  
  
                                    [1X[5XDigraphs[105X[101X
  
  
                                 Version 1.6.1
  
  
                                  Jan De Beule
  
                                 Julius Jonušas
  
                               James D. Mitchell
  
                                 Maria Tsalakou
  
                                 Wilf A. Wilson
  
                                Michael C. Young
  
                        Marina Anagnostopoulou-Merkouri
  
                                   Finn Buck
  
                                 Stuart Burrell
  
                                 Reinis Cirpons
  
                                Tom Conti-Leslie
  
                                  Luke Elliott
  
                                 Ewan Gilligan
  
                                    Max Horn
  
                             Christopher Jefferson
  
                                Markus Pfeiffer
  
                                   Lea Racine
  
                              Christopher Russell
  
                                   Finn Smith
  
                                   Ben Spiers
  
                                  Murray White
  
  
  
  Jan De Beule
      Email:    [7Xmailto:jdebeule@cage.ugent.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~jdbeule[107X
  Julius Jonušas
      Email:    [7Xmailto:julius.jonusas@tuwien.ac.at[107X
      Homepage: [7Xhttp://julius.jonusas.work[107X
  James D. Mitchell
      Email:    [7Xmailto:jdm3@st-and.ac.uk[107X
      Homepage: [7Xhttps://jdbm.me[107X
  Maria Tsalakou
      Email:    [7Xmailto:mt200@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://mariatsalakou.github.io[107X
  Wilf A. Wilson
      Email:    [7Xmailto:gap@wilf-wilson.net[107X
      Homepage: [7Xhttps://wilf.me[107X
  Michael C. Young
      Email:    [7Xmailto:mct25@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://mct25.host.cs.st-andrews.ac.uk[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XDigraphs[105X  package  is  a  [5XGAP[105X  package  containing  methods for graphs,
  digraphs, and multidigraphs.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  2014-22  by  Jan  De  Beule,  Julius  Jonušas, James D. Mitchell, Wilf A.
  Wilson, Michael Young et al.[133X
  
  [33X[0;0Y[5XDigraphs[105X  is  free  software; you can redistribute it and/or modify it under
  the      terms      of      the      GNU      General     Public     License
  ([7Xhttps://www.fsf.org/licenses/gpl.html[107X)  as  published  by the Free Software
  Foundation;  either  version 3 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe would like to thank Christopher Jefferson for his help in including [5Xbliss[105X
  in  [5XDigraphs[105X. This package's methods for computing digraph homomorphisms are
  based on work by Max Neunhöffer, and independently Artur Schäfer.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (digraphs)[101X
  
  1 [33X[0;0YThe [5XDigraphs[105X package[133X
    1.1 [33X[0;0YIntroduction[133X
      1.1-1 [33X[0;0YDefinitions[133X
  2 [33X[0;0YInstalling [5XDigraphs[105X[133X
    2.1 [33X[0;0YFor those in a hurry[133X
      2.1-1 [33X[0;0YConfiguration options[133X
    2.2 [33X[0;0YOptional package dependencies[133X
      2.2-1 [33X[0;0YThe Grape package[133X
    2.3 [33X[0;0YCompiling the kernel module[133X
    2.4 [33X[0;0YRebuilding the documentation[133X
      2.4-1 DigraphsMakeDoc
    2.5 [33X[0;0YTesting your installation[133X
      2.5-1 DigraphsTestInstall
      2.5-2 DigraphsTestStandard
      2.5-3 DigraphsTestExtreme
  3 [33X[0;0YCreating digraphs[133X
    3.1 [33X[0;0YCreating digraphs[133X
      3.1-1 IsDigraph
      3.1-2 IsMutableDigraph
      3.1-3 IsImmutableDigraph
      3.1-4 IsCayleyDigraph
      3.1-5 IsDigraphWithAdjacencyFunction
      3.1-6 DigraphByOutNeighboursType
      3.1-7 Digraph
      3.1-8 DigraphByAdjacencyMatrix
      3.1-9 DigraphByEdges
      3.1-10 EdgeOrbitsDigraph
      3.1-11 DigraphByInNeighbours
      3.1-12 CayleyDigraph
      3.1-13 ListNamedDigraphs
    3.2 [33X[0;0YChanging representations[133X
      3.2-1 AsBinaryRelation
      3.2-2 AsDigraph
      3.2-3 Graph
      3.2-4 AsGraph
      3.2-5 AsTransformation
    3.3 [33X[0;0YNew digraphs from old[133X
      3.3-1 DigraphImmutableCopy
      3.3-2 DigraphImmutableCopyIfImmutable
      3.3-3 InducedSubdigraph
      3.3-4 ReducedDigraph
      3.3-5 MaximalSymmetricSubdigraph
      3.3-6 MaximalAntiSymmetricSubdigraph
      3.3-7 UndirectedSpanningForest
      3.3-8 DigraphShortestPathSpanningTree
      3.3-9 QuotientDigraph
      3.3-10 DigraphReverse
      3.3-11 DigraphDual
      3.3-12 DigraphSymmetricClosure
      3.3-13 DigraphTransitiveClosure
      3.3-14 DigraphTransitiveReduction
      3.3-15 DigraphAddVertex
      3.3-16 DigraphAddVertices
      3.3-17 DigraphAddEdge
      3.3-18 DigraphAddEdgeOrbit
      3.3-19 DigraphAddEdges
      3.3-20 DigraphRemoveVertex
      3.3-21 DigraphRemoveVertices
      3.3-22 DigraphRemoveEdge
      3.3-23 DigraphRemoveEdgeOrbit
      3.3-24 DigraphRemoveEdges
      3.3-25 DigraphRemoveLoops
      3.3-26 DigraphRemoveAllMultipleEdges
      3.3-27 DigraphReverseEdges
      3.3-28 DigraphDisjointUnion
      3.3-29 DigraphEdgeUnion
      3.3-30 DigraphJoin
      3.3-31 DigraphCartesianProduct
      3.3-32 DigraphDirectProduct
      3.3-33 ConormalProduct
      3.3-34 HomomorphicProduct
      3.3-35 LexicographicProduct
      3.3-36 ModularProduct
      3.3-37 StrongProduct
      3.3-38 DigraphCartesianProductProjections
      3.3-39 DigraphDirectProductProjections
      3.3-40 LineDigraph
      3.3-41 LineUndirectedDigraph
      3.3-42 DoubleDigraph
      3.3-43 BipartiteDoubleDigraph
      3.3-44 DigraphAddAllLoops
      3.3-45 DistanceDigraph
      3.3-46 DigraphClosure
      3.3-47 DigraphMycielskian
    3.4 [33X[0;0YRandom digraphs[133X
      3.4-1 RandomDigraph
      3.4-2 RandomMultiDigraph
      3.4-3 RandomTournament
      3.4-4 RandomLattice
    3.5 [33X[0;0YStandard examples[133X
      3.5-1 AndrasfaiGraph
      3.5-2 BananaTree
      3.5-3 BinaryTree
      3.5-4 BinomialTreeGraph
      3.5-5 BishopsGraph
      3.5-6 BondyGraph
      3.5-7 BookGraph
      3.5-8 StackedBookGraph
      3.5-9 ChainDigraph
      3.5-10 CirculantGraph
      3.5-11 CompleteDigraph
      3.5-12 CompleteBipartiteDigraph
      3.5-13 CompleteMultipartiteDigraph
      3.5-14 CycleDigraph
      3.5-15 CycleGraph
      3.5-16 EmptyDigraph
      3.5-17 GearGraph
      3.5-18 HaarGraph
      3.5-19 HalvedCubeGraph
      3.5-20 HanoiGraph
      3.5-21 HelmGraph
      3.5-22 HypercubeGraph
      3.5-23 JohnsonDigraph
      3.5-24 KellerGraph
      3.5-25 KingsGraph
      3.5-26 KneserGraph
      3.5-27 KnightsGraph
      3.5-28 LindgrenSousselierGraph
      3.5-29 LollipopGraph
      3.5-30 MobiusLadderGraph
      3.5-31 MycielskiGraph
      3.5-32 OddGraph
      3.5-33 PathGraph
      3.5-34 PermutationStarGraph
      3.5-35 PetersenGraph
      3.5-36 GeneralisedPetersenGraph
      3.5-37 PrismGraph
      3.5-38 StackedPrismGraph
      3.5-39 QueensGraph
      3.5-40 RooksGraph
      3.5-41 SquareGridGraph
      3.5-42 TriangularGridGraph
      3.5-43 StarGraph
      3.5-44 TadpoleGraph
      3.5-45 WalshHadamardGraph
      3.5-46 WebGraph
      3.5-47 WheelGraph
      3.5-48 WindmillGraph
  4 [33X[0;0YOperators[133X
    4.1 [33X[0;0YOperators for digraphs[133X
      4.1-1 IsSubdigraph
      4.1-2 IsUndirectedSpanningTree
  5 [33X[0;0YAttributes and operations[133X
    5.1 [33X[0;0YVertices and edges[133X
      5.1-1 DigraphVertices
      5.1-2 DigraphNrVertices
      5.1-3 DigraphEdges
      5.1-4 DigraphNrEdges
      5.1-5 DigraphNrLoops
      5.1-6 DigraphSinks
      5.1-7 DigraphSources
      5.1-8 DigraphTopologicalSort
      5.1-9 DigraphVertexLabel
      5.1-10 DigraphVertexLabels
      5.1-11 DigraphEdgeLabel
      5.1-12 DigraphEdgeLabels
      5.1-13 DigraphInEdges
      5.1-14 DigraphOutEdges
      5.1-15 IsDigraphEdge
      5.1-16 IsMatching
      5.1-17 DigraphMaximalMatching
      5.1-18 DigraphMaximumMatching
    5.2 [33X[0;0YNeighbours and degree[133X
      5.2-1 AdjacencyMatrix
      5.2-2 CharacteristicPolynomial
      5.2-3 BooleanAdjacencyMatrix
      5.2-4 DigraphAdjacencyFunction
      5.2-5 DigraphRange
      5.2-6 OutNeighbours
      5.2-7 InNeighbours
      5.2-8 OutDegrees
      5.2-9 InDegrees
      5.2-10 OutDegreeOfVertex
      5.2-11 OutNeighboursOfVertex
      5.2-12 InDegreeOfVertex
      5.2-13 InNeighboursOfVertex
      5.2-14 DigraphLoops
      5.2-15 DegreeMatrix
      5.2-16 LaplacianMatrix
    5.3 [33X[0;0YOrders[133X
      5.3-1 PartialOrderDigraphMeetOfVertices
      5.3-2 NonUpperSemimodularPair
    5.4 [33X[0;0YReachability and connectivity[133X
      5.4-1 DigraphDiameter
      5.4-2 DigraphShortestDistance
      5.4-3 DigraphShortestDistances
      5.4-4 DigraphLongestDistanceFromVertex
      5.4-5 DigraphDistanceSet
      5.4-6 DigraphGirth
      5.4-7 DigraphOddGirth
      5.4-8 DigraphUndirectedGirth
      5.4-9 DigraphConnectedComponents
      5.4-10 DigraphConnectedComponent
      5.4-11 DigraphStronglyConnectedComponents
      5.4-12 DigraphStronglyConnectedComponent
      5.4-13 DigraphBicomponents
      5.4-14 ArticulationPoints
      5.4-15 Bridges
      5.4-16 StrongOrientation
      5.4-17 DigraphPeriod
      5.4-18 DigraphFloydWarshall
      5.4-19 IsReachable
      5.4-20 IsDigraphPath
      5.4-21 VerticesReachableFrom
      5.4-22 DigraphPath
      5.4-23 DigraphShortestPath
      5.4-24 DigraphRandomWalk
      5.4-25 Dominators
      5.4-26 DominatorTree
      5.4-27 IteratorOfPaths
      5.4-28 DigraphAllSimpleCircuits
      5.4-29 DigraphLongestSimpleCircuit
      5.4-30 DigraphLayers
      5.4-31 DigraphDegeneracy
      5.4-32 DigraphDegeneracyOrdering
      5.4-33 HamiltonianPath
      5.4-34 NrSpanningTrees
      5.4-35 DigraphDijkstra
    5.5 [33X[0;0YCayley graphs of groups[133X
      5.5-1 GroupOfCayleyDigraph
      5.5-2 GeneratorsOfCayleyDigraph
    5.6 [33X[0;0YAssociated semigroups[133X
      5.6-1 AsSemigroup
      5.6-2 AsSemigroup
    5.7 [33X[0;0YPlanarity[133X
      5.7-1 KuratowskiPlanarSubdigraph
      5.7-2 KuratowskiOuterPlanarSubdigraph
      5.7-3 PlanarEmbedding
      5.7-4 OuterPlanarEmbedding
      5.7-5 SubdigraphHomeomorphicToK23
  6 [33X[0;0YProperties of digraphs[133X
    6.1 [33X[0;0YVertex properties[133X
      6.1-1 DigraphHasAVertex
      6.1-2 DigraphHasNoVertices
    6.2 [33X[0;0YEdge properties[133X
      6.2-1 DigraphHasLoops
      6.2-2 IsAntiSymmetricDigraph
      6.2-3 IsBipartiteDigraph
      6.2-4 IsCompleteBipartiteDigraph
      6.2-5 IsCompleteDigraph
      6.2-6 IsCompleteMultipartiteDigraph
      6.2-7 IsEmptyDigraph
      6.2-8 IsEquivalenceDigraph
      6.2-9 IsFunctionalDigraph
      6.2-10 IsPermutationDigraph
      6.2-11 IsMultiDigraph
      6.2-12 IsNonemptyDigraph
      6.2-13 IsReflexiveDigraph
      6.2-14 IsSymmetricDigraph
      6.2-15 IsTournament
      6.2-16 IsTransitiveDigraph
    6.3 [33X[0;0YOrders[133X
      6.3-1 IsPreorderDigraph
      6.3-2 IsPartialOrderDigraph
      6.3-3 IsMeetSemilatticeDigraph
      6.3-4 DigraphMeetTable
      6.3-5 IsUpperSemimodularDigraph
      6.3-6 IsDistributiveLatticeDigraph
    6.4 [33X[0;0YRegularity[133X
      6.4-1 IsInRegularDigraph
      6.4-2 IsOutRegularDigraph
      6.4-3 IsRegularDigraph
      6.4-4 IsDistanceRegularDigraph
    6.5 [33X[0;0YConnectivity and cycles[133X
      6.5-1 IsAcyclicDigraph
      6.5-2 IsChainDigraph
      6.5-3 IsConnectedDigraph
      6.5-4 IsBiconnectedDigraph
      6.5-5 IsBridgelessDigraph
      6.5-6 IsStronglyConnectedDigraph
      6.5-7 IsAperiodicDigraph
      6.5-8 IsDirectedTree
      6.5-9 IsUndirectedTree
      6.5-10 IsEulerianDigraph
      6.5-11 IsHamiltonianDigraph
      6.5-12 IsCycleDigraph
    6.6 [33X[0;0YPlanarity[133X
      6.6-1 IsPlanarDigraph
      6.6-2 IsOuterPlanarDigraph
    6.7 [33X[0;0YHomomorphisms and transformations[133X
      6.7-1 IsDigraphCore
      6.7-2 IsEdgeTransitive
      6.7-3 IsVertexTransitive
  7 [33X[0;0YHomomorphisms[133X
    7.1 [33X[0;0YActing on digraphs[133X
      7.1-1 OnDigraphs
      7.1-2 OnMultiDigraphs
      7.1-3 OnTuplesDigraphs
    7.2 [33X[0;0YIsomorphisms and canonical labellings[133X
      7.2-1 DigraphsUseNauty
      7.2-2 AutomorphismGroup
      7.2-3 BlissAutomorphismGroup
      7.2-4 NautyAutomorphismGroup
      7.2-5 AutomorphismGroup
      7.2-6 AutomorphismGroup
      7.2-7 BlissCanonicalLabelling
      7.2-8 BlissCanonicalLabelling
      7.2-9 BlissCanonicalDigraph
      7.2-10 DigraphGroup
      7.2-11 DigraphOrbits
      7.2-12 DigraphOrbitReps
      7.2-13 DigraphSchreierVector
      7.2-14 DigraphStabilizer
      7.2-15 IsIsomorphicDigraph
      7.2-16 IsIsomorphicDigraph
      7.2-17 IsomorphismDigraphs
      7.2-18 IsomorphismDigraphs
      7.2-19 RepresentativeOutNeighbours
      7.2-20 IsDigraphIsomorphism
      7.2-21 IsDigraphColouring
      7.2-22 MaximalCommonSubdigraph
      7.2-23 MinimalCommonSuperdigraph
    7.3 [33X[0;0YHomomorphisms of digraphs[133X
      7.3-1 HomomorphismDigraphsFinder
      7.3-2 DigraphHomomorphism
      7.3-3 HomomorphismsDigraphs
      7.3-4 DigraphMonomorphism
      7.3-5 MonomorphismsDigraphs
      7.3-6 DigraphEpimorphism
      7.3-7 EpimorphismsDigraphs
      7.3-8 DigraphEmbedding
      7.3-9 EmbeddingsDigraphs
      7.3-10 IsDigraphHomomorphism
      7.3-11 IsDigraphEmbedding
      7.3-12 DigraphsRespectsColouring
      7.3-13 GeneratorsOfEndomorphismMonoid
      7.3-14 DigraphColouring
      7.3-15 DigraphGreedyColouring
      7.3-16 DigraphWelshPowellOrder
      7.3-17 ChromaticNumber
      7.3-18 DigraphCore
      7.3-19 LatticeDigraphEmbedding
      7.3-20 IsLatticeHomomorphism
  8 [33X[0;0YFinding cliques and independent sets[133X
    8.1 [33X[0;0YFinding cliques[133X
      8.1-1 IsClique
      8.1-2 CliquesFinder
      8.1-3 DigraphClique
      8.1-4 DigraphMaximalCliques
      8.1-5 CliqueNumber
    8.2 [33X[0;0YFinding independent sets[133X
      8.2-1 IsIndependentSet
      8.2-2 DigraphIndependentSet
      8.2-3 DigraphMaximalIndependentSets
  9 [33X[0;0YVisualising and IO[133X
    9.1 [33X[0;0YVisualising a digraph[133X
      9.1-1 Splash
      9.1-2 DotDigraph
      9.1-3 DotSymmetricDigraph
      9.1-4 DotPartialOrderDigraph
      9.1-5 DotPreorderDigraph
      9.1-6 DotHighlightedDigraph
    9.2 [33X[0;0YReading and writing digraphs to a file[133X
      9.2-1 String
      9.2-2 DigraphFromGraph6String
      9.2-3 Graph6String
      9.2-4 DigraphFile
      9.2-5 ReadDigraphs
      9.2-6 WriteDigraphs
      9.2-7 IteratorFromDigraphFile
      9.2-8 DigraphPlainTextLineEncoder
      9.2-9 TournamentLineDecoder
      9.2-10 AdjacencyMatrixUpperTriangleLineDecoder
      9.2-11 TCodeDecoder
      9.2-12 PlainTextString
      9.2-13 WritePlainTextDigraph
      9.2-14 WriteDIMACSDigraph
  A [33X[0;0YGrape to Digraphs Command Map[133X
    A.1 [33X[0;0YFunctions to construct and modify graphs[133X
    A.2 [33X[0;0YFunctions to inspect graphs, vertices and edges[133X
    A.3 [33X[0;0YFunctions to determine regularity properties of graphs[133X
    A.4 [33X[0;0YSome special vertex subsets of a graph[133X
    A.5 [33X[0;0YFunctions to construct new graphs from old[133X
    A.6 [33X[0;0YVertex-Colouring and Complete Subgraphs[133X
    A.7 [33X[0;0YAutomorphism groups and isomorphism testing for graphs[133X
  
  
  [32X
