  
  [1X3 [33X[0;0YCreating digraphs[133X[101X
  
  [33X[0;0YIn this chapter we describe how to create digraphs.[133X
  
  
  [1X3.1 [33X[0;0YCreating digraphs[133X[101X
  
  [1X3.1-1 IsDigraph[101X
  
  [33X[1;0Y[29X[2XIsDigraph[102X[32X Category[133X
  
  [33X[0;0YEvery  digraph  in  [5XDigraphs[105X  belongs  to the category [10XIsDigraph[110X. Some basic
  attributes   and   operations  for  digraphs  are  [2XDigraphVertices[102X  ([14X5.1-1[114X),
  [2XDigraphEdges[102X ([14X5.1-3[114X), and [2XOutNeighbours[102X ([14X5.2-6[114X).[133X
  
  [1X3.1-2 IsMutableDigraph[101X
  
  [33X[1;0Y[29X[2XIsMutableDigraph[102X[32X Category[133X
  
  [33X[0;0Y[10XIsMutableDigraph[110X   is   a   synonym  for  [2XIsDigraph[102X  ([14X3.1-1[114X)  and  [2XIsMutable[102X
  ([14XReference: IsMutable[114X). A mutable digraph may be changed in-place by methods
  in   the   [5XDigraphs[105X   package,   and   is   not   attribute-storing   –  see
  [2XIsAttributeStoringRep[102X ([14XReference: IsAttributeStoringRep[114X).[133X
  
  [33X[0;0YA  mutable  digraph  may  be  converted  into an immutable attribute-storing
  digraph by calling [2XMakeImmutable[102X ([14XReference: MakeImmutable[114X) on the digraph.[133X
  
  [1X3.1-3 IsImmutableDigraph[101X
  
  [33X[1;0Y[29X[2XIsImmutableDigraph[102X[32X Category[133X
  
  [33X[0;0Y[10XIsImmutableDigraph[110X  is a subcategory of [2XIsDigraph[102X ([14X3.1-1[114X). Digraphs that lie
  in  [10XIsImmutableDigraph[110X  are  immutable and attribute-storing. In particular,
  they lie in [2XIsAttributeStoringRep[102X ([14XReference: IsAttributeStoringRep[114X).[133X
  
  [33X[0;0YA  mutable digraph may be converted to an immutable digraph that lies in the
  category    [10XIsImmutableDigraph[110X    by   calling   [2XMakeImmutable[102X   ([14XReference:
  MakeImmutable[114X) on the digraph.[133X
  
  [33X[0;0YThe  operation [2XDigraphMutableCopy[102X ([14X3.3-1[114X) can be used to construct a mutable
  copy  of  an  immutable  digraph.  It  is however not possible to convert an
  immutable digraph into a mutable digraph in-place.[133X
  
  [1X3.1-4 IsCayleyDigraph[101X
  
  [33X[1;0Y[29X[2XIsCayleyDigraph[102X[32X Category[133X
  
  [33X[0;0Y[10XIsCayleyDigraph[110X  is  a  subcategory  of  [10XIsDigraph[110X. Digraphs that are Cayley
  digraphs  of a group and that are constructed by the operation [2XCayleyDigraph[102X
  ([14X3.1-12[114X) are constructed in this category, and are always immutable.[133X
  
  [1X3.1-5 IsDigraphWithAdjacencyFunction[101X
  
  [33X[1;0Y[29X[2XIsDigraphWithAdjacencyFunction[102X[32X Category[133X
  
  [33X[0;0Y[10XIsDigraphWithAdjacencyFunction[110X  is a subcategory of [10XIsDigraph[110X. Digraphs that
  are [13Xcreated[113X using an adjacency function are constructed in this category.[133X
  
  [1X3.1-6 DigraphByOutNeighboursType[101X
  
  [33X[1;0Y[29X[2XDigraphByOutNeighboursType[102X[32X global variable[133X
  [33X[1;0Y[29X[2XDigraphFamily[102X[32X family[133X
  
  [33X[0;0YThe  type  of  all digraphs is [10XDigraphByOutNeighboursType[110X. The family of all
  digraphs is [10XDigraphFamily[110X.[133X
  
  [1X3.1-7 Digraph[101X
  
  [33X[1;0Y[29X[2XDigraph[102X( [[3Xfilt[103X, ][3Xobj[103X[, [3Xsource[103X, [3Xrange[103X] ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraph[102X( [[3Xfilt[103X, ][3Xlist[103X, [3Xfunc[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraph[102X( [[3Xfilt[103X, ][3XG[103X, [3Xlist[103X, [3Xact[103X, [3Xadj[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [8Xfor a list (i.e. an adjacency list)[108X
        [33X[0;6Yif  [3Xobj[103X is a list of lists of positive integers in the range from [10X1[110X to
        [10XLength([3Xobj[103X[10X)[110X,  then this function returns the digraph with vertices [22XE ^
        0 =[122X[10X[1 .. Length([3Xobj[103X[10X)][110X, and edges corresponding to the entries of [3Xobj[103X.[133X
  
        [33X[0;6YMore  precisely,  there is an edge from vertex [10Xi[110X to [10Xj[110X if and only if [10Xj[110X
        is  in  [10X[3Xobj[103X[10X[i][110X;  the source of this edge is [10Xi[110X and the range is [10Xj[110X. If [10Xj[110X
        occurs in [10X[3Xobj[103X[10X[i][110X with multiplicity [10Xk[110X, then there are [10Xk[110X edges from [10Xi[110X to
        [10Xj[110X.[133X
  
  [8Xfor three lists[108X
        [33X[0;6Yif  [3Xobj[103X  is  a  duplicate-free list, and [3Xsource[103X and [3Xrange[103X are lists of
        equal  length  consisting  of  positive  integers  in  the  list [10X[1 ..
        Length([3Xobj[103X[10X)][110X, then this function returns a digraph with vertices [22XE ^ 0
        =[122X[10X[1  ..  Length([3Xobj[103X[10X)][110X,  and  [10XLength([3Xsource[103X[10X)[110X edges. For each [10Xi[110X in [10X[1 ..
        Length([3Xsource[103X[10X)][110X  there exists an edge with source vertex [10Xsource[i][110X and
        range  vertex  [10Xrange[i][110X.  See  [2XDigraphSource[102X  ([14X5.2-5[114X) and [2XDigraphRange[102X
        ([14X5.2-5[114X).[133X
  
        [33X[0;6YThe vertices of the digraph will be labelled by the elements of [3Xobj[103X.[133X
  
  [8Xfor an integer, and two lists[108X
        [33X[0;6Yif  [3Xobj[103X  is an integer, and [3Xsource[103X and [3Xrange[103X are lists of equal length
        consisting  of  positive  integers  in  the list [10X[1 .. [3Xobj[103X[10X][110X, then this
        function  returns  a  digraph  with  vertices  [22XE  ^ 0 =[122X[10X[1 .. [3Xobj[103X[10X][110X, and
        [10XLength([3Xsource[103X[10X)[110X edges. For each [10Xi[110X in [10X[1 .. Length([3Xsource[103X[10X)][110X there exists
        an  edge  with  source vertex [10Xsource[i][110X and range vertex [10Xrange[i][110X. See
        [2XDigraphSource[102X ([14X5.2-5[114X) and [2XDigraphRange[102X ([14X5.2-5[114X).[133X
  
  [8Xfor a list and a function[108X
        [33X[0;6Yif  [3Xlist[103X  is a list and [3Xfunc[103X is a function taking 2 arguments that are
        elements  of [3Xlist[103X, and [3Xfunc[103X returns [9Xtrue[109X or [9Xfalse[109X, then this operation
        creates  a  digraph with vertices [10X[1 .. Length([3Xlist[103X[10X)][110X and an edge from
        vertex  [10Xi[110X  to  vertex  [10Xj[110X if and only if [10X[3Xfunc[103X[10X([3Xlist[103X[10X[i], [3Xlist[103X[10X[j])[110X returns
        [9Xtrue[109X.[133X
  
  [8Xfor a group, a list, and two functions[108X
        [33X[0;6YThe arguments will be [3XG, list, act, adj[103X.[133X
  
        [33X[0;6YLet [3XG[103X be a group acting on the objects in [3Xlist[103X via the action [3Xact[103X, and
        let  [3Xadj[103X  be  a function taking two objects from [3Xlist[103X as arguments and
        returning  [10Xtrue[110X or [10Xfalse[110X. The function [3Xadj[103X will describe the adjacency
        between  objects  from [3Xlist[103X, which is invariant under the action of [3XG[103X.
        This  variant  of  the constructor returns a digraph with vertices the
        objects of [3Xlist[103X and directed edges [10X[x, y][110X when [10Xf(x, y)[110X is [10Xtrue[110X.[133X
  
        [33X[0;6YThe  action  of  the  group  [3XG[103X on the objects in [3Xlist[103X is stored in the
        attribute  [2XDigraphGroup[102X  ([14X7.2-10[114X),  and is used to speed up operations
        like [2XDigraphDiameter[102X ([14X5.3-1[114X).[133X
  
  [8Xfor a Grape package graph[108X
        [33X[0;6Yif  [3Xobj[103X is a [5XGrape[105X package graph (i.e. a record for which the function
        [10XIsGraph[110X returns [9Xtrue[109X), then this function returns a digraph isomorphic
        to [3Xobj[103X.[133X
  
  [8Xfor a binary relation[108X
        [33X[0;6Yif  [3Xobj[103X is a binary relation on the points [10X[1 .. n][110X for some posititve
        integer  [22Xn[122X,  then  this  function  returns the digraph defined by [3Xobj[103X.
        Specifically,  this  function  returns a digraph which has [22Xn[122X vertices,
        and  which  has an edge with source [10Xi[110X and range [10Xj[110X if and only if [10X[i,j][110X
        is a pair in the binary relation [3Xobj[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[127X[104X
    [4X[25X>[125X [27X[2, 5, 8, 10], [2, 3, 4, 2, 5, 6, 8, 9, 10], [1],[127X[104X
    [4X[25X>[125X [27X[3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],[127X[104X
    [4X[25X>[125X [27X[1, 5, 9], [1, 2, 7, 8], [3, 5]]);[127X[104X
    [4X[28X<immutable multidigraph with 10 vertices, 38 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(["a", "b", "c"], ["a"], ["b"]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(5, [1, 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, 1]);[127X[104X
    [4X[28X<immutable multidigraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XPetersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,[127X[104X
    [4X[25X>[125X [27Xfunction(x, y) return Intersection(x, y) = []; end);;[127X[104X
    [4X[25Xgap>[125X [27XDigraph(Petersen);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([1 .. 10], ReturnTrue);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 100 edges>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  example  illustrates the uses of the fourth and fifth variants of
  this  constructor. The resulting digraph is a strongly regular graph, and it
  is  actually  the  point  graph  of the van Lint-Schrijver partial geometry,
  [vLS81].  The  algebraic  description  is  taken  from  the seminal paper of
  Calderbank and Kantor [CK86].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := GF(3 ^ 4);[127X[104X
    [4X[28XGF(3^4)[128X[104X
    [4X[25Xgap>[125X [27Xgamma := First(f, x -> Order(x) = 5);[127X[104X
    [4X[28XZ(3^4)^64[128X[104X
    [4X[25Xgap>[125X [27XL := Union([Zero(f)], List(Group(gamma)));[127X[104X
    [4X[28X[ 0*Z(3), Z(3)^0, Z(3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ][128X[104X
    [4X[25Xgap>[125X [27Xomega := Union(List(L, x -> List(Difference(L, [x]), y -> x - y)));[127X[104X
    [4X[28X[ 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, [128X[104X
    [4X[28X  Z(3^4)^16, Z(3^4)^21, Z(3^4)^23, Z(3^4)^24, Z(3^4)^29, Z(3^4)^31, [128X[104X
    [4X[28X  Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4)^45, Z(3^4)^47, Z(3^4)^48, [128X[104X
    [4X[28X  Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(3^4)^63, Z(3^4)^64, [128X[104X
    [4X[28X  Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 ][128X[104X
    [4X[25Xgap>[125X [27Xadj := function(x, y)[127X[104X
    [4X[25X>[125X [27X  return x - y in omega;[127X[104X
    [4X[25X>[125X [27Xend;[127X[104X
    [4X[28Xfunction( x, y ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := Digraph(AsList(f), adj);[127X[104X
    [4X[28X<immutable digraph with 81 vertices, 2430 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgroup := Group(Z(3));[127X[104X
    [4X[28X<group with 1 generators>[128X[104X
    [4X[25Xgap>[125X [27Xact := \*;[127X[104X
    [4X[28X<Operation "*">[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := Digraph(group, List(f), act, adj);[127X[104X
    [4X[28X<immutable digraph with 81 vertices, 2430 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-8 DigraphByAdjacencyMatrix[101X
  
  [33X[1;0Y[29X[2XDigraphByAdjacencyMatrix[102X( [[3Xfilt[103X, ][3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf [3Xlist[103X is the adjacency matrix of a digraph in the sense of [2XAdjacencyMatrix[102X
  ([14X5.2-1[114X), then this operation returns the digraph which is defined by [3Xlist[103X.[133X
  
  [33X[0;0YAlternatively,  if  [3Xlist[103X  is  a  square  boolean matrix, then this operation
  returns  the  digraph with [10XLength([110X[3Xlist[103X[10X)[110X vertices which has the edge [10X[i,j][110X if
  and only if [3Xlist[103X[10X[i][j][110X is [9Xtrue[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphByAdjacencyMatrix([[127X[104X
    [4X[25X>[125X [27X[0, 1, 0, 2, 0],[127X[104X
    [4X[25X>[125X [27X[1, 1, 1, 0, 1],[127X[104X
    [4X[25X>[125X [27X[0, 3, 2, 1, 1],[127X[104X
    [4X[25X>[125X [27X[0, 0, 1, 0, 1],[127X[104X
    [4X[25X>[125X [27X[2, 0, 0, 0, 0]]);[127X[104X
    [4X[28X<immutable multidigraph with 5 vertices, 18 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphByAdjacencyMatrix([[127X[104X
    [4X[25X>[125X [27X[true, false, true],[127X[104X
    [4X[25X>[125X [27X[false, false, true],[127X[104X
    [4X[25X>[125X [27X[false, true, false]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 1, 3 ], [ 3 ], [ 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := DigraphByAdjacencyMatrix(IsMutableDigraph, [127X[104X
    [4X[25X>[125X [27X[[true, false, true],[127X[104X
    [4X[25X>[125X [27X [false, false, true],[127X[104X
    [4X[25X>[125X [27X [false, true, false]]);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 4 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-9 DigraphByEdges[101X
  
  [33X[1;0Y[29X[2XDigraphByEdges[102X( [[3Xfilt[103X, ][3Xlist[103X[, [3Xn[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf  [3Xlist[103X  is  list of pairs of positive integers, then this function returns
  the  digraph  with the minimum number of vertices [10Xm[110X such that its list equal
  [3Xlist[103X.[133X
  
  [33X[0;0YIf  the optional second argument [3Xn[103X is a positive integer with [10X[3Xn[103X[10X >= m[110X (with [10Xm[110X
  defined  as  above),  then this function returns the digraph with [3Xn[103X vertices
  and list [3Xlist[103X.[133X
  
  [33X[0;0YSee [2XDigraphEdges[102X ([14X5.1-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphByEdges([127X[104X
    [4X[25X>[125X [27X[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],[127X[104X
    [4X[25X>[125X [27X [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]]);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphByEdges([127X[104X
    [4X[25X>[125X [27X[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],[127X[104X
    [4X[25X>[125X [27X [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);[127X[104X
    [4X[28X<immutable digraph with 12 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphByEdges(IsMutableDigraph, [127X[104X
    [4X[25X>[125X [27X[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],[127X[104X
    [4X[25X>[125X [27X [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);[127X[104X
    [4X[28X<mutable digraph with 12 vertices, 10 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-10 EdgeOrbitsDigraph[101X
  
  [33X[1;0Y[29X[2XEdgeOrbitsDigraph[102X( [3XG[103X, [3Xedges[103X[, [3Xn[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn immutable digraph.[133X
  
  [33X[0;0YIf  [3XG[103X  is a permutation group, [3Xedges[103X is an edge or list of edges, and [3Xn[103X is a
  non-negative integer such that [3XG[103X fixes [10X[1 .. [3Xn[103X[10X][110X setwise, then this operation
  returns  an immutable digraph with [3Xn[103X vertices and the union of the orbits of
  the edges in [3X edges [103X under the action of the permutation group [3XG[103X. An edge in
  this context is simply a pair of positive integers.[133X
  
  [33X[0;0YIf  the  optional  third  argument  [3Xn[103X is not present, then the largest moved
  point of the permutation group [3XG[103X is used by default.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),[127X[104X
    [4X[25X>[125X [27X                                [[1, 2], [4, 5]], 5);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(digraph);[127X[104X
    [4X[28X[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentativeOutNeighbours(digraph);[127X[104X
    [4X[28X[ [ 2, 4, 5 ], [  ] ][128X[104X
  [4X[32X[104X
  
  [1X3.1-11 DigraphByInNeighbours[101X
  
  [33X[1;0Y[29X[2XDigraphByInNeighbours[102X( [[3Xfilt[103X, ][3Xlist[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphByInNeighbors[102X( [[3Xfilt[103X, ][3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf  [3Xlist[103X  is  a  list  of  lists  of  positive integers list the range [10X[1 ..
  Length([3Xlist[103X[10X)][110X,  then  this function returns the digraph with vertices [22XE^0=[122X[10X[1
  ..  Length([3Xlist[103X[10X)][110X,  and  edges  corresponding  to  the entries of [3Xlist[103X. More
  precisely,  there is an edge with source vertex [10Xi[110X and range vertex [10Xj[110X if [10Xi[110X is
  list [10X[3Xlist[103X[10X[j][110X.[133X
  
  [33X[0;0YIf  [10Xi[110X  occurs  list  [10X[3Xlist[103X[10X[j][110X  with multiplicity [10Xk[110X, then there are [10Xk[110X multiple
  edges from [10Xi[110X to [10Xj[110X.[133X
  
  [33X[0;0YSee [2XInNeighbours[102X ([14X5.2-7[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphByInNeighbours([[127X[104X
    [4X[25X>[125X [27X[2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10],[127X[104X
    [4X[25X>[125X [27X[1], [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],[127X[104X
    [4X[25X>[125X [27X[1, 5, 9], [1, 2, 7, 8], [3, 5]]);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 37 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphByInNeighbours([[2, 3, 2], [1], [1, 2, 3]]);[127X[104X
    [4X[28X<immutable multidigraph with 3 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphByInNeighbours(IsMutableDigraph, [127X[104X
    [4X[25X>[125X [27X                              [[2, 3, 2], [1], [1, 2, 3]]);[127X[104X
    [4X[28X<mutable multidigraph with 3 vertices, 7 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-12 CayleyDigraph[101X
  
  [33X[1;0Y[29X[2XCayleyDigraph[102X( [3XG[103X[, [3Xgens[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn immutable digraph.[133X
  
  [33X[0;0YLet  [3XG[103X  be any group and let [3Xgens[103X be a list of elements of [3XG[103X. This operation
  returns  an immutable digraph that corresponds to the Cayley graph of [3XG[103X with
  respect  [3Xgens[103X. The vertices are the elements of [3XG[103X. There exists an edge from
  the  vertex  [10Xu[110X  to the vertex [10Xv[110X if and only if there exists a generator [10Xg[110X in
  [3Xgens[103X such that [10Xx * g = y[110X.[133X
  
  [33X[0;0YIf  the optional second argument [3Xgens[103X is not present, then the generators of
  [3XG[103X are used by default.[133X
  
  [33X[0;0YThe   digraph   created   by   this   operation   belongs  to  the  category
  [2XIsCayleyDigraph[102X ([14X3.1-4[114X), the group [3XG[103X can be recovered from the digraph using
  [2XGroupOfCayleyDigraph[102X  ([14X5.4-1[114X), and the generators [3Xgens[103X can be obtained using
  [2XGeneratorsOfCayleyDigraph[102X ([14X5.4-2[114X).[133X
  
  [33X[0;0YNote that this function can only return an immutable digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralGroup(8);[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XCayleyDigraph(G);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := DihedralGroup(IsPermGroup, 8);[127X[104X
    [4X[28XGroup([ (1,2,3,4), (2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XCayleyDigraph(G);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 16 edges>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := CayleyDigraph(G, [()]);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XGroupOfCayleyDigraph(digraph) = G;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfCayleyDigraph(digraph);[127X[104X
    [4X[28X[ () ][128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YChanging representations[133X[101X
  
  [1X3.2-1 AsBinaryRelation[101X
  
  [33X[1;0Y[29X[2XAsBinaryRelation[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA binary relation.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph  with  a  positive  number of vertices [22Xn[122X, and no
  multiple  edges, then this operation returns a binary relation on the points
  [10X[1..n][110X.  The pair [10X[i,j][110X is in the binary relation if and only if [10X[i,j][110X is an
  edge in [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[3, 2], [1, 2], [2], [3, 4]]);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsBinaryRelation(D);[127X[104X
    [4X[28XBinary Relation on 4 points[128X[104X
  [4X[32X[104X
  
  [1X3.2-2 AsDigraph[101X
  
  [33X[1;0Y[29X[2XAsDigraph[102X( [[3Xfilt[103X, ][3Xtrans[103X[, [3Xn[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph, or [9Xfail[109X.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf  [3Xtrans[103X is a transformation, and [3Xn[103X is a non-negative integer such that the
  restriction  of [3Xtrans[103X to [10X[1 .. [3Xn[103X[10X][110X defines a transformation of [10X[1 .. [3Xn[103X[10X][110X, then
  [10XAsDigraph[110X  returns  the functional digraph with [3Xn[103X vertices defined by [3Xtrans[103X.
  See [2XIsFunctionalDigraph[102X ([14X6.1-9[114X).[133X
  
  [33X[0;0YSpecifically, the digraph returned by [10XAsDigraph[110X has [3Xn[103X edges: for each vertex
  [10Xx[110X  in  [10X[1  ..  [3Xn[103X[10X][110X, there is a unique edge with source [10Xx[110X; this edge has range
  [10Xx^[3Xtrans[103X[10X[110X.[133X
  
  [33X[0;0YIf  the  optional  second argument [3Xn[103X is not supplied, then the degree of the
  transformation  [3Xtrans[103X  is used by default. If the restriction of [3Xtrans[103X to [10X[1
  .. [3Xn[103X[10X][110X does not define a transformation of [10X[1 .. [3Xn[103X[10X][110X, then [10XAsDigraph([3Xtrans[103X[10X, [3Xn[103X[10X)[110X
  returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);[127X[104X
    [4X[28XTransformation( [ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XAsDigraph(f);[127X[104X
    [4X[28X<immutable functional digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XAsDigraph(f, 4);[127X[104X
    [4X[28X<immutable functional digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27XAsDigraph(f, 5);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xb := BinaryRelationOnPoints([127X[104X
    [4X[25X>[125X [27X[[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);[127X[104X
    [4X[28XBinary Relation on 5 points[128X[104X
    [4X[25Xgap>[125X [27XD := AsDigraph(b);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 11 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.2-3 Graph[101X
  
  [33X[1;0Y[29X[2XGraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA [5XGrape[105X package graph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a mutable or immutable digraph without multiple edges, then
  this operation returns a [5XGrape[105X package graph that is isomorphic to [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  multidigraph, then since [5XGrape[105X does not support multiple
  edges, the multiple edges will be reduced to a single edge in the result. In
  order  words,  for  a  multidigraph  this  operation will return the same as
  [10XGraph(DigraphRemoveAllMultipleEdges([110X[3Xdigraph[103X[10X))[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPetersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,[127X[104X
    [4X[25X>[125X [27Xfunction(x, y) return Intersection(x, y) = []; end);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(Petersen);[127X[104X
    [4X[28Xrec([128X[104X
    [4X[28X  adjacencies := [ [ 3, 5, 8 ] ],[128X[104X
    [4X[28X  group := [128X[104X
    [4X[28X   Group( [ ( 1, 2, 3, 5, 7)( 4, 6, 8, 9,10), ( 2, 4)( 6, 9)( 7,10) [128X[104X
    [4X[28X     ] ),[128X[104X
    [4X[28X  isGraph := true,[128X[104X
    [4X[28X  names := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], [128X[104X
    [4X[28X      [ 2, 4 ], [ 1, 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],[128X[104X
    [4X[28X  order := 10,[128X[104X
    [4X[28X  representatives := [ 1 ],[128X[104X
    [4X[28X  schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27XDigraph(Petersen);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XGraph(last) = Petersen;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.2-4 AsGraph[101X
  
  [33X[1;0Y[29X[2XAsGraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA [5XGrape[105X package graph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph, then this method returns the same as [2XGraph[102X ([14X3.2-3[114X),
  except  that  if  [3Xdigraph[103X  is immutable, then the result will be stored as a
  mutable  attribute of [3Xdigraph[103X. In this latter case, when [10XAsGraph([110X[3Xdigraph[103X[10X)[110X is
  called subsequently, the same [5XGAP[105X object will be returned as before.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2], [3], []]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := AsGraph(D);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 1, 2 ], [ 3 ], [  ] ], group := Group(()), [128X[104X
    [4X[28X  isGraph := true, names := [ 1 .. 3 ], order := 3, [128X[104X
    [4X[28X  representatives := [ 1, 2, 3 ], schreierVector := [ -1, -2, -3 ] )[128X[104X
  [4X[32X[104X
  
  [1X3.2-5 AsTransformation[101X
  
  [33X[1;0Y[29X[2XAsTransformation[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  functional  digraph,  then  [10XAsTransformation[110X returns the
  transformation which is defined by [3Xdigraph[103X. See [2XIsFunctionalDigraph[102X ([14X6.1-9[114X).
  Otherwise, [10XAsTransformation([110X[3Xdigraph[103X[10X)[110X returns [9Xfail[109X.[133X
  
  [33X[0;0YIf    [3Xdigraph[103X    is   a   functional   digraph   with   [22Xn[122X   vertices,   then
  [10XAsTransformation([110X[3Xdigraph[103X[10X)[110X will return the transformation [10Xf[110X of degree at most
  [22Xn[122X  where  for  each [22X1 ≤ i ≤ n[122X, [10Xi ^ f[110X is equal to the unique out-neighbour of
  vertex [10Xi[110X in [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1], [3], [2]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsTransformation(D);[127X[104X
    [4X[28XTransformation( [ 1, 3, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsTransformation(D);[127X[104X
    [4X[28XTransformation( [ 2, 3, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XAsPermutation(last);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 3], [], []]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsTransformation(D);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YNew digraphs from old[133X[101X
  
  [1X3.3-1 DigraphImmutableCopy[101X
  
  [33X[1;0Y[29X[2XDigraphImmutableCopy[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphMutableCopy[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphCopySameMutability[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphCopy[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YEach  of  these operations returns a new copy of [3Xdigraph[103X, of the appropriate
  mutability, retaining none of the attributes or properties of [3Xdigraph[103X.[133X
  
  [33X[0;0Y[10XDigraphCopy[110X is a synonym for [10XDigraphCopySameMutability[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(10);[127X[104X
    [4X[28X<immutable cycle digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XDigraphCopy(D) = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(DigraphCopy(D), D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphMutableCopy(D);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-2 DigraphImmutableCopyIfImmutable[101X
  
  [33X[1;0Y[29X[2XDigraphImmutableCopyIfImmutable[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphImmutableCopyIfMutable[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphMutableCopyIfMutable[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphMutableCopyIfImmutable[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YEach  of these operations returns either the original argument [3Xdigraph[103X, or a
  new  copy  of  [3Xdigraph[103X  of  the  appropriate  mutability,  according  to the
  mutability of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC := CycleDigraph(10);[127X[104X
    [4X[28X<immutable cycle digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphImmutableCopyIfImmutable(C);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(C, D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XC = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphImmutableCopyIfMutable(C);[127X[104X
    [4X[28X<immutable cycle digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(C, D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XC = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopyIfMutable(C);[127X[104X
    [4X[28X<immutable cycle digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsMutableDigraph(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopyIfImmutable(C);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsMutableDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XC := CycleDigraph(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphImmutableCopyIfImmutable(C);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(C, D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XC = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphImmutableCopyIfMutable(C);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(C, D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XC = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopyIfMutable(C);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsMutableDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopyIfImmutable(C);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(C, D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMutableDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-3 InducedSubdigraph[101X
  
  [33X[1;0Y[29X[2XInducedSubdigraph[102X( [3Xdigraph[103X, [3Xverts[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph, and [3Xverts[103X is a subset of the vertices of [3Xdigraph[103X,
  then  this operation returns a digraph constructed from [3Xdigraph[103X by retaining
  precisely  those  vertices  in [3Xverts[103X, and those edges whose source and range
  vertices are both contained in [3Xverts[103X.[133X
  
  [33X[0;0YThe  vertices  of  the  induced  subdigraph  are  [10X[1..Length(verts)][110X but the
  original vertex labels can be accessed via [2XDigraphVertexLabels[102X ([14X5.1-9[114X).[133X
  
  [33X[0;0YIf  [3Xdigraph[103X belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then [3Xdigraph[103X is modified in
  place.  If  [3Xdigraph[103X  belongs  to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), a new immutable
  digraph containing the appropriate vertices and edges is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 1, 2, 3, 4, 4], [1, 3, 4], [3, 1], [1, 1]]);[127X[104X
    [4X[28X<immutable multidigraph with 4 vertices, 13 edges>[128X[104X
    [4X[25Xgap>[125X [27XInducedSubdigraph(D, [1, 3, 4]);[127X[104X
    [4X[28X<immutable multidigraph with 3 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(last);[127X[104X
    [4X[28X[ 1 .. 3 ][128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopy(D);[127X[104X
    [4X[28X<mutable multidigraph with 4 vertices, 13 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := InducedSubdigraph(D, [1, 3, 4]);[127X[104X
    [4X[28X<mutable multidigraph with 3 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-4 ReducedDigraph[101X
  
  [33X[1;0Y[29X[2XReducedDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XReducedDigraphAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThis  function  returns  a  digraph  isomorphic to the subdigraph of [3Xdigraph[103X
  induced  by the set of non-isolated vertices, i.e. the set of those vertices
  of  [3Xdigraph[103X  which  are  the  source  or  range of some edge in [3Xdigraph[103X. See
  [2XInducedSubdigraph[102X ([14X3.3-3[114X).[133X
  
  [33X[0;0YThe  ordering  of the remaining vertices of [3Xdigraph[103X is preserved, as are the
  labels  of the remaining vertices and edges; see [2XDigraphVertexLabels[102X ([14X5.1-9[114X)
  and  [2XDigraphEdgeLabels[102X ([14X5.1-11[114X). This can allow one to match a vertex in the
  reduced digraph to the corresponding vertex in [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the isolated vertices of the mutable digraph [3Xdigraph[103X are removed
  in-place.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2], [], [], [1, 4], []]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XR := ReducedDigraph(D);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(R);[127X[104X
    [4X[28X[ [ 1, 2 ], [  ], [ 1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(R);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabel(R, 3);[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabel(R, 2);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[], [3], [2]]);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XReducedDigraph(D);[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-5 MaximalSymmetricSubdigraph[101X
  
  [33X[1;0Y[29X[2XMaximalSymmetricSubdigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMaximalSymmetricSubdigraphAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XMaximalSymmetricSubdigraphWithoutLoops[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMaximalSymmetricSubdigraphWithoutLoopsAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph, then [10XMaximalSymmetricSubdigraph[110X returns a symmetric
  digraph without multiple edges which has the same vertex set as [3Xdigraph[103X, and
  whose  edge  list  is  formed  from  [3Xdigraph[103X by ignoring the multiplicity of
  edges,  and  by  ignoring edges [10X[u,v][110X for which there does not exist an edge
  [10X[v,u][110X.[133X
  
  [33X[0;0YThe  digraph returned by [10XMaximalSymmetricSubdigraphWithoutLoops[110X is the same,
  except that loops are removed.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  mutable  digraph  [3Xdigraph[103X  is  changed in-place into such a
  digraph described above.[133X
  
  [33X[0;0YSee    [2XIsSymmetricDigraph[102X    ([14X6.1-12[114X),    [2XIsMultiDigraph[102X    ([14X6.1-10[114X),    and
  [2XDigraphHasLoops[102X ([14X6.1-1[114X) for more information.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 2], [1, 3], [4], [3, 1]]);[127X[104X
    [4X[28X<immutable multidigraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnot IsSymmetricDigraph(D) and IsMultiDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XS := MaximalSymmetricSubdigraph(D);[127X[104X
    [4X[28X<immutable symmetric digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(S) and not IsMultiDigraph(S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(S);[127X[104X
    [4X[28X[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XMaximalSymmetricSubdigraph(D);[127X[104X
    [4X[28X<mutable empty digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable empty digraph with 3 vertices>[128X[104X
  [4X[32X[104X
  
  [1X3.3-6 MaximalAntiSymmetricSubdigraph[101X
  
  [33X[1;0Y[29X[2XMaximalAntiSymmetricSubdigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMaximalAntiSymmetricSubdigraphAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then [10XMaximalAntiSymmetricSubdigraph[110X returns an
  anti-symmetric  subdigraph  of  [3Xdigraph[103X  formed by retaining the vertices of
  [3Xdigraph[103X,  discarding  any  duplicate edges, and discarding any edge [10X[i,j][110X of
  [3Xdigraph[103X  where  [10Xi  >  j[110X and the reverse edge [10X[j,i][110X is an edge of [3Xdigraph[103X. In
  other  words,  for every symmetric pair of edges [10X[i,j][110X and [10X[j,i][110X in [3Xdigraph[103X,
  where [10Xi[110X and [10Xj[110X are distinct, it discards the the edge [22X[max(i,j),min(i,j)][122X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise, the mutable digraph [3Xdigraph[103X is changed in-place.[133X
  
  [33X[0;0YSee [2XIsAntisymmetricDigraph[102X ([14X6.1-2[114X) for more information.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 2], [1, 3], [4], [3, 1]]);[127X[104X
    [4X[28X<immutable multidigraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnot IsAntiSymmetricDigraph(D) and IsMultiDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := MaximalAntiSymmetricSubdigraph(D);[127X[104X
    [4X[28X<immutable antisymmetric digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAntiSymmetricDigraph(D) and not IsMultiDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[2], [1]]);[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XMaximalAntiSymmetricSubdigraph(D);[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 1 edge>[128X[104X
  [4X[32X[104X
  
  [1X3.3-7 UndirectedSpanningForest[101X
  
  [33X[1;0Y[29X[2XUndirectedSpanningForest[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XUndirectedSpanningForestAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XUndirectedSpanningTree[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XUndirectedSpanningTreeAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph, or [9Xfail[109X.[133X
  
  [33X[0;0YIf    [3Xdigraph[103X    is   a   digraph   with   at   least   one   vertex,   then
  [10XUndirectedSpanningForest[110X  returns  an undirected spanning forest of [3Xdigraph[103X,
  otherwise   this  attribute  returns  [9Xfail[109X.  See  [2XIsUndirectedSpanningForest[102X
  ([14X4.1-2[114X) for the definition of an undirected spanning forest.[133X
  
  [33X[0;0YIf   [3Xdigraph[103X   is   a   digraph   with   at   least  one  vertex  and  whose
  [2XMaximalSymmetricSubdigraph[102X  ([14X3.3-5[114X)  is  connected  (see  [2XIsConnectedDigraph[102X
  ([14X6.3-3[114X)), then [10XUndirectedSpanningTree[110X returns an undirected spanning tree of
  [3Xdigraph[103X, otherwise this attribute returns [9Xfail[109X. See [2XIsUndirectedSpanningTree[102X
  ([14X4.1-2[114X) for the definition of an undirected spanning tree.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is immutable, then an immutable digraph is returned. Otherwise,
  the  mutable digraph [3Xdigraph[103X is changed in-place into an undirected spanning
  tree of [3Xdigraph[103X.[133X
  
  [33X[0;0YNote  that for an immutable digraph that has known undirected spanning tree,
  the  attribute  [10XUndirectedSpanningTree[110X  returns  the  same  digraph  as  the
  attribute [10XUndirectedSpanningForest[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);[127X[104X
    [4X[28X<immutable multidigraph with 4 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningTree(D);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xforest := UndirectedSpanningForest(D);[127X[104X
    [4X[28X<immutable symmetric digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(forest);[127X[104X
    [4X[28X[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedSpanningForest(D, forest);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphConnectedComponents(forest).comps;[127X[104X
    [4X[28X[ [ 1, 2 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphConnectedComponents(MaximalSymmetricSubdigraph(D)).comps;[127X[104X
    [4X[28X[ [ 1, 2 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningForest(MaximalSymmetricSubdigraph(D))[127X[104X
    [4X[25X>[125X [27X= forest;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(4);[127X[104X
    [4X[28X<immutable complete digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xtree := UndirectedSpanningTree(D);[127X[104X
    [4X[28X<immutable symmetric digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedSpanningTree(D, tree);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xtree = UndirectedSpanningForest(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningForest(EmptyDigraph(0));[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XD := PetersenGraph(IsMutableDigraph);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningTree(D);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 18 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 18 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-8 QuotientDigraph[101X
  
  [33X[1;0Y[29X[2XQuotientDigraph[102X( [3Xdigraph[103X, [3Xp[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a digraph, and [3Xp[103X is a partition of the vertices of [3Xdigraph[103X,
  then  this  operation  returns  a  digraph  constructed  by amalgamating all
  vertices of [3Xdigraph[103X which lie in the same part of [3Xp[103X.[133X
  
  [33X[0;0YA  partition  of  the  vertices  of  [3Xdigraph[103X is a list of non-empty disjoint
  lists,  such  that  the union of all the sub-lists is equal to vertex set of
  [3Xdigraph[103X. In particular, each vertex must appear in precisely one sub-list.[133X
  
  [33X[0;0YThe vertices of [3Xdigraph[103X in part [10Xi[110X of [3Xp[103X will become vertex [10Xi[110X in the quotient,
  and  there  exists  some  edge in [3Xdigraph[103X with source in part [10Xi[110X and range in
  part  [10Xj[110X  if  and  only  if  there is an edge from [10Xi[110X to [10Xj[110X in the quotient. In
  particular, this means that the quotient of a digraph has no multiple edges.
  which was a change introduced in version 1.0.0 of the [5XDigraphs[105X package.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then [3Xdigraph[103X is modified in
  place.  If  [3Xdigraph[103X  belongs  to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), a new immutable
  digraph with the above properties is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 1], [4], [1], [1, 3, 4]]);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(D);[127X[104X
    [4X[28X[ 1 .. 4 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], [128X[104X
    [4X[28X  [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xp := [[1], [2, 4], [3]];[127X[104X
    [4X[28X[ [ 1 ], [ 2, 4 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xquo := QuotientDigraph(D, p);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(quo);[127X[104X
    [4X[28X[ 1 .. 3 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(quo);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XQuotientDigraph(EmptyDigraph(0), []);[127X[104X
    [4X[28X<immutable empty digraph with 0 vertices>[128X[104X
  [4X[32X[104X
  
  [1X3.3-9 DigraphReverse[101X
  
  [33X[1;0Y[29X[2XDigraphReverse[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphReverseAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  reverse of a digraph is the digraph formed by reversing the orientation
  of  each  of its edges, i.e. for every edge [10X[i, j][110X of a digraph, the reverse
  contains the corresponding edge [10X[j, i][110X.[133X
  
  [33X[0;0Y[10XDigraphReverse[110X  returns  the  reverse  of the digraph [3Xdigraph[103X. If [3Xdigraph[103X is
  immutable,  then a new immutable digraph is returned. Otherwise, the mutable
  digraph [3Xdigraph[103X is changed in-place into its reverse.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphReverse(D);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(last);[127X[104X
    [4X[28X[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 4], [1], [4], [3, 4]]);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(DigraphReverse(D));[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphReverse(D);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 3 ], [ 1 ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-10 DigraphDual[101X
  
  [33X[1;0Y[29X[2XDigraphDual[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphDualAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  [13Xdual[113X  of [3Xdigraph[103X has the same vertices as [3Xdigraph[103X, and there is an edge
  in  the  dual  from [10Xi[110X to [10Xj[110X whenever there is no edge from [10Xi[110X to [10Xj[110X in [3Xdigraph[103X.
  The [13Xdual[113X is sometimes called the [13Xcomplement[113X.[133X
  
  [33X[0;0Y[10XDigraphDual[110X  returns  the  dual  of  the  digraph  [3Xdigraph[103X. If [3Xdigraph[103X is an
  immutable  digraph, then a new immutable digraph is returned. Otherwise, the
  mutable digraph [3Xdigraph[103X is changed in-place into its dual.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 3], [], [4, 6], [5], [],[127X[104X
    [4X[25X>[125X [27X[7, 8, 9], [], [], []]);[127X[104X
    [4X[28X<immutable digraph with 9 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphDual(D);[127X[104X
    [4X[28X<immutable digraph with 9 vertices, 73 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphDual(D);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 6 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-11 DigraphSymmetricClosure[101X
  
  [33X[1;0Y[29X[2XDigraphSymmetricClosure[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphSymmetricClosureAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph, then this attribute gives the minimal symmetric
  digraph which has the same vertices and contains all the edges of [3Xdigraph[103X.[133X
  
  [33X[0;0YA  digraph  is  [13Xsymmetric[113X if its adjacency matrix [2XAdjacencyMatrix[102X ([14X5.2-1[114X) is
  symmetric.  For  a digraph with multiple edges this means that there are the
  same number of edges from a vertex [10Xu[110X to a vertex [10Xv[110X as there are from [10Xv[110X to [10Xu[110X;
  see [2XIsSymmetricDigraph[102X ([14X6.1-12[114X).[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  mutable  digraph  [3Xdigraph[103X  is  changed  in-place  into  its
  symmetric closure.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphSymmetricClosure(D);[127X[104X
    [4X[28X<immutable symmetric digraph with 4 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList(OutNeighbours(D), AsSet);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 2], [1]]);[127X[104X
    [4X[28X<immutable multidigraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphSymmetricClosure(D);[127X[104X
    [4X[28X<immutable symmetric multidigraph with 2 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphSymmetricClosure(D);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 6 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-12 DigraphTransitiveClosure[101X
  
  [33X[1;0Y[29X[2XDigraphTransitiveClosure[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphTransitiveClosureAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDigraphReflexiveTransitiveClosure[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphReflexiveTransitiveClosureAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph with no multiple edges, then these attributes return
  the (reflexive) transitive closure of [3Xdigraph[103X.[133X
  
  [33X[0;0YA  digraph  is  [13Xreflexive[113X  if  it  has  a  loop  at  every vertex, and it is
  [13Xtransitive[113X  if  whenever [10X[i,j][110X and [10X[j,k][110X are edges of [3Xdigraph[103X, [10X[i,k][110X is also
  an  edge.  The  [13X(reflexive)  transitive  closure[113X of a digraph [3Xdigraph[103X is the
  least (reflexive and) transitive digraph containing [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  mutable  digraph  [3Xdigraph[103X  is  changed  in-place  into  its
  (reflexive) transitive closure.[133X
  
  [33X[0;0YLet  [22Xn[122X  be  the  number  of  vertices of [3Xdigraph[103X, and let [22Xm[122X be the number of
  edges.  For an arbitrary digraph, these attributes will use a version of the
  Floyd-Warshall   algorithm,   with   complexity   [22XO(n^3)[122X.   However,  for  a
  topologically  sortable  digraph [see [2XDigraphTopologicalSort[102X ([14X5.1-7[114X)], these
  attributes  will  use  methods with complexity [22XO(m + n + m ⋅ n)[122X when this is
  faster.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphFromDiSparse6String(".H`eOWR`Ul^");[127X[104X
    [4X[28X<immutable digraph with 9 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsReflexiveDigraph(D) or IsTransitiveDigraph(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 4, 6 ], [ 1, 3 ], [  ], [ 5 ], [  ], [ 7, 8, 9 ], [  ], [  ], [128X[104X
    [4X[28X  [  ] ][128X[104X
    [4X[25Xgap>[125X [27XT := DigraphTransitiveClosure(D);[127X[104X
    [4X[28X<immutable transitive digraph with 9 vertices, 18 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(T);[127X[104X
    [4X[28X[ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [  ], [ 5 ], [128X[104X
    [4X[28X  [  ], [ 7, 8, 9 ], [  ], [  ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27XRT := DigraphReflexiveTransitiveClosure(D);[127X[104X
    [4X[28X<immutable preorder digraph with 9 vertices, 27 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(RT);[127X[104X
    [4X[28X[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7, 8, 9, 2 ], [ 3 ], [128X[104X
    [4X[28X  [ 5, 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphReflexiveTransitiveClosure(D);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 9 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-13 DigraphTransitiveReduction[101X
  
  [33X[1;0Y[29X[2XDigraphTransitiveReduction[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphTransitiveReductionAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDigraphReflexiveTransitiveReduction[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphReflexiveTransitiveReductionAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a topologically sortable digraph [see [2XDigraphTopologicalSort[102X
  ([14X5.1-7[114X)]   with   no  multiple  edges,  then  these  operations  return  the
  (reflexive) transitive reduction of [3Xdigraph[103X.[133X
  
  [33X[0;0YThe  (reflexive)  transitive reduction of such a digraph is the unique least
  subgraph  such  that  the  (reflexive) transitive closure of the subgraph is
  equal    to   the   (reflexive)   transitive   closure   of   [3Xdigraph[103X   [see
  [2XDigraphReflexiveTransitiveClosure[102X ([14X3.3-12[114X)]. In order words, it is the least
  subgraph of [3Xdigraph[103X which retains the same reachability as [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  mutable  digraph  [3Xdigraph[103X  is  changed  in-place  into  its
  (reflexive) transitive reduction.[133X
  
  [33X[0;0YLet  [22Xn[122X  be  the number of vertices of an arbitrary digraph, and let [22Xm[122X be the
  number of edges. Then these operations use methods with complexity [22XO(m + n +
  m ⋅ n)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 3], [3], [3]]);;[127X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD1 := DigraphReflexiveTransitiveReduction(D);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(D1);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D1);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27XD2 := DigraphTransitiveReduction(D);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(D2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D2);[127X[104X
    [4X[28X[ [ 2, 1 ], [ 3 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphReflexiveTransitiveClosure(D)[127X[104X
    [4X[25X>[125X [27X = DigraphReflexiveTransitiveClosure(D1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphTransitiveClosure(D)[127X[104X
    [4X[25X>[125X [27X = DigraphTransitiveClosure(D2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[1], [1], [1, 2, 3]]);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphReflexiveTransitiveReduction(D);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 2 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-14 DigraphAddVertex[101X
  
  [33X[1;0Y[29X[2XDigraphAddVertex[102X( [3Xdigraph[103X[, [3Xlabel[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  operation returns a digraph constructed from [3Xdigraph[103X by adding a single
  new vertex, and no new edges.[133X
  
  [33X[0;0YIf  the  optional second argument [3Xlabel[103X is a [5XGAP[105X object, then the new vertex
  will be labelled [3Xlabel[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs  to  [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the vertex is added
  directly  to  [3Xdigraph[103X.  If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), an
  immutable copy of [3Xdigraph[103X with the additional vertex is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(3);[127X[104X
    [4X[28X<immutable complete digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertex(D);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(new);[127X[104X
    [4X[28X[ 1 .. 4 ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertex(D, Group([(1, 2)]));[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ 1, 2, 3, Group([ (1,2) ]) ][128X[104X
    [4X[25Xgap>[125X [27XD := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertex(D);[127X[104X
    [4X[28X<mutable digraph with 6 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-15 DigraphAddVertices[101X
  
  [33X[1;0Y[29X[2XDigraphAddVertices[102X( [3Xdigraph[103X, [3Xm[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphAddVertices[102X( [3Xdigraph[103X, [3Xlabels[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFor  a  non-negative integer [3Xm[103X, this operation returns a digraph constructed
  from [3Xdigraph[103X by adding [3Xm[103X new vertices.[133X
  
  [33X[0;0YOtherwise,  if  [3Xlabels[103X  is  a  list  consisting  of [10Xk[110X [5XGAP[105X objects, then this
  operation  returns  a  digraph  constructed  from  [3Xdigraph[103X  by  adding [10Xk[110X new
  vertices, which are labelled according to this list.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the vertices are added
  directly  to  [3Xdigraph[103X,  which  is  changed  in-place.  If [3Xdigraph[103X belongs to
  [2XIsImmutableDigraph[102X  ([14X3.1-3[114X),  then  the  result  is returned as an immutable
  digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(3);[127X[104X
    [4X[28X<immutable complete digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertices(D, 3);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(new);[127X[104X
    [4X[28X[ 1 .. 6 ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertices(D, [Group([(1, 2)]), "d"]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ 1, 2, 3, Group([ (1,2) ]), "d" ][128X[104X
    [4X[25Xgap>[125X [27XDigraphAddVertices(D, 0) = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertices(D, 4);[127X[104X
    [4X[28X<mutable digraph with 9 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-16 DigraphAddEdge[101X
  
  [33X[1;0Y[29X[2XDigraphAddEdge[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphAddEdge[102X( [3Xdigraph[103X, [3Xsrc[103X, [3Xran[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xedge[103X  is  a  pair of vertices of [3Xdigraph[103X, or [3Xsrc[103X and [3Xran[103X are vertices of
  [3Xdigraph[103X,  then  this operation returns a digraph constructed from [3Xdigraph[103X by
  adding a new edge with source [3Xedge[103X[10X[1][110X [[3Xsrc[103X] and range [3Xedge[103X[10X[2][110X [[3Xran[103X].[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs  to  [2XIsMutableDigraph[102X  ([14X3.1-2[114X),  then the edge is added
  directly  to [3Xdigraph[103X. If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then
  an immutable copy of [3Xdigraph[103X with the additional edge is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD1 := Digraph([[2], [3], []]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D1);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XD2 := DigraphAddEdge(D1, [3, 1]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD3 := DigraphAddEdge(D2, [2, 3]);[127X[104X
    [4X[28X<immutable multidigraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D3);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 4);[127X[104X
    [4X[28X<mutable digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddEdge(D, [1, 3]);[127X[104X
    [4X[28X<mutable digraph with 4 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(new);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-17 DigraphAddEdgeOrbit[101X
  
  [33X[1;0Y[29X[2XDigraphAddEdgeOrbit[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA new digraph.[133X
  
  [33X[0;0YThis  operation  returns  a  new digraph with the same vertices and edges as
  [3Xdigraph[103X  and  with additional edges consisting of the orbit of the edge [3Xedge[103X
  under the action of the [2XDigraphGroup[102X ([14X7.2-10[114X) of [3Xdigraph[103X. If [3Xedge[103X is already
  an edge in [3Xdigraph[103X, then [3Xdigraph[103X is returned unchanged. The argument [3Xdigraph[103X
  must be an immutable digraph.[133X
  
  [33X[0;0YAn edge is simply a pair of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := CayleyDigraph(DihedralGroup(8));[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 32 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr1);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], [128X[104X
    [4X[28X  [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [128X[104X
    [4X[28X  [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], [128X[104X
    [4X[28X  [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], [128X[104X
    [4X[28X  [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], [128X[104X
    [4X[28X  [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], [128X[104X
    [4X[28X  [ 8, 5 ], [ 8, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr3 = gr1;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-18 DigraphAddEdges[101X
  
  [33X[1;0Y[29X[2XDigraphAddEdges[102X( [3Xdigraph[103X, [3Xedges[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xedges[103X  is  a (possibly empty) list of pairs of vertices of [3Xdigraph[103X, then
  this  operation  returns  a  digraph  constructed from [3Xdigraph[103X by adding the
  edges  specified  by  [3Xedges[103X.  More precisely, for every [10Xedge[110X in [3Xedges[103X, a new
  edge will be added with source [10Xedge[1][110X and range [10Xedges[2][110X.[133X
  
  [33X[0;0YIf an edge is included in [3Xedges[103X with multiplicity [10Xk[110X, then it will be added [10Xk[110X
  times.  If  [3Xdigraph[103X  belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the edges are
  added directly to [3Xdigraph[103X. If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X),
  then the result is returned as an immutable digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfunc := function(n)[127X[104X
    [4X[25X>[125X [27X local source, range, i;[127X[104X
    [4X[25X>[125X [27X source := [];[127X[104X
    [4X[25X>[125X [27X range  := [];[127X[104X
    [4X[25X>[125X [27X for i in [1 .. n - 2] do[127X[104X
    [4X[25X>[125X [27X   Add(source, i);[127X[104X
    [4X[25X>[125X [27X   Add(range, i + 1);[127X[104X
    [4X[25X>[125X [27X od;[127X[104X
    [4X[25X>[125X [27X return Digraph(n, source, range);[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27XD := func(1024);[127X[104X
    [4X[28X<immutable digraph with 1024 vertices, 1022 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddEdges(D,[127X[104X
    [4X[25X>[125X [27X[[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);[127X[104X
    [4X[28X<immutable multidigraph with 1024 vertices, 1026 edges>[128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XD2 := DigraphMutableCopy(func(1024));[127X[104X
    [4X[28X<mutable digraph with 1024 vertices, 1022 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddEdges(D2,[127X[104X
    [4X[25X>[125X [27X[[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);[127X[104X
    [4X[28X<mutable multidigraph with 1024 vertices, 1026 edges>[128X[104X
    [4X[25Xgap>[125X [27XD2 = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-19 DigraphRemoveVertex[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveVertex[102X( [3Xdigraph[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xv[103X  is  a  vertex  of  [3Xdigraph[103X,  then  this  operation  returns a digraph
  constructed  from  [3Xdigraph[103X  by  removing vertex [3Xv[103X, along with any edge whose
  source or range vertex is [3Xv[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  has  [10Xn[110X  vertices, then the vertices of the returned digraph are
  [10X[1..n-1][110X,  but  the  original labels can be accessed via [2XDigraphVertexLabels[102X
  ([14X5.1-9[114X).[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the vertex is removed
  directly  from [3Xdigraph[103X. If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), an
  immutable copy of [3Xdigraph[103X without the vertex is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph(["a", "b", "c"],[127X[104X
    [4X[25X>[125X [27X                 ["a", "a", "b", "c", "c"],[127X[104X
    [4X[25X>[125X [27X                 ["b", "c", "a", "a", "c"]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(D);[127X[104X
    [4X[28X[ "a", "b", "c" ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 3, 1 ], [ 3, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveVertex(D, 2);[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ "a", "c" ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 5);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveVertex(D, 1);[127X[104X
    [4X[28X<mutable digraph with 4 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(D);[127X[104X
    [4X[28X[ 2, 3, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-20 DigraphRemoveVertices[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveVertices[102X( [3Xdigraph[103X, [3Xverts[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xverts[103X  is a (possibly empty) duplicate-free list of vertices of [3Xdigraph[103X,
  then  this  operation returns a digraph constructed from [3Xdigraph[103X by removing
  every  vertex  in [3Xverts[103X, along with any edge whose source or range vertex is
  in [3Xverts[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  has  [10Xn[110X vertices, then the vertices of the new digraph are [10X[1 ..
  n-Length([3Xverts[103X[10X)][110X,   but   the   original   labels   can   be   accessed  via
  [2XDigraphVertexLabels[102X ([14X5.1-9[114X).[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs  to  [2XIsMutableDigraph[102X  ([14X3.1-2[114X),  then  the vertices are
  removed  directly  from  [3Xdigraph[103X.  If  [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X
  ([14X3.1-3[114X), an immutable copy of [3Xdigraph[103X without the vertices is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XSetDigraphVertexLabels(D, ["a", "b", "c", "d", "e"]);[127X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveVertices(D, [2, 4]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ "a", "c", "e" ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 5);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveVertices(D, [1, 3]);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(D);[127X[104X
    [4X[28X[ 2, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-21 DigraphRemoveEdge[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveEdge[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphRemoveEdge[102X( [3Xdigraph[103X, [3Xsrc[103X, [3Xran[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph  with  no  multiple  edges and [3Xedge[103X is a pair of
  vertices  of  [3Xdigraph[103X,  or  [3Xsrc[103X  and  [3Xran[103X are vertices of [3Xdigraph[103X, then this
  operation  returns  a  digraph constructed from [3Xdigraph[103X by removing the edge
  specified by [3Xedge[103X or [3X[src, ran][103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs  to  [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the edge is removed
  directly  from [3Xdigraph[103X. If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), an
  immutable copy of [3Xdigraph[103X without the edge is returned.[133X
  
  [33X[0;0YNote  that if [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then a new copy
  of  [3Xdigraph[103X  will  be returned even if [3Xedge[103X or [3X[src, ran][103X does not define an
  edge of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(250000);[127X[104X
    [4X[28X<immutable cycle digraph with 250000 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphRemoveEdge(D, [250000, 1]);[127X[104X
    [4X[28X<immutable digraph with 250000 vertices, 249999 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveEdge(D, [25000, 2]);;[127X[104X
    [4X[25Xgap>[125X [27Xnew = D;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(new, D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopy(D);;[127X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveEdge(D, 2500, 2);;[127X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj(new, D);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-22 DigraphRemoveEdgeOrbit[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveEdgeOrbit[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA new digraph.[133X
  
  [33X[0;0YThis  operation  returns a new digraph with the same vertices as [3Xdigraph[103X and
  with  the  orbit  of  the  edge  [3Xedge[103X  (under the action of the [2XDigraphGroup[102X
  ([14X7.2-10[114X)  of  [3Xdigraph[103X)  removed.  If  [3Xedge[103X  is  not an edge in [3Xdigraph[103X, then
  [3Xdigraph[103X  is  returned  unchanged.  The argument [3Xdigraph[103X must be an immutable
  digraph.[133X
  
  [33X[0;0YAn edge is simply a pair of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := CayleyDigraph(DihedralGroup(8));[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 32 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr1);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], [128X[104X
    [4X[28X  [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [128X[104X
    [4X[28X  [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], [128X[104X
    [4X[28X  [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], [128X[104X
    [4X[28X  [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], [128X[104X
    [4X[28X  [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], [128X[104X
    [4X[28X  [ 8, 5 ], [ 8, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr3 = gr1;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-23 DigraphRemoveEdges[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveEdges[102X( [3Xdigraph[103X, [3Xedges[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf one of the following holds:[133X
  
  [30X    [33X[0;6Y[3Xdigraph[103X  is  a  digraph with no multiple edges, and [3Xedges[103X is a list of
        pairs of vertices of [3Xdigraph[103X, or[133X
  
  [30X    [33X[0;6Y[3Xdigraph[103X is a digraph and [3Xedges[103X is an empty list[133X
  
  [33X[0;0Ythen  this  operation returns a digraph constructed from [3Xdigraph[103X by removing
  all of the edges specified by [3Xedges[103X (see [2XDigraphRemoveEdge[102X ([14X3.3-21[114X)).[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs  to  [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the edge is removed
  directly from [3Xdigraph[103X. If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), the
  edge  is  removed  from an immutable copy of [3Xdigraph[103X and this new digraph is
  returned.[133X
  
  [33X[0;0YNote  that if [3Xedges[103X is empty, then this operation will always return [3Xdigraph[103X
  rather  than a copy. Also, if any element of [3Xedges[103X is invalid (i.e. does not
  define an edge of [3Xdigraph[103X) then that element will simply be ignored.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(250000);[127X[104X
    [4X[28X<immutable cycle digraph with 250000 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphRemoveEdges(D, [[250000, 1]]);[127X[104X
    [4X[28X<immutable digraph with 250000 vertices, 249999 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopy(D);[127X[104X
    [4X[28X<mutable digraph with 250000 vertices, 249999 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveEdges(D, [[1, 2], [2, 3], [3, 100]]);[127X[104X
    [4X[28X<mutable digraph with 250000 vertices, 249997 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew = D;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-24 DigraphRemoveLoops[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveLoops[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphRemoveLoopsAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph, then this operation returns a digraph constructed
  from [3Xdigraph[103X by removing every loop. A loop is an edge with equal source and
  range.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise, the loops are removed from the mutable digraph [3Xdigraph[103X in-place.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphRemoveLoops(D);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[1, 2], [1]]);[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphRemoveLoops(D);[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-25 DigraphRemoveAllMultipleEdges[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveAllMultipleEdges[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphRemoveAllMultipleEdgesAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph, then this operation returns a digraph constructed
  from  [3Xdigraph[103X  by  removing  all  multiple  edges. The result is the largest
  subdigraph of [3Xdigraph[103X which does not contain multiple edges.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  multiple  edges  of the mutable digraph [3Xdigraph[103X are removed
  in-place.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);[127X[104X
    [4X[28X<immutable multidigraph with 3 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XD2 := DigraphRemoveAllMultipleEdges(D1);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D2);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[2, 2], [1]]);[127X[104X
    [4X[28X<mutable multidigraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphRemoveAllMultipleEdges(D);[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 2 vertices, 2 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-26 DigraphReverseEdges[101X
  
  [33X[1;0Y[29X[2XDigraphReverseEdges[102X( [3Xdigraph[103X, [3Xedges[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphReverseEdge[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphReverseEdge[102X( [3Xdigraph[103X, [3Xsrc[103X, [3Xran[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph without multiple edges, and [3Xedges[103X is a list of pairs
  of vertices of [3Xdigraph[103X (the entries of each pair corresponding to the source
  and  the range of an edge, respectively), then [10XDigraphReverseEdges[110X returns a
  digraph  constructed from [3Xdigraph[103X by reversing the orientation of every edge
  specified   by   [3Xedges[103X.   If   only   one  edge  is  to  be  reversed,  then
  [10XDigraphReverseEdge[110X  can  be  used instead. In this case, the second argument
  should  just  be  a  single  vertex-pair,  or the second and third arguments
  should be the source and range of an edge respectively.[133X
  
  [33X[0;0YNote  that  even  though  [3Xdigraph[103X cannot have multiple edges, the output may
  have multiple edges.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the edges are reversed
  in  [3Xdigraph[103X.  If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), an immutable
  copy of [3Xdigraph[103X with the specified edges reversed is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphFromDiSparse6String(".Tg?i@s?t_e?_qEsC");[127X[104X
    [4X[28X<immutable digraph with 21 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(D);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], [128X[104X
    [4X[28X  [ 11, 2 ], [ 21, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphReverseEdge(D, [7, 19]);[127X[104X
    [4X[28X<immutable digraph with 21 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(new);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, 1 ], [ 11, 2 ], [128X[104X
    [4X[28X  [ 19, 7 ], [ 21, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD2 := DigraphMutableCopy(new);;[127X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphReverseEdges(D2, [[19, 7]]);;[127X[104X
    [4X[25Xgap>[125X [27XD2 = new;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD = new;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-27 DigraphDisjointUnion[101X
  
  [33X[1;0Y[29X[2XDigraphDisjointUnion[102X( [3XD1[103X, [3XD2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphDisjointUnion[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn  the  first form, if [3XD1[103X, [3XD2[103X, etc. are digraphs, then [10XDigraphDisjointUnion[110X
  returns  their  disjoint  union.  In the second form, if [3Xlist[103X is a non-empty
  list  of  digraphs,  then [10XDigraphDisjointUnion[110X returns the disjoint union of
  the digraphs contained in the list.[133X
  
  [33X[0;0YFor  a  disjoint  union of digraphs, the vertex set is the disjoint union of
  the vertex sets, and the edge list is the disjoint union of the edge lists.[133X
  
  [33X[0;0YMore  specifically,  for  a collection of digraphs [3XD1[103X, [3XD2[103X, [10X...[110X, the disjoint
  union   with   have  [10XDigraphNrVertices([110X[3XD1[103X[10X)[110X  [10X+[110X  [10XDigraphNrVertices([110X[3XD2[103X[10X)[110X  [10X+[110X  [10X...[110X
  vertices. The edges of [3XD1[103X will remain unchanged, whilst the edges of the [10Xi[110Xth
  digraph,  [3XD[103X[10X[i][110X,  will  be changed so that they belong to the vertices of the
  disjoint  union corresponding to [3XD[103X[10X[i][110X. In particular, the edges of [3XD[103X[10X[i][110X will
  have  their  source  and  range  increased  by [10XDigraphNrVertices([110X[3XD1[103X[10X)[110X [10X+[110X [10X...[110X [10X+[110X
  [10XDigraphNrVertices([110X[3XD[103X[10X[i-1])[110X.[133X
  
  [33X[0;0YNote that previously set [2XDigraphVertexLabels[102X ([14X5.1-9[114X) will be lost.[133X
  
  [33X[0;0YIf  the first digraph [3XD1[103X [[3Xlist[1][103X] belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then
  [3XD1[103X  [[3Xlist[1][103X]  is  modified in place to contain the appropriate vertices and
  edges.  If  [3Xdigraph[103X  belongs  to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), a new immutable
  digraph containing the appropriate vertices and edges is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD1 := CycleDigraph(3);[127X[104X
    [4X[28X<immutable cycle digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D1);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD2 := CompleteDigraph(3);[127X[104X
    [4X[28X<immutable complete digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D2);[127X[104X
    [4X[28X[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xunion := DigraphDisjointUnion(D1, D2);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(union);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-28 DigraphEdgeUnion[101X
  
  [33X[1;0Y[29X[2XDigraphEdgeUnion[102X( [3XD1[103X, [3XD2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphEdgeUnion[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn  the  first  form,  if  [3XD1[103X,  [3XD2[103X, etc. are digraphs, then [10XDigraphEdgeUnion[110X
  returns their edge union. In the second form, if [3Xlist[103X is a non-empty list of
  digraphs,  then  [10XDigraphEdgeUnion[110X  returns  the  edge  union of the digraphs
  contained in the list.[133X
  
  [33X[0;0YThe vertex set of the edge union of a collection of digraphs is the [13Xunion[113X of
  the vertex sets, whilst the edge list of the edge union is the [13Xconcatenation[113X
  of  the edge lists. The number of vertices of the edge union is equal to the
  [13Xmaximum[113X  number  of  vertices  of  one of the digraphs, whilst the number of
  edges  of  the  edge union will equal the [13Xsum[113X of the number of edges of each
  digraph.[133X
  
  [33X[0;0YNote that previously set [2XDigraphVertexLabels[102X ([14X5.1-9[114X) will be lost.[133X
  
  [33X[0;0YIf  the first digraph [3XD1[103X [[3Xlist[1][103X] belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then
  [3XD1[103X  [[3Xlist[1][103X]  is  modified in place to contain the appropriate vertices and
  edges.  If  [3Xdigraph[103X  belongs  to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), a new immutable
  digraph containing the appropriate vertices and edges is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(10);[127X[104X
    [4X[28X<immutable cycle digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdgeUnion(D, D);[127X[104X
    [4X[28X<immutable multidigraph with 10 vertices, 20 edges>[128X[104X
    [4X[25Xgap>[125X [27XD1 := Digraph([[2], [1]]);[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XD2 := Digraph([[2, 3], [2], [1]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27Xunion := DigraphEdgeUnion(D1, D2);[127X[104X
    [4X[28X<immutable multidigraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(union);[127X[104X
    [4X[28X[ [ 2, 2, 3 ], [ 1, 2 ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xunion = DigraphByEdges([127X[104X
    [4X[25X>[125X [27XConcatenation(DigraphEdges(D1), DigraphEdges(D2)));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-29 DigraphJoin[101X
  
  [33X[1;0Y[29X[2XDigraphJoin[102X( [3XD1[103X, [3XD2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphJoin[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn  the  first  form, if [3XD1[103X, [3XD2[103X, etc. are digraphs, then [10XDigraphJoin[110X returns
  their  join.  In  the  second form, if [3Xlist[103X is a non-empty list of digraphs,
  then [10XDigraphJoin[110X returns the join of the digraphs contained in the list.[133X
  
  [33X[0;0YThe  join  of a collection of digraphs [3XD1[103X, [3XD2[103X, [10X...[110X is formed by first taking
  the  [2XDigraphDisjointUnion[102X ([14X3.3-27[114X) of the collection. In the disjoint union,
  if  [22Xi ≠ j[122X then there are no edges between vertices corresponding to digraphs
  [3XD[103X[10X[i][110X  and  [3XD[103X[10X[j][110X in the collection; the join is created by including all such
  edges.[133X
  
  [33X[0;0YFor example, the join of two empty digraphs is a complete bipartite digraph.[133X
  
  [33X[0;0YNote that previously set [2XDigraphVertexLabels[102X ([14X5.1-9[114X) will be lost.[133X
  
  [33X[0;0YIf  the first digraph [3XD1[103X [[3Xlist[1][103X] belongs to [2XIsMutableDigraph[102X ([14X3.1-2[114X), then
  [3XD1[103X  [[3Xlist[1][103X]  is  modified in place to contain the appropriate vertices and
  edges.  If  [3Xdigraph[103X  belongs  to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), a new immutable
  digraph containing the appropriate vertices and edges is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(3);[127X[104X
    [4X[28X<immutable complete digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteDigraph(DigraphJoin(D, D));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD2 := CycleDigraph(3);[127X[104X
    [4X[28X<immutable cycle digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XDigraphJoin(D, D2);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 27 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-30 DigraphCartesianProduct[101X
  
  [33X[1;0Y[29X[2XDigraphCartesianProduct[102X( [3Xgr1[103X, [3Xgr2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphCartesianProduct[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn    the   first   form,   if   [3Xgr1[103X,   [3Xgr2[103X,   etc.   are   digraphs,   then
  [10XDigraphCartesianProduct[110X  returns  a  digraph  isomorphic  to their Cartesian
  product.[133X
  
  [33X[0;0YIn  the  second  form,  if  [3Xlist[103X  is  a  non-empty  list  of  digraphs, then
  [10XDigraphCartesianProduct[110X  returns  a  digraph  isomorphic  to  the  Cartesian
  product of the digraphs contained in the list.[133X
  
  [33X[0;0YMathematically, the Cartesian product of two digraphs [10XG[110X, [10XH[110X is a digraph with
  vertex set [10XCartesian(DigraphVertices(G), DigraphVertices(H))[110X such that there
  is  an edge from [10X[u, u'][110X to [10X[v, v'][110X iff [10X u = v [110X and there is an edge from [10Xu'[110X
  to [10Xv'[110X in [10XH[110X or [10X u' = v'[110X and there is an edge from [10Xu[110X to [10Xv[110X in [10XG[110X.[133X
  
  [33X[0;0YDue  to technical reasons, the digraph [10XD[110X returned by [10XDigraphCartesianProduct[110X
  has vertex set [10X[1 .. DigraphNrVertices(G)*DigraphNrVertices(H)][110X instead, and
  the   exact   method   of  encoding  pairs  of  vertices  into  integers  is
  implementation  specific.  The original vertex pair can be somewhat regained
  by   using   [2XDigraphCartesianProductProjections[102X   ([14X3.3-32[114X).   In   addition,
  [2XDigraphVertexLabels[102X ([14X5.1-9[114X) are preserved: if vertex pair [10X[u,u'][110X was encoded
  as  [10Xi[110X  then the vertex label of [10Xi[110X will be the pair of vertex labels of [10Xu[110X and
  [10Xu'[110X      i.e.     [10XDigraphVertexLabel(D,i)     =     [DigraphVertexLabel(G,u),
  DigraphVertexLabel(H,u')][110X.[133X
  
  [33X[0;0YAs  the  Cartesian  product  is  associative,  the  Cartesian  product  of a
  collection of digraphs [3Xgr1[103X, [3Xgr2[103X, [10X...[110X is computed in the obvious fashion.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := ChainDigraph(4);[127X[104X
    [4X[28X<immutable chain digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := CycleDigraph(3);[127X[104X
    [4X[28X<immutable cycle digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphCartesianProduct(gr, gr2);[127X[104X
    [4X[28X<immutable digraph with 12 vertices, 21 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(gr3, [127X[104X
    [4X[25X>[125X [27XDigraph([[2, 5], [3, 6], [4, 7], [8], [127X[104X
    [4X[25X>[125X [27X         [6, 9], [7, 10], [8, 11], [12],[127X[104X
    [4X[25X>[125X [27X         [10, 1], [11, 2], [12, 3], [4]]));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-31 DigraphDirectProduct[101X
  
  [33X[1;0Y[29X[2XDigraphDirectProduct[102X( [3Xgr1[103X, [3Xgr2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphDirectProduct[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn the first form, if [3Xgr1[103X, [3Xgr2[103X, etc. are digraphs, then [10XDigraphDirectProduct[110X
  returns a digraph isomorphic to their direct product.[133X
  
  [33X[0;0YIn  the  second  form,  if  [3Xlist[103X  is  a  non-empty  list  of  digraphs, then
  [10XDigraphDirectProduct[110X  returns  a digraph isomorphic to the direct product of
  the digraphs contained in the list.[133X
  
  [33X[0;0YMathematically,  the  direct  product of two digraphs [10XG[110X, [10XH[110X is a digraph with
  vertex set [10XCartesian(DigraphVertices(G), DigraphVertices(H))[110X such that there
  is an edge from [10X[u, u'][110X to [10X[v, v'][110X iff there is an edge from [10Xu[110X to [10Xv[110X in [10XG[110X and
  an edge from [10Xu'[110X to [10Xv'[110X in [10XH[110X.[133X
  
  [33X[0;0YDue to technical reasons, the digraph [10XD[110X returned by [10XDigraphDirectProduct[110X has
  vertex set [10X[1 .. DigraphNrVertices(G)*DigraphNrVertices(H)][110X instead, and the
  exact  method  of encoding pairs of vertices into integers is implementation
  specific.  The  original  vertex  pair  can  be  somewhat  regained by using
  [2XDigraphDirectProductProjections[102X  ([14X3.3-33[114X).  In  addition [2XDigraphVertexLabels[102X
  ([14X5.1-9[114X)  are  preserved:  if  vertex  pair  [10X[u,u'][110X was encoded as [10Xi[110X then the
  vertex  label  of  [10Xi[110X  will  be  the  pair  of vertex labels of [10Xu[110X and [10Xu'[110X i.e.
  [10XDigraphVertexLabel(D,i)              =             [DigraphVertexLabel(G,u),
  DigraphVertexLabel(H,u')][110X.[133X
  
  [33X[0;0YAs  the direct product is associative, the direct product of a collection of
  digraphs [3Xgr1[103X, [3Xgr2[103X, [10X...[110X is computed in the obvious fashion.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := ChainDigraph(4);[127X[104X
    [4X[28X<immutable chain digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := CycleDigraph(3);[127X[104X
    [4X[28X<immutable cycle digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphDirectProduct(gr, gr2);[127X[104X
    [4X[28X<immutable digraph with 12 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(gr3, [127X[104X
    [4X[25X>[125X [27XDigraph([[6], [7], [8], [], [127X[104X
    [4X[25X>[125X [27X         [10], [11], [12], [],[127X[104X
    [4X[25X>[125X [27X         [2], [3], [4], []]));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-32 DigraphCartesianProductProjections[101X
  
  [33X[1;0Y[29X[2XDigraphCartesianProductProjections[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0YIf    [3Xdigraph[103X    is    a    Cartesian    product    digraph,    [10Xdigraph    =
  DigraphCartesianProduct(gr_1,         gr_2,        ...        )[110X,        then
  [10XDigraphCartesianProductProjections[110X  returns a list [10Xproj[110X such that [10Xproj[i][110X is
  the projection onto the [10Xi[110X-th coordinate of the product.[133X
  
  [33X[0;0YA  projection is an idempotent endomorphism of [3Xdigraph[103X. If [10Xgr1, gr2, ...[110X are
  all  loopless  digraphs, then the induced subdigraph of [3Xdigraph[103X on the image
  of [10Xproj[i][110X is isomorphic to [10Xgr_i[110X.[133X
  
  [33X[0;0YCurrently  this attribute is only set upon creating an immutable digraph via
  [10XDigraphCartesianProduct[110X  and  there  is  no  way of calculating it for other
  digraphs.[133X
  
  [33X[0;0YFor more information see [2XDigraphCartesianProduct[102X ([14X3.3-30[114X)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphCartesianProduct(ChainDigraph(3), CycleDigraph(4),[127X[104X
    [4X[25X>[125X [27XDigraph([[2], [2]]));;[127X[104X
    [4X[25Xgap>[125X [27XHasDigraphCartesianProductProjections(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xproj := DigraphCartesianProductProjections(D);; Length(proj);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XIsIdempotent(proj[2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRankOfTransformation(proj[3]);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XS := ImageSetOfTransformation(proj[2]);;[127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(CycleDigraph(4), InducedSubdigraph(D, S));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-33 DigraphDirectProductProjections[101X
  
  [33X[1;0Y[29X[2XDigraphDirectProductProjections[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a direct product digraph, [10Xdigraph = DigraphDirectProduct(gr_1,
  gr_2,  ...  )[110X, then [10XDigraphDirectProductProjections[110X returns a list [10Xproj[110X such
  that [10Xproj[i][110X is the projection onto the [10Xi[110X-th coordinate of the product.[133X
  
  [33X[0;0YA  projection is an idempotent endomorphism of [3Xdigraph[103X. If [10Xgr1, gr2, ...[110X are
  all loopless digraphs, then the image of [3Xdigraph[103X under [10Xproj[i][110X is isomorphic
  to [10Xgr_i[110X.[133X
  
  [33X[0;0YCurrently  this attribute is only set upon creating an immutable digraph via
  [10XDigraphDirectProduct[110X  and  there  is  no  way  of  calculating  it for other
  digraphs.[133X
  
  [33X[0;0YFor more information, see [2XDigraphDirectProduct[102X ([14X3.3-31[114X)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphDirectProduct(ChainDigraph(3), CycleDigraph(4),[127X[104X
    [4X[25X>[125X [27XDigraph([[2], [2]]));;[127X[104X
    [4X[25Xgap>[125X [27XHasDigraphDirectProductProjections(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xproj := DigraphDirectProductProjections(D);; Length(proj);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XIsIdempotent(proj[2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRankOfTransformation(proj[3]);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XP := DigraphRemoveAllMultipleEdges([127X[104X
    [4X[25X>[125X [27XReducedDigraph(OnDigraphs(D, proj[2])));; [127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(CycleDigraph(4), P);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-34 LineDigraph[101X
  
  [33X[1;0Y[29X[2XLineDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XEdgeDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YGiven  a  digraph  [3Xdigraph[103X,  the  operation  returns the digraph obtained by
  associating  a vertex with each edge of [3Xdigraph[103X, and creating an edge from a
  vertex  [9Xv[109X  to  a  vertex  [9Xu[109X  if  and only if the terminal vertex of the edge
  associated with [9Xv[109X is the start vertex of the edge associated with [9Xu[109X.[133X
  
  [33X[0;0YNote  that  the  returned digraph is always a new immutable digraph, and the
  argument [3Xdigraph[103X is never modified.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLineDigraph(CompleteDigraph(3));[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XLineDigraph(ChainDigraph(3));[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 1 edge>[128X[104X
  [4X[32X[104X
  
  [1X3.3-35 LineUndirectedDigraph[101X
  
  [33X[1;0Y[29X[2XLineUndirectedDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XEdgeUndirectedDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YGiven  a  symmetric  digraph  [3Xdigraph[103X,  the  operation returns the symmetric
  digraph obtained by associating a vertex with each edge of [3Xdigraph[103X, ignoring
  directions and multiplicites, and adding an edge between two vertices if and
  only if the corresponding edges have a vertex in common.[133X
  
  [33X[0;0YNote  that  the  returned digraph is always a new immutable digraph, and the
  argument [3Xdigraph[103X is never modified.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLineUndirectedDigraph(CompleteDigraph(3));[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XLineUndirectedDigraph(DigraphSymmetricClosure(ChainDigraph(3)));[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 2 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-36 DoubleDigraph[101X
  
  [33X[1;0Y[29X[2XDoubleDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YLet [3Xdigraph[103X be a digraph with vertex set [10XV[110X. This function returns the double
  digraph  of  [3Xdigraph[103X.  The  vertex  set of the double digraph is the orginal
  vertex  set  together  with  a duplicate. The edges are [10X[u_1, v_2][110X and [10X[u_2,
  v_1][110X if and only if [10X[u, v][110X is an edge in [3Xdigraph[103X, together with the original
  edges and their duplicates.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  mutable,  then  [3Xdigraph[103X is modified in-place. If [3Xdigraph[103X is
  immutable,  then  a  new immutable digraph constructed as described above is
  returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgamma := Digraph([[2], [3], [1]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDoubleDigraph(gamma);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 12 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-37 BipartiteDoubleDigraph[101X
  
  [33X[1;0Y[29X[2XBipartiteDoubleDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YLet  [3Xdigraph[103X  be  a  digraph  with  vertex  set [10XV[110X. This function returns the
  bipartite double digraph of [3Xdigraph[103X. The vertex set of the double digraph is
  the  original vertex set together with a duplicate. The edges are [10X[u_1, v_2][110X
  and  [10X[u_2,  v_1][110X  if and only if [10X[u, v][110X is an edge in [3Xdigraph[103X. The resulting
  graph  is  bipartite,  since  the  orignal  edges  are  not  included in the
  resulting digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  mutable,  then  [3Xdigraph[103X is modified in-place. If [3Xdigraph[103X is
  immutable,  then  a  new immutable digraph constructed as described above is
  returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgamma := Digraph([[2], [3], [1]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XBipartiteDoubleDigraph(gamma);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 6 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-38 DigraphAddAllLoops[101X
  
  [33X[1;0Y[29X[2XDigraphAddAllLoops[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphAddAllLoopsAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFor  a digraph [3Xdigraph[103X this operation returns a new digraph constructed from
  [3Xdigraph[103X,  such  that  a  loop is added for every vertex which did not have a
  loop in [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  loops  are  added  to  the loopless vertices of the mutable
  digraph [3Xdigraph[103X in-place.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := EmptyDigraph(13);[127X[104X
    [4X[28X<immutable empty digraph with 13 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphAddAllLoops(D);[127X[104X
    [4X[28X<immutable reflexive digraph with 13 vertices, 13 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ], [ 8 ], [ 9 ], [128X[104X
    [4X[28X  [ 10 ], [ 11 ], [ 12 ], [ 13 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 3], [1, 3], [1]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphAddAllLoops(D);[127X[104X
    [4X[28X<immutable reflexive digraph with 3 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 3);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphAddAllLoops(D);[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 3 vertices, 6 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-39 DistanceDigraph[101X
  
  [33X[1;0Y[29X[2XDistanceDigraph[102X( [3Xdigraph[103X, [3Xi[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDistanceDigraph[102X( [3Xdigraph[103X, [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  first  argument  is  a  digraph,  the second argument is a non-negative
  integer  or a list of positive integers. This operation returns a digraph on
  the same set of vertices as [3Xdigraph[103X, with two vertices being adjacent if and
  only  if  the  distance  between  them in [3Xdigraph[103X equals [3Xi[103X or is a number in
  [3Xlist[103X. See [2XDigraphShortestDistance[102X ([14X5.3-2[114X).[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  mutable,  then  [3Xdigraph[103X is modified in-place. If [3Xdigraph[103X is
  immutable,  then  a  new immutable digraph constructed as described above is
  returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph := DigraphFromSparse6String([127X[104X
    [4X[25X>[125X [27X":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfHMjKM");[127X[104X
    [4X[28X<immutable digraph with 30 vertices, 90 edges>[128X[104X
    [4X[25Xgap>[125X [27XDistanceDigraph(digraph, 1);[127X[104X
    [4X[28X<immutable digraph with 30 vertices, 90 edges>[128X[104X
    [4X[25Xgap>[125X [27XDistanceDigraph(digraph, [1, 2]);[127X[104X
    [4X[28X<immutable digraph with 30 vertices, 270 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-40 DigraphClosure[101X
  
  [33X[1;0Y[29X[2XDigraphClosure[102X( [3Xdigraph[103X, [3Xk[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YGiven  a  symmetric  loopless  digraph  with  no multiple edges [3Xdigraph[103X, the
  [13X[3Xk[103X-closure of [3Xdigraph[103X[113X is defined to be the unique smallest symmetric loopless
  digraph  [10XC[110X  with  no multiple edges on the vertices of [3Xdigraph[103X that contains
  all  the  edges  of  [3Xdigraph[103X  and satsifies the property that the sum of the
  degrees  of  every  two  non-adjacenct  vertices  in  [10XC[110X  is less than [3Xk[103X. See
  [2XIsSymmetricDigraph[102X   ([14X6.1-12[114X),   [2XDigraphHasLoops[102X   ([14X6.1-1[114X),   [2XIsMultiDigraph[102X
  ([14X6.1-10[114X), and [2XOutDegreeOfVertex[102X ([14X5.2-10[114X).[133X
  
  [33X[0;0YThe operation [10XDigraphClosure[110X returns the [3Xk[103X-closure of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(6);[127X[104X
    [4X[28X<immutable complete digraph with 6 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 28 edges>[128X[104X
    [4X[25Xgap>[125X [27Xclosure := DigraphClosure(D, 6);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteDigraph(closure);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-41 DigraphMycielskian[101X
  
  [33X[1;0Y[29X[2XDigraphMycielskian[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphMycielskianAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  symmetric  digraph,  then [10XDigraphMycielskian[110X returns the
  Mycielskian of [3Xdigraph[103X.[133X
  
  [33X[0;0YThe  Mycielskian  of  a  symmetric  digraph  is  a  larger symmetric digraph
  constructed  from  it,  which  has  a  larger  chromatic number. For further
  information, see [7Xhttps://en.wikipedia.org/wiki/Mycielskian[107X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable,  then  a  new  immutable  digraph  is  returned.
  Otherwise,  the  mutable  digraph  [3Xdigraph[103X  is  changed  in-place  into  its
  Mycielskian.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(2);[127X[104X
    [4X[28X<immutable cycle digraph with 2 vertices>[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(D);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMycielskian(D);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(D);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMycielskian(D);[127X[104X
    [4X[28X<immutable digraph with 11 vertices, 40 edges>[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(D);[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XD := CompleteBipartiteDigraph(IsMutable, 2, 3);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphMycielskian(D);[127X[104X
    [4X[28X<mutable digraph with 11 vertices, 46 edges>[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<mutable digraph with 11 vertices, 46 edges>[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YRandom digraphs[133X[101X
  
  [1X3.4-1 RandomDigraph[101X
  
  [33X[1;0Y[29X[2XRandomDigraph[102X( [[3Xfilt[103X, ][3Xn[103X[, [3Xp[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf [3Xn[103X is a positive integer, then this function returns a random digraph with
  [3Xn[103X vertices and without multiple edges. The result may or may not have loops.[133X
  
  [33X[0;0YIf the optional second argument [3Xp[103X is a float with value [22X0 ≤[122X [3X p [103X [22X≤ 1[122X, then an
  edge will exist between each pair of vertices with probability approximately
  [3Xp[103X.  If [3Xp[103X is not specified, then a random probability will be assumed (chosen
  with uniform probability).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomDigraph(1000);[127X[104X
    [4X[28X<immutable digraph with 1000 vertices, 364444 edges>[128X[104X
    [4X[25Xgap>[125X [27XRandomDigraph(10000, 0.023);[127X[104X
    [4X[28X<immutable digraph with 10000 vertices, 2300438 edges>[128X[104X
    [4X[25Xgap>[125X [27XRandomDigraph(IsMutableDigraph, 1000, 1 / 2);[127X[104X
    [4X[28X<mutable digraph with 1000 vertices, 499739 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.4-2 RandomMultiDigraph[101X
  
  [33X[1;0Y[29X[2XRandomMultiDigraph[102X( [3Xn[103X[, [3Xm[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xn[103X is a positive integer, then this function returns a random digraph with
  [3Xn[103X  vertices.  If  the optional second argument [3Xm[103X is a positive integer, then
  the  digraph  will  have  [3Xm[103X edges. If [3Xm[103X is not specified, then the number of
  edges  will  be chosen randomly (with uniform probability) from the range [10X[1
  .. [110X[22Xn choose 2[122X[10X][110X.[133X
  
  [33X[0;0YThe  method  used  by  this  function  chooses each edge from the set of all
  possible edges with uniform probability. No effort is made to avoid creating
  multiple  edges, so it is possible (but not guaranteed) that the result will
  have multiple edges. The result may or may not have loops.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomMultiDigraph(1000);[127X[104X
    [4X[28X<immutable multidigraph with 1000 vertices, 216659 edges>[128X[104X
    [4X[25Xgap>[125X [27XRandomMultiDigraph(1000, 950);[127X[104X
    [4X[28X<immutable multidigraph with 1000 vertices, 950 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.4-3 RandomTournament[101X
  
  [33X[1;0Y[29X[2XRandomTournament[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a non-negative integer, this function returns a random tournament
  with [3Xn[103X vertices. See [2XIsTournament[102X ([14X6.1-13[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomTournament(10);[127X[104X
    [4X[28X<immutable tournament with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XRandomTournament(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable digraph with 1000 vertices, 500601 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.4-4 RandomLattice[101X
  
  [33X[1;0Y[29X[2XRandomLattice[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  positive integer, this function return a random lattice with [10Xm[110X
  vertices,  where  it  is  guaranteed  that  [10Xm[110X  is  between  [3Xn[103X and [10X2 * [3Xn[103X[10X[110X. See
  [2XIsLatticeDigraph[102X ([14X6.1-17[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomLattice(10);[127X[104X
    [4X[28X<immutable lattice digraph with 10 vertices, 39 edges>[128X[104X
    [4X[25Xgap>[125X [27XRandomLattice(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable digraph with 12 vertices, 52 edges>[128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YStandard examples[133X[101X
  
  [1X3.5-1 ChainDigraph[101X
  
  [33X[1;0Y[29X[2XChainDigraph[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a positive integer, this function returns a chain with [3Xn[103X vertices
  and  [10X[3Xn[103X[10X  -  1[110X edges. Specifically, for each vertex [10Xi[110X (with [10Xi[110X < [10Xn[110X), there is a
  directed edge with source [10Xi[110X and range [10Xi + 1[110X.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YThe [2XDigraphReflexiveTransitiveClosure[102X ([14X3.3-12[114X) of a chain represents a total
  order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XChainDigraph(42);[127X[104X
    [4X[28X<immutable chain digraph with 42 vertices>[128X[104X
    [4X[25Xgap>[125X [27XChainDigraph(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 9 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-2 CompleteDigraph[101X
  
  [33X[1;0Y[29X[2XCompleteDigraph[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is a non-negative integer, this function returns the complete digraph
  with [3Xn[103X vertices. See [2XIsCompleteDigraph[102X ([14X6.1-5[114X).[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompleteDigraph(20);[127X[104X
    [4X[28X<immutable complete digraph with 20 vertices>[128X[104X
    [4X[25Xgap>[125X [27XCompleteDigraph(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 90 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-3 CompleteBipartiteDigraph[101X
  
  [33X[1;0Y[29X[2XCompleteBipartiteDigraph[102X( [[3Xfilt[103X, ][3Xm[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YA  complete bipartite digraph is a digraph whose vertices can be partitioned
  into  two non-empty vertex sets, such there exists a unique edge with source
  [10Xi[110X and range [10Xj[110X if and only if [10Xi[110X and [10Xj[110X lie in different vertex sets.[133X
  
  [33X[0;0YIf  [3Xm[103X  and  [3Xn[103X  are  positive  integers,  this  function returns the complete
  bipartite digraph with vertex sets of sizes [3Xm[103X (containing the vertices [10X[1 ..
  m][110X) and [3Xn[103X (containing the vertices [10X[m + 1 .. m + n][110X).[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompleteBipartiteDigraph(2, 3);[127X[104X
    [4X[28X<immutable complete bipartite digraph with bicomponent sizes 2 and 3>[128X[104X
    [4X[25Xgap>[125X [27XCompleteBipartiteDigraph(IsMutableDigraph, 3, 2);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 12 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-4 CompleteMultipartiteDigraph[101X
  
  [33X[1;0Y[29X[2XCompleteMultipartiteDigraph[102X( [[3Xfilt[103X, ][3Xorders[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFor  a list [3Xorders[103X of [10Xn[110X positive integers, this function returns the digraph
  containing  [10Xn[110X  independent  sets  of  vertices  of  orders  [10X[[3Xl[103X[10X[1]  .. [3Xl[103X[10X[n]][110X.
  Moreover,  each  vertex is adjacent to every other not contained in the same
  independent set.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompleteMultipartiteDigraph([5, 4, 2]);[127X[104X
    [4X[28X<immutable complete multipartite digraph with 11 vertices, 76 edges>[128X[104X
    [4X[25Xgap>[125X [27XCompleteMultipartiteDigraph(IsMutableDigraph, [5, 4, 2]);[127X[104X
    [4X[28X<mutable digraph with 11 vertices, 76 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-5 CycleDigraph[101X
  
  [33X[1;0Y[29X[2XCycleDigraph[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  positive integer, this function returns a [13Xcycle[113X digraph with [3Xn[103X
  vertices and [3Xn[103X edges. Specifically, for each vertex [10Xi[110X (with [10Xi[110X < [10Xn[110X), there is
  a directed edge with source [10Xi[110X and range [10Xi + 1[110X. In addition, there is an edge
  with source [10Xn[110X and range [10X1[110X.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCycleDigraph(1);[127X[104X
    [4X[28X<immutable digraph with 1 vertex, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XCycleDigraph(123);[127X[104X
    [4X[28X<immutable cycle digraph with 123 vertices>[128X[104X
    [4X[25Xgap>[125X [27XCycleDigraph(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 10 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-6 EmptyDigraph[101X
  
  [33X[1;0Y[29X[2XEmptyDigraph[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNullDigraph[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  non-negative  integer, this function returns the [13Xempty[113X or [13Xnull[113X
  digraph with [3Xn[103X vertices. An empty digraph is one with no edges.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0Y[10XNullDigraph[110X is a synonym for [10XEmptyDigraph[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEmptyDigraph(20);[127X[104X
    [4X[28X<immutable empty digraph with 20 vertices>[128X[104X
    [4X[25Xgap>[125X [27XNullDigraph(10);[127X[104X
    [4X[28X<immutable empty digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XEmptyDigraph(IsMutableDigraph, 10);[127X[104X
    [4X[28X<mutable empty digraph with 10 vertices>[128X[104X
  [4X[32X[104X
  
  [1X3.5-7 JohnsonDigraph[101X
  
  [33X[1;0Y[29X[2XJohnsonDigraph[102X( [[3Xfilt[103X, ][3Xn[103X, [3Xk[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  and  [3Xk[103X  are  non-negative  integers,  then  this  operation returns a
  symmetric digraph which corresponds to the undirected [13XJohnson graph[113X [22XJ(n, k)[122X.[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YThe  [13XJohnson  graph[113X  [22XJ(n,  k)[122X has vertices given by all the [3Xk[103X-subsets of the
  range  [10X[1  ..  [3Xn[103X[10X][110X,  and  two  vertices  are  connected  by an edge iff their
  intersection has size [22X[3Xk[103X - 1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := JohnsonDigraph(3, 1);[127X[104X
    [4X[28X<immutable symmetric digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr := JohnsonDigraph(4, 2);[127X[104X
    [4X[28X<immutable symmetric digraph with 6 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [ 1, 2, 5, 6 ], [ 1, 2, 5, 6 ], [128X[104X
    [4X[28X  [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XJohnsonDigraph(1, 0);[127X[104X
    [4X[28X<immutable empty digraph with 1 vertex>[128X[104X
    [4X[25Xgap>[125X [27XJohnsonDigraph(IsMutableDigraph, 1, 0);[127X[104X
    [4X[28X<mutable empty digraph with 1 vertex>[128X[104X
  [4X[32X[104X
  
  [1X3.5-8 PetersenGraph[101X
  
  [33X[1;0Y[29X[2XPetersenGraph[102X( [[3Xfilt[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFrom [7Xhttps://en.wikipedia.org/wiki/Petersen_graph[107X:[133X
  
  [33X[0;0Y[21XThe  Petersen graph is an undirected graph with 10 vertices and 15 edges. It
  is a small graph that serves as a useful example and counterexample for many
  problems in graph theory. The Petersen graph is named after Julius Petersen,
  who in 1898 constructed it to be the smallest bridgeless cubic graph with no
  three-edge-coloring.[121X[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YSee also [2XGeneralisedPetersenGraph[102X ([14X3.5-9[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(PetersenGraph());[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XPetersenGraph(IsMutableDigraph);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 30 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-9 GeneralisedPetersenGraph[101X
  
  [33X[1;0Y[29X[2XGeneralisedPetersenGraph[102X( [[3Xfilt[103X, ][3Xn[103X, [3Xk[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X is a positive integer and [3Xk[103X is a non-negative integer less than [10X[3Xn[103X[10X / 2[110X,
  then this operation returns the [13Xgeneralised Petersen graph[113X [22XGPG([3Xn[103X, k)[122X.[133X
  
  [33X[0;0YFrom [7Xhttps://en.wikipedia.org/wiki/Generalized_Petersen_graph[107X:[133X
  
  [33X[0;0Y[21XThe  generalized  Petersen  graphs  are  a  family of cubic graphs formed by
  connecting  the  vertices of a regular polygon to the corresponding vertices
  of a star polygon. They include the Petersen graph and generalize one of the
  ways  of  constructing  the  Petersen  graph. The generalized Petersen graph
  family  was introduced in 1950 by H. S. M. Coxeter and was given its name in
  1969 by Mark Watkins.[121X[133X
  
  [33X[0;0YIf the optional first argument [3Xfilt[103X is present, then this should specify the
  category  or  representation  the  digraph being created will belong to. For
  example, if [3Xfilt[103X is [2XIsMutableDigraph[102X ([14X3.1-2[114X), then the digraph being created
  will  be  mutable,  if  [3Xfilt[103X is [2XIsImmutableDigraph[102X ([14X3.1-3[114X), then the digraph
  will  be immutable. If the optional first argument [3Xfilt[103X is not present, then
  [2XIsImmutableDigraph[102X ([14X3.1-3[114X) is used by default.[133X
  
  [33X[0;0YSee also [2XPetersenGraph[102X ([14X3.5-8[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGeneralisedPetersenGraph(7, 2);[127X[104X
    [4X[28X<immutable symmetric digraph with 14 vertices, 42 edges>[128X[104X
    [4X[25Xgap>[125X [27XGeneralisedPetersenGraph(40, 1);[127X[104X
    [4X[28X<immutable symmetric digraph with 80 vertices, 240 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := GeneralisedPetersenGraph(5, 2);[127X[104X
    [4X[28X<immutable symmetric digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(D, PetersenGraph());[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XGeneralisedPetersenGraph(IsMutableDigraph, 9, 4);[127X[104X
    [4X[28X<mutable digraph with 18 vertices, 54 edges>[128X[104X
  [4X[32X[104X
  
