exists an edge with source reps[i] and range j in\n \ digraph.\n
\n\n If is trivial\ , then \n is returned.\n\n D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2,\ 3, 4]]);\n\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 := Digraph(IsMutableDigra\ ph, [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);\n\ngap> DigraphGroup(D);\nGroup(())\ngap> R\ epresentativeOutNeighbours(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)\ ;;\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 ] ]]]>\n \n\n\n \n <\ Oper Name=\"IsDigraphIsomorphism\" Arg=\"src, ran, x\" \n Label=\"for d\ igraphs and transformation or permutation\"/>\n \n \n \n true or false.\n \n \ IsDigraphIsomorphism returns true if the permutation or\n tra\ nsformation x is an isomorphism from the digraph src to\n the\ digraph ran.\n
\n\n IsDigraphAutomorphism returns \ true if the permutation or\n transformation x is an automorphism\ of the digraph digraph.\n
\n\n A permutation or transformati\ on x is an isomorphism from a\n digraph src to a digra\ ph ran if the following hold:\n \n \n x is a bijection from the vertices of src to those of\n ran<\ /A>;\n \n \n [u ^ x, v ^ x]\ is an edge of\n ran if and only if [u, v] is an\n \ edge of src; and\n \n \n x fixes \ every i which is not a vertex of src.\n \n \n See also .
\n \n If col1 and col2, or col, are given, \ then they must\n represent vertex colourings; see \n \n for detail\ s of the permissible values for\n these arguments. The homomorphism must th\ en also have the property:\n \n \n \n col1[i] =\ col2[i ^ x] for all vertices i of src,\n for IsDi\ graphIsomorphism. \n \n col[i] = col[i ^ x] for all vertices i of digraph,\n for IsDigraphAutomo\ rphism. \n \n\n For some digraphs, it can be faster to\ use IsDigraphAutomorphism\n than to test membership in the automorp\ hism group of digraph.\n\n src := Digraph([\ [1], [1, 2], [1, 3]]);\n\ngap> IsD\ igraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigraphAutomorphism(src, (2\ , 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 1, 1]);\ntrue\ngap> \ IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ngap> IsDigraphAutomorph\ ism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphism(src, (1, 4));\nfals\ e\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran := Digraph([[2, 1], [2\ ], [2, 3]]);\n\ngap> IsDigraphIsom\ orphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ \ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntrue\ngap> IsDigraphIsom\ orphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nfalse\ngap> IsDigraphIs\ omorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ngap> IsDigraphIsomor\ phism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n]]>\n \n\n\n \n \n \n true\ or false. \n \n The operation IsDigr\ aphColouring verifies whether or not \n the list list describe\ s a proper colouring of the digraph \n digraph. \n
\n\n \ A list list describes a proper colouring of a digraph\n \ digraph if list consists of positive integers, the length of\n \ list equals the number of vertices in digraph, and for any\ \n vertices u, v of digraph if u and v are ad\ jacent,\n then list[u] >< list[v].\n
\ \n\n A transformation t describes a proper colouring of a digraph\ \n digraph, if ImageListOfTransformation(t,\n Digra\ phNrVertices(digraph)) is a proper colouring of \n digraph<\ /A>.
\n\n See also .\n \n \ D := JohnsonDigraph(5, 3);\n\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]);\n\ false\ngap> IsDigraphColouring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphColouring(\ D, IdentityTransformation);\ntrue\n]]>\n \n \n\n
\n\n
Homomorphisms of digraphs\n\n The following methods 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 digraph will be treated as if it has no multiple edges.\ \n\n \n \n The argument user_param.\n \n This function finds homomorphisms from the digrap\ h D1 to the digraph\n D2 subject to the conditions imposed by\ the other arguments as\n described below.
\n\n If f and g<\ /C> are homomorphisms found by\n HomomorphismDigraphsFinder, then f cannot be obtained from\n g by right multiplying by an automo\ rphism of D2 in\n aut_grp.\n\n \n hook\ \n \n This argument should be a function or fa\ il.
\n\n If hook is a function, then it must have two arg\ uments\n user_param (see below) and a transformation t. T\ he\n function hook(user_param, t) is called every \ time\n a new homomorphism t is found by\n Homomorphism\ DigraphsFinder. If the function returns true,\n then Homo\ morphismDigraphsFinder stops and does not find any\n further homomo\ rphisms. This feature might be useful if you are\n searching for a homo\ morphism that satisfies some condition that you\n cannot specify via th\ e other arguments to\n HomomorphismDigraphsFinder.\n
\ \n\n If hook is fail, then a default function is used whi\ ch\n simply adds every new homomorphism found by\n Homomorphi\ smDigraphsFinder to user_param, which must be a\n mutable li\ st in this case.\n \n\n user_param\n \ \n If hook is a function, then user_param can be a\ ny &GAP;\n object. The object user_param is used as the first ar\ gument of\n the function hook. For example, user_param mi\ ght be a\n transformation semigroup, and hook(user_param, t)\n might set user_param to be the closure of user_p\ aram and\n t.
\n\n If the value of hook is\ fail, then the value of\n user_param must be a mutable l\ ist.\n \n\n max_results\n \n \ This argument should be a positive integer or infinity.\n \ HomomorphismDigraphsFinder will return after it has found\n m\ ax_results homomorphisms or the search is complete, whichever\n hap\ pens first.\n \n\n hint\n \n \ This argument should be a positive integer or fail.
\n\n \ If hint is a positive integer, then only homorphisms of rank\n \ hint are found.
\n\n If hint is fail, then no\ restriction is put on the rank\n of homomorphisms found.\n \n\n injective\n \n This argument \ should be 0, 1, or 2. If it is\n 2, then o\ nly embeddings are found, if it is 1, then only\n injective homo\ morphisms are found, and if it is 0 there are no\n restrictions \ imposed by this argument.\n
\n\n For backwards compatibility\ , injective can also be false\n or true which corr\ espond to the values 0 and 1\n described in the previous \ paragraph, respectively. \n \n\n image\n \ \n This argument should be a subset of the vertices of the g\ raph D2.\n HomomorphismDigraphsFinder only finds homomorp\ hisms from\n D1 to the subgraph of D2 induced by the vert\ ices\n image.\n \n\n partial_map\n \n This argument should be a partial map from D1<\ /A> to D2,\n that is, a (not necessarily dense) list of vertices\ of the digraph\n D2 of length no greater than the number vertic\ es in the digraph\n D1. HomomorphismDigraphsFinder only f\ inds homomorphisms\n extending partial_map (if any).\n \n \n colors1\n \n This sho\ uld be a list representing possible colours of vertices in the\n digrap\ h D1; see \n \n for details of the permis\ sible values for this argument.\n \n \n colors2\ \n \n This should be a list representing possible\ colours of vertices in the\n digraph D2; see \n \n for details of the permissible values for this argument.\n \ \n order\n \n The optiona\ l argument order specifies the order the\n vertices in D1\ appear in the search for homomorphisms. \n The value of this parameter\ can have a large impact\n on the runtime of the function. It seems in \ many cases to be a good\n idea for this to be the , i.e.\n vertices ordered from highest to lowest degr\ ee.\n \n \n The optional argument aut_grp\ should be a subgroup of the\n automorphism group of D2. This fu\ nction returns unique\n representatives of the homomorphisms found up t\ o right multiplication\n by aut_grp. If this argument is not spe\ cific, it defaults to the\n full automorphism group of D2, which\ may be costly to calculate.\n \n \n\n 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 \ := CompleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, fail, [], 1, f\ ail, 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], fail, fail);\n[ T\ ransformation( [ 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 Transformatio\ n( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFinder(NullD\ igraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], fail, fa\ il, fail, fail,\n> Group(()));\n[ IdentityTransformation, Transformation( [ 1,\ 3, 3 ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), \n \ Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> Homomorph\ ismDigraphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail,\ 1, [1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityTransform\ ation, Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ) ]]]>\n \n\n\n \n \n A transformati\ on, or fail.\n \n A homomorphism from dig\ raph1 to digraph2 is a mapping from\n the vertex set of digra\ ph1 to a subset of the vertices of\n digraph2, such that every p\ air of vertices [i,j] which has\n an edge i->j is mapped to a\ pair of vertices [a,b] which\n has an edge a->b. Note that \ non-adjacent vertices can still be\n mapped to adjacent vertices.
\n\n \ DigraphHomomorphism returns a single homomorphism between\n di\ graph1 and digraph2 if it exists, otherwise it returns\n fail\ .\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph(\ [[3, 5], [2], [3, 1], [], [4]]);\n\ \ngap> DigraphHomomorphism(gr1, gr1);\nIdentityTransformation\ngap> map := Dig\ raphHomomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap> IsDigra\ phHomomorphism(gr1, gr2, map);\ntrue\n]]>\n \n\n\n \n \n \n A list of transformations.\n \ \n HomomorphismsDigraphsRepresentatives finds every\n \ between digraph1 and\n digr\ aph2, up to right multiplication by an element of the\n of digraph2.\n In other w\ ords, every homomorphism f between digraph1 and\n digraph2\ can be written as the composition f = g * x,\n where g i\ s one of the HomomorphismsDigraphsRepresentatives\n and x is \ an automorphism of digraph2.\n
\n\n HomomorphismsDigraphs<\ /C> returns all homomorphisms between\n digraph1 and digraph2\ .\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph(\ [[3, 5], [2], [3, 1], [], [4]]);\n\ \ngap> HomomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ] ), Tran\ sformation( [ 1, 3, 3 ] ), \n Transformation( [ 1, 5, 4, 4, 5 ] ), Transforma\ tion( [ 2, 2, 2 ] ), \n Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1\ , 5, 4, 5 ] ), \n Transformation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] \ ) ]\ngap> HomomorphismsDigraphsRepresentatives(gr1, CompleteDigraph(3));\n[ Tr\ ansformation( [ 2, 1 ] ), Transformation( [ 2, 1, 2 ] ) ]\n]]>\n \n\n\n \n \n A transformation, or fail<\ /K>.\n \n DigraphMonomorphism returns a sing\ le injective between digr\ aph1 and digraph2\n if one exists, otherwise it returns fail<\ /K>.\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digra\ ph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> DigraphMonomorphism(gr1, gr1);\nIdentityTransformation\ngap> Digraph\ Monomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\n]]>\n \ \n\n\n \n \n \n A list of transfor\ mations.\n \n These operations behave the same as <\ Ref Oper=\"HomomorphismsDigraphs\"/> and\n , except they only return\n injective homomorphi\ sms.\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digra\ ph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> MonomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 5, 4, 4, 5 \ ] ), \n Transformation( [ 3, 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigraphsRepr\ esentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ) ]\n]]>\n \n\n\n \n \n A transformation, \ or fail.\n \n DigraphEpimorphism retu\ rns a single surjective betw\ een digraph1 and digraph2\n if one exists, otherwise it retur\ ns fail.\n gr1 := DigraphReverse(ChainDigra\ ph(4));\n\ngap> gr2 := DigraphRemo\ veEdge(CompleteDigraph(3), [1, 2]);\n\ngap> DigraphEpimorphism(gr2, gr1);\nfail\ngap> DigraphEpimorphism(gr1, gr\ 2);\nTransformation( [ 3, 1, 2, 3 ] )\n]]>\n \n\n\n \n \n \n A list of transformations.\n \n These operations behave the same as and\n , exc\ ept they only return\n surjective homomorphisms.\n gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(3));\n\ngap> EpimorphismsDigraphs\ Representatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Transformation\ ( [ 3, 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> EpimorphismsD\ igraphs(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 Transformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] ), \n Tr\ ansformation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), \n Transfor\ mation( [ 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]]>\n \n<\ /ManSection>\n\n \n \n A transformation, or fail.\n \ \n An embedding of a digraph digraph1 into another dig\ raph\n digraph2 is a from dig\ raph1\n to digraph2 which has the additional property that a pai\ r of\n vertices [i, j] which have no edge i -> j in digrap\ h1 are\n\n mapped to a pair of vertices [a, b] which have no edg\ e a->b in\n digraph2.
\n\n In other words, an embedding\ t is an isomorphism from\n digraph1 to the of digraph2\n on the image of t.
\n\n \ DigraphEmbedding returns a single embedding if one exists, otherwise\n\ it returns fail.\n gr := ChainDigraph(3\ );\n\ngap> DigraphEmbedding(gr, Compl\ eteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Digraph([[3], [1, 4], [1], [3\ ]]));\nTransformation( [ 2, 4, 3, 4 ] )\n]]>\n \n\n\n \n \n \n \ A list of transformations.\n \n These oper\ ations behave the same as and\n , except they only return\n \ embeddings of D1 into D2.
\n\n See also .\n D1 := NullDigraph(2);\n\ngap> D2 := CycleDigraph(5);\n\ngap> EmbeddingsDigraphsRepresentatives(D1, D2);\ \n[ Transformation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ) ]\ngap> Em\ beddingsDigraphs(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 Transformatio\ n( [ 3, 5, 5, 1, 2 ] ), \n Transformation( [ 4, 1, 1, 2, 3 ] ), \n Transform\ ation( [ 4, 2, 1, 2, 3 ] ), \n Transformation( [ 5, 2, 2, 3, 4 ] ), \n Trans\ formation( [ 5, 3, 2, 3, 4 ] ) ]\n]]>\n \n\n\n \n \n \ \n \n \n \n \n \n \n true or <\ K>false.\n \n IsDigraphHomomorphism retu\ rns true if the permutation\n or transformation x is a homomo\ rphism from the digraph\n src to the digraph ran.\n
\n\ \n IsDigraphEpimorphism returns true if the permutation\n \ or transformation x is a surjective homomorphism from the digraph\n \ src to the digraph ran.\n
\n\n IsDigraphMonomorphis\ m returns true if the permutation\n or transformation x i\ s an injective homomorphism from the digraph\n src to the digraph ran.\n
\n\n IsDigraphEndomorphism returns true if\ the permutation\n or transformation x is an endomorphism of the dig\ raph\n digraph.\n
\n\n A permutation or transformation \ x is a homomorphism from a\n digraph src to a digraph \ ran if the following hold:\n \n \n [u ^ x\ , v ^ x] is an edge of\n ran whenever [u, v] is an\n edge of src; and \n \n x<\ /A> fixes every i which is not a vertex of src.\n \n\ \n See also .
\ \n\n\n If col1 and col2, or col, are given, then they \ must\n represent vertex colourings; see \n \n for details of the p\ ermissible values for these argument. The\n homomorphism must then also hav\ e the property:\n \n \n \n col[i] = col[i ^ x]<\ /C> for all vertices i of digraph,\n in the case of Is\ DigraphEndomorphism.\n \n col1[i] = col2[i ^ x\ ] for all vertices i of src,\n in the cases of the ot\ her operations.\n \n\n See also .\n \n src := Digraph([[1], [1, \ 2], [1, 3]]);\n\ngap> ran := Digra\ ph([[1], [1, 2]]);\n\ngap> IsDigra\ phHomomorphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDigraphHomo\ morphism(src, ran, Transformation([2, 1, 2]));\nfalse\ngap> IsDigraphHomomorph\ ism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphHomomorphism(s\ rc, src, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphHomomorphism(src, ra\ n, 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> IsDigraphEndomorphism(s\ rc, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphEndomorphism(src, Transfo\ rmation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsDigraphEndomorphism(src, Transfo\ rmation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> IsDigraphEpimorphism(src, ran, Tr\ ansformation([3, 3, 3]));\nfalse\ngap> IsDigraphMonomorphism(src, ran, Transfo\ rmation([1, 2, 2]));\nfalse\ngap> IsDigraphEpimorphism(src, ran, Transformatio\ n([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphism(ran, src, ());\ntrue]]>\n \n\n\n \n \n \n true or false.\n \n IsDigraphEmbedding returns true if the permutat\ ion\n or transformation x is a embedding of the digraph\n src<\ /A> into the digraph ran, while respecting the colourings\n col1<\ /A> and col2 if given.\n
\n\n A permutation or transformation\ x is a embedding of a digraph\n src into a digraph ran if x is a monomorphism from\n src to ran and \ the inverse of x is a monomorphism from\n the subdigraph of ran induced by the image of x to\n src.\n\n See also .
\n\n src := Di\ graph([[1], [1, 2]]);\n\ngap> ran \ := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(s\ rc, 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, IdentityTransformation);\ntru\ e\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\nfalse]]>\n \n\n\n \n \n true or false. \n \n The operation \ DigraphsRespectsColouring verifies whether or not \n the permutati\ on or transformation x respects the vertex colourings\n col1 and col2 of the digraphs src and range.\n That is,\ DigraphsRespectsColouring returns true if and only if for\n \ all vertices i of src, col1[i] = col2[i ^ x].\n \n\n src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]\ );\n\ngap> DigraphsRespectsColouri\ ng(src, ran, (1, 2), [2, 1], [1, 2, 1]);\ntrue\ngap> DigraphsRespectsColouring\ (src, ran, (1, 2), [2, 1], [2, 1, 1]);\nfalse\n]]>\n \n \n\n \n \n \n A list of transformation\ s.\n \n An endomorphism of digraph is a homo\ morphism\n from digraph back to \ itself.\n\n GeneratorsOfEndomorphismMonoid, called with a single arg\ ument,\n returns a generating set for the monoid of all endomorphisms of\n \ digraph. If digraph belongs to , then the value of\n GeneratorsOfEndomorphismMonoid wil\ l not be recomputed on future\n calls.
\n\n If the colors argu\ ment is specified, then\n GeneratorsOfEndomorphismMonoid will return\ a generating\n set for the monoid of endomorphisms which respect the given\ colouring.\n The colouring colors can be in one of two forms:
\ \n\n \n \n A list of positive integers of size the \ number of vertices of\n digraph, where colors[i] i\ s the colour of vertex\n i.\n \n \n \ A list of lists, such that colors[i] is a list of all\n \ vertices with colour i.\n \n \n\n If the l\ imit argument is specified, then it will return only the\n first lim\ it homomorphisms, where limit must be a positive\n integer or infinity.
\n\n gr := Digraph(List([1 .. 3\ ], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ Transformati\ on( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation, Tra\ nsformation( [ 1, 2, 1 ] ), \n Transformation( [ 1, 2, 2 ] ), Transformation(\ [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 ] ) ]\ngap> GeneratorsOfEndomorp\ hismMonoid(gr, 3);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] \ ), \n IdentityTransformation ]\ngap> gr := CompleteDigraph(3);;\ngap> Generat\ orsOfEndomorphismMonoid(gr);\n[ Transformation( [ 2, 3, 1 ] ), Transformation(\ [ 2, 1 ] ), \n IdentityTransformation ]\ngap> GeneratorsOfEndomorphismMonoid\ (gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]\ng\ ap> GeneratorsOfEndomorphismMonoid(gr, [[1], [2, 3]]);\n[ Transformation( [ 1,\ 3, 2 ] ), IdentityTransformation ]\n]]>\n \n\n\n \n \n A trans\ formation, or fail.\n \n A proper colouri\ ng of a digraph is a labelling of its vertices in\n such a way that adj\ acent vertices have different labels. A proper\n n-colouring is a proper colouring that uses exactly n\n colours. Equivalently\ , a proper (n-)colouring of a digraph can be\n defined to be a from a digraph onto the\n complete digraph (w\ ith n vertices); see . Note that\ a digraph with loops () does not have a\ proper n-colouring for\n any value n.\n
\n\n If digraph is a digraph and n is a non-negative integer, then\n DigraphColouring(digraph, n) returns an epimorphism\n fr\ om digraph onto the complete digraph with n vertices if one\n \ exists, else it returns fail.
\n\n See also and \n
\n\n Note that a \ digraph with at least two vertices has a 2-colouring if and only\n if it is\ bipartite, see .\n\n DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphColouring(C\ hainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := DigraphCol\ ouring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigr\ aphColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransformation( [\ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n]]>\n \n\n\n \n \n \n \n\ A transformation, or fail.\n \n \ A proper colouring of a digraph is a labelling of its vertices in\n \ such a way that adjacent vertices have different labels. Note that a digraph \ \n with loops () does not have any proper \n\ colouring.\n
\n \n If digraph is a digraph and orde\ r is a dense list consisting \n of all of the vertices of digraph (in any order), then \n DigraphGreedyColouring\n uses a greedy\ algorithm with the specified order to obtain some proper\n colouring of digraph, which may not use the minimal number of\n colours.
\n \ \n If digraph is a digraph and func is a function whose argum\ ent \n is a digraph, and that returns a dense list order, then \n \ DigraphGreedyColouring(digraph, func) returns\n Di\ graphGreedyColouring(digraph, func(digraph)).\n \n\n If the optional second argument (either a list or a function), is no\ t \n specified, then is used by def\ ault. \n
\n \n See also\n .\n
\n\n 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]]>\n \n\n\n \n \n A list of the vertices.\ \n \n DigraphWelshPowellOrder returns a list\ of all of the vertices of \n the digraph digraph ordered according \ to the sum of the number of \n out- and in-neighbours, from highest to lowe\ st. \n
\n\n 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]]>\n \n\n\n \ \n \n \ A non-negative integer.\n \n A proper colouring<\ /E> of a digraph is a labelling of its\n vertices in such a way that adjace\ nt vertices have different labels.\n Equivalently, a proper digraph colouri\ ng can be defined to be a from a dig\ raph onto a complete digraph.
\n\n If digraph is a digraph witho\ ut loops (see , then ChromaticNumber<\ /C> returns the least\n non-negative integer n such that there is a \ proper colouring of\n digraph with n colours. In other words\ , for a digraph with at\n least one vertex, ChromaticNumber returns \ the least number n\n such that DigraphColouring(digraph, n\ ) does not return\n fail. See .\n\n ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(CompleteDi\ graph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));\n2\ngap\ > ChromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> ChromaticNum\ ber(NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> ChromaticNumber(D);\n3\n]]>\n \n\n\n \n \n A list of positive integers.\n \n If D is a digraph, then DigraphCore \ returns a list of vertices\n corresponding to the core of D\ . In particular, the subdigraph\n of D induced by this list is iso\ morphic to the core of D.
\n\n The core of a digraph D<\ /C> is the minimal subdigraph C of\n D which is a homomorphic\ image of D. The core of a digraph \n is unique up to isomorphism.\n\ D := DigraphSymmetricClosure(CycleDigraph(8));\ \n\ngap> DigraphCore(D)\ ;\n[ 1, 2 ]\ngap> D := PetersenGraph();\n\ngap> DigraphCore(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMutableDigra\ ph, [[3], [3], [4], [5], [2]]);\n\ng\ ap> DigraphCore(D);\n[ 2, 3, 4, 5 ]\n]]>\n \n \n\n
\n\n\n \n \n Finding cliques and\ independent sets\n\n In &Digraphs;, a clique of a digraph\ is a set of mutually adjacent\n vertices of the digraph, and an indepen\ dent set is a set of mutually\n non-adjacent vertices of the digraph. A\ maximal clique is a clique\n which is not properly contained in ano\ ther clique, and a maximal\n independent set is an independent set\ which is not properly contained\n in another independent set. Using this \ definition in &Digraphs;, cliques\n and independent sets are both permitted\ , but not required, to contain vertices\n at which there is a loop. Anothe\ r name for a clique is a complete\n subgraph.\n
\n\n &Di\ graphs; provides extensive functionality for computing cliques and\n indepe\ ndent sets of a digraph, whether maximal or not. The fundamental\n algorit\ hm used in most of the methods in &Digraphs; to calculate cliques\n and ind\ ependent sets is a version of the Bron-Kerbosch algorithm.\n Calculating th\ e cliques and independent sets of a digraph is a well-known\n and hard prob\ lem, and searching for cliques or independent sets in a\n digraph can be ve\ ry lengthy, even for a digraph with a small number of\n vertices. &Digraph\ s; uses several strategies to increase the performance\n of these calculati\ ons.\n
\n\n From the definition of cliques and independent sets, it \ follows that the\n presence of loops and multiple edges in a digraph is irr\ elevant to the\n existence of cliques and independent sets in the digraph. \ See and \ for more\n information about these properties. Therefore given a digraph\ \n digraph, the cliques and independent sets of digraph are\n\ equal to the cliques and independent sets of the digraph:\n\n \n \ \n DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(<\ A>digraph)).\n \n \n\n See and f\ or more information about these\n attributes. Furthermore, the cliques of \ this digraph are equal to the\n cliques of the digraph formed by removing a\ ny edge [u,v] for which\n the corresponding reverse edge [v,u] is not present. Therefore,\n overall, the cliques of digraph are \ equal to the cliques of the\n symmetric digraph:\n\n \n \n MaximalSymmetricSubdigraphWithoutLoops(digraph).\n \n \n\n See for more\n information about this. The of this symmetric digraph con\ tains the\n automorphism group of digraph as a subgroup. By perform\ ing the\n search for maximal cliques with the help of this larger automorph\ ism group\n to reduce the search space, the computation time may be reduced\ . The\n functions and attributes which return representatives of cliques o\ f\n digraph will return orbit representatives of cliques under the\n\ action of the automorphism group of the maximal symmetric subdigraph\n \ without loops on sets of vertices.
\n\n The independent sets of\ a digraph are equal to the independent sets of the\n . Therefore, overall, the independent\n sets of digra\ ph are equal to the independent sets of the symmetric\n digraph:\n\n \ \n \n DigraphSymmetricClosure(DigraphRemoveLoops(D\ igraphRemoveAllMultipleEdges(\n digraph))).\n \n \n\n Again, the automorphism group of this symmetric digr\ aph contains the\n automorphism group of digraph. Since a search for\ independent sets\n is equal to a search for cliques in the , the\n methods used in &Digraphs; usually transform a search f\ or independent sets\n into a search for cliques, as described above. The f\ unctions and\n attributes which return representatives of independent sets \ of\n digraph will return orbit representatives of independent sets u\ nder\n the action of the automorphism group of the symmetric closure\ of the\n digraph formed by removing loops and multiple edges.
\n\n P\ lease note that in &Digraphs;, cliques and independent sets are not\n requi\ red to be maximal. Some authors use the word clique to mean\n maximal clique, and some authors use the phrase independent set to\n mean maxi\ mal independent set, but please note that &Digraphs; does\n not use thi\ s definition.\n
\n\n
Finding cliques\n \ \n\n\ntrue or false.\n\n If digraph is a digraph and l is a duplic\ ate-free list of\n vertices of digraph, then\n IsClique(dig\ raph,l) returns true\n if l is a \ clique of digraph and false if it is\n not. Similarly,\n\ IsMaximalClique(digraph,l) returns\n \ true if l is a maximal clique of digraph and\n \ false if it is not.
\n\n A clique of a digraph is a set\ of mutually adjacent vertices of the\n digraph. A maximal clique is\ a clique that is not properly\n contained in another clique. A clique is p\ ermitted, but not required, to\n contain vertices at which there is a loop.\ \n D := CompleteDigraph(4);;\ngap> IsClique(D, [1,\ 3, 2]);\ntrue\ngap> IsMaximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMaximalCliq\ ue(D, DigraphVertices(D));\ntrue\ngap> D := Digraph([[1, 2, 4, 4], [1, 3, 4], \ [2, 1], [1, 2]]);\n\ngap> Is\ Clique(D, [2, 3, 4]);\nfalse\ngap> IsMaximalClique(D, [1, 2, 4]);\ntrue\ngap> \ D := CompleteDigraph(IsMutableDigraph, 4);;\ngap> IsClique(D, [1, 3, 2]);\ntru\ e]]>\n \n\n\n \n \n The argument user_param.\n \n This function finds cliques of the digraph digraph\ subject to the\n conditions imposed by the other arguments as describe\ d below. Note\n that a clique is represented by the immutable list of the v\ ertices that\n it contains.\n
\n\n Let G denote the automo\ rphism group of the maximal symmetric\n subdigraph of digraph withou\ t loops (see \ and ).\n\n \n hook\n \n This argument shoul\ d be a function or fail.
\n\n If hook is a function, t\ hen it should have two arguments\n user_param (see below) and a \ clique c. The function\n hook(user_param, c) is called every time a new\n clique c is found by CliquesFi\ nder.
\n\n If hook is fail, then a default functio\ n is used that\n simply adds every new clique found by CliquesFinder\ to\n user_param, which must be a list in this case.\n \ \n\n user_param\n \n If h\ ook is a function, then user_param can be any &GAP;\n object\ . The object user_param is used as the first argument for\n the \ function hook. For example, user_param might be a\n list,\ and hook(user_param, c) might add the size\n of t\ he clique c to the list user_param.
\n\n If the value\ of hook is fail, then the value of\n user_param m\ ust be a list.\n \n\n limit\n \ \n This argument should be a positive integer or infinity.\n \ CliquesFinder will return after it has found\n limit \ cliques or the search is complete.\n \n\n include and exclude\n \n These arguments should each\ be a (possibly empty) duplicate-free list\n of vertices of digraph<\ /A> (i.e. positive integers less than the\n number of vertices of di\ graph).
\n\n CliquesFinder will only look for cliques co\ ntaining all of the\n vertices in include and containing none of\ the vertices in\n exclude.
\n\n Note that the search\ may be much more efficient if each of these lists\n is invariant under\ the action of G on sets of vertices.\n \n\n \ max\n \n This argument should be true or <\ K>false. If max is\n true then CliquesFinder will on\ ly search for maximal\n cliques. If max is false t\ hen non-maximal cliques may be\n found.\n \n\n <\ A>size\n \n This argument should be fail o\ r a positive integer.\n If size is a positive integer then Cl\ iquesFinder will\n only search for cliques that contain precisely <\ A>size vertices.\n If size is fail then cliques of an\ y size may be found.\n \n\n reps\n <\ Item>\n This argument should be true or false.
\n\n \ If reps is true then the arguments include and\n \ exclude are each required to be invariant under the action of\n \ G on sets of vertices. In this case, CliquesFinder will\n \ find representatives of the orbits of the desired cliques under the\n \ action of G, although representatives may be returned that\n \ are in the same orbit.\n If reps is false then Cli\ quesFinder will not take this into\n consideration.
\n\n \ For a digraph such that G is non-trivial, the search for\n cliqu\ e representatives can be much more efficient than the search for\n all \ cliques.\n \n \n\n This function uses a version of th\ e Bron-Kerbosch algorithm.\n D := CompleteDigraph(\ 5);\n\ngap> user_param := [];;\nga\ p> f := function(a, b) # Calculate size of clique\n> AddSet(user_param, Siz\ e(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], [], false,\ fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFinder(D, fa\ il, [], 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, [], inf\ inity, [], [], 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,\ false);\n[ ]\ngap> D := CompleteDigraph(IsMutableDigraph, 5);\n\ngap> user_param := [];;\ngap> f := function(a,\ b) # Calculate size of clique\n> AddSet(user_param, Size(b));\n> end;;\nga\ p> CliquesFinder(D, f, user_param, 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> CliquesFinder(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, [], infinity, [1, 3], [], true, 3, false);\n[ ]]]>\n \n\n\n \n \n \ \n An immutable list of positive integers, or fail.\ \n \n If digraph is a digraph, then these functions re\ turns a clique\n of digraph if one exists that satisfies the argumen\ ts, else it\n returns fail. A clique is defined by the set of verti\ ces that it\n contains; see and .
\n\n The optional arguments include and exclude\ must each be a\n (possibly empty) duplicate-free list of vertices of <\ A>digraph, and\n the optional argument size must be a positive i\ nteger. By default,\n include and exclude are empty. These f\ unctions will search for\n a clique of digraph that includes the ver\ tices of include\n but does not include any vertices in exclude; if the argument\n size is supplied, then additionally the clique\ will be required to\n contain precisely size vertices.
\n\n I\ f include is not a clique, then these functions return fail.\n \ Otherwise, the functions behave in the following way, depending on the\n \ number of arguments:\n\n \n One or two arguments\n \ \n If one or two arguments are supplied, then DigraphCliq\ ue and\n DigraphMaximalClique greedily enlarge the clique include\n until it can not continue. The result is guaranteed to \ be a maximal\n clique. This may or may not return an answer more quickl\ y than using\n . with a limit of \ 1.\n \n Three arguments\n \n I\ f three arguments are supplied, then DigraphClique greedily\n en\ larges the clique include until it can not continue, although\n \ this clique may not be maximal.
\n\n Given three arguments, Digra\ phMaximalClique returns the maximal\n clique returned by Digraph\ MaximalCliques(digraph,\n include, e\ xclude, 1) if one exists,\n else fail.\n \ \n Four arguments\n \n If four arguments a\ re supplied, then DigraphClique returns the\n clique returned by\ DigraphCliques(digraph,\n include, exclude, 1,\n size) if one exists, else <\ K>fail. This clique may\n not be maximal.
Given four arguments,\ DigraphMaximalClique\n returns the maximal clique returned by\n\ DigraphMaximalCliques(digraph, include,\ \n exclude, 1, size) if one exists, els\ e\n fail.\n \n \n\n D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> Digrap\ hClique(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 ]\ng\ ap> 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 ]]]>\n \n\n\n \n \ \n \n \n \n \n \n An immutable list of lists \ of positive integers.\n \n If digraph is dig\ raph, then these functions and attributes use to return cliques of digraph. A\n clique is defined by the set \ of vertices that it contains; see and .
\n\n The optional arguments include and \ exclude must each be a\n (possibly empty) list of vertices of dig\ raph, the optional argument\n limit must be either a positive in\ teger or infinity, and the\n optional argument size must be a\ positive integer. If not\n specified, then include and exclude<\ /A> are chosen to be empty\n lists, and limit is set to infinity<\ /C>.
\n\n The functions will return as many suitable cliques as possibl\ e, up to the\n number limit. These functions will find cliques that\ contain all\n of the vertices of include but do not contain any of \ the\n vertices of exclude. The argument size restricts the s\ earch\n to those cliques that contain precisely size vertices.\n \ If the function or attribute has Maximal in its name, then only\n ma\ ximal cliques will be returned; otherwise non-maximal cliques may be\n retu\ rned.
\n\n Let G denote the automorphism group of maximal symmet\ ric subdigraph\n of digraph without loops (see and ).\n\n \n Distinct cliques<\ /Mark>\n \n DigraphMaximalCliques and DigraphCliqu\ es each return a\n duplicate-free list of at most limit cliq\ ues of digraph\n that satisfy the arguments.
\n\n The \ computation may be significantly faster if include and\n excl\ ude are invariant under the action of G\n on sets of vertice\ s.\n \n\n Orbit representatives of cliques\n \ \n To use DigraphMaximalCliquesReps or DigraphCliques\ Reps,\n the arguments include and exclude must each b\ e invariant\n under the action of G on sets of vertices.
\n\n\ If this is the case, then DigraphMaximalCliquesReps and\n \ DigraphCliquesReps each return a duplicate-free list of at most\n \ limit orbits representatives (under the action of G on\n \ sets vertices) of cliques of digraph that satisfy the\n argu\ ments.
\n\n The representatives are not guaranteed to be in distinct\ orbits.\n However, if fewer than lim results are returned, then\ there\n will be at least one representative from each orbit of maximal\ cliques.\n \n \n\n D := Dig\ raph([\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> Digra\ phMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ngap> DigraphMaxi\ malCliquesReps(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> Digrap\ hMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCliques(D, [1], [\ 5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph(IsMutableDigra\ ph, [\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> DigraphM\ aximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]]]>\n \n\n\n \n \n A non-negative integer.\n \n If digraph is a digraph, then CliqueNumber(digraph)\n returns the largest integer n such that digraph contains \ a\n clique with n vertices as an induced subdigraph.\n
\n\n \ A clique of a digraph is a set of mutually adjacent vertices of the\n\ digraph. Loops and multiple edges are ignored for the purpose of\n dete\ rmining the clique number of a digraph. \n D := Co\ mpleteDigraph(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(IsMutableDigraph, 4);\ ;\ngap> CliqueNumber(D);\n4]]>\n\n\n\n <\ /Section>\n\n
Finding independent sets\n \n \n \n true or false.\n \n If digraph is a digraph and\ l is a duplicate-free list of\n vertices of digraph, then\n \ IsIndependentSet(digraph,l) returns\n \ true if l is an independent set of digraph and\n \ false if it is not. Similarly,\n IsMaximalIndependentSet(\ digraph,l)\n returns true if l is\ a maximal independent set of\n digraph and false if i\ t is not.
\n\n An independent set of a digraph is a set of mutu\ ally non-adjacent\n vertices of the digraph. A maximal independent set is an independent\n set that is not properly contained in another indepe\ ndent set. An\n independent set is permitted, but not required, to contain \ vertices at\n which there is a loop.\n D := Cyc\ leDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsMaximalIndependen\ tSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfalse\ngap> IsInde\ pendentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, [2, 4]);\ntrue\n\ gap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndependentSet(D, [1]);\ \ntrue]]>\n \n\n\n \n \n \n An immutable list of positive\ integers, or fail.\n \n If digraph i\ s a digraph, then these functions returns an independent\n set of digrap\ h if one exists that satisfies the arguments, else it\n returns fail\ . An independent set is defined by the set of vertices\n that it conta\ ins; see and .
\n\n The optional arguments include and exclud\ e must each be a\n (possibly empty) duplicate-free list of vertices of \ digraph, and the\n optional argument size must be a positive \ integer. By default,\n include and exclude are empty. These \ functions will search\n for an independent set of digraph that inclu\ des the vertices of\n include but does not include any vertices in <\ A>exclude;\n if the argument size is supplied, then additionally\ the independent\n set will be required to contain precisely size ve\ rtices.
\n\n If include is not an independent set, then these fun\ ctions return\n fail. Otherwise, the functions behave in the follow\ ing way,\n depending on the number of arguments:\n\n \n One or two arguments\n \n If one or two arguments ar\ e supplied, then DigraphIndependentSet\n and DigraphMaximalIn\ dependentSet greedily enlarge the\n independent set include \ until it can not continue. The result\n is guaranteed to be a maximal \ independent set. This may or may not\n return an answer more quickly th\ an using . with a lim\ it of 1.\n \n Three arguments\n \n \ If three arguments are supplied, then DigraphIndependentSet\n \ greedily enlarges the independent set include until it can not\n \ continue, although this independent set may not be maximal.
\n\n \ Given three arguments, DigraphMaximalIndependentSet returns the\n \ maximal independent set returned by\n DigraphMaximalIndependentSe\ ts(digraph,\n include, exclude, 1) if one exists,\n else fail.\n \n Four arguments\n \n If four arguments are supplied,\ then DigraphIndependentSet\n returns the independent set return\ ed by\n DigraphIndependentSets(digraph, include\ ,\n exclude, 1, size) if one exi\ sts, else\n fail. This independent set may not be maximal.
G\ iven four\n arguments, DigraphMaximalIndependentSet returns the \ maximal\n independent set returned by\n DigraphMaximalIndepen\ dentSets(digraph,\n include, exclu\ de, 1,\n size) if one exists, else fail\ .\n \n \n D := ChainDigraph(\ 6);\n\ngap> DigraphIndependentSet(D);\ \n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> Digrap\ hIndependentSet(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> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndepe\ ndentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(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> DigraphMaximal\ IndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, \ 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> Digra\ phIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet\ (D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ \ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ngap> DigraphMaximal\ IndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]]]>\n \n\n\n \n \n \n \n \n\ \n \n A\ n immutable list of lists of positive integers.\n \n \ If digraph is digraph, then these functions and attributes use to return independent sets of digraph. A\ n\n independent set is defined by the set of vertices that it contains; see\ \n and .
\n\n The optional arguments include and exclude must each be a\n (possibly empty) list of vertices of digraph, th\ e optional argument\n limit must be either a positive integer or \ infinity, and the\n optional argument size must be a positive in\ teger. If not\n specified, then include and exclude are chos\ en to be empty\n lists, and limit is set to infinity.
\n\ \n The functions will return as many suitable independent sets as possible,\ up\n to the number limit. These functions will find independent se\ ts\n that contain all of the vertices of include but do not\n con\ tain any of the vertices of exclude The argument size\n restr\ icts the search to those cliques that contain precisely size\n verti\ ces. If the function or attribute has Maximal in its name,\n then o\ nly maximal independent sets will be returned; otherwise non-maximal\n inde\ pendent sets may be returned.\n
\n\n Let G denote the \n of the of the digraph formed from\n digraph by rem\ oving loops and ignoring the multiplicity of edges.\n\n \n Distinct independent sets\n \n DigraphMaximalInde\ pendentSets and DigraphIndependentSets\n each return a dupli\ cate-free list of at most limit independent\n sets of digraph\ that satisfy the arguments.
\n\n The computation may be signifi\ cantly faster if include and\n exclude are invariant unde\ r the action of G on sets of\n vertices.\n \n\n \ Representatives of distinct orbits of independent sets\n \n To use DigraphMaximalIndependentSetsReps or\n D\ igraphIndependentSetsReps, the arguments include and\n ex\ clude must each be invariant under the action of G on\n sets\ of vertices.
\n\n If this is the case, then DigraphMaximalIndepe\ ndentSetsReps and\n DigraphIndependentSetsReps each return a\ list of\n at most limit orbits representatives (under the actio\ n of\n G on sets of vertices) of independent sets of digraph<\ /A>\n that satisfy the arguments.
\n\n The representatives a\ re not guaranteed to be in distinct orbits.\n However, if lim is\ not specified, or fewer than lim\n results are returned, then t\ here will be at least one representative\n from each orbit of maximal i\ ndependent sets.\n \n \n\n D\ := CycleDigraph(5);\n\ngap> DigraphM\ aximalIndependentSetsReps(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> DigraphMaximalIndependentSe\ ts(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> DigraphMaximalIndependentSetsReps(D\ );\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ], [ 1, 3 ] ]\nga\ p> 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 ] ]]]>\ \n \n\n\n
\n\n \n Visualising and\ IO\n\n
Visualising a digraph\n \n \n Noth\ ing.\n \n This function attempts to convert the\ string str into a pdf\n document and open this document, i.e. to \ splash it all over your monitor.
\n\n The string str must corre\ spond to a valid dot or\n LaTeX text file and you must have\ have GraphViz and\n pdflatex installed on your computer. \ For details about these file\n formats, see http://www.latex-project\ .org and\n http://www.graphviz.org.
\n\n This fu\ nction is provided to allow convenient, immediate viewing of the\n pictur\ es produced by the function .
\n\n The optio\ nal second argument opts should be a record with\n components corr\ esponding to various options, given below.\n\n \n path\ \n \n this should be a string representing the pa\ th to the directory where\n you want Splash to do its work. Th\ e default value of this\n option is \"~/\".\n \n\ \n directory\n \n this should be a s\ tring representing the name of the directory in\n path where y\ ou want Splash to do its work. This function\n will create thi\ s directory if does not already exist.
\n\n The default value of \ this option is \"tmp.viz\" if the option\n path is pres\ ent, and the result of\n is used otherwise.\n \n\n filename\n\ \n this should be a string representing the name of the\ file where\n str will be written. The default value of this \ option is\n \"vizpicture\".\n \n\n \ viewer\n \n this should be a string representing \ the name of the program which\n should open the files produced by \ GraphViz or pdflatex.\n \n\n type\ \n \n this option can be used to specify that the string\ str contains\n a &LaTeX; or dot document. You can spec\ ify this option in\n str directly by making the first line \ \"%latex\" or\n \"//dot\". There is no default value for \ this option, this\n option must be specified in str or in o\ pt.type.\n \n\n engine\n \n\ this option can be used to specify the GraphViz engine to use\n \ to render a dot document. The valid choices are \"dot\",\n \ \"neato\", \"circo\", \"twopi\", \"fdp\",\ \n \"sfdp\", and \"patchwork\". Please refer to the\n \ GraphViz documentation for details on these engines.\n The de\ fault value for this option is \"dot\", and it\n must be speci\ fied in opt.engine.\n \n\n filetype\n\ \n this should be a string representing the type of fil\ e which\n Splash should produce. For &LaTeX; files, this optio\ n is\n ignored and the default value \"pdf\" is used.\n \ \n \n\n This function was written by Attila Egri-Nagy\ and Manuel Delgado with some\n minor changes by J. D. Mitchell.
\n\n \ Splash(DotDigraph(RandomDigraph(4)));\n]]>\n \ \n \n\n \n \n \n A string.\n \n Dot\ Digraph produces a graphical representation of the digraph\n digra\ ph. Vertices are displayed as circles, numbered consistently\n with <\ A>digraph. Edges are displayed as arrowed lines between\n vertices, w\ ith the arrowhead of each line pointing towards the range\n of the edge.<\ P/>\n\n DotVertexLabelledDigraph differs from DotDigraph on\ ly in\n that the values in are used t\ o label\n the vertices in the produced picture rather than the numbers 1 to\n the number of vertices of the digraph.
\n\n The outp\ ut is in dot format (also known as GraphViz)\n format. For \ details about this file format, and information about how to\n display or\ edit this format see http://www.graphviz.org.
\n\n The st\ ring returned by DotDigraph or\n DotVertexLabelledDigraph c\ an be written to a file using\n the command .
\n\n 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\ngap> FileString(\"dot/k4.dot\", D\ otDigraph(gr));\n154]]>\n \n \n\n \n \n A string.\n \n This function produces a gr\ aphical representation of the symmetric\n digraph digraph. DotS\ ymmetricDigraph will return an\n error if digraph is not a sym\ metric digraph. See\n .
\n\n Ve\ rtices are displayed as circles, numbered consistently with\n digraph<\ /A>. Since digraph is symmetric, for every non-loop\n edge there i\ s a complementary edge with opposite source and range.\n DotSymmetricD\ igraph displays each pair of complementary edges\n as a single line b\ etween the relevant vertices, with no arrowhead.
\n\n The output is in\ dot format (also known as GraphViz)\n format. For details \ about this file format, and information about how\n to display or edit th\ is format see\n http://www.graphviz.org.
\n\n The str\ ing returned by DotSymmetricDigraph can be written to a\n file usi\ ng the command .
\n\n star := Digraph([[2, 2, 3, 4], [1, 1], [1], [1, 4]]);\n\ngap> IsSymmetricDigraph(star);\ \ntrue\ngap> FileString(\"dot/star.dot\", DotSymmetricDigraph(gr));\n83]]>\n \n \n\n \n \n A string.\ \n \n This function produces a graphical representation o\ f a partial order\n digraph digraph. DotPartialOrderDigraph\ will return an error\n if digraph is not a partial order digraph.\ See .
\n\n Since dig\ raph is a partial order, it is both reflexive and\n transitive. The o\ utput of DotPartialOrderDigraph is the\n of the\n of digr\ aph.
\n\n The output is in dot format (also known as Gra\ phViz)\n format. For details about this file format, and information \ about how\n to display or edit this format see\n http://www.gra\ phviz.org.
\n\n The string returned by DotPartialOrderDigrap\ h can be written to a\n file using the command .
\n\n poset := Digraph([\ [1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigraph(gr);\ntrue\ngap> Fil\ eString(\"dot/poset.dot\", DotPartialOrderDigraph(gr));\n83]]>\n \n \n\n \n \n \n A string.\n \n This fun\ ction produces a graphical representation of a preorder\n digraph digr\ aph. DotPreorderDigraph will return an error\n if digraph is not a preorder digraph. See .<\ P/>\n\n A preorder digraph is reflexive and transitive but in general it \ is\n not anti-symmetric and may have strongly connected components\n \ containing more than one vertex. The \n <\ A>Q obtained by forming the quotient of digraph by the\n parti\ tion of its vertices into the strongly connected components\n satisfies <\ Ref Prop=\"IsPartialOrderDigraph\"/>. Thus every vertex of\n Q cor\ responds to a strongly connected component of digraph.\n The outpu\ t of DotPreorderDigraph displays the \n of Q with\n vertices displayed as rounde\ d rectangles labelled by all of the vertices\n of digraph in the c\ orresponding strongly connected component.
\n\n The output is in d\ ot format (also known as GraphViz)\n format. For details about\ this file format, and information about how\n to display or edit this fo\ rmat see\n http://www.graphviz.org.
\n\n The string r\ eturned by DotPreorderDigraph can be written to a\n file using the\ command .
\n\n 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\ \", DotProrderDigraph(gr));\n83]]>\n \n \ \n\n \n \n A string.\n \n DotHighlightedDigraph produces a graphical represenation\ of the\n digraph digraph, where the vertices in the list verts\ , and\n edges between them, are drawn with colour colour1 and \ all other\n vertices and edges in digraph are drawn with colour colour2.\n If colour1 and colour2 are not given then\n\ DotHighlightedDigraph uses black and grey respectively.
\n\n \ Note that DotHighlightedDigraph does not validate the colours\n \ colour1 and colour2 - consult the GraphViz documentation to\n\ see what is available.\n\n See for more \ details on the output.
\n\n digraph := Digraph([[\ 2, 3], [2], [1, 3]]);\n\ngap> FileString(\"d\ ot/my_digraph.dot\",\n> DotHighlightedDigraph(digraph, [1, 2], \"red\", \"blac\ k\"));\n264]]>\n \n \n\n
\n\n \
Reading and writing digraphs to a file\n This \ section describes different ways to store and read graphs\n from a file in \ the &Digraphs; package.\n \n Graph6 \n \ \n\t Graph6 is a graph data format for storing undirected graphs\n\twi\ th no multiple edges nor loops of size up to 2^{36} - 1 in\n\tprintab\ le chracters. The format consists of two parts. The first part\n\tuses one to \ eight bytes to store the number of vertices. And the second\n\tpart is the upp\ er half of the adjacency matrix converted into ASCII\n\tcharacters. For a more\ detail description see \n\thttp://cs.anu.edu.au/~bdm/data/\ formats.txt.\n \n\n Sparse6 \n \n\t Sparse6 is a graph data format for storing undirected graphs\n\t\ with possibly multiple edges or loops. The maximal number of vertices\n\tallow\ ed is 2^{36} - 1 . The format consists of two parts. The\n\tfirst part\ uses one to eight bytes to store the number of vertices.\n\tAnd the second pa\ rt only stores information about the edges.\n\tTherefore, the Sparse6 \ format return a more compact encoding\n\tthen Graph6 for sparse grap\ h, i.e. graphs where the number\n\tof edges is much less than the number of v\ ertices squared. For a more\n\tdetail description see \n \ http://cs.anu.edu.au/~bdm/data/formats.txt.\n \n\n \ Digraph6 \n \n\tDigraph6 is a new format ba\ sed on Graph6 , but designed\n\tfor digraphs. The entire adjacency mat\ rix is stored, and\n\ttherefore there is support for directed edges and single\ -vertex loops.\n\tHowever, multiple edges are not supported.\n \n\ \n DiSparse6 \n \n\tDiSparse6 is a new fo\ rmat based on Sparse6 , but\n\tdesigned for digraphs. In this format t\ he list of edges is\n\tpartitioned into inceasing and decreasing edges, depend\ ing whether the\n\tedge has its source bigger than the range. Then both sets o\ f edges are\n\twritten separetly in Sparse6 format with a separation \ symbol\n\tin between.\n \n \n \n \n \n \n \n A digraph.\n \n \ If str is a string encoding a graph in Graph6, Digraph6, Sparse6 or\n\ DiSparse6 format, then the corresponding function returns a digraph. In th\ e\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 i\ n\n the opposite direction is also present.
\n\n Each of these functi\ ons takes an optional first argument filt,\n which should be either \ \n or ,\n\ and which specifies whether the output digraph shall\n be mutable or im\ mutable.\n If no first argument is provided, then an immutable\n digraph\ is returned by default.\n DigraphFromGraph6String\ (\"?\");\n\ngap> DigraphFromGraph6Str\ ing(\"C]\");\n\ngap> DigraphFromGr\ aph6String(\"H?AAEM{\");\n\ngap> \ DigraphFromDigraph6String(\"&?\");\n\ \ngap> DigraphFromDigraph6String(IsMutableDigraph, \"&DOOOW?\");\n\ngap> DigraphFromDigraph6String(\"&CQFG\");\n\ngap> DigraphFromDigraph6String(\"&\ IM[SrKLc~lhesbU[F_\");\n\ngap> D\ igraphFromDiSparse6String(\".CaWBGA?b\");\n\n]]>\n \n\n\n \n \n \n \ \n \n A string.\n \n These four functions return a highly compressed s\ tring fully describing the\n digraph digraph.
\n\n Graph6 and\ Digraph6 are formats best used on small, dense graphs, if\n applicable. Fo\ r larger, sparse graphs use Sparse6 and\n Disparse6 (this lat\ ter also preserves multiple edges).
\n\n See .\n gr := Digraph([[2, 3], [1], [1]]);\n\ngap> Sparse6String(gr);\n\":Bc\"\ngap>\ DiSparse6String(gr);\n\".Bc{f\"\n]]>\n \n\n\n \n \n An IO file object.\n \n If \ filename is a string representing the name of a file, then\n Digr\ aphFile returns an &IO; package file object for that file.\n
\n\n \ If the optional argument coder is specified\n and is a function wh\ ich either encodes a digraph as a string, or decodes a\n string into a digr\ aph, then this function will be used when reading or\n writing to the retur\ ned file object. If the optional argument coder\n is not specified, \ then the encoding of the digraphs in the returned file\n object must be spe\ cified in the the file extension. The file extension must\n be one of: .\ g6, .s6, .d6, .ds6, .txt,\n .p, or <\ C>.pickle; more details of these file formats is given\n below.
\n\ \n If the optional argument mode is specified, then it must be one o\ f:\n \"w\" (for write), \"a\" (for append), or \"r\" (\ for read).\n If mode is not specified, then \"r\" is used by \ default.
\n\n If filename ends in one of: .gz, .bz2, or\n .xz, then the digraphs which are read from, or written to, t\ he\n returned file object are decompressed, or compressed, appropriately.\n\
\n\n The file object returned by DigraphFile can be given as\ the first\n argument for either of the functions or . The purpose of this is to reduce\ the overhead of\n recreating the file object inside the functions \n or when, for example, \ reading or writing many\n digraphs in a loop.\n
\n\n The currentl\ y supported file formats, and associated filename extensions,\n are:\n <\ List>\n graph6 (.g6)\n \n\tA standard and widely-\ used format for undirected graphs, with no support\n\tfor loops or multiple ed\ ges. Only symmetric graphs are allowed -- each\n\tedge is combined with its c\ onverse edge to produce a single undirected\n\tedge. This format is best used\ for \"dense\" graphs -- those with many\n\tedges per vertex.\n \n\ sparse6 (.s6)\n \n\tUnlike graph6, sparse6 has s\ upport for loops and multiple edges.\n\tHowever, its use is still limited to s\ ymmetric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- thos\ e with few edges per vertex.\n \n digraph6 (.d6)\ \n \n\tThis format is based on graph6, but stores direction informa\ tion -\n\ttherefore is not limited to symmetric graphs. Loops are allowed, bu\ t\n\tmultiple edges are not. Best compression with \"dense\" graphs.\n <\ /Item>\n disparse6 (.ds6)\n \n\tAny type of digra\ ph can be encoded in disparse6: directions, loops, and\n\tmultiple edges are a\ ll allowed. Similar to sparse6, this has the best\n\tcompression rate with \"\ sparse\" graphs.\n \n plain text (.txt)\n <\ Item>\n\tThis is a human-readable format which stores graphs in the form\n \ 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 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.
\n\n See for a more flexible way to store\n digraphs in a plain\ text file.
\n \n\n pickled (.p or .pickl\ e)\n \n Digraphs are pickled using the &IO; packa\ ge. This is particularly good\n when the i\ s non-trivial.\n \n \n\n fil\ ename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap> file :=\ DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> WriteDigraphs(f\ ile, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\ngap> file\ := DigraphFile(filename, \"r\");;\ngap> ReadDigraphs(file, 9);\n]]>\n \n\n\n \n \n A digraph, or a list of digraphs.\n \n If filename is a string containing the name of a file contain\ ing\n encoded digraphs or an &IO; file object created using , then ReadDigraphs returns the digraphs\n encode\ d in the file as a list. Note that if filename is a\n compressed fi\ le, which has been compressed appropriately to give a filename\n extension \ of .gz, .bz2, or .xz, then\n ReadDigraphs can r\ ead filename without it first needing to be\n decompressed.\n
\n\n If the optional argument decoder is specified\n and is a fu\ nction which decodes a string into a digraph,\n then ReadDigraphs wi\ ll use decoder to decode the digraphs\n contained in filename\ .
\n\n If the optional argument n is specified, then ReadDigra\ phs\n returns the nth digraph encoded in the file filename.\n
\n\n If the optional argument decoder is not specified, \ then\n ReadDigraphs will deduce which decoder to use based on the fi\ lename\n extension of filename (after removing the compression-relat\ ed\n filename extensions .gz, .bz2, and .xz). For exa\ mple,\n if the filename extension is .g6, then ReadDigraphs w\ ill use\n the graph6 decoder .\n \
\n\n The currently supported file formats, and associated filename exte\ nsions,\n are:\n \n graph6 (.g6)\n \n\ \tA standard and widely-used format for undirected graphs, with no support\n\t\ for loops or multiple edges. Only symmetric graphs are allowed -- each\n\tedg\ e is combined with its converse edge to produce a single undirected\n\tedge. \ This format is best used for \"dense\" graphs -- those with many\n\tedges per \ vertex.\n \n sparse6 (.s6)\n \n\tUnli\ ke graph6, sparse6 has support for loops and multiple edges.\n\tHowever, its u\ se is still limited to symmetric graphs. This format is\n\tbetter-suited to \ \"sparse\" graphs -- those with few edges per vertex.\n \n digraph6 (.d6)\n \n\tThis format is based on graph6, but\ stores direction information -\n\ttherefore is not limited to symmetric graph\ s. Loops are allowed, but\n\tmultiple edges are not. Best compression with \ \"dense\" graphs.\n \n disparse6 (.ds6)\n <\ Item>\n\tAny type of digraph can be encoded in disparse6: directions, loops, a\ nd\n\tmultiple edges are all allowed. Similar to sparse6, this has the best\n\ \tcompression rate with \"sparse\" graphs.\n \n plain t\ ext (.txt)\n \n\tThis is a human-readable format which store\ s graphs in the form\n 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 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.
\n\n \ See for a more flexible way to store\n \ digraphs in a plain text file.
\n\n \n\n pickled (.p \ or .pickle)\n \n Digraphs are pickled using th\ e &IO; package. This is particularly good\n when the is non-trivial.\n \n \n\n ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"\ ), 10);\n\ngap> ReadDigraphs(\n> C\ oncatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(\ ), \"/data/tree9.4.txt\"));\n[ , \ \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n ]]]>\n \n\n\n\ \n \n \n If digraphs is a list of digr\ aphs or a digraph and filename\n is a string or an &IO; file object \ created using ,\n then WriteDigraphs writ\ es the digraphs to the file represented by\n filename. If the suppl\ ied filename ends in one of the extensions\n .gz, .bz2, or .xz, then the file will be compressed\n appropriately. Excluding thes\ e extensions, if the file ends with an\n extension in the list below, the c\ orresponding graph format will be used to\n encode it. If such an extensio\ n is not included, an appropriate format\n will be chosen intelligently, an\ d an extension appended, to minimise file\n size.\n
\n\n \n\n For more verbose information on the progress of\ the function, set the info\n level of InfoDigraphs to 1 or higher, \ using SetInfoLevel.
\n\n The currently supported file formats are\ :\n \n graph6 (.g6)\n \n\tA standard and\ widely-used format for undirected graphs, with no support\n\tfor loops or mul\ tiple edges. Only symmetric graphs are allowed -- each\n\tedge is combined wi\ th its converse edge to produce a single undirected\n\tedge. This format is b\ est used for \"dense\" graphs -- those with many\n\tedges per vertex.\n <\ /Item>\n sparse6 (.s6)\n \n\tUnlike graph6, spars\ e6 has support for loops and multiple edges.\n\tHowever, its use is still limi\ ted to symmetric graphs. This format is\n\tbetter-suited to \"sparse\" graphs\ -- those with few edges per vertex.\n \n digraph6 (.d6\ )\n \n\tThis format is based on graph6, but stores direction\ information -\n\ttherefore is not limited to symmetric graphs. Loops are all\ owed, but\n\tmultiple edges are not. Best compression with \"dense\" graphs.\ \n \n disparse6 (.ds6)\n \n\tAny type\ of digraph can be encoded in disparse6: directions, loops, and\n\tmultiple ed\ ges are all allowed. Similar to sparse6, this has the best\n\tcompression rat\ e with \"sparse\" graphs.\n \n\n plain text (.txt)\n \n\tThis is a human-readable format which stores graphs in the\ form\n 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 i.e. pairs of ver\ tices\n describing edges in a graph. More specifically, the vertices ma\ king up\n one edge must be separated by a single space, and pairs of ve\ rtices must\n be separated by two spaces.
\n\n See for a more flexible way to store\n digraphs\ in a plain text file.
\n\n \n pickled (.p or .pickle)\n\ \n Digraphs are pickled using the &IO; package. This is par\ ticularly good\n when the is non-trivial.\ \n \n \n\n 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 := Concatenation(DIGRAPHS_D\ ir(), \"/tst/out/man.d6.gz\");;\ngap> WriteDigraphs(filename, grs, \"w\");\nIO\ _OK\ngap> ReadDigraphs(filename);\n[ \ , \n , \n ]]]>\n \n\n\n\ \n \n An iterator.\n \n If fil\ ename is a string representing the name of a file containing\n encoded \ digraphs, then IteratorFromDigraphFile returns an iterator\n for whi\ ch the value of is the\n next\ digraph encoded in the file.\n
\n\n If the optional argument dec\ oder is specified and is a function\n which decodes a string into a dig\ raph, then IteratorFromDigraphFile\n will use decoder to deco\ de the digraphs contained in filename.\n
\n\n The purpose of \ this function is to easily 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 .\n\n 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_Close(file);;\ngap> iter := IteratorFromDigraphFile(filenam\ e);\n\ngap> for x in iter do od;\n]]>\n \n<\ /ManSection>\n\n \n \n \n A string.\n \n These two functions return a function which enc\ odes or decodes a digraph in\n a plain text format.
\n\n DigraphPl\ ainTextLineEncoder returns a function which takes a single\n digraph as\ an argument. The function returns a string describing the edges\n of that\ digraph; each edge is written as a pair of integers separated by the\n str\ ing delimiter2, and the edges themselves are separated by the\n stri\ ng delimiter1. DigraphPlainTextLineDecoder returns the\n cor\ responding decoder function, which takes a string argument in this format\n \ and returns a digraph.
\n\n If only one delimiter is passed as an argum\ ent to\n DigraphPlainTextLineDecoder, it will return a function whic\ h decodes\n a single edge, returning its contents as a list of integers.
\n\n The argument offset should be an integer, which will describe \ a\n number to be added to each vertex before it is encoded, or after it is\ \n decoded. This may be used, for example, to label vertices starting at 0\ \n instead of 1.
\n\n Note that the number of vertices of a digraph i\ s not stored, and so vertices\n which are not connected to any edge may be \ lost.\n\n gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEnco\ der(\" \", \" \", -1);;\ngap> dec := DigraphPlainTextLineDecoder(\" \", \" \ \", 1);;\ngap> enc(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> dec(last);\n\n]]>\n \n\n\n \n \n\ A digraph.\n \n This function takes a st\ ring str, decodes it, and then returns the\n tournament [see ] which it defines, according to\n the following rules\ .
\n\n The characters of the string str represent the entries in\ the upper\n triangle of a tournament's adjacency matrix. The number of ver\ tices\n n will be detected from the length of the string and will be\ as\n large as possible.
\n\n The first character represents the po\ ssible edge 1 -> 2, the second\n represents 1 -> 3 and so on \ until 1 -> n; then the following\n character represents 2 -> 3, and so on up to the character which\n represents the edge n-1 -> n.
\n\n If a character of the string with corresponding edge i -> j<\ /C> is equal\n to 1, then the edge i -> j is present in the t\ ournament.\n Otherwise, the edge i -> j is present instead. In this\ way, all the\n possible edges are encoded one-by-one.\n\n gr := TournamentLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1\ , 2 ] ]\n]]>\n \n\n\n \n \ \n A digraph.\n \n This function takes a string str, decodes it, and then returns the\n topologically sorted digraph [\ see ]\n which it defines, according t\ o the following rules.
\n\n The characters of the string str rep\ resent the entries in the upper\n triangle of a digraph's adjacency matrix.\ The number of vertices n\n will be detected from the length of the \ string and will be as large as\n possible.
\n\n The first character\ represents the possible edge 1 -> 2, the second\n represents 1 -\ > 3 and so on until 1 -> n; then the following\n character repre\ sents 2 -> 3, and so on up to the character which\n represents the e\ dge n-1 -> n. If a character of the string with\n corresponding edg\ e i -> j is equal to 1, then this edge is\n present in the di\ graph. Otherwise, it is not present. In this way, all the\n possible edges\ are encoded one-by-one.
\n\n In particular, note that there exists no \ edge [i, j] if j \\leq\n i. In order words, the digraph wi\ ll be topologically sorted.\n\n gr := AdjacencyMat\ rixUpperTriangleLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 4 ], [ ] ]\ngap> gr := Ad\ jacencyMatrixUpperTriangleLineDecoder(\"111111x111\");\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 3, 4 ], [\ 4, 5 ], [ 5 ], [ ] ]\n]]>\n \n\n\n \ \n \n A digrap\ h.\n \n If str is a string consisting of at \ least two non-negative integers\n separated by spaces, then this function w\ ill attempt to return the digraph\n which it defines as a TCode string.
\n\n The first integer of the string defines the number of vertices v in\n the digraph, and the second defines the number of edges e. \ The\n following 2e integers should be vertex numbers in the range [0 ..\n v-1]. These integers are read in pairs and define the digrap\ h's\n edges. This function will return an error if str has fewer th\ an\n 2e+2 entries.
\n\n Note that the vertex numbers will be \ incremented by 1 in the digraph\n returned. Hence the string fragment 0\ 6 will describe the edge\n [1,7].\n \ gr := TCodeDecoder(\"3 2 0 2 2 1\");\n\ngap> OutNeighbours(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeDecode\ r(\"12 3 0 10 5 2 8 8\");\n\ngap>\ OutNeighbours(gr);\n[ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ], [ 9 \ ], [ ], \n [ ], [ ] ]\n]]>\n \n\n\n \ \n \n \n A string.\ \n \n PlainTextString takes a single digraph, and retu\ rns a string\n describing the edges of that digraph. DigraphFromPlainTe\ xtString\n takes such a string and returns the digraph which it describ\ es. Each edge\n is written as a pair of integers separated by a single spa\ ce. The edges\n themselves are separated by a double space. Vertex number\ s are reduced by\n 1 when they are encoded, so that vertices in the string \ are labelled\n starting at 0.
\n\n Note that the number of vertices o\ f a digraph is not stored, and so vertices\n which are not connected to any\ edge may be lost.
\n\n \n The operation DigraphFromPlainTextStrin\ g\n takes an optional first argument \n\ or , 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 gr := Digraph([[2, 3], [1], [1]]);\n\ngap> PlainTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFr\ omPlainTextString(last);\n\n]]>\n \n\n\n \n \n \n\ \n These functions write and read a single digraph in a huma\ n-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 \ delimiter.
\n\n filename should be the name of a file whic\ h will be written to or\n read from, and offset should be an integer\ which is added to each\n vertex number as it is written or read. For exam\ ple, if\n WritePlainTextDigraph is called with offset -1, then\n the vertices will be numbered in the file starting from 0 instea\ d of 1 -\n ReadPlainTextDigraph would then need to be called with offset\n 1 to convert back to the original graph.
\n\n ignore should be a list of characters which will be ignored when\n rea\ ding the graph.\n gr := Digraph([[1, 2, 3], [1, 1]\ , [2]]);\n\ngap> filename := \ Concatenation(DIGRAPHS_Dir(), \"/tst/out/plain.txt\");;\ngap> WritePlainTextDi\ graph(filename, gr, \",\", -1);\ngap> ReadPlainTextDigraph(filename, \",\", 1,\ ['/', '%']);\n\n]]>\n \n\n\n \n \n \n \n These operations write or read the si\ ngle symmetric digraph digraph\n to or from a file in DIMACS format,\ as appropriate. The operation\n WriteDIMACSDigraph records the vert\ ices and edges of digraph.\n The vertex labels of digraph wil\ l be recorded only if they are\n integers. See and .
\n\n The first ar\ gument filename should be the name of the file which will\n be writt\ en to or read from. A file can contain one symmetric digraph in\n DIMACS f\ ormat. If filename ends in one of .gz, .bz2,\n or .\ xz, then the file is compressed, or decompressed, appropriately.\n
\ \n\n The DIMACS format is described as follows. Each line in the DIMACS fi\ le has\n one of four types:\n \n \n A line begin\ ning with c and followed by any number of characters\n is a comm\ ent line, and is ignored.\n \n \n A line beginni\ ng with p defines the numbers of vertices and edges\n the digrap\ h. This line has the format p edge <nr_vertices>\n <nr_e\ dges>, where <nr_vertices> and\n <nr_edges>\ ; are replaced by the relevant integers. There\n must be exactly on\ e such line in the file, and it must occur before any\n of the followin\ g kinds of line.
\n\n Although it is required to be present, the val\ ue of\n <nr_edges> will be ignored. The correct number of \ edges\n will be deduced from the rest of the information in the file.\n\ \n \n A line of the form e <v> <w&g\ t;, where <v>\n and <w> are integers in t\ he range [1 ..\n <nr_vertices>], specifies that there is\ a (symmetric) edge\n in the digraph between the vertices <v><\ /C> and\n <w>. A symmetric edge only needs to be defined o\ nce; an\n additional line e <v> <w>, or e <w&g\ t;\n <v>, will be interpreted as an additional, multiple, e\ dge.\n Loops are permitted.\n \n \n A lin\ e of the form n <v> <label>, where\n <v> is an integer in the range [1 ..\n <nr_vertices>] and \ <label> is an integer,\n signifies that the vertex <\ v> has the label\n <label> in the digraph. If a lab\ el is not specified for a\n vertex, then ReadDIMACSDigraph will \ assign the label\n 1, according to the DIMACS specification.\n \ \n \n\n A detailed definition of the DIMACS format can\ be found at\n http://mat.gsia.cmu.edu/COLOR/general/ccformat.ps\ , in Section\n 2.1. Note that optional descriptor lines, as described in Se\ ction 2.1, will\n be ignored.\n
\n\n\n g\ r := Digraph([[2], [1, 3, 4], [2, 4], [2, 3]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigraph(filen\ ame, gr);;\ngap> ReadDIMACSDigraph(filename);\n]]>\n \n\n\n
\n<\ /Chapter>\n \n\n\n\n \n \ Grape to Digraphs Command Map\n \n Below is a table of &Grape; com\ mands with the &Digraphs; counterparts.\n The sections in this chapter corres\ pond to the chapters in the &Grape;\n manual.\n\n
\n \n\ Functions to construct and modify graphs\n \n \n \n The table in this section contains more informat\ ion when viewed in\n html format.\n \n \n\n \n \n \n \n \ &Grape; command\n \n \n &Digraphs; command\ \n \n \n \n \ Comments\n \n \n \n \n\ \n \n \n Graph\n \n \ \n \n \n \n\ \n Used with the arguments of Graph in &Grape;, will return an equivalent \ digraph.
\n \n \n \n \n\n \n \n EdgeOrbitsGraph\n <\ /Item>\n \n \n <\ /Item>\n \n \n The operation\ behaves the same as\n EdgeOrbit\ sDigraph in &Grape;.\n \n \n \n \ \n\n \n \n NullGraph\n \ \n \n \n \ \n or\n \ \n \n \n \n \n Note that, unlike in &Grape;, the arguments of and are just\n \ non-negative integers. Since digraphs in &Digraphs; are immutable,\n \ there is no reason (although it is possible) to create a null digrap\ h\n and then add edges, as you might be used to doing in &Grape;.\n\ \n \n \n \n\n \n \n CompleteGraph\n \n <\ Item>\n \n \n \n \n Note that, unlike in &Grape;, \ the argument to is simply a non\ -negative integer\n indicating the number of vertices. By default a\ complete digraph in\n &Digraphs; does not have any loops. If you w\ ant a complete digraph\n with loops at every vertex, then use .\n \n \n \n \n\n \n \n \ JohnsonGraph\n \n \n \n \n \n \n The &Grape; function JohnsonGraph is equivalent to\n \ .\n \n \n\ \n \n\n \n \n Ca\ yleyGraph\n \n \n \n \n \n \n \ The operation behaves the same as\n \ CayleyGraph in &Grape;.\n \n \n \ \n \n\n \n \n AddEd\ geOrbit\n \n \n \n \n \n \ \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 \n \ \n \n \n\n \n \n \ RemoveEdgeOrbit\n \n \n \n \n \n <\ Alt Only=\"HTML\">\n Note that, unlike in &Grape;, digraphs in &Dig\ raphs; are immutable,\n and this operation returns a new digraph, i\ .e. it does not modify the\n digraph given as an argument.\n \ \n \n \n \n\n \n \ \n AssignVertexNames\n \n \n \n \n and\n \n \ \n \n \n \ \n Use to set an individual\ \n vertex label and to set a\ ll of\n the vertex labels.\n \n \n \ \n \n \n
\n\n
\n \n Functions to inspect graphs, vertices and edges\n \ \n \n \n The table in this section contain\ s more information when viewed in\n html format.\n \n \n\n \n \n\n \n <\ Item>\n &Grape; command\n \n \n &\ Digraphs; command\n \n \n \n Comments\n \n \n \n \ \n\n \n \n IsGraph\n \ \n \n \n \n \n \n is a category and not a function like\n IsGraph.\ \n \n \n \n \n\n <\ Row>\n \n OrderGraph\n \n \n \n \n \n \n The operation behaves the same as\n OrderGraph in &Grap\ e;.\n \n \n \n \n\n \ \n \n IsVertex(graph, v)\n \ \n \n v in DigraphVertices(dig\ raph)\n \n \n \ \n The attribute can be used to ret\ urn\n the list of vertices of a digraph, or if v is known to\ be\n a positive integer, then you can simply do\n v <= DigraphNrVertices(digraph)\n \n \ \n \n \n\n \n \n \ VertexName\n \n \n \n \n \n \n Unlike VertexName in &Grape;, the return value o\ f\n is mutable.\n \n \n \n \n\n \n \n VertexNames\n \n \n \ \n \n \n \ \n Unlike VertexNames in &Grape;, the r\ eturn value of\n is mutable.\n \ \n \n \n \n\n \n \n Vertices\n \n \ \n \n \n \n\ \n The vertices of a digraph in &Digra\ phs; is always the range [1\n .. n] where n is the \ number of vertices of the\n digraph. The operation behaves the\n same as Vertices in &Grape;.\ \n \n \n \n \n\n <\ Row>\n \n VertexDegree\n \n \ \n \n \n (see also: )\n \n \n \n \n \ The operation behaves the same as\n \ VertexDegree in &Grape;.\n \n \n \ \n \n\n \n \n VertexD\ egrees\n \n \n \n \n (see also:\n ,\n ,\n \ ,\n ,\n )\n \n \n \n \n The operation behaves the same as\n VertexDegrees in &Grape;.\n \n \n \n \n\n \n \n IsLoopy\n \ \n \n \n \n\ \n \n The operation behaves the same as\n IsLoopy in \ &Grape;.\n \n \n \n \n\ \n \n \n IsSimpleGraph\n \ \n \n \n \n and not \n \ \n \n \n \n \n \n \n The &Grape; function IsSimpleGraph is equi\ valent to\n \n and not\ .\n \n \n <\ /Row>\n \n\n \n \n Adjacency\ \n \n \n \n \n (see also )\n \n \n \n \ \n The operation behaves the same\n as Adjacency in &Grape;.\n \ \n \n \n \n\n \n \n IsEdge\n \n \n \ \n \ \n \n \n The argum\ ents of\n \n can be a digraph and an edge, or a digraph and two vertices.\ In the\n first form the operation\n \n behaves the same as \ IsEdge in &Grape;.\n \n \n \n \ \n\n \n \n DirectedEdges\n \ \n \n \n \ \n \n \n The attri\ bute behaves the same\n as Directed\ Edges in &Grape;.\n \n \n \n \ \n\n \n \n UndirectedEdges\n \ \n \n None\n \n \n \n There is no analogous function i\ n &Digraphs;, and no plans to\n include one.\n \n \ \n \n \n\n \n \n \ Distance\n \n \n \n \n \n \n \ The operation behaves the same as\n Dis\ tance in &Grape;.\n \n \n \n \ \n\n \n \n Diameter\n \ \n \n \n \n \n \n The attribut\ e behaves the same\n as Diameter\ in &Grape;.\n \n \n \n \n\n \n \n Girth\n \ \n \n \n \n \n \n The attribut\ e behaves the same\n as G\ irth in &Grape; except that in the case that the\n argument is \ a forest, infinity is returned by &Digraphs;\n whereas -1\ is returned by &Grape;.\n \n \n \ \n \n\n \n \n IsConnectedGra\ ph\n \n \n \n \n \n \n The attribute behave\ s the\n same as IsConnectedGraph in &Grape;.\n \n \n \n \n\n \n \n IsBipartite\n \n \n \ \n \n \ and \n \n \n \ \n \n The &Grape; function\ IsBipartite is equivalent to\n and .\n \n \n \n \n\n \n \n IsNullGraph\n \n \n \ \n \n or <\ Ref Prop=\"IsEmptyDigraph\"/>\n \n \n \n \n The &Grape; function IsNullG\ raph is equivalent to\n and its \ synonym .\n \n \ \n \n \n\n \n \n \ IsCompleteGraph\n \n \n \n \n \n \n The &Grape; function IsCompleteGraph is ro\ ughly\n equivalent to\n except that in &Digraphs;\n a complete digraph has no loops.\n \n \n \n \n\n\n <\ /Table>\n \n\n
\n \n Functions to determ\ ine regularity properties of graphs\n \n \n \ \n The table in this section contains more information when vie\ wed in\n html format.\n \n \n\n
\n \n\n \n \n &Grape; co\ mmand\n \n \n &Digraphs; command\n \ \n \n \n Comments\n\ \n \n \n \n\n \n \n IsRegularGraph\n \n \ \n \n \n (see also\n \ \n and\n )\n \n \n \n \n \ The &Grape; function IsRegularGraph is\n equivalent to <\ Ref Prop=\"IsOutRegularDigraph\"/>.\n \n \n \ \n \n\n \n \n LocalPar\ ameters\n \n \n None\n \ \n \n \n The &Grape; functi\ on LocalParameters has no equivalent\n function in &Digraphs\ ;, although this might change in the future.\n \n \n \n \n\n \n \n GlobalParameters\n \n \n None\n \ \n \n \n The &Grap\ e; function GlobalParameters has no equivalent\n function in\ &Digraphs;, although this might change in the future.\n \n \ \n \n \n\n \n \n \ IsDistanceRegular\n \n \n <\ Ref Prop=\"IsDistanceRegularDigraph\"/>\n \n \n \ \n The &Grape; function IsDistanceRegul\ ar is\n equivalent to .\ \n \n \n \n \n\n <\ Row>\n \n CollapsedAdjacencyMat\n \ \n \n None\n \n \n \n The &Grape; function CollapsedAdjacencyMat has no equivalent\n function in &Digraphs;, although this might \ change in the future.\n \n \n \n \ \n\n \n \n OrbitalGraphColadjMats\ \n \n \n None\n \n \ \n \n The &Grape; function Or\ bitalGraphColadjMats has no equivalent\n function in &Digraphs;\ , although this might change in the future.\n \n \ \n \n \n\n \n \n \ VertexTransitiveDRGs\n \n \n None\n \ \n \n \n The &G\ rape; function VertexTransitiveDRGs has no equivalent\n func\ tion in &Digraphs;, although this might change in the future.\n \n \n \n \n\n
\n\n \n\n
\n \n Some special vertex subsets of a gra\ ph\n \n \n \n The table in th\ is section contains more information when viewed in\n html format.\n \ \n \n\n \n \n\n \ \n \n &Grape; command\n \n <\ Item>\n &Digraphs; command\n \n \n \ \n Comments\n \n \ \n \n \n\n \n \n \ ConnectedComponent\n \n \n \n \n \n \n When applied to a simple di/graph and a vertex,\ \n ConnectedComponent in &Grape; is equivalent to\n \ .\n can also be applied to\n non-simple graphs (\ in the language of &Grape;).\n \n \n \ \n \n\n \n \n ConnectedCompo\ nents\n \n \n \n \n \n \n When applied to a simple di/graph,\n Connected\ Components in &Grape; is equivalent to\n .\n \ can also be applied to\n non-simple graphs (in the language of &Gra\ pe;).\n \n \n \n \n\n \ \n \n Bicomponents\n \n \ \n \n \n \ \n \n When applied to a si\ mple bipartite di/graph,\n Bicomponents in &Grape; is equiva\ lent to\n .\n The def\ inition of does not require\n it\ s argument to be simple (in the language of &Grape;), i.e. the\n bi\ partite graphs in &Grape; are a proper subset of the bipartite\n di\ graphs in &Digraphs;.\n \n \n \n \ \n\n \n \n DistanceSet\n \ \n \n \n \n \ \n \n DistanceSet i\ n &Grape; is equivalent to\n in &Digraphs;.\n \ \n \n \n \n\n \ \n \n Layers\n \n \n \ \n \n \n \ \n Layers in &Grape; is equivalent to\ \n in &Digraphs;.\n \n\ \n \n \n\n \n \ \n IndependentSet\n \n \n \ \n \n \n \ \n IndependentSet in &Grape; is equiv\ alent to\n in &Digraphs;.\n \ \n \n \n \n\n \n\n \n\n
\n \n Functions to construct \ new graphs from old\n \n \n \n \ The table in this section contains more information when viewed in\n \ html format.\n \n \n\n
\n <\ HorLine/>\n\n \n \n &Grape; command\n <\ /Item>\n \n &Digraphs; command\n \n \ \n \n Comments\n \ \n \n \n \n\n \n \n InducedSubgraph\n \n \n \ \n \n \n \ \n in &Digr\ aphs; does not allow a third\n argument G specifying a group\ of automorphisms on the\n resultant graph. This may be included in\ a future release.\n \n \n \n \n\n \n \n DistanceSetInduced\n \ \n \n None\n \n \n \n There is no analogous function i\ n &Digraphs;, and no plans to\n include one.\n \n \ \n \n \n\n \n \n \ DistanceGraph\n \n \n \n \n \ \n \n DistanceGraph\ in &Grape; is equivalent to\n in\n &Digraphs;.\n \n \ \n \n \n\n \n \n \ ComplementGraph\n \n \n \n \n \n \n in &Digraphs; is equivalen\ t to\n ComplementGraph in &Grape; with the optional argument\ \n comploops set to true (loops/nonloops are\n \ complemented).\n \n \n \n \n\n \n \n PointGraph\n \ \n \n None\n \n \n \ \n There is no analogous function in &Digr\ aphs;, and no plans to\n include one.\n \n \n \n \n\n \n \n \ EdgeGraph\n \n \n \n \n \n \n EdgeGraph in &Grape; is equivalent to\n \ in &Digraphs;.\n \n \ \n \n \n\n \n \n \ SwitchedGraph\n \n \n None\ \n \n \n \n \ There is no analogous function in &Digraphs;, and no plans to\n inc\ lude one.\n \n \n \n \n\ \n \n \n UnderlyingGraph\n \n \n \n \ \n \n \n Underl\ yingGraph in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \n \n\n \n \n QuotientGrap\ h\n \n \n \n \n \n \n \ in &Digraphs; takes a partition\n \ p of the vertices of the digraph, rather than a set of\n \ generating pairs. Hence, the equivalence might not be\n group-inva\ riant. Also note that in &Digraphs; multiple edges may be\n create\ d, whereas in &Grape; they are effectively combined into a\n single\ edge.\n \n \n \n \n\n \ \n \n BipartiteDouble\n \ \n \n \n \n \n \n Bipartite\ Double in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \n\ \n\n \n \n GeodesicsGraph\n \n \n None\n \n \ \n \n There is no analogous funct\ ion in &Digraphs;, and no plans to\n include one.\n \ \n \n \n \n\n \n \n CollapsedIndependentOrbitsGraph\n \n \ \n None\n \n \n \n There is no analogous function in &Digraphs;, although \ this feature\n might be implemented in the future.\n \n \n \n \n\n \n \n CollapsedCompleteOrbitsGraph\n \n <\ Item>\n None\n \n \n \n There is no analogous function in &Digraphs;, although th\ is feature\n might be implemented in the future.\n \ \n \n \n \n\n \n \n NewGroupGraph\n \n \n \ None\n \n \n \n \ There is no analogous function in &Digraphs;. However, the same\n \ results would be possible by calling on\n \ the digraph to produce a copy gr, and then applying\n \ SetDigraphGroup(gr, G).\n \n \n \n \n\n
\n\n
\n\n
\n \n Vertex-Colouring and Complete Subgraphs\n \n \n \n The table in this section contains more inf\ ormation when viewed in\n html format.\n \n \n\n <\ Table Align=\"|l|l|l|\">\n \n\n \n \n \ &Grape; command\n \n \n &Digraphs; \ command\n \n \n \n \ Comments\n \n \n \n \n\n \n \n VertexColouring\n \ \n \n \n \n \n \n VertexColouring in &Grape; is equivalent to in &Digra\ phs;,\n except it returns a transformation rather than a list of ve\ rtex\n colors.\n \n \n \n \ \n\n \n \n CompleteSubgraphs<\ /C>\n \n \n \ \n \n \n \n \ The functionality of CompleteSubgraphs in &Grape; is provided\n \ by and \n in &Digraphs;.\n \n \n \ \n \n\n \n \n CompleteSubgra\ phsOfGivenSize\n \n \n \n \n \n \n Much of the functionality of CompleteSubgraphsOfGivenSize in\n &Grape; is provided by and in &Digraphs;, except that \ the\n functionality relating to vertex weights and partial colourin\ gs is\n not implemented.\n \n \n \ \n \n\n \n
\n\n
\n \n Automorphism groups and isomorphism testing for graphs\n \n \n \n The table in this section c\ ontains more information when viewed in\n html format.\n \n \ \n\n \n \n\n \n \ \n &Grape; command\n \n \n \ &Digraphs; command\n \n \n \n Comments\n \n \n \n \n\n \n \n AutGroupGrap\ h\n \n \n \n \n \n \n AutGroupGraph in &Grape; is equivalent to\ \n in &Di\ graphs;.\n \n \n \n \n\ \n \n \n GraphIsomorphism\n \n \n \n \n \n \n \ GraphIsomorphism in &Grape; is equivalent to in &Digraphs;.\n \ Note however that GraphIsomorphism's optional argument\n \ firstunbindcanon is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ \n \n \n \n\n \n \n IsIsomorphicGraph\n \n \n \ \n \n \n \n IsIsomorphic\ Graph in &Grape; is equivalent to in &Digraphs;.\n Note however that \ IsIsomorphicGraph's optional argument\n firstunbindcanon<\ /A> is not applicable, since the canonical\n labellings of a digrap\ h are not portable between sessions.\n \n \n \ \n \n\n \n \n GraphIs\ omorphismClassRepresentatives\n \n \n \ None\n \n \n \n \ There is no analogous function in &Digraphs;, although this feature\n \ might be implemented in the future.\n \n \ \n \n \n\n
\n
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"chap5_mj.html#X87D5C60D7B0C1309" ], ("AsSemigroup (for a filter and a digraph)") := [ "5.5-1", "chap5_mj.html#X87D5C60D7B0C1309" ], ("AsSemigroup (for a filter, semilattice digraph, and two lists)") := [ "5.5-2", "chap5_mj.html#X7C6D5EC27C51066B" ], AsTransformation := [ "3.2-5", "chap3_mj.html#X7C5360B2799943F3" ], ("Attributes and operations") := [ "5", "chap5_mj.html#X8739F6CD78C90B14" ], ("AutomorphismGroup (for a digraph and a homogeneous list)") := [ "7.2-5", "chap7_mj.html#X877732B1783C391B" ], ("AutomorphismGroup (for a digraph)") := [ "7.2-2", "chap7_mj.html#X858C32127A190175" ], ("AutomorphismGroup (for a digraph, homogeneous list, and list)") := [ "7.2-6", "chap7_mj.html#X7DA2C4FE837FFE01" ], BipartiteDoubleDigraph := [ "3.3-37", "chap3_mj.html#X7C6E6CB284982C7A" ], ("BlissAutomorphismGroup (for a digraph and homogenous list)") := [ "7.2-3", "chap7_mj.html#X7E7B0D88865A89F6" ], ("BlissAutomorphismGroup (for a digraph)") := [ "7.2-3", "chap7_mj.html#X7E7B0D88865A89F6" ], 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1 \n The Digraphs package\n \n
1.1\ Introduction\n\n
\n
  1.1-1 De\ finitions\n\n
\n
\n
<\ a href=\"chap2_mj.html#X817088B27F90D596\">2 Installin\ g Digraphs\n
\ 2.1 For those in a hurry\n\n
\n
2.2 \ Optional package dependencies\n\n\n
&\ nbsp; 2.2-1 The Grape package\n\n
\n
2.3 Compiling th\ e kernel module\n\n
\n
2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
<\ span class=\"nocss\">  2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\n
  \ 2.5-3 DigraphsTestExtreme\n
\n
\n
3 Cr\ eating digraphs\n
3\ .1 Creating digraphs\n\n
\n
&nb\ sp;3.1-1 IsDigraph\n
  3.1-2 IsMutableDigraph\n
  3.1-3 IsImmutableDigraph\n
  3.1-4 IsCayleyDigraph\n
  3.1-5 IsDigraphWithAdjacencyFunction\n
  3.1-6 DigraphByOutNeighboursType\n
  3.1-7 Digraph\n
  3.1-8 DigraphByAdjacencyMatrix\n
  3.1-9 DigraphByEdges\n
\   3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n
3.2 Changing representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2\ -2 AsDigraph\n
&n\ bsp; 3.2-3 Graph\n
  <\ a href=\"chap3_mj.html#X7C4F13E080EC16B0\">3.2-4 AsGraph\n
  3.2-5 AsTransformation\n
\n 3.3 New\ digraphs from old\n\n
\n
  3.3-1 DigraphImmutableCopy\n
  3.3-2 DigraphImmutableCopyIfImmutable\n
  3.3-3 InducedSubdigraph\n
  3.3-4 ReducedDigraph\n
  3.3-5 MaximalSymmetricSubdigraph\n
  3.3-6 MaximalAntiSymmetricSubdigraph\n
  3.3-7 UndirectedSpanningForest\n
  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 Digr\ aphAddEdgeOrbit\n
  3.3-18 Digraph\ AddEdges\n
\ 3.3-19 DigraphRemoveV\ ertex\n
&nb\ sp;3.3-20 DigraphRemoveVert\ ices\n
&nbs\ p;3.3-21 DigraphRemoveEdge<\ /a>\n
  3.3-22 DigraphRemoveEdgeOrbit<\ /a>\n
  3.3-23 DigraphRemoveEdges<\ /span>\n
  \ 3.3-24 DigraphRemoveLoops\n
  3.3-25 DigraphRemoveAllMultipleEdges\n
  3.3-26 DigraphReverseEdges<\ /span>\n
  \ 3.3-27 DigraphDisjointUnion\n
  3.3-28 DigraphEdgeUnion\n\
  3.3-29 DigraphJoin\n
  3.3-30 DigraphCartesianProduct\n
  3.3-31 DigraphDirectProduct\n
  3.3-32 DigraphCartesianProductProjections\n\
  3.3-33 DigraphDirectProductProjections\ \n
  3.3-34 LineDigraph\n
  3.3-35 LineUndirectedDigraph\n
  3.3-36 DoubleDigraph\n
  3.3-37 BipartiteDoubleDigraph\n
  3.3-38 DigraphAddAllLoops\n
  3.3-39 DistanceDigraph\n\
  3.3-40 DigraphClosure\n
  3.3-41 DigraphMycielskian\n
\n
3.4 Random digraphs\ \n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 R\ andomMultiDigraph\n
  3.4-3 Random\ Tournament\n
&nbs\ p; 3.4-4 RandomLattice\ \n
\n
3.\ 5 Standard examples\n\n
\n
  \ ;3.5-1 ChainDigraph\n
  3.5-2 CompleteDigraph\n
  3.5-3 CompleteBipartiteDigraph\n
  3.5-4 CompleteMultipartiteDigraph\ \n
  3.5-5 CycleDigraph\n
  3.5-6 EmptyDigraph\n
  3.5-7 JohnsonDigraph\n
\   3.5-8 PetersenGraph\n
  3.5\ -9 GeneralisedPetersenGraph\n
\n
\n
4 \ Operators\n
4.1 Operators for digraphs\n\n
\n
  \ ;4.1-1 IsSubdigraph\n
  4.1-2 IsUndirectedSpanningTree\n
\n
\n
5 Attributes and operations\n
&nbs\ p;5.1 Vertices and edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 DigraphNrEdges\n
  5.1-\ 5 DigraphSinks\n
\   5.1-6 DigraphSo\ urces\n
&nb\ sp;5.1-7 DigraphTopological\ Sort\n
&nbs\ p;5.1-8 DigraphVertexLabel<\ /a>\n
  5.1-9 DigraphVertexLabels<\ /span>\n
  \ 5.1-10 DigraphEdgeLabel\ \n
  5.1-11 DigraphEdgeLabels\n
  5.1-12 DigraphInEdges\n
  5.1-13 DigraphOutEdges\n
  5.1-14 IsDigraphEdge\n
  5.1-15 IsMatching\n
\n
<\ span class=\"tocline\"> 5.2 Neighbours and degree\n\n
\n
  5.2-1 AdjacencyMatrix\n
  5.2-\ 2 CharacteristicPolynomial\n
  5.2\ -3 BooleanAdjacencyMatrix\n
  5.2-\ 4 DigraphAdjacencyFunction\n
  5.2\ -5 DigraphRange\n
  5.2-6 OutNeigh\ bours\n
&nb\ sp;5.2-7 InNeighbours\n
  <\ a href=\"chap5_mj.html#X7F5ACE807D1BC2E2\">5.2-8 OutDegrees\n
  5.2-9 InDegrees\n
  5.2-10 OutDegreeOfVertex\n  5.2-11 OutNeighboursOfVertex\n  5.2-12 InDegreeOfVertex\n
  5.2-13 InNeighboursOfVertex\n
  5.2-14 DigraphLoops\n
  5.2-15 \ PartialOrderDigraphMeetOfVertices\n
  5.2-16 DegreeMatrix\n
  5.2-1\ 7 LaplacianMatrix\n
\n
5.3 Reachability and connectivity\n\n
\n
  5.3-1 DigraphDiameter\n
  5.3-2\ DigraphShortestDistance\n
  5.3-3\ DigraphShortestDistances\n
  5.3-\ 4 DigraphLongestDistanceFromVertex\n
  5.3-5 DigraphDistanceSet\n
  \ 5.3-6 DigraphGirth\n
  5.3-7 Digra\ phOddGirth\n
&nbs\ p; 5.3-8 DigraphUndire\ ctedGirth\n
\ ; 5.3-9 DigraphConnect\ edComponents\n
&n\ bsp; 5.3-10 DigraphCon\ nectedComponent\n
  5.3-11 Digraph\ StronglyConnectedComponents\n
  5.\ 3-12 DigraphStronglyConnectedComponent\n
  5.3-13 DigraphBicomponents\n
<\ span class=\"nocss\">  5.3-14 ArticulationPoints\n
  5.3-15 DigraphPeriod\n
  5.3-1\ 6 DigraphFloydWarshall\n
  5.3-17 \ IsReachable\n
&nb\ sp; 5.3-18 DigraphPath\ \n
  5.3-19 DigraphShortestPath\n
  5.3-20 IteratorOfPaths\n
  5.3-21 DigraphAllSimpleCircuits\n
  5.3-22 DigraphLongestSimpleCircuit\n
  5.3-23 DigraphLayers\ \n
  5.3-24 DigraphDegeneracy\n
  5.3-25 DigraphDegeneracyOrdering\n\
  5.3-26 HamiltonianPath\n
  5.3-27 NrSpanningTrees\n
\n\
5.4 C\ ayley graphs of groups\n\n
\n
  5.4-1 GroupOfCayleyDigraph\n
  5.4-2 GeneratorsOfCayleyDigraph\n
\
\n
\ 5.5 Associated semigroups\n\n
\ \n
  5.5-1 AsSemigroup\n
  5.5-2 AsSemigroup\n
\n
5.6 Planarity\ \n\n
\n
  5.6-1 KuratowskiPlanarSubdigraph\n<\ br />  5.6-2 KuratowskiOuterPlanarSubdigraph\n
  5.6-3 PlanarEmbedding\n<\ br />  5.6-4 OuterPlanarEmbedding\n
  5.6-5 SubdigraphHomeomorphicToK23\n
\n
\n<\ div class=\"ContChap\">6 Properties of digraphs\n
6.1 Edge properties\n<\ /span>\n
\n
  6.1-1\ DigraphHasLoops\n
  6.1-2 IsAnti\ SymmetricDigraph\n
  6.1-3 IsBipa\ rtiteDigraph\n
&n\ bsp; 6.1-4 IsCompleteB\ ipartiteDigraph\n
  6.1-5 IsComple\ teDigraph\n
\ ; 6.1-6 IsCompleteMult\ ipartiteDigraph\n
  6.1-7 IsEmptyD\ igraph\n
&n\ bsp;6.1-8 IsEquivalenceDigr\ aph\n
  \ ;6.1-9 IsFunctionalDigraph<\ /a>\n
  6.1-10 IsMultiDigraph\n
  6.1-11 IsReflexiveDigraph\n\
  6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\n
  6.1-14 IsTransitiveDigraph\n
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n\
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regulari\ ty\n\n
\n  6.2-1 IsInRegularDigraph\n
<\ span class=\"nocss\">  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n
  6.2-\ 4 IsDistanceRegularDigraph\n
\n
<\ span class=\"tocline\"> 6.3 Connectivity and cycles\n\n
\n
<\ span class=\"nocss\">  6.3-1 IsAcyclicDigraph\n
  6\ .3-2 IsChainDigraph\n
  6.3-3 IsCo\ nnectedDigraph\n
\   6.3-4 IsBiconne\ ctedDigraph\n
&nb\ sp; 6.3-5 IsStronglyCo\ nnectedDigraph\n
\   6.3-6 IsAperiod\ icDigraph\n
\ ; 6.3-7 IsDirectedTree\ \n
  6.3-8 IsUndirectedTree\n
  6.3-9 IsEulerianDigraph\n\
  6.3-10 IsHamiltonianDigraph\n
  6.3-11 IsCycleDigraph\n
\n
&nbs\ p;6.4 Planarity\n\n
\n
  6.4-1 IsPlanarDigraph\n\
  6.4-2 IsOuterPlanarDigraph\n
\n
6.5 Homomorphism\ s and transformations\n\n
\n
  6.5-1 IsDigraphCore\n
  6.5-2 IsEdgeTransitive\n\
  6.5-3 IsVertexTransitive\n
\n
\n
7 Homomorphisms\n
7.1 Acting on digraphs\n\n\n
&\ nbsp; 7.1-1 OnDigraphs\ \n
  7.1-2 OnMultiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n<\ div class=\"ContSSBlock\">\n
  7.2-1 Digraphs\ UseNauty\n
\ 7.2-2 AutomorphismGro\ up\n
  \ 7.2-3 BlissAutomorphismGrou\ p\n
  <\ /span>7.2-4 NautyAutomorphismGroup\ \n
  7.2-5 AutomorphismGroup\n
  <\ a href=\"chap7_mj.html#X7DA2C4FE837FFE01\">7.2-6 AutomorphismGroup\ \n
  7.2-7 BlissCanonicalLabelling\ \n
  7.2-8 BlissCanonicalLabelling\ \n
  7.2-9 BlissCanonicalDigraph\n\
  7.2-10 DigraphGroup\n
  7.2-11 DigraphOrbits\n
  7.2-12 DigraphOrbitReps\n  7.2-13 DigraphSchreierVector\n  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  \ 7.2-17 IsomorphismDigraphs\n
  7.2\ -18 IsomorphismDigraphs\n
  7.2-1\ 9 RepresentativeOutNeighbours\n
  \ 7.2-20 IsDigraphIsomorphism\n
  7.\ 2-21 IsDigraphColouring\n
\n
7.3 Homomorphisms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7\ .3-5 MonomorphismsDigraphs\n
  7.3\ -6 DigraphEpimorphism\n
  7.3-7 Ep\ imorphismsDigraphs\n
  7.3-8 Digra\ phEmbedding\n
&nb\ sp; 7.3-9 EmbeddingsDi\ graphs\n
&n\ bsp;7.3-10 IsDigraphHomomor\ phism\n
&nb\ sp;7.3-11 IsDigraphEmbeddin\ g\n
  <\ /span>7.3-12 DigraphsRespectsColou\ ring\n
&nbs\ p;7.3-13 GeneratorsOfEndomo\ rphismMonoid\n
&n\ bsp; 7.3-14 DigraphCol\ ouring\n
&n\ bsp;7.3-15 DigraphGreedyCol\ ouring\n
&n\ bsp;7.3-16 DigraphWelshPowe\ llOrder\n
&\ nbsp;7.3-17 ChromaticNumber\ \n
  7.3-18 DigraphCore\ \n
\n
\n
8 Finding cliques and independent sets\n
8.1 Finding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8.1-3 DigraphClique\n
  8\ .1-4 DigraphMaximalCliques\n
  8.1\ -5 CliqueNumber\n
\n
8.2 Finding independent sets\n\ \n
\n
  8.2-\ 1 IsIndependentSet\n
  8.2-2 Digra\ phIndependentSet\n
  8.2-3 Digrap\ hMaximalIndependentSets\n
\n
\n
9 Visu\ alising and IO\n
9.\ 1 Visualising a digraph\n\n
\n
&\ nbsp;9.1-1 Splash\n
  9.1-2 DotDigraph\n
  9.1-3 DotSymmetricDigraph\n
  9.1-4 DotPartialOrderDigraph\n
  9.1-5 DotPreorderDigraph\n
  9.1-6 DotHighlightedDigraph\n
\n
9.2 Reading a\ nd writing digraphs to a file\n\n
\ \n
  9.2-1 DigraphFromGraph6String\ \n
  9.2-2 Graph6String\n
  9.2-3 DigraphFile\n
  9.2-4 ReadDigraphs\n
  9.2-5 WriteDigraphs\n
  9.2-6\ IteratorFromDigraphFile\n
  9.2-7\ DigraphPlainTextLineEncoder\n
  9\ .2-8 TournamentLineDecoder\n
  9.2\ -9 AdjacencyMatrixUpperTriangleLineDecoder\n\
  9.2-10 TCodeDecoder\n
  9.2-11 PlainTextString\n
  9.2-\ 12 WritePlainTextDigraph\n
  9.2-1\ 3 WriteDIMACSDigraph\n
\n
\n
A \n G\ rape to Digraphs Command Map\n \n
A.1 \n Functions to construct and\ modify graphs\n \n\n
\n
A.2 \n Functions to inspect \ graphs, vertices and edges\n \n\n
\n
A.3 \n Functions\ to determine regularity properties of graphs\n \n\n\n
\ ;A.4 \n Some special vertex subsets of a graph\n \n\n<\ /div>\n
&\ nbsp;A.5 \n Functions to construct new graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring and Complete Subgraphs\n \ \n\n
\n
A.7 \n Automorphism groups and isomorphism testing for grap\ hs\n \n\n
\n
\n
References
\n
Index
\n
1 \n The Digraphs package\n \n
<\ span class=\"nocss\"> \ 1.1 Introduction\n\n
\n
  1.1-1 Definitions\n\n
\n
\n
2 Insta\ lling Digraphs\n
2.1 For those in a hurry<\ /span>\n\n
\n
<\ span class=\"nocss\"> \ 2.2 Optional package dependencies\n\ \n
\n
  2.2-1 The Grape package\n\n
\n
<\ a href=\"chap2_mj.html#X849F6196875A6DF5\">2.3 Compili\ ng the kernel module\n\n
\n
2.4 Rebuilding the documentation<\ /span>\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installat\ ion\n\n
\n<\ br />  2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\n
  2.5-3 DigraphsTestExtreme\n
\n
\n
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
\ ; 3.1-1 IsDigraph<\ /span>\n
  \ 3.1-2 IsMutableDigraph\ \n
  3.1-3 IsImmutableDigraph\n
  3.1-4 IsCayleyDigraph\n
  3.1-5 IsDigraphWithAdjacencyFunction\n
  3.1-6 DigraphByOutNeighboursType\n<\ span class=\"ContSS\">
  3.1-7 Digraph\n
  3.1-8 DigraphByAdjacencyMatrix\n
  3.1-9 DigraphByEdges\n<\ br />  3.1-10 EdgeOrbitsDigraph\n
\   3.1-11 DigraphByInNeighbours\n
<\ span class=\"nocss\">  3.1-12 CayleyDigraph\n
\n
<\ span class=\"tocline\"> 3.2 Changing representations\n\n
\n
\   3.2-1 AsBinaryRelation\n
  \ 3.2-2 AsDigraph\n
  3.2-3 Graph\n
  3.2-4 AsGraph\n
  3.2-5 AsTransformation\n
\ \n
\ 3.3 New digraphs from old\n\n
\n
  3.3-1 DigraphImmutableCopy\n
  3.3-2 DigraphImmutableCopyIfImmutable\n\
  3.3-3 InducedSubdigraph\n
  3.3-4 ReducedDigraph\n
  3.3-5 MaximalSymmetricSubdigraph\n
  3.3-6 MaximalAntiSymmetricSubdigraph\n
  3.3-7 UndirectedSpanningForest\n
  3.3-8 QuotientDigraph\n
  3.3-9 DigraphReverse\n
  3.3-10 DigraphDual\n
  3.3-11 DigraphSymmetricClosure\n
<\ span class=\"nocss\">  3.3-12 DigraphTransitiveClosure\n
  3.3-13 DigraphTransitiveReduction\n<\ br />  3.3-14 DigraphAddVertex\n
<\ span class=\"nocss\">  3.3-15 DigraphAddVertices\n
  3.3-16 DigraphAddEdge\n
  3.3-1\ 7 DigraphAddEdgeOrbit\n
  3.3-18 D\ igraphAddEdges\n
\   3.3-19 DigraphR\ emoveVertex\n
&nb\ sp; 3.3-20 DigraphRemo\ veVertices\n
&nbs\ p; 3.3-21 DigraphRemov\ eEdge\n
&nb\ sp;3.3-22 DigraphRemoveEdge\ Orbit\n
&nb\ sp;3.3-23 DigraphRemoveEdge\ s\n
  <\ /span>3.3-24 DigraphRemoveLoops\n
  3.3-25 DigraphRemoveAllMultipleE\ dges\n
&nbs\ p;3.3-26 DigraphReverseEdge\ s\n
  <\ /span>3.3-27 DigraphDisjointUnion<\ /a>\n
  3.3-28 DigraphEdgeUnion\n
  3.3-29 DigraphJoin\n
  3.3-30 DigraphCartesianProduct\n
  3.3-31 DigraphDirectProduct\n
  3.3-32 DigraphCartesianProductProjections\n
  3.3-33 DigraphDirectProductProjectio\ ns\n
  \ 3.3-34 LineDigraph\n
  3.3-35 LineUndirectedDigraph\n
  3.3-36 DoubleDigraph\n
  3.3-37 BipartiteDoubleDigraph\n
  3.3-38 DigraphAddAllLoops\n
  3.3-39 DistanceDigraph\n
  3.3-40 DigraphClosure\n  3.3-41 DigraphMycielskian\n
\n
3.4 Random digraphs<\ /span>\n\n
\n
  3.4-1 RandomDigraph\n
  3.\ 4-2 RandomMultiDigraph\n
  3.4-3 R\ andomTournament\n
  3.4-4 RandomLa\ ttice\n
\n
3.5 Standard examples\n\n
\n
\ ; 3.5-1 ChainDigraph\n
  3.5-2 CompleteDigraph\n
  3.5-3 CompleteBipartiteDigraph\n
  3.5-4 CompleteMultipartiteDigraph<\ /span>\n
  \ 3.5-5 CycleDigraph\n
  3.5-6 EmptyDigraph\n
  3.5-7 JohnsonDigraph\n\
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n
4 Operators\n
4\ .1 Operators for digraphs\n\n
\n
\ ; 4.1-1 IsSubdigraph\n
  4.1-2 IsUndirectedSpanningTree<\ /a>\n
\n
\n
5 Attributes and operations\n
5.1 Vertices and edges\n\n
\ \n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n\
  5.1-4 DigraphNrEdges\n
  5.1-5 DigraphSinks\n
  5.1-6 D\ igraphSources\n
&\ nbsp; 5.1-7 DigraphTop\ ologicalSort\n
&n\ bsp; 5.1-8 DigraphVert\ exLabel\n
&\ nbsp;5.1-9 DigraphVertexLab\ els\n
  \ ;5.1-10 DigraphEdgeLabel\n
  5.1-11 DigraphEdgeLabels\n
  5.1-12 DigraphInEdges\n
  5.1-13 DigraphOutEdges\n
  5.1-14 IsDigraphEdge\n
  5.1-15 IsMatching\n
\n
5.2 Neighbours and deg\ ree\n\n
\n<\ br />  5.2-1 AdjacencyMatrix\n
  5.2-2 CharacteristicPolynomial\n
  5.2-3 BooleanAdjacencyMatrix\n
  5.2-4 DigraphAdjacencyFunction\n
  5.2-5 DigraphRange\n
  5.2-6\ OutNeighbours\n
\   5.2-7 InNeighbo\ urs\n
  \ ;5.2-8 OutDegrees\n
  5.2-9 InDegrees\n
  5.2-10 OutDegreeOfVertex\n
  5.2-11 OutNeighboursOfVertex\n
  5.2-12 InDegreeOfVertex\n\
  5.2-13 InNeighboursOfVertex\n  5.2-14 DigraphLoops\n
  5.2-15 PartialOrderDigraphMeetOfVertices\n<\ br />  5.2-16 DegreeMatrix\n
  5.2-17 LaplacianMatrix\n
\n
5.3 Reachability and connectivi\ ty\n\n
\n  5.3-1 DigraphDiameter\n
  5.3-2 DigraphShortestDistance\n
  5.3-3 DigraphShortestDistances\n
  5.3-4 DigraphLongestDistanceFromVertex\n\
  5.3-5 DigraphDistanceSet\n
  5.3-6 DigraphGirth\n
  5.3\ -7 DigraphOddGirth\n
  5.3-8 Digra\ phUndirectedGirth\n
  5.3-9 Digrap\ hConnectedComponents\n
  5.3-10 Di\ graphConnectedComponent\n
  5.3-1\ 1 DigraphStronglyConnectedComponents\n
\   5.3-12 DigraphStronglyConnectedComponent\n
  5.3-13 DigraphBicomponents\n
  5.3-14 ArticulationPoints\n<\ br />  5.3-15 DigraphPeriod\n
  5.3-16 DigraphFloydWarshall\n
  5.3-17 IsReachable\n
  5.3-18 Dig\ raphPath\n
\ 5.3-19 DigraphShortes\ tPath\n
&nb\ sp;5.3-20 IteratorOfPaths\n
  5.3-21 DigraphAllSimpleCircuits\ \n
  5.3-22 DigraphLongestSimpleCi\ rcuit\n
&nb\ sp;5.3-23 DigraphLayers\ \n
  5.3-24 DigraphDegeneracy\n
  5.3-25 DigraphDegeneracyOrdering\n
  <\ a href=\"chap5_mj.html#X863FDFC4839A3B82\">5.3-26 HamiltonianPath\n\
  5.3-27 NrSpanningTrees\n
<\ /div>\n
&\ nbsp;5.4 Cayley graphs of groups\n\n
\ \n
  5.4-1 GroupOfCayleyDigraph\n<\ span class=\"ContSS\">
  5.4-2 GeneratorsOfCayleyDigraph\ \n
\n
5.5 Associated semigroups\n\n
\n
  5.5-1 AsSemigroup\n
  5.5-2 AsSemigroup\n
\n<\ div class=\"ContSect\"> 5.6 Pl\ anarity\n\n
\n
  5.6-1 KuratowskiPlanarSubdigraph\n
  5.6-2 KuratowskiOuterPlanarSubdigraph\n
  5.6-3 PlanarEmbedding\n
  5.6-4 OuterPlanarEmbedding\n
  5.6-5 SubdigraphHomeomorphicToK23\n
\n\n
6 Properties of digraphs\n
6.1 Edge properties\ \n\n
\n
  6.1-1 DigraphHasLoops\n
  6.1-2\ IsAntiSymmetricDigraph\n
  6.1-3\ IsBipartiteDigraph\n
  6.1-4 IsCo\ mpleteBipartiteDigraph\n
  6.1-5 I\ sCompleteDigraph\n
  6.1-6 IsComp\ leteMultipartiteDigraph\n
  6.1-7\ IsEmptyDigraph\n
  6.1-8 IsEquiva\ lenceDigraph\n
&n\ bsp; 6.1-9 IsFunctiona\ lDigraph\n
\ 6.1-10 IsMultiDigraph\ \n
  6.1-11 IsReflexiveDigraph\ \n
  6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\n
  6.1-14 IsTransitiveDigraph\n
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
&nbs\ p;6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n\
  6.2-3 IsRegularDigraph\n
<\ span class=\"nocss\">  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectivity and\ cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  \ 6.3-3 IsConnectedDigraph\n
  6.3-4\ IsBiconnectedDigraph\n
  6.3-5 Is\ StronglyConnectedDigraph\n
  6.3-6\ IsAperiodicDigraph\n
  6.3-7 IsDi\ rectedTree\n
&nbs\ p; 6.3-8 IsUndirectedT\ ree\n
  \ ;6.3-9 IsEulerianDigraph\n
  6.3-10 IsHamiltonianDigraph<\ /span>\n
  \ 6.3-11 IsCycleDigraph\n\
\n
6.4 Planarity\n\n
\n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n<\ div class=\"ContSect\"> 6.5 Ho\ momorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\n
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\n\n
7 Homomorphisms\n
7.1 Acting on digraphs\n<\ /span>\n
\n
  7.1-1\ OnDigraphs\n
&nb\ sp; 7.1-2 OnMultiDigra\ phs\n
\n
\ 7.2 Isomorphisms and canonical labellings\ \n\n
\n
  7.\ 2-1 DigraphsUseNauty\n
  7.2-2 Aut\ omorphismGroup\n
\   7.2-3 BlissAuto\ morphismGroup\n
&\ nbsp; 7.2-4 NautyAutom\ orphismGroup\n
&n\ bsp; 7.2-5 Automorphis\ mGroup\n
&n\ bsp;7.2-6 AutomorphismGroup\ \n
  7.2-7 BlissCanonicalLabelling\ \n
  7.2-8 BlissCanonicalLabelling\ \n
  7.2-9 BlissCanonicalDigraph\n
  7.2-10 DigraphGroup\ \n
  7.2-11 DigraphOrbits\n
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n\
  7.2-17 IsomorphismDigraphs\n  7.2-18 IsomorphismDigraphs\n
\   7.2-19 RepresentativeOutNeighbours\n<\ br />  7.2-20 IsDigraphIsomorphism\n  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphisms of \ digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n
  7.3-7 EpimorphismsDigraphs\n
  \ 7.3-8 DigraphEmbedding\n
  7.3-9 E\ mbeddingsDigraphs\n
  7.3-10 IsDig\ raphHomomorphism\n
  7.3-11 IsDig\ raphEmbedding\n
&\ nbsp; 7.3-12 DigraphsR\ espectsColouring\n
  7.3-13 Gener\ atorsOfEndomorphismMonoid\n
  7.3-\ 14 DigraphColouring\n
  7.3-15 Dig\ raphGreedyColouring\n
  7.3-16 Dig\ raphWelshPowellOrder\n
  7.3-17 Ch\ romaticNumber\n
&\ nbsp; 7.3-18 DigraphCo\ re\n
\n
\n
8 Finding cliques and indep\ endent sets\n
8.1 <\ span class=\"Heading\">Finding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8.1-3 DigraphClique\n<\ br />  8.1-4 DigraphMaximalCliques\n  8.1-5 CliqueNumber\n
\n
8.2 Finding independent sets\ \n\n
\n
  8.2-1 IsIndependentSet\n
  8.2-2 DigraphIndependentSet\n
  8.2-3 DigraphMaximalIndependentSets\n
\n
\n
9 Visualising and IO\n
9.1 Visualising a digraph\n\n
\n
  9.1-1 Spl\ ash\n
  \ ;9.1-2 DotDigraph\n
  9.1-3 DotSymmetricDigraph\n<\ span class=\"ContSS\">
  9.1-4 DotPartialOrderDigraph\n<\ span class=\"ContSS\">
  9.1-5 DotPreorderDigraph\n
  9.1-6 DotHighlightedDigraph\n
<\ /div>\n
&\ nbsp;9.2 Reading and writing digraphs to a file\n\n
\n
&nb\ sp;9.2-1 DigraphFromGraph6S\ tring\n
&nb\ sp;9.2-2 Graph6String\n
  <\ a href=\"chap9_mj.html#X7845ACDA7D4D333D\">9.2-3 DigraphFile\n
  9.2-4 ReadDigraphs\n
  9.2-5 WriteDigraphs\n  9.2-6 IteratorFromDigraphFile\n  9.2-7 DigraphPlainTextLineEncoder\n
  9.2-8 TournamentLineDecoder\n<\ br />  9.2-9 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-10 TCodeDecoder\n
  9.2-11 PlainTextString\n  9.2-12 WritePlainTextDigraph\n  9.2-13 WriteDIMACSDigraph\n
\n
\n
A \n Grape to Digraphs Command Map\n \n
A.1 \n Functions t\ o construct and modify graphs\n \n\n
\n
A.2 \n Functi\ ons to inspect graphs, vertices and edges\n \n\n
\n<\ div class=\"ContSect\"> A.3 \n\ Functions to determine regularity properties of graphs\n \ \n\n
\n
A.4 \n Some special vertex subsets of a graph\n <\ /a>\n\n
\n
A.5 \n Functions to construct new graphs from old\n <\ /span>\n\n
\n
<\ span class=\"nocss\"> \ A.6 \n Vertex-Colouring and Complete Subgraphs\n \ \n\n
\n
A.7 \n Automorphism groups and isomorphism te\ sting for graphs\n \n\n
\n
\n
References
\n
Index
\n" ) gap> gap> ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── 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). Homepage: https://gap-packages.github.io/orb ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape ───────────────────────────────────────────────────────────────────────────── ─────────────��─────────────��──────────────────────────��─────────────��──────── Loading Digraphs 1.1.1 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (http://www-groups.mcs.st-andrews.ac.uk/~jamesm), Michael Torpey (https://mtorpey.github.io), and Wilf Wilson (http://wilf.me). Homepage: https://gap-packages.github.io/Digraphs ──────��──────────────────────────��─────────────��──────────────────────────��── #I File: /builddir/build/BUILD/digraphs-1.1.1/../pkg/digraphs-1.1.1/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.AWrGFG + umask 022 + cd /builddir/build/BUILD + '[' /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x '!=' / ']' + rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x ++ dirname /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x + mkdir -p /builddir/build/BUILDROOT + mkdir /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x + cd digraphs-1.1.1 + mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap/pkg/digraphs-1.1.1/doc + cp -a bin data gap tst VERSIONS PackageInfo.g init.g makedoc.g read.g /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap/pkg/digraphs-1.1.1 + 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-1.1.1-1.fc32.s390x/usr/lib/gap/pkg/digraphs-1.1.1/doc + /usr/lib/rpm/find-debuginfo.sh -j2 --strict-build-id -m -i --build-id-seed 1.1.1-1.fc32 --unique-debug-suffix -1.1.1-1.fc32.s390x --unique-debug-src-base gap-pkg-digraphs-1.1.1-1.fc32.s390x --run-dwz --dwz-low-mem-die-limit 10000000 --dwz-max-die-limit 50000000 -S debugsourcefiles.list /builddir/build/BUILD/digraphs-1.1.1 explicitly decompress any DWARF compressed ELF sections in /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap/pkg/digraphs-1.1.1/bin/s390x-ibm-linux-gnu-default64-kv3/digraphs.so extracting debug info from /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap/pkg/digraphs-1.1.1/bin/s390x-ibm-linux-gnu-default64-kv3/digraphs.so Exception caught while booting Guile. /usr/bin/gdb.minimal: warning: Could not complete Guile gdb module initialization from: /usr/share/gdb/guile/gdb/boot.scm. Limited Guile support is available. Suggest passing --data-directory=/path/to/gdb/data-directory. Error in function "open-file": No such file or directory: "/usr/share/gdb/guile/gdb/boot.scm"original debug info size: 3572kB, size after compression: 3392kB /usr/lib/rpm/sepdebugcrcfix: Updated 1 CRC32s, 0 CRC32s did match. 980 blocks + /usr/lib/rpm/check-buildroot + /usr/lib/rpm/redhat/brp-ldconfig + /usr/lib/rpm/brp-compress + /usr/lib/rpm/brp-strip-static-archive /usr/bin/strip + /usr/lib/rpm/redhat/brp-python-bytecompile /usr/bin/python 1 0 + /usr/lib/rpm/brp-python-hardlink + /usr/lib/rpm/redhat/brp-mangle-shebangs *** WARNING: ./usr/src/debug/gap-pkg-digraphs-1.1.1-1.fc32.s390x/extern/bliss-0.73/graph.cc is executable but has no shebang, removing executable bit Executing(%check): /bin/sh -e /var/tmp/rpm-tmp.Nc5dAF + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.1.1 + export LC_ALL=C.UTF-8 + LC_ALL=C.UTF-8 + cd tst + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.10.2 of 19-Jun-2019 │ GAP │ https://www.gap-system.org └───────┘ Architecture: s390x-ibm-linux-gnu-default64-kv3 Configuration: gmp 6.1.2, 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 1.6.3, IO 4.7.0, PrimGrp 3.4.0, SmallGrp 1.4.1, TransGrp 2.0.4 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── 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). Homepage: https://gap-packages.github.io/orb ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.1.1 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (http://www-groups.mcs.st-andrews.ac.uk/~jamesm), Michael Torpey (https://mtorpey.github.io), and Wilf Wilson (http://wilf.me). Homepage: https://gap-packages.github.io/Digraphs ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: testinstall.tst msecs: 156 true gap> Digraphs package: standard/attr.tst msecs: 2524 true gap> Digraphs package: standard/cliques.tst msecs: 278 true gap> Digraphs package: standard/constructors.tst msecs: 29 true gap> Digraphs package: standard/display.tst msecs: 10 true gap> Digraphs package: standard/examples.tst msecs: 0 true gap> Digraphs package: standard/grahom.tst msecs: 417 true gap> Digraphs package: standard/grape.tst msecs: 10 true gap> Digraphs package: standard/io.tst msecs: 1774 true gap> Digraphs package: standard/isomorph.tst msecs: 284 true gap> Digraphs package: standard/labels.tst msecs: 0 true gap> Digraphs package: standard/oper.tst msecs: 539 true gap> Digraphs package: standard/orbits.tst msecs: 0 true gap> Digraphs package: standard/planar.tst msecs: 33 true gap> Digraphs package: standard/prop.tst msecs: 2532 true gap> gap> + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.10.2 of 19-Jun-2019 │ GAP │ https://www.gap-system.org └───────┘ Architecture: s390x-ibm-linux-gnu-default64-kv3 Configuration: gmp 6.1.2, 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 1.6.3, IO 4.7.0, PrimGrp 3.4.0, SmallGrp 1.4.1, TransGrp 2.0.4 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── 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). Homepage: https://gap-packages.github.io/orb ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.1.1 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (http://www-groups.mcs.st-andrews.ac.uk/~jamesm), Michael Torpey (https://mtorpey.github.io), and Wilf Wilson (http://wilf.me). Homepage: https://gap-packages.github.io/Digraphs ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: standard/digraph.tst msecs: 218 gap> + find /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/lib/gap/pkg/digraphs-1.1.1 -size 0 -delete + RPM_EC=0 ++ jobs -p + exit 0 Processing files: gap-pkg-digraphs-1.1.1-1.fc32.s390x Executing(%doc): /bin/sh -e /var/tmp/rpm-tmp.28XOFI + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.1.1 + DOCDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/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-1.1.1-1.fc32.s390x/usr/share/doc/gap-pkg-digraphs + cp -pr CHANGELOG.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/share/doc/gap-pkg-digraphs + cp -pr README.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/share/doc/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.DGLWWH + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.1.1 + LICENSEDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/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-1.1.1-1.fc32.s390x/usr/share/licenses/gap-pkg-digraphs + cp -pr GPL /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/share/licenses/gap-pkg-digraphs + cp -pr LICENSE /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x/usr/share/licenses/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Provides: bundled(bliss) = 0.73 gap-pkg-digraphs = 1.1.1-1.fc32 gap-pkg-digraphs(s390-64) = 1.1.1-1.fc32 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Requires: libc.so.6()(64bit) libc.so.6(GLIBC_2.11)(64bit) libc.so.6(GLIBC_2.2)(64bit) libc.so.6(GLIBC_2.4)(64bit) libc.so.6(GLIBC_2.7)(64bit) libgcc_s.so.1()(64bit) libgcc_s.so.1(GCC_3.0)(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) rtld(GNU_HASH) Recommends: gap-pkg-grape(s390-64) gap-pkg-nautytracesinterface(s390-64) Processing files: gap-pkg-digraphs-doc-1.1.1-1.fc32.noarch Provides: gap-pkg-digraphs-doc = 1.1.1-1.fc32 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-1.1.1-1.fc32.s390x Provides: gap-pkg-digraphs-debugsource = 1.1.1-1.fc32 gap-pkg-digraphs-debugsource(s390-64) = 1.1.1-1.fc32 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-1.1.1-1.fc32.s390x Provides: debuginfo(build-id) = c0be799dbbbb391a8231097a8514dd58fbef12ae gap-pkg-digraphs-debuginfo = 1.1.1-1.fc32 gap-pkg-digraphs-debuginfo(s390-64) = 1.1.1-1.fc32 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Recommends: gap-pkg-digraphs-debugsource(s390-64) = 1.1.1-1.fc32 Checking for unpackaged file(s): /usr/lib/rpm/check-files /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x Wrote: /builddir/build/RPMS/gap-pkg-digraphs-1.1.1-1.fc32.s390x.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debugsource-1.1.1-1.fc32.s390x.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-doc-1.1.1-1.fc32.noarch.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debuginfo-1.1.1-1.fc32.s390x.rpm Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.NkxH7G + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.1.1 + /usr/bin/rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.1.1-1.fc32.s390x + RPM_EC=0 ++ jobs -p + exit 0 Child return code was: 0