  
  [1X7 [33X[0;0YHomomorphisms[133X[101X
  
  
  [1X7.1 [33X[0;0YActing on digraphs[133X[101X
  
  [1X7.1-1 OnDigraphs[101X
  
  [33X[1;0Y[29X[2XOnDigraphs[102X( [3Xdigraph[103X, [3Xperm[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOnDigraphs[102X( [3Xdigraph[103X, [3Xtrans[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a digraph, and the second argument [3Xperm[103X is a [13Xpermutation[113X of
  the  vertices  of [3Xdigraph[103X, then this operation returns a digraph constructed
  by  relabelling  the vertices of [3Xdigraph[103X according to [3Xperm[103X. Note that for an
  automorphism [10Xf[110X of a digraph, we have [10XOnDigraphs([3Xdigraph[103X[10X, f) = [110X[3Xdigraph[103X.[133X
  
  [33X[0;0YIf the second argument is a [13Xtransformation[113X [3Xtrans[103X of the vertices of [3Xdigraph[103X,
  then this operation returns a digraph constructed by transforming the source
  and  range  of  each  edge  according to [3Xtrans[103X. Thus a vertex which does not
  appear  in  the image of [3Xtrans[103X will be isolated in the returned digraph, and
  the  returned  digraph may contain multiple edges, even if [3Xdigraph[103X does not.
  If  [3Xtrans[103X  is  mathematically  a permutation, then the result coincides with
  [10XOnDigraphs([3Xdigraph[103X[10X, AsPermutation([3Xtrans[103X[10X))[110X.[133X
  
  [33X[0;0YThe  [2XDigraphVertexLabels[102X  ([14X5.1-10[114X)  of  [3Xdigraph[103X  will not be retained in the
  returned digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  belongs  to  [2XIsMutableDigraph[102X  ([14X3.1-2[114X), then relabelling of the
  vertices   is   performed   directly  on  [3Xdigraph[103X.  If  [3Xdigraph[103X  belongs  to
  [2XIsImmutableDigraph[102X  ([14X3.1-3[114X),  an immutable copy of [3Xdigraph[103X with the vertices
  relabelled 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 [27Xnew := OnDigraphs(D, (1, 2));[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(new);[127X[104X
    [4X[28X[ [ 2, 3, 5 ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2], [], [2]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27Xt := Transformation([1, 2, 1]);;[127X[104X
    [4X[25Xgap>[125X [27Xnew := OnDigraphs(D, t);[127X[104X
    [4X[28X<immutable multidigraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(new);[127X[104X
    [4X[28X[ [ 2, 2 ], [  ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27XForAll(DigraphEdges(D),[127X[104X
    [4X[25X>[125X [27X e -> IsDigraphEdge(new, [e[1] ^ t, e[2] ^ t]));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.1-2 OnMultiDigraphs[101X
  
  [33X[1;0Y[29X[2XOnMultiDigraphs[102X( [3Xdigraph[103X, [3Xpair[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOnMultiDigraphs[102X( [3Xdigraph[103X, [3Xperm1[103X, [3Xperm2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph, and [3Xpair[103X is a pair consisting of a permutation of
  the  vertices and a permutation of the edges of [3Xdigraph[103X, then this operation
  returns  a  digraph  constructed  by  relabelling  the vertices and edges of
  [3Xdigraph[103X according to [3Xperm[1][103X and [3Xperm[2][103X, respectively.[133X
  
  [33X[0;0YIn  its  second  form,  [10XOnMultiDigraphs[110X  returns a digraph with vertices and
  edges permuted by [3Xperm1[103X and [3Xperm2[103X, respectively.[133X
  
  [33X[0;0YNote  that  [10XOnDigraphs([3Xdigraph[103X[10X,  perm)=OnMultiDigraphs([3Xdigraph[103X[10X,  [perm, ()])[110X
  where  [10Xperm[110X  is  a  permutation  of the vertices of [3Xdigraph[103X. If you are only
  interested in the action of a permutation on the vertices of a digraph, then
  you  can  use  [10XOnDigraphs[110X  instead of [10XOnMultiDigraphs[110X. If [3Xdigraph[103X belongs to
  [2XIsMutableDigraph[102X  ([14X3.1-2[114X),  then  relabelling  of  the vertices is performed
  directly  on  [3Xdigraph[103X.  If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X), an
  immutable copy of [3Xdigraph[103X with the vertices relabelled is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD1 := Digraph([[127X[104X
    [4X[25X>[125X [27X[3, 6, 3], [], [3], [9, 10], [9], [],  [], [10, 4, 10], [], []]);[127X[104X
    [4X[28X<immutable multidigraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27Xp := BlissCanonicalLabelling(D1);[127X[104X
    [4X[28X[ (1,7)(3,6)(4,10)(5,9), () ][128X[104X
    [4X[25Xgap>[125X [27XD2 := OnMultiDigraphs(D1, p);[127X[104X
    [4X[28X<immutable multidigraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D2);[127X[104X
    [4X[28X[ [  ], [  ], [  ], [  ], [  ], [ 6 ], [ 6, 3, 6 ], [ 4, 10, 4 ], [128X[104X
    [4X[28X  [ 5 ], [ 5, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X7.1-3 OnTuplesDigraphs[101X
  
  [33X[1;0Y[29X[2XOnTuplesDigraphs[102X( [3Xlist[103X, [3Xperm[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOnSetsDigraphs[102X( [3Xlist[103X, [3Xperm[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list or set of digraphs.[133X
  
  [33X[0;0YIf  [3Xlist[103X is a list of digraphs, and [3Xperm[103X is a [13Xpermutation[113X of the vertices of
  the  digraphs  in [3Xlist[103X, then [2XOnTuplesDigraphs[102X returns a new list constructed
  by applying [3Xperm[103X via [2XOnDigraphs[102X ([14X7.1-1[114X) to a copy (with the same mutability)
  of each entry of [3Xlist[103X in turn.[133X
  
  [33X[0;0YMore   precisely,   [10XOnTuplesDigraphs([3Xlist[103X[10X,[3Xperm[103X[10X)[110X   is   a   list   of  length
  [10XLength([3Xlist[103X[10X)[110X, whose [10Xi[110X-th entry is [10XOnDigraphs(DigraphCopy([3Xlist[103X[10X[i]), [3Xperm[103X[10X)[110X.[133X
  
  [33X[0;0YIf  [3Xlist[103X  is  moreover  a  [5XGAP[105X set (i.e. a duplicate-free sorted list), then
  [2XOnSetsDigraphs[102X  returns  the  sorted  output  of  [2XOnTuplesDigraphs[102X, which is
  therefore again a set.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlist := [CycleDigraph(IsMutableDigraph, 6),[127X[104X
    [4X[25X>[125X [27X            DigraphReverse(CycleDigraph(6))];[127X[104X
    [4X[28X[ <mutable digraph with 6 vertices, 6 edges>, [128X[104X
    [4X[28X  <immutable digraph with 6 vertices, 6 edges> ][128X[104X
    [4X[25Xgap>[125X [27Xp := (1, 6)(2, 5)(3, 4);;[127X[104X
    [4X[25Xgap>[125X [27Xresult_tuples := OnTuplesDigraphs(list, p);[127X[104X
    [4X[28X[ <mutable digraph with 6 vertices, 6 edges>, [128X[104X
    [4X[28X  <immutable digraph with 6 vertices, 6 edges> ][128X[104X
    [4X[25Xgap>[125X [27Xresult_tuples[2] = OnDigraphs(list[2], p);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xresult_tuples = list;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xresult_tuples = Reversed(list);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xresult_sets := OnSetsDigraphs(list, p);[127X[104X
    [4X[28X[ <immutable digraph with 6 vertices, 6 edges>, [128X[104X
    [4X[28X  <mutable digraph with 6 vertices, 6 edges> ][128X[104X
    [4X[25Xgap>[125X [27Xresult_sets = list;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X7.2 [33X[0;0YIsomorphisms and canonical labellings[133X[101X
  
  [33X[0;0YFrom  version 0.11.0 of [5XDigraphs[105X it is possible to use either [5Xbliss[105X or [5Xnauty[105X
  (via   [5XNautyTracesInterface[105X)   to   calculate   canonical   labellings   and
  automorphism  groups  of  digraphs;  see  [JK07] and [MP14] for more details
  about [5Xbliss[105X and [5Xnauty[105X, respectively.[133X
  
  [1X7.2-1 DigraphsUseNauty[101X
  
  [33X[1;0Y[29X[2XDigraphsUseNauty[102X(  ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphsUseBliss[102X(  ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YNothing.[133X
  
  [33X[0;0YThese functions can be used to specify whether [5Xnauty[105X or [5Xbliss[105X should be used
  by default by [5XDigraphs[105X. If [5XNautyTracesInterface[105X is not available, then these
  functions  do  nothing.  Otherwise,  by  calling [10XDigraphsUseNauty[110X subsequent
  computations will default to using [5Xnauty[105X rather than [5Xbliss[105X, where possible.[133X
  
  [33X[0;0YYou can call these functions at any point in a [5XGAP[105X session, as many times as
  you  like,  it  is  guaranteed that existing digraphs remain valid, and that
  comparison   of   existing   digraphs   and   newly   created  digraphs  via
  [2XIsIsomorphicDigraph[102X       ([14X7.2-15[114X),       [2XIsIsomorphicDigraph[102X      ([14X7.2-16[114X),
  [2XIsomorphismDigraphs[102X  ([14X7.2-17[114X),  and  [2XIsomorphismDigraphs[102X  ([14X7.2-18[114X)  are also
  valid.[133X
  
  [33X[0;0YIt  is also possible to compute the automorphism group of a specific digraph
  using   both  [5Xnauty[105X  and  [5Xbliss[105X  using  [2XNautyAutomorphismGroup[102X  ([14X7.2-4[114X)  and
  [2XBlissAutomorphismGroup[102X ([14X7.2-3[114X), respectively.[133X
  
  [1X7.2-2 AutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XAutomorphismGroup[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then  this  attribute  contains  the  group of
  automorphisms  of [3Xdigraph[103X. An [13Xautomorphism[113X of [3Xdigraph[103X is an isomorphism from
  [3Xdigraph[103X  to  itself.  See  [2XIsomorphismDigraphs[102X ([14X7.2-17[114X) for more information
  about isomorphisms of digraphs.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is not a multidigraph then the automorphism group is returned as
  a group of permutations on the set of vertices of [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is a multidigraph then the automorphism group is returned as the
  direct  product of a group of permutations on the set of vertices of [3Xdigraph[103X
  with  a  group  of permutations on the set of edges of [3Xdigraph[103X. These groups
  can  be  accessed using [2XProjection[102X ([14XReference: Projection for a domain and a
  positive integer[114X) on the returned group.[133X
  
  [33X[0;0YBy  default,  the  automorphism group is found using [5Xbliss[105X by Tommi Junttila
  and  Petteri  Kaski.  If  [5XNautyTracesInterface[105X  is  available, then [5Xnauty[105X by
  Brendan    Mckay   and   Adolfo   Piperno   can   be   used   instead;   see
  [2XBlissAutomorphismGroup[102X      ([14X7.2-3[114X),     [2XNautyAutomorphismGroup[102X     ([14X7.2-4[114X),
  [2XDigraphsUseBliss[102X ([14X7.2-1[114X), and [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [33X[0;0YIf  the argument [3Xdigraph[103X is mutable, then the return value of this attribute
  is recomputed every time it is called.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xjohnson := DigraphFromGraph6String("E}lw");[127X[104X
    [4X[28X<immutable symmetric digraph with 6 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := AutomorphismGroup(johnson);[127X[104X
    [4X[28XGroup([ (3,4), (2,3)(4,5), (1,2)(5,6) ])[128X[104X
    [4X[25Xgap>[125X [27Xcycle := CycleDigraph(9);[127X[104X
    [4X[28X<immutable cycle digraph with 9 vertices>[128X[104X
    [4X[25Xgap>[125X [27XG := AutomorphismGroup(cycle);[127X[104X
    [4X[28XGroup([ (1,2,3,4,5,6,7,8,9) ])[128X[104X
    [4X[25Xgap>[125X [27XIsCyclic(G) and Size(G) = 9;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-3 BlissAutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XBlissAutomorphismGroup[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XBlissAutomorphismGroup[102X( [3Xdigraph[103X, [3Xvertex_colours[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XBlissAutomorphismGroup[102X( [3Xdigraph[103X, [3Xvertex_colours[103X, [3Xedge_colours[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then  this  attribute  contains  the  group of
  automorphisms  of  [3Xdigraph[103X  as  calculated using [5Xbliss[105X by Tommi Junttila and
  Petteri Kaski.[133X
  
  [33X[0;0YThe  attribute  [2XAutomorphismGroup[102X  ([14X7.2-2[114X)  and  operation [2XAutomorphismGroup[102X
  ([14X7.2-5[114X)    returns   the   value   of   either   [10XBlissAutomorphismGroup[110X   or
  [2XNautyAutomorphismGroup[102X ([14X7.2-4[114X). These groups are, of course, equal but their
  generating sets may differ.[133X
  
  [33X[0;0YThe    attribute    [2XAutomorphismGroup[102X   ([14X7.2-6[114X)   returns   the   value   of
  [10XBlissAutomorphismGroup[110X  as  it is not implemented for [5Xnauty[105X The requirements
  for the optional arguments [3Xvertex_colours[103X and [3Xedge_colours[103X are documented in
  [2XAutomorphismGroup[102X   ([14X7.2-6[114X).   See   also   [2XDigraphsUseBliss[102X   ([14X7.2-1[114X),  and
  [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [33X[0;0YIf  the argument [3Xdigraph[103X is mutable, then the return value of this attribute
  is recomputed every time it is called.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := BlissAutomorphismGroup(JohnsonDigraph(5, 2));;[127X[104X
    [4X[25Xgap>[125X [27XIsSymmetricGroup(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(G);[127X[104X
    [4X[28X120[128X[104X
  [4X[32X[104X
  
  [1X7.2-4 NautyAutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XNautyAutomorphismGroup[102X( [3Xdigraph[103X[, [3Xvert_colours[103X] ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then  this  attribute  contains  the  group of
  automorphisms  of  [3Xdigraph[103X  as  calculated  using [5Xnauty[105X by Brendan Mckay and
  Adolfo  Piperno  via  [5XNautyTracesInterface[105X.  The attribute [2XAutomorphismGroup[102X
  ([14X7.2-2[114X)  and operation [2XAutomorphismGroup[102X ([14X7.2-5[114X) returns the value of either
  [10XNautyAutomorphismGroup[110X  or [2XBlissAutomorphismGroup[102X ([14X7.2-3[114X). These groups are,
  of course, equal but their generating sets may differ.[133X
  
  [33X[0;0YSee also [2XDigraphsUseBliss[102X ([14X7.2-1[114X), and [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [33X[0;0YIf  the argument [3Xdigraph[103X is mutable, then the return value of this attribute
  is recomputed every time it is called.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNautyAutomorphismGroup(JohnsonDigraph(5, 2));[127X[104X
    [4X[28XGroup([ (3,4)(6,7)(8,9), (2,3)(5,6)(9,10), (2,5)(3,6)(4,7),[128X[104X
    [4X[28X (1,2)(6,8)(7,9) ])[128X[104X
  [4X[32X[104X
  
  [1X7.2-5 AutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XAutomorphismGroup[102X( [3Xdigraph[103X, [3Xvert_colours[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0YThis operation computes the automorphism group of a vertex-coloured digraph.
  A vertex-coloured digraph can be specified by its underlying digraph [3Xdigraph[103X
  and  its colouring [3Xvert_colours[103X. Let [10Xn[110X be the number of vertices of [3Xdigraph[103X.
  The colouring [3Xvert_colours[103X may have one of the following two forms:[133X
  
  [30X    [33X[0;6Ya list of [10Xn[110X integers, where [3Xvert_colours[103X[10X[i][110X is the colour of vertex [10Xi[110X,
        using the colours [10X[1 .. m][110X for some [10Xm <= n[110X; or[133X
  
  [30X    [33X[0;6Ya    list    of    non-empty    disjoint    lists   whose   union   is
        [10XDigraphVertices([3Xdigraph[103X[10X)[110X, such that [3Xvert_colours[103X[10X[i][110X is the list of all
        vertices with colour [10Xi[110X.[133X
  
  [33X[0;0YThe  [13Xautomorphism  group[113X  of  a  coloured  digraph  [3Xdigraph[103X  with  colouring
  [3Xvert_colours[103X  is  the group consisting of its automorphisms; an [13Xautomorphism[113X
  of  [3Xdigraph[103X  is  an isomorphism of coloured digraphs from [3Xdigraph[103X to itself.
  This group is equal to the subgroup of [10XAutomorphismGroup([3Xdigraph[103X[10X)[110X consisting
  of   those   automorphisms   that   preserve   the  colouring  specified  by
  [3Xvert_colours[103X.  See  [2XAutomorphismGroup[102X  ([14X7.2-2[114X),  and see [2XIsomorphismDigraphs[102X
  ([14X7.2-18[114X) for more information about isomorphisms of coloured digraphs.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is not a multidigraph then the automorphism group is returned as
  a group of permutations on the set of vertices of [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is a multidigraph then the automorphism group is returned as the
  direct  product of a group of permutations on the set of vertices of [3Xdigraph[103X
  with  a  group  of permutations on the set of edges of [3Xdigraph[103X. These groups
  can  be  accessed using [2XProjection[102X ([14XReference: Projection for a domain and a
  positive integer[114X) on the returned group.[133X
  
  [33X[0;0YBy  default,  the  automorphism group is found using [5Xbliss[105X by Tommi Junttila
  and  Petteri  Kaski.  If  [5XNautyTracesInterface[105X  is  available, then [5Xnauty[105X by
  Brendan    Mckay   and   Adolfo   Piperno   can   be   used   instead;   see
  [2XBlissAutomorphismGroup[102X      ([14X7.2-3[114X),     [2XNautyAutomorphismGroup[102X     ([14X7.2-4[114X),
  [2XDigraphsUseBliss[102X ([14X7.2-1[114X), and [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcycle := CycleDigraph(9);[127X[104X
    [4X[28X<immutable cycle digraph with 9 vertices>[128X[104X
    [4X[25Xgap>[125X [27XG := AutomorphismGroup(cycle);;[127X[104X
    [4X[25Xgap>[125X [27XIsCyclic(G) and Size(G) = 9;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcolours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;[127X[104X
    [4X[25Xgap>[125X [27XH := AutomorphismGroup(cycle, colours);;[127X[104X
    [4X[25Xgap>[125X [27XSize(H);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XH = AutomorphismGroup(cycle, [1, 2, 3, 1, 2, 3, 1, 2, 3]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XH = SubgroupByProperty(G, p -> OnTuplesSets(colours, p) = colours);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsTrivial(AutomorphismGroup(cycle, [1, 1, 2, 2, 2, 2, 2, 2, 2]));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-6 AutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XAutomorphismGroup[102X( [3Xdigraph[103X, [3Xvert_colours[103X, [3Xedge_colours[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0YThis   operation  computes  the  automorphism  group  of  a  vertex-  and/or
  edge-coloured digraph. A coloured digraph can be specified by its underlying
  digraph  [3Xdigraph[103X  and  colourings  [3Xvert_colours[103X,  [3Xedge_colours[103X. Let [10Xn[110X be the
  number of vertices of [3Xdigraph[103X. The colourings must have the following forms:[133X
  
  [30X    [33X[0;6Y[3Xvert_colours[103X   must   be   [9Xfail[109X   or  a  list  of  [10Xn[110X  integers,  where
        [3Xvert_colours[103X[10X[i][110X  is the colour of vertex [10Xi[110X, using the colours [10X[1 .. m][110X
        for some [10Xm <= n[110X;[133X
  
  [30X    [33X[0;6Y[3Xedge_colours[103X must be [9Xfail[109X or a list of [10Xn[110X lists of integers of the same
        shape  as  [10XOutNeighbours(digraph)[110X,  where  [3Xedge_colours[103X[10X[i][j][110X  is  the
        colour  of the edge [10XOutNeighbours(digraph)[i][j][110X, using the colours [10X[1
        .. k][110X for some [10Xk <= n[110X;[133X
  
  [33X[0;0YGiving  [3Xvert_colours[103X  [[3Xedge_colours[103X]  as  [10Xfail[110X  is equivalent to setting all
  vertices [edges] to be the same colour.[133X
  
  [33X[0;0YUnlike  [2XAutomorphismGroup[102X ([14X7.2-2[114X), it is possible to obtain the automorphism
  group of an edge-coloured multidigraph (see [2XIsMultiDigraph[102X ([14X6.2-11[114X)) when no
  two  edges  share the same source, range, and colour. The [13Xautomorphism group[113X
  of  a  vertex/edge-coloured  digraph  [3Xdigraph[103X  with colouring [3Xc[103X is the group
  consisting   of   its   vertex/edge-colour   preserving   automorphisms;  an
  [13Xautomorphism[113X  of  [3Xdigraph[103X is an isomorphism of vertex/edge-coloured digraphs
  from   [3Xdigraph[103X   to   itself.  This  group  is  equal  to  the  subgroup  of
  [10XAutomorphismGroup([3Xdigraph[103X[10X)[110X  consisting  of those automorphisms that preserve
  the  colouring  specified by [3Xcolours[103X. See [2XAutomorphismGroup[102X ([14X7.2-2[114X), and see
  [2XIsomorphismDigraphs[102X  ([14X7.2-18[114X)  for  more  information  about isomorphisms of
  coloured digraphs.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is not a multidigraph then the automorphism group is returned as
  a group of permutations on the set of vertices of [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is a multidigraph then the automorphism group is returned as the
  direct  product of a group of permutations on the set of vertices of [3Xdigraph[103X
  with  a  group  of permutations on the set of edges of [3Xdigraph[103X. These groups
  can  be  accessed using [2XProjection[102X ([14XReference: Projection for a domain and a
  positive integer[114X) on the returned group.[133X
  
  [33X[0;0YBy  default,  the  automorphism group is found using [5Xbliss[105X by Tommi Junttila
  and  Petteri  Kaski.  If  [5XNautyTracesInterface[105X  is  available, then [5Xnauty[105X by
  Brendan    Mckay   and   Adolfo   Piperno   can   be   used   instead;   see
  [2XBlissAutomorphismGroup[102X      ([14X7.2-3[114X),     [2XNautyAutomorphismGroup[102X     ([14X7.2-4[114X),
  [2XDigraphsUseBliss[102X ([14X7.2-1[114X), and [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcycle := CycleDigraph(12);[127X[104X
    [4X[28X<immutable cycle digraph with 12 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xvert_colours := List([1 .. 12], x -> x mod 3 + 1);;[127X[104X
    [4X[25Xgap>[125X [27Xedge_colours := List([1 .. 12], x -> [x mod 2 + 1]);;[127X[104X
    [4X[25Xgap>[125X [27XSize(AutomorphismGroup(cycle));[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XSize(AutomorphismGroup(cycle, vert_colours));[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XSize(AutomorphismGroup(cycle, fail, edge_colours));[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XSize(AutomorphismGroup(cycle, vert_colours, edge_colours));[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XIsTrivial(AutomorphismGroup(cycle,[127X[104X
    [4X[25X>[125X [27Xvert_colours, List([1 .. 12], x -> [x mod 4 + 1])));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-7 BlissCanonicalLabelling[101X
  
  [33X[1;0Y[29X[2XBlissCanonicalLabelling[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XNautyCanonicalLabelling[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA permutation, or a list of two permutations.[133X
  
  [33X[0;0YA  function [22Xρ[122X that maps a digraph to a digraph is a [13Xcanonical representative
  map[113X if the following two conditions hold for all digraphs [22XG[122X and [22XH[122X:[133X
  
  [30X    [33X[0;6Y[22Xρ(G)[122X and [22XG[122X are isomorphic as digraphs; and[133X
  
  [30X    [33X[0;6Y[22Xρ(G)=ρ(H)[122X if and only if [22XG[122X and [22XH[122X are isomorphic as digraphs.[133X
  
  [33X[0;0YA  [13Xcanonical  labelling[113X of a digraph [22XG[122X (under [22Xρ[122X) is an isomorphism of [22XG[122X onto
  its  [13Xcanonical  representative[113X,  [22Xρ(G)[122X.  See [2XIsomorphismDigraphs[102X ([14X7.2-17[114X) for
  more information about isomorphisms of digraphs.[133X
  
  [33X[0;0Y[10XBlissCanonicalLabelling[110X returns a canonical labelling of the digraph [3Xdigraph[103X
  found    using    [5Xbliss[105X    by    Tommi    Junttila    and   Petteri   Kaski.
  [10XNautyCanonicalLabelling[110X returns a canonical labelling of the digraph [3Xdigraph[103X
  found  using  [5Xnauty[105X  by  Brendan  McKay  and  Adolfo  Piperno. Note that the
  canonical  labellings  returned  by [5Xbliss[105X and [5Xnauty[105X are not usually the same
  (and may depend of the version used).[133X
  
  [33X[0;0Y[10XBlissCanonicalLabelling[110X  can  only  be  computed  if [3Xdigraph[103X has no multiple
  edges; see [2XIsMultiDigraph[102X ([14X6.2-11[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := DigraphFromDiSparse6String(".ImNS_AiB?qRN");[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XBlissCanonicalLabelling(digraph1);[127X[104X
    [4X[28X(1,9,5,7)(3,6,4,10)[128X[104X
    [4X[25Xgap>[125X [27Xp := (1, 2, 7, 5)(3, 9)(6, 10, 8);;[127X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := OnDigraphs(digraph1, p);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph1 = digraph2;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph1, BlissCanonicalLabelling(digraph1)) =[127X[104X
    [4X[25X>[125X [27X   OnDigraphs(digraph2, BlissCanonicalLabelling(digraph2));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-8 BlissCanonicalLabelling[101X
  
  [33X[1;0Y[29X[2XBlissCanonicalLabelling[102X( [3Xdigraph[103X, [3Xcolours[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNautyCanonicalLabelling[102X( [3Xdigraph[103X, [3Xcolours[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation.[133X
  
  [33X[0;0YA  function  [22Xρ[122X  that  maps  a  coloured  digraph  to a coloured digraph is a
  [13Xcanonical  representative  map[113X  if the following two conditions hold for all
  coloured digraphs [22XG[122X and [22XH[122X:[133X
  
  [30X    [33X[0;6Y[22Xρ(G)[122X and [22XG[122X are isomorphic as coloured digraphs; and[133X
  
  [30X    [33X[0;6Y[22Xρ(G)=ρ(H)[122X if and only if [22XG[122X and [22XH[122X are isomorphic as coloured digraphs.[133X
  
  [33X[0;0YA [13Xcanonical labelling[113X of a coloured digraph [22XG[122X (under [22Xρ[122X) is an isomorphism of
  [22XG[122X  onto its [13Xcanonical representative[113X, [22Xρ(G)[122X. See [2XIsomorphismDigraphs[102X ([14X7.2-18[114X)
  for more information about isomorphisms of coloured digraphs.[133X
  
  [33X[0;0YA  coloured  digraph  can be specified by its underlying digraph [3Xdigraph[103X and
  its  colouring  [3Xcolours[103X.  Let  [10Xn[110X  be  the number of vertices of [3Xdigraph[103X. The
  colouring [3Xcolours[103X may have one of the following two forms:[133X
  
  [30X    [33X[0;6Ya  list  of  [10Xn[110X  integers,  where [3Xcolours[103X[10X[i][110X is the colour of vertex [10Xi[110X,
        using the colours [10X[1 .. m][110X for some [10Xm <= n[110X; or[133X
  
  [30X    [33X[0;6Ya    list    of    non-empty    disjoint    lists   whose   union   is
        [10XDigraphVertices([3Xdigraph[103X[10X)[110X,  such  that  [3Xcolours[103X[10X[i][110X  is  the list of all
        vertices with colour [10Xi[110X.[133X
  
  [33X[0;0YIf    [3Xdigraph[103X    and    [3Xcolours[103X    together   form   a   coloured   digraph,
  [10XBlissCanonicalLabelling[110X returns a canonical labelling of the digraph [3Xdigraph[103X
  found   using   [5Xbliss[105X  by  Tommi  Junttila  and  Petteri  Kaski.  Similarly,
  [10XNautyCanonicalLabelling[110X returns a canonical labelling of the digraph [3Xdigraph[103X
  found  using  [5Xnauty[105X  by  Brendan  McKay  and  Adolfo  Piperno. Note that the
  canonical  labellings  returned  by [5Xbliss[105X and [5Xnauty[105X are not usually the same
  (and may depend of the version used).[133X
  
  [33X[0;0Y[10XBlissCanonicalLabelling[110X  can  only  be  computed  if [3Xdigraph[103X has no multiple
  edges;  see  [2XIsMultiDigraph[102X  ([14X6.2-11[114X). The canonical labelling of [3Xdigraph[103X is
  given  as  a  permutation  of  its vertices. The canonical representative of
  [3Xdigraph[103X  can  be created from [3Xdigraph[103X and its canonical labelling [10Xp[110X by using
  the operation [2XOnDigraphs[102X ([14X7.1-1[114X):[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph, p);[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  colouring  of the canonical representative can easily be constructed. A
  vertex  [10Xv[110X  (in [3Xdigraph[103X) has colour [10Xi[110X if and only if the vertex [10Xv ^ p[110X (in the
  canonical  representative)  has  colour [10Xi[110X, where [10Xp[110X is the permutation of the
  canonical  labelling that acts on the vertices of [3Xdigraph[103X. In particular, if
  [3Xcolours[103X  has  the  first form that is described above, then the colouring of
  the canonical representative is given by:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList(DigraphVertices(digraph), i -> colours[i / p]);[127X[104X
  [4X[32X[104X
  
  [33X[0;0YOn  the other hand, if [3Xcolours[103X has the second form above, then the canonical
  representative has colouring:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOnTuplesSets(colours, p);[127X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph := DigraphFromDiSparse6String(".ImNS_AiB?qRN");[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27Xcolours := [[1, 2, 8, 9, 10], [3, 4, 5, 6, 7]];;[127X[104X
    [4X[25Xgap>[125X [27Xp := BlissCanonicalLabelling(digraph, colours);[127X[104X
    [4X[28X(1,5,8,4,10,3,9)(6,7)[128X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph, p);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XOnTuplesSets(colours, p);[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcolours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;[127X[104X
    [4X[25Xgap>[125X [27Xp := BlissCanonicalLabelling(digraph, colours);[127X[104X
    [4X[28X(1,6,9,7)(3,4,5,8,10)[128X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph, p);[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XList(DigraphVertices(digraph), i -> colours[i / p]);[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ][128X[104X
  [4X[32X[104X
  
  [1X7.2-9 BlissCanonicalDigraph[101X
  
  [33X[1;0Y[29X[2XBlissCanonicalDigraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XNautyCanonicalDigraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  attribute  [10XBlissCanonicalLabelling[110X returns the canonical representative
  found  by  applying [2XBlissCanonicalLabelling[102X ([14X7.2-7[114X). The digraph returned is
  canonical in the sense that[133X
  
  [30X    [33X[0;6Y[10XBlissCanonicalDigraph([3Xdigraph[103X[10X)[110X and [3Xdigraph[103X are isomorphic as digraphs;
        and[133X
  
  [30X    [33X[0;6YIf    [10Xgr[110X   is   any   digraph   then   [10XBlissCanonicalDigraph(gr)[110X   and
        [10XBlissCanonicalDigraph([3Xdigraph[103X[10X)[110X are equal if and only if [10Xgr[110X and [3Xdigraph[103X
        are isomorphic as digraphs.[133X
  
  [33X[0;0YAnalogously,  the  attribute  [10XNautyCanonicalLabelling[110X  returns the canonical
  representative found by applying [2XNautyCanonicalLabelling[102X ([14X7.2-7[114X).[133X
  
  [33X[0;0YIf  the argument [3Xdigraph[103X is mutable, then the return value of this attribute
  is recomputed every time it is called.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27Xcanon := BlissCanonicalDigraph(digraph);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(canon);[127X[104X
    [4X[28X[ [ 1 ], [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X7.2-10 DigraphGroup[101X
  
  [33X[1;0Y[29X[2XDigraphGroup[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  immutable  and  was  created  knowing  a  subgroup  of  its
  automorphism group, then this group is stored in the attribute [10XDigraphGroup[110X.
  If  [3Xdigraph[103X  is  mutable,  or  was  not  created  knowing  a subgroup of its
  automorphism  group, then [10XDigraphGroup[110X returns the entire automorphism group
  of  [3Xdigraph[103X. Note that if [3Xdigraph[103X is mutable, then the automorphism group is
  recomputed every time this function is called.[133X
  
  [33X[0;0YNote  that  certain  other  constructor  operations  such  as  [2XCayleyDigraph[102X
  ([14X3.1-12[114X),  [2XBipartiteDoubleDigraph[102X  ([14X3.3-43[114X), and [2XDoubleDigraph[102X ([14X3.3-42[114X), may
  not  require  a  group  as  one  of  the  arguments,  but  use  the standard
  constructor  method  using a group, and hence set the [10XDigraphGroup[110X attribute
  for the resulting digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := 4;;[127X[104X
    [4X[25Xgap>[125X [27Xadj := function(x, y)[127X[104X
    [4X[25X>[125X [27X     return (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);[127X[104X
    [4X[25X>[125X [27X   end;;[127X[104X
    [4X[25Xgap>[125X [27Xgroup := CyclicGroup(IsPermGroup, n);[127X[104X
    [4X[28XGroup([ (1,2,3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, group, [1 .. n], \^, adj);[127X[104X
    [4X[28X<mutable digraph with 4 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XHasDigraphGroup(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphGroup(D);[127X[104X
    [4X[28XGroup([ (2,4), (1,2)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XAutomorphismGroup(D);[127X[104X
    [4X[28XGroup([ (2,4), (1,2)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(group, [1 .. n], \^, adj);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XHasDigraphGroup(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphGroup(D);[127X[104X
    [4X[28XGroup([ (1,2,3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XD := DoubleDigraph(D);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 32 edges>[128X[104X
    [4X[25Xgap>[125X [27XHasDigraphGroup(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphGroup(D);[127X[104X
    [4X[28XGroup([ (1,2,3,4)(5,6,7,8), (1,5)(2,6)(3,7)(4,8) ])[128X[104X
    [4X[25Xgap>[125X [27XAutomorphismGroup(D) =[127X[104X
    [4X[25X>[125X [27XGroup([(6, 8), (5, 7), (4, 6), (3, 5), (2, 4),[127X[104X
    [4X[25X>[125X [27X       (1, 2)(3, 4)(5, 6)(7, 8)]);[127X[104X
    [4X[28Xtrue[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 [27XHasDigraphGroup(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XHasAutomorphismGroup(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphGroup(D);[127X[104X
    [4X[28XGroup([ (2,3) ])[128X[104X
    [4X[25Xgap>[125X [27XHasAutomorphismGroup(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgroup := DihedralGroup(8);[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XD := CayleyDigraph(group);[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XHasDigraphGroup(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(DigraphGroup(D));[127X[104X
    [4X[28X[ (1,2)(3,5)(4,6)(7,8), (1,7,4,3)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ][128X[104X
  [4X[32X[104X
  
  [1X7.2-11 DigraphOrbits[101X
  
  [33X[1;0Y[29X[2XDigraphOrbits[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of lists of integers.[133X
  
  [33X[0;0Y[10XDigraphOrbits[110X  returns the orbits of the action of the [2XDigraphGroup[102X ([14X7.2-10[114X)
  on the set of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := Group([(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;[127X[104X
    [4X[25Xgap>[125X [27XD := EdgeOrbitsDigraph(G, [1, 2]);[127X[104X
    [4X[28X<immutable digraph with 9 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbits(D);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopy(D);[127X[104X
    [4X[28X<mutable digraph with 9 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbits(D);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 4, 5, 6, 7, 8, 9 ] ][128X[104X
  [4X[32X[104X
  
  [1X7.2-12 DigraphOrbitReps[101X
  
  [33X[1;0Y[29X[2XDigraphOrbitReps[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of integers.[133X
  
  [33X[0;0Y[10XDigraphOrbitReps[110X  returns  a  list of orbit representatives of the action of
  the [2XDigraphGroup[102X ([14X7.2-10[114X) on the set of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CayleyDigraph(AlternatingGroup(4));[127X[104X
    [4X[28X<immutable digraph with 12 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbitReps(D);[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopy(D);[127X[104X
    [4X[28X<mutable digraph with 12 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbitReps(D);[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XD := DigraphFromDigraph6String("&IGO??S?`?_@?a?CK?O");[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 14 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbitReps(D);[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbitReps(DigraphMutableCopy(D));[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ][128X[104X
  [4X[32X[104X
  
  [1X7.2-13 DigraphSchreierVector[101X
  
  [33X[1;0Y[29X[2XDigraphSchreierVector[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of integers.[133X
  
  [33X[0;0Y[10XDigraphSchreierVector[110X returns the so-called [13XSchreier vector[113X of the action of
  the  [2XDigraphGroup[102X  ([14X7.2-10[114X)  on the set of vertices of [3Xdigraph[103X. The Schreier
  vector  is  a  list  [10Xsch[110X  of integers with length [10XDigraphNrVertices([3Xdigraph[103X[10X)[110X
  where:[133X
  
  [8X[10Xsch[i] < 0:[110X[8X[108X
        [33X[0;6Yimplies     that     [10Xi[110X     is     an    orbit    representative    and
        [10XDigraphOrbitReps([3Xdigraph[103X[10X)[-sch[i]] = i[110X.[133X
  
  [8X[10Xsch[i] > 0:[110X[8X[108X
        [33X[0;6Yimplies  that  [10Xi  /  gens[sch[i]][110X  is  one step closer to the root (or
        representative)   of  the  tree,  where  [10Xgens[110X  is  the  generators  of
        [10XDigraphGroup([3Xdigraph[103X[10X)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := 4;;[127X[104X
    [4X[25Xgap>[125X [27Xadj := function(x, y)[127X[104X
    [4X[25X>[125X [27X     return (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);[127X[104X
    [4X[25X>[125X [27X   end;;[127X[104X
    [4X[25Xgap>[125X [27Xgroup := CyclicGroup(IsPermGroup, n);[127X[104X
    [4X[28XGroup([ (1,2,3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, group, [1 .. n], \^, adj);[127X[104X
    [4X[28X<mutable digraph with 4 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27Xsch := DigraphSchreierVector(D);[127X[104X
    [4X[28X[ -1, 2, 2, 1 ][128X[104X
    [4X[25Xgap>[125X [27XD := CayleyDigraph(AlternatingGroup(4));[127X[104X
    [4X[28X<immutable digraph with 12 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xsch := DigraphSchreierVector(D);[127X[104X
    [4X[28X[ -1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphOrbitReps(D);[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27Xgens := GeneratorsOfGroup(DigraphGroup(D));[127X[104X
    [4X[28X[ (1,7,5)(2,10,9)(3,4,11)(6,8,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12) ][128X[104X
    [4X[25Xgap>[125X [27X10 / gens[sch[10]];[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27X7 / gens[sch[7]];[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27X5 / gens[sch[5]];[127X[104X
    [4X[28X7[128X[104X
  [4X[32X[104X
  
  [1X7.2-14 DigraphStabilizer[101X
  
  [33X[1;0Y[29X[2XDigraphStabilizer[102X( [3Xdigraph[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation group.[133X
  
  [33X[0;0Y[10XDigraphStabilizer[110X returns the stabilizer of the vertex [3Xv[103X under of the action
  of the [2XDigraphGroup[102X ([14X7.2-10[114X) on the set of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphFromDigraph6String("&GYHPQgWTIIPW");[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphStabilizer(D, 8);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XDigraphStabilizer(D, 2);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphMutableCopy(D);[127X[104X
    [4X[28X<mutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphStabilizer(D, 8);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XDigraphStabilizer(D, 2);[127X[104X
    [4X[28XGroup(())[128X[104X
  [4X[32X[104X
  
  [1X7.2-15 IsIsomorphicDigraph[101X
  
  [33X[1;0Y[29X[2XIsIsomorphicDigraph[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  operation returns [9Xtrue[109X if there exists an isomorphism from the digraph
  [3Xdigraph1[103X  to the digraph [3Xdigraph2[103X. See [2XIsomorphismDigraphs[102X ([14X7.2-17[114X) for more
  information about isomorphisms of digraphs.[133X
  
  [33X[0;0YBy  default,  an  isomorphism is found using the canonical labellings of the
  digraphs  obtained  from  [5Xbliss[105X  by  Tommi  Junttila  and  Petteri Kaski. If
  [5XNautyTracesInterface[105X  is  available,  then [5Xnauty[105X by Brendan Mckay and Adolfo
  Piperno   can   be   used   instead;   see   [2XDigraphsUseBliss[102X  ([14X7.2-1[114X),  and
  [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := CycleDigraph(4);[127X[104X
    [4X[28X<immutable cycle digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := CycleDigraph(5);[127X[104X
    [4X[28X<immutable cycle digraph with 5 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := DigraphReverse(digraph1);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := Digraph([[3], [], []]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := Digraph([[], [], [2]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-16 IsIsomorphicDigraph[101X
  
  [33X[1;0Y[29X[2XIsIsomorphicDigraph[102X( [3Xdigraph1[103X, [3Xdigraph2[103X, [3Xcolours1[103X, [3Xcolours2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  operation  tests  for  isomorphism  of  coloured  digraphs. A coloured
  digraph  can  be  specified  by  its  underlying  digraph  [3Xdigraph1[103X  and its
  colouring  [3Xcolours1[103X.  Let  [10Xn[110X  be  the  number  of  vertices of [3Xdigraph1[103X. The
  colouring [3Xcolours1[103X may have one of the following two forms:[133X
  
  [30X    [33X[0;6Ya  list  of  [10Xn[110X  integers,  where [3Xcolours[103X[10X[i][110X is the colour of vertex [10Xi[110X,
        using the colours [10X[1 .. m][110X for some [10Xm <= n[110X; or[133X
  
  [30X    [33X[0;6Ya    list    of    non-empty    disjoint    lists   whose   union   is
        [10XDigraphVertices([3Xdigraph[103X[10X)[110X,  such  that  [3Xcolours[103X[10X[i][110X  is  the list of all
        vertices with colour [10Xi[110X.[133X
  
  [33X[0;0YIf  [3Xdigraph1[103X  and [3Xdigraph2[103X are digraphs without multiple edges, and [3Xcolours1[103X
  and  [3Xcolours2[103X  are  colourings  of [3Xdigraph1[103X and [3Xdigraph2[103X, respectively, then
  this operation returns [9Xtrue[109X if there exists an isomorphism between these two
  coloured  digraphs.  See  [2XIsomorphismDigraphs[102X  ([14X7.2-18[114X) for more information
  about isomorphisms of coloured digraphs.[133X
  
  [33X[0;0YBy  default,  an  isomorphism is found using the canonical labellings of the
  digraphs  obtained  from  [5Xbliss[105X  by  Tommi  Junttila  and  Petteri Kaski. If
  [5XNautyTracesInterface[105X  is  available,  then [5Xnauty[105X by Brendan Mckay and Adolfo
  Piperno   can   be   used   instead;   see   [2XDigraphsUseBliss[102X  ([14X7.2-1[114X),  and
  [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := ChainDigraph(4);[127X[104X
    [4X[28X<immutable chain digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := ChainDigraph(3);[127X[104X
    [4X[28X<immutable chain digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2,[127X[104X
    [4X[25X>[125X [27X [[1, 4], [2, 3]], [[1, 2], [3]]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := DigraphReverse(digraph1);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2,[127X[104X
    [4X[25X>[125X [27X [1, 1, 1, 1], [1, 1, 1, 1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2,[127X[104X
    [4X[25X>[125X [27X [1, 2, 2, 1], [1, 2, 2, 1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicDigraph(digraph1, digraph2,[127X[104X
    [4X[25X>[125X [27X [1, 1, 2, 2], [1, 1, 2, 2]);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X7.2-17 IsomorphismDigraphs[101X
  
  [33X[1;0Y[29X[2XIsomorphismDigraphs[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation, or a pair of permutations, or [9Xfail[109X.[133X
  
  [33X[0;0YThis  operation  returns  an  isomorphism  between the digraphs [3Xdigraph1[103X and
  [3Xdigraph2[103X if one exists, else this operation returns [9Xfail[109X.[133X
  
  [33X[0;0YAn  [13Xisomorphism[113X from a digraph [3Xdigraph1[103X to a digraph [3Xdigraph2[103X is a bijection
  [10Xp[110X  from  the  vertices  of  [3Xdigraph1[103X  to  the  vertices of [3Xdigraph2[103X with the
  following  property: for all vertices [10Xi[110X and [10Xj[110X of [3Xdigraph1[103X, [10X[i, j][110X is an edge
  of [3Xdigraph1[103X if and only if [10X[i ^ p, j ^ p][110X is an edge of [3Xdigraph2[103X.[133X
  
  [33X[0;0YIf  there  exists  such an isomorphism, then this operation returns one. The
  form of this isomorphism is a permutation [10Xp[110X of the vertices of [3Xdigraph1[103X such
  that[133X
  
  [33X[0;0Y[10XOnDigraphs([3Xdigraph1[103X[10X,  p)  =  digraph2[110X.  By  default, an isomorphism is found
  using  the canonical labellings of the digraphs obtained from [5Xbliss[105X by Tommi
  Junttila and Petteri Kaski. If [5XNautyTracesInterface[105X is available, then [5Xnauty[105X
  by   Brendan   Mckay   and   Adolfo   Piperno   can  be  used  instead;  see
  [2XDigraphsUseBliss[102X ([14X7.2-1[114X), and [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := CycleDigraph(4);[127X[104X
    [4X[28X<immutable cycle digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := CycleDigraph(5);[127X[104X
    [4X[28X<immutable cycle digraph with 5 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismDigraphs(digraph1, digraph2);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := CompleteBipartiteDigraph(10, 5);[127X[104X
    [4X[28X<immutable complete bipartite digraph with bicomponent sizes 10 and 5>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := CompleteBipartiteDigraph(5, 10);[127X[104X
    [4X[28X<immutable complete bipartite digraph with bicomponent sizes 5 and 10>[128X[104X
    [4X[25Xgap>[125X [27Xp := IsomorphismDigraphs(digraph1, digraph2);[127X[104X
    [4X[28X(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)[128X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph1, p) = digraph2;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-18 IsomorphismDigraphs[101X
  
  [33X[1;0Y[29X[2XIsomorphismDigraphs[102X( [3Xdigraph1[103X, [3Xdigraph2[103X, [3Xcolours1[103X, [3Xcolours2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA permutation, or [9Xfail[109X.[133X
  
  [33X[0;0YThis  operation  searches  for  an  isomorphism between coloured digraphs. A
  coloured digraph can be specified by its underlying digraph [3Xdigraph1[103X and its
  colouring  [3Xcolours1[103X.  Let  [10Xn[110X  be  the  number  of  vertices of [3Xdigraph1[103X. The
  colouring [3Xcolours1[103X may have one of the following two forms:[133X
  
  [30X    [33X[0;6Ya  list  of  [10Xn[110X  integers,  where [3Xcolours[103X[10X[i][110X is the colour of vertex [10Xi[110X,
        using the colours [10X[1 .. m][110X for some [10Xm <= n[110X; or[133X
  
  [30X    [33X[0;6Ya    list    of    non-empty    disjoint    lists   whose   union   is
        [10XDigraphVertices([3Xdigraph[103X[10X)[110X,  such  that  [3Xcolours[103X[10X[i][110X  is  the list of all
        vertices with colour [10Xi[110X.[133X
  
  [33X[0;0YAn  [13Xisomorphism[113X  between  coloured  digraphs  is  an isomorphism between the
  underlying  digraphs  that preserves the colourings. See [2XIsomorphismDigraphs[102X
  ([14X7.2-17[114X)   for   more  information  about  isomorphisms  of  digraphs.  More
  precisely,  let  [10Xf[110X  be  an isomorphism of digraphs from the digraph [3Xdigraph1[103X
  (with colouring [3Xcolours1[103X) to the digraph [3Xdigraph2[103X (with colouring [3Xcolours2[103X),
  and let [10Xp[110X be the permutation of the vertices of [3Xdigraph1[103X that corresponds to
  [10Xf[110X.  Then  [10Xf[110X preserves the colourings of [3Xdigraph1[103X and [3Xdigraph2[103X – and hence is
  an  isomorphism  of coloured digraphs – if [10X[3Xcolours1[103X[10X[i] = [3Xcolours2[103X[10X[i ^ p][110X for
  all vertices [10Xi[110X in [3Xdigraph1[103X.[133X
  
  [33X[0;0YThis  operation  returns  such an isomorphism if one exists, else it returns
  [9Xfail[109X.[133X
  
  [33X[0;0YBy  default,  an  isomorphism is found using the canonical labellings of the
  digraphs  obtained  from  [5Xbliss[105X  by  Tommi  Junttila  and  Petteri Kaski. If
  [5XNautyTracesInterface[105X  is  available,  then [5Xnauty[105X by Brendan Mckay and Adolfo
  Piperno   can   be   used   instead;   see   [2XDigraphsUseBliss[102X  ([14X7.2-1[114X),  and
  [2XDigraphsUseNauty[102X ([14X7.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph1 := ChainDigraph(4);[127X[104X
    [4X[28X<immutable chain digraph with 4 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := ChainDigraph(3);[127X[104X
    [4X[28X<immutable chain digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismDigraphs(digraph1, digraph2,[127X[104X
    [4X[25X>[125X [27X [[1, 4], [2, 3]], [[1, 2], [3]]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xdigraph2 := DigraphReverse(digraph1);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xcolours1 := [1, 1, 1, 1];;[127X[104X
    [4X[25Xgap>[125X [27Xcolours2 := [1, 1, 1, 1];;[127X[104X
    [4X[25Xgap>[125X [27Xp := IsomorphismDigraphs(digraph1, digraph2, colours1, colours2);[127X[104X
    [4X[28X(1,4)(2,3)[128X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph1, p) = digraph2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList(DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcolours1 := [1, 1, 2, 2];;[127X[104X
    [4X[25Xgap>[125X [27Xcolours2 := [2, 2, 1, 1];;[127X[104X
    [4X[25Xgap>[125X [27Xp := IsomorphismDigraphs(digraph1, digraph2, colours1, colours2);[127X[104X
    [4X[28X(1,4)(2,3)[128X[104X
    [4X[25Xgap>[125X [27XOnDigraphs(digraph1, p) = digraph2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList(DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismDigraphs(digraph1, digraph2,[127X[104X
    [4X[25X>[125X [27X [1, 1, 2, 2], [1, 1, 2, 2]);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X7.2-19 RepresentativeOutNeighbours[101X
  
  [33X[1;0Y[29X[2XRepresentativeOutNeighbours[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of lists.[133X
  
  [33X[0;0YThis  function returns the list [10Xout[110X of [13Xout-neighbours[113X of each representative
  of  the  orbits  of the action of [2XDigraphGroup[102X ([14X7.2-10[114X) on the vertex set of
  the digraph [3Xdigraph[103X.[133X
  
  [33X[0;0YMore  specifically,  if  [10Xreps[110X  is  the list of orbit representatives, then a
  vertex  [10Xj[110X  appears  in  [10Xout[i][110X  each  time  there exists an edge with source
  [10Xreps[i][110X and range [10Xj[110X in [3Xdigraph[103X.[133X
  
  [33X[0;0YIf [2XDigraphGroup[102X ([14X7.2-10[114X) is trivial, then [2XOutNeighbours[102X ([14X5.2-6[114X) is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[127X[104X
    [4X[25X>[125X [27X [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 16 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphGroup(D);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XRepresentativeOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[127X[104X
    [4X[25X>[125X [27X [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 16 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphGroup(D);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XRepresentativeOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := DigraphFromDigraph6String("&GYHPQgWTIIPW");[127X[104X
    [4X[28X<immutable digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := DigraphGroup(D);;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(G);[127X[104X
    [4X[28X[ (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1,5)(2,6)(3,8)(4,7) ][128X[104X
    [4X[25Xgap>[125X [27XSet(RepresentativeOutNeighbours(D), Set);[127X[104X
    [4X[28X[ [ 2, 3, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X7.2-20 IsDigraphIsomorphism[101X
  
  [33X[1;0Y[29X[2XIsDigraphIsomorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphIsomorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X, [3Xcol1[103X, [3Xcol2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphAutomorphism[102X( [3Xdigraph[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphAutomorphism[102X( [3Xdigraph[103X, [3Xx[103X, [3Xcol[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0Y[10XIsDigraphIsomorphism[110X  returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is
  an isomorphism from the digraph [3Xsrc[103X to the digraph [3Xran[103X.[133X
  
  [33X[0;0Y[10XIsDigraphAutomorphism[110X returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is
  an automorphism of the digraph [3Xdigraph[103X.[133X
  
  [33X[0;0YA  permutation or transformation [3Xx[103X is an [13Xisomorphism[113X from a digraph [3Xsrc[103X to a
  digraph [3Xran[103X if the following hold:[133X
  
  [30X    [33X[0;6Y[3Xx[103X is a bijection from the vertices of [3Xsrc[103X to those of [3Xran[103X;[133X
  
  [30X    [33X[0;6Y[10X[u  ^  [3Xx[103X[10X, v ^ [3Xx[103X[10X][110X is an edge of [3Xran[103X if and only if [10X[u, v][110X is an edge of
        [3Xsrc[103X; and[133X
  
  [30X    [33X[0;6Y[3Xx[103X fixes every [10Xi[110X which is not a vertex of [3Xsrc[103X.[133X
  
  [33X[0;0YSee also [2XAutomorphismGroup[102X ([14X7.2-2[114X).[133X
  
  [33X[0;0YIf  [3Xcol1[103X  and  [3Xcol2[103X,  or  [3Xcol[103X,  are  given,  then they must represent vertex
  colourings;  see  [2XAutomorphismGroup[102X  ([14X7.2-5[114X)  for details of the permissible
  values  for  these  arguments.  The  homomorphism  must  then  also have the
  property:[133X
  
  [30X    [33X[0;6Y[10Xcol1[i]   =   col2[i   ^   x][110X   for   all   vertices  [10Xi[110X  of  [3Xsrc[103X,  for
        [10XIsDigraphIsomorphism[110X.[133X
  
  [30X    [33X[0;6Y[10Xcol[i]   =   col[i   ^   x][110X   for  all  vertices  [10Xi[110X  of  [3Xdigraph[103X,  for
        [10XIsDigraphAutomorphism[110X.[133X
  
  [33X[0;0YFor  some  digraphs,  it  can be faster to use [10XIsDigraphAutomorphism[110X than to
  test membership in the automorphism group of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsrc := Digraph([[1], [1, 2], [1, 3]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, (1, 2, 3));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, (2, 3));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, (2, 3), [2, 1, 1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, (2, 3)(4, 5));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, (1, 4));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphAutomorphism(src, ());[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xran := Digraph([[2, 1], [2], [2, 3]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphIsomorphism(src, ran, (1, 2));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphIsomorphism(ran, src, (1, 2));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphIsomorphism(ran, src, (1, 2));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X7.2-21 IsDigraphColouring[101X
  
  [33X[1;0Y[29X[2XIsDigraphColouring[102X( [3Xdigraph[103X, [3Xlist[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphColouring[102X( [3Xdigraph[103X, [3Xt[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThe  operation  [10XIsDigraphColouring[110X  verifies  whether  or  not the list [3Xlist[103X
  describes a proper colouring of the digraph [3Xdigraph[103X.[133X
  
  [33X[0;0YA  list  [3Xlist[103X  describes  a  [13Xproper  colouring[113X  of a digraph [3Xdigraph[103X if [3Xlist[103X
  consists  of  positive  integers,  the  length  of [3Xlist[103X equals the number of
  vertices  in  [3Xdigraph[103X,  and  for any vertices [10Xu, v[110X of [3Xdigraph[103X if [10Xu[110X and [10Xv[110X are
  adjacent, then [10X[3Xlist[103X[10X[u] >< [3Xlist[103X[10X[v][110X.[133X
  
  [33X[0;0YA  transformation  [3Xt[103X  describes  a proper colouring of a digraph [3Xdigraph[103X, if
  [10XImageListOfTransformation([3Xt[103X[10X,   DigraphNrVertices([3Xdigraph[103X[10X))[110X   is   a   proper
  colouring of [3Xdigraph[103X.[133X
  
  [33X[0;0YSee also [2XIsDigraphHomomorphism[102X ([14X7.3-10[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JohnsonDigraph(5, 3);[127X[104X
    [4X[28X<immutable symmetric digraph with 10 vertices, 60 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, 7]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphColouring(D, [1, 2, 3]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphColouring(D, IdentityTransformation);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-22 MaximalCommonSubdigraph[101X
  
  [33X[1;0Y[29X[2XMaximalCommonSubdigraph[102X( [3XD1[103X, [3XD2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list containing a digraph and two transformations.[133X
  
  [33X[0;0YIf    [3XD1[103X    and    [3XD2[103X    are   digraphs   without   multiple   edges,   then
  [10XMaximalCommonSubdigraph[110X  returns  a  maximal  common subgraph [10XM[110X of [3XD1[103X and [3XD2[103X
  with  the  maximum  number  of vertices. So [10XM[110X is a digraph which embeds into
  both  [3XD1[103X  and [3XD2[103X and has the largest number of vertices among such digraphs.
  It  returns  a list [10X[M, t1, t2][110X where [10XM[110X is the maximal common subdigraph and
  [10Xt1, t2[110X are transformations embedding [10XM[110X into [3XD1[103X and [3XD2[103X respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximalCommonSubdigraph(PetersenGraph(), CompleteDigraph(10));[127X[104X
    [4X[28X[ <immutable digraph with 2 vertices, 2 edges>, [128X[104X
    [4X[28X  IdentityTransformation, IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27XMaximalCommonSubdigraph(PetersenGraph(),[127X[104X
    [4X[25X>[125X [27XDigraphSymmetricClosure(CycleDigraph(5)));[127X[104X
    [4X[28X[ <immutable digraph with 5 vertices, 10 edges>, [128X[104X
    [4X[28X  IdentityTransformation, IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27XMaximalCommonSubdigraph(NullDigraph(0), CompleteDigraph(10));[127X[104X
    [4X[28X[ <immutable empty digraph with 0 vertices>, IdentityTransformation, [128X[104X
    [4X[28X  IdentityTransformation ][128X[104X
  [4X[32X[104X
  
  [1X7.2-23 MinimalCommonSuperdigraph[101X
  
  [33X[1;0Y[29X[2XMinimalCommonSuperdigraph[102X( [3XD1[103X, [3XD2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list containing a digraph and two transformations.[133X
  
  [33X[0;0YIf    [3XD1[103X    and    [3XD2[103X    are   digraphs   without   multiple   edges,   then
  [10XMinimalCommonSuperdigraph[110X  returns a minimal common superdigraph [10XM[110X of [3XD1[103X and
  [3XD2[103X with the minimum number of vertices. So [10XM[110X is a digraph into which both [3XD1[103X
  and [3XD2[103X embed and has the smallest number of vertices among such digraphs. It
  returns  a  list  [10X[M, t1, t2][110X where [10XM[110X is the minimal common superdigraph and
  [10Xt1, t2[110X are transformations embedding [3XD1[103X and [3XD2[103X respectively into [10XM[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalCommonSuperdigraph(PetersenGraph(), CompleteDigraph(10));[127X[104X
    [4X[28X[ <immutable digraph with 18 vertices, 118 edges>, [128X[104X
    [4X[28X  IdentityTransformation, [128X[104X
    [4X[28X  Transformation( [ 1, 2, 11, 12, 13, 14, 15, 16, 17, 18, 11, 12, 13,[128X[104X
    [4X[28X      14, 15, 16, 17, 18 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XMinimalCommonSuperdigraph(PetersenGraph(),[127X[104X
    [4X[25X>[125X [27XDigraphSymmetricClosure(CycleDigraph(5)));[127X[104X
    [4X[28X[ <immutable digraph with 10 vertices, 30 edges>, [128X[104X
    [4X[28X  IdentityTransformation, IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27XMinimalCommonSuperdigraph(NullDigraph(0), CompleteDigraph(10));[127X[104X
    [4X[28X[ <immutable digraph with 10 vertices, 90 edges>, [128X[104X
    [4X[28X  IdentityTransformation, IdentityTransformation ][128X[104X
  [4X[32X[104X
  
  
  [1X7.3 [33X[0;0YHomomorphisms of digraphs[133X[101X
  
  [33X[0;0YThe  following  methods  exist to find homomorphisms between digraphs. If an
  argument  to one of these methods is a digraph with multiple edges, then the
  multiplicity  of  edges will be ignored in order to perform the calculation;
  the digraph will be treated as if it has no multiple edges.[133X
  
  [1X7.3-1 HomomorphismDigraphsFinder[101X
  
  [33X[1;0Y[29X[2XHomomorphismDigraphsFinder[102X( [3XD1[103X, [3XD2[103X, [3Xhook[103X, [3Xuser_param[103X, [3Xmax_results[103X, [3Xhint[103X, [3Xinjective[103X, [3Ximage[103X, [3Xpartial_map[103X, [3Xcolors1[103X, [3Xcolors2[103X[, [3Xorder[103X, [3Xaut_grp[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe argument [3Xuser_param[103X.[133X
  
  [33X[0;0YThis  function  finds  homomorphisms  from  the digraph [3XD1[103X to the digraph [3XD2[103X
  subject to the conditions imposed by the other arguments as described below.[133X
  
  [33X[0;0YIf  [10Xf[110X  and  [10Xg[110X  are homomorphisms found by [10XHomomorphismDigraphsFinder[110X, then [10Xf[110X
  cannot  be  obtained from [10Xg[110X by right multiplying by an automorphism of [3XD2[103X in
  [3Xaut_grp[103X.[133X
  
  [8X[3Xhook[103X[8X[108X
        [33X[0;6YThis argument should be a function or [9Xfail[109X.[133X
  
        [33X[0;6YIf [3Xhook[103X is a function, then it must have two arguments [3Xuser_param[103X (see
        below)  and  a  transformation  [10Xt[110X. The function [10X[3Xhook[103X[10X([3Xuser_param[103X[10X, t)[110X is
        called    every    time   a   new   homomorphism   [10Xt[110X   is   found   by
        [10XHomomorphismDigraphsFinder[110X.   If   the  function  returns  [9Xtrue[109X,  then
        [10XHomomorphismDigraphsFinder[110X   stops  and  does  not  find  any  further
        homomorphisms. This feature might be useful if you are searching for a
        homomorphism that satisfies some condition that you cannot specify via
        the other arguments to [10XHomomorphismDigraphsFinder[110X.[133X
  
        [33X[0;6YIf  [3Xhook[103X  is  [9Xfail[109X,  then a default function is used which simply adds
        every   new   homomorphism   found  by  [10XHomomorphismDigraphsFinder[110X  to
        [3Xuser_param[103X, which must be a mutable list in this case.[133X
  
  [8X[3Xuser_param[103X[8X[108X
        [33X[0;6YIf  [3Xhook[103X  is  a  function,  then [3Xuser_param[103X can be any [5XGAP[105X object. The
        object  [3Xuser_param[103X is used as the first argument of the function [3Xhook[103X.
        For  example,  [3Xuser_param[103X  might  be  a  transformation semigroup, and
        [10X[3Xhook[103X[10X([3Xuser_param[103X[10X,  t)[110X  might  set  [3Xuser_param[103X  to  be  the  closure  of
        [3Xuser_param[103X and [10Xt[110X.[133X
  
        [33X[0;6YIf  the  value of [3Xhook[103X is [9Xfail[109X, then the value of [3Xuser_param[103X must be a
        mutable list.[133X
  
  [8X[3Xmax_results[103X[8X[108X
        [33X[0;6YThis   argument   should   be   a   positive   integer   or  [9Xinfinity[109X.
        [10XHomomorphismDigraphsFinder[110X  will return after it has found [3Xmax_results[103X
        homomorphisms or the search is complete, whichever happens first.[133X
  
  [8X[3Xhint[103X[8X[108X
        [33X[0;6YThis argument should be a positive integer or [9Xfail[109X.[133X
  
        [33X[0;6YIf  [3Xhint[103X is a positive integer, then only homorphisms of rank [3Xhint[103X are
        found.[133X
  
        [33X[0;6YIf  [3Xhint[103X  is  [9Xfail[109X,  then  no  restriction  is  put  on  the  rank  of
        homomorphisms found.[133X
  
  [8X[3Xinjective[103X[8X[108X
        [33X[0;6YThis  argument  should be [10X0[110X, [10X1[110X, or [10X2[110X. If it is [10X2[110X, then only embeddings
        are  found,  if  it is [10X1[110X, then only injective homomorphisms are found,
        and if it is [10X0[110X there are no restrictions imposed by this argument.[133X
  
        [33X[0;6YFor backwards compatibility, [3Xinjective[103X can also be [9Xfalse[109X or [9Xtrue[109X which
        correspond  to the values [10X0[110X and [10X1[110X described in the previous paragraph,
        respectively.[133X
  
  [8X[3Ximage[103X[8X[108X
        [33X[0;6YThis  argument  should  be  a  subset of the vertices of the graph [3XD2[103X.
        [10XHomomorphismDigraphsFinder[110X  only  finds  homomorphisms  from [3XD1[103X to the
        subgraph of [3XD2[103X induced by the vertices [3Ximage[103X.[133X
  
  [8X[3Xpartial_map[103X[8X[108X
        [33X[0;6YThis  argument  should be a partial map from [3XD1[103X to [3XD2[103X, that is, a (not
        necessarily  dense)  list  of  vertices of the digraph [3XD2[103X of length no
        greater    than    the    number   vertices   in   the   digraph   [3XD1[103X.
        [10XHomomorphismDigraphsFinder[110X    only   finds   homomorphisms   extending
        [3Xpartial_map[103X (if any).[133X
  
  [8X[3Xcolors1[103X[8X[108X
        [33X[0;6YThis should be a list representing possible colours of vertices in the
        digraph   [3XD1[103X;   see  [2XAutomorphismGroup[102X  ([14X7.2-5[114X)  for  details  of  the
        permissible values for this argument.[133X
  
  [8X[3Xcolors2[103X[8X[108X
        [33X[0;6YThis should be a list representing possible colours of vertices in the
        digraph   [3XD2[103X;   see  [2XAutomorphismGroup[102X  ([14X7.2-5[114X)  for  details  of  the
        permissible values for this argument.[133X
  
  [8X[3Xorder[103X[8X[108X
        [33X[0;6YThe  optional  argument  [3Xorder[103X  specifies the order the vertices in [3XD1[103X
        appear  in  the  search for homomorphisms. The value of this parameter
        can  have  a  large impact on the runtime of the function. It seems in
        many    cases   to   be   a   good   idea   for   this   to   be   the
        [2XDigraphWelshPowellOrder[102X  ([14X7.3-16[114X),  i.e. vertices ordered from highest
        to lowest degree.[133X
  
        [33X[0;6YThe optional argument [3Xaut_grp[103X should be a subgroup of the automorphism
        group  of  [3XD2[103X.  This  function  returns  unique representatives of the
        homomorphisms  found  up  to  right multiplication by [3Xaut_grp[103X. If this
        argument  is  not specific, it defaults to the full automorphism group
        of [3XD2[103X, which may be costly to calculate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := ChainDigraph(10);[127X[104X
    [4X[28X<immutable chain digraph with 10 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphSymmetricClosure(D);[127X[104X
    [4X[28X<immutable symmetric digraph with 10 vertices, 18 edges>[128X[104X
    [4X[25Xgap>[125X [27XHomomorphismDigraphsFinder(D, D, fail, [], infinity, 2, 0,[127X[104X
    [4X[25X>[125X [27X[3, 4], [], fail, fail);[127X[104X
    [4X[28X[ Transformation( [ 3, 4, 3, 4, 3, 4, 3, 4, 3, 4 ] ), [128X[104X
    [4X[28X  Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XD2 := CompleteDigraph(6);;[127X[104X
    [4X[25Xgap>[125X [27XHomomorphismDigraphsFinder(D, D2, fail, [], 1, fail, 0,[127X[104X
    [4X[25X>[125X [27X[1 .. 6], [1, 2, 1], fail, fail);[127X[104X
    [4X[28X[ Transformation( [ 1, 2, 1, 3, 4, 5, 6, 1, 2, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xfunc := function(user_param, t)[127X[104X
    [4X[25X>[125X [27XAdd(user_param, t * user_param[1]);[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27XHomomorphismDigraphsFinder(D, D2, func, [Transformation([2, 2])],[127X[104X
    [4X[25X>[125X [27X3, fail, 0, [1 .. 6], [1, 2, 1], fail, fail);[127X[104X
    [4X[28X[ Transformation( [ 2, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XHomomorphismDigraphsFinder(NullDigraph(2), NullDigraph(3), fail,[127X[104X
    [4X[25X>[125X [27X[], infinity, fail, 1, [1, 2, 3], fail, fail, fail, fail,[127X[104X
    [4X[25X>[125X [27XGroup(()));[127X[104X
    [4X[28X[ IdentityTransformation, Transformation( [ 1, 3, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XHomomorphismDigraphsFinder(NullDigraph(2), NullDigraph(3), fail,[127X[104X
    [4X[25X>[125X [27X[], infinity, fail, 1, [1, 2, 3], fail, fail, fail, fail,[127X[104X
    [4X[25X>[125X [27XGroup((1, 2)));[127X[104X
    [4X[28X[ IdentityTransformation, Transformation( [ 1, 3, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 3 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X7.3-2 DigraphHomomorphism[101X
  
  [33X[1;0Y[29X[2XDigraphHomomorphism[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0YA homomorphism from [3Xdigraph1[103X to [3Xdigraph2[103X is a mapping from the vertex set of
  [3Xdigraph1[103X  to  a  subset of the vertices of [3Xdigraph2[103X, such that every pair of
  vertices  [10X[i,j][110X which has an edge [10Xi->j[110X is mapped to a pair of vertices [10X[a,b][110X
  which  has an edge [10Xa->b[110X. Note that non-adjacent vertices can still be mapped
  to adjacent vertices.[133X
  
  [33X[0;0Y[10XDigraphHomomorphism[110X  returns  a  single  homomorphism  between  [3Xdigraph1[103X and
  [3Xdigraph2[103X if it exists, otherwise it returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := ChainDigraph(3);;[127X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphHomomorphism(gr1, gr1);[127X[104X
    [4X[28XIdentityTransformation[128X[104X
    [4X[25Xgap>[125X [27Xmap := DigraphHomomorphism(gr1, gr2);[127X[104X
    [4X[28XTransformation( [ 3, 1, 5, 4, 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(gr1, gr2, map);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.3-3 HomomorphismsDigraphs[101X
  
  [33X[1;0Y[29X[2XHomomorphismsDigraphs[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHomomorphismsDigraphsRepresentatives[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0Y[10XHomomorphismsDigraphsRepresentatives[110X finds every [2XDigraphHomomorphism[102X ([14X7.3-2[114X)
  between  [3Xdigraph1[103X  and [3Xdigraph2[103X, up to right multiplication by an element of
  the   [2XAutomorphismGroup[102X   ([14X7.2-2[114X)   of   [3Xdigraph2[103X.  In  other  words,  every
  homomorphism  [10Xf[110X  between  [3Xdigraph1[103X  and  [3Xdigraph2[103X  can  be  written  as  the
  composition    [10Xf    =    g    *    x[110X,    where    [10Xg[110X    is    one    of   the
  [10XHomomorphismsDigraphsRepresentatives[110X and [10Xx[110X is an automorphism of [3Xdigraph2[103X.[133X
  
  [33X[0;0Y[10XHomomorphismsDigraphs[110X   returns   all  homomorphisms  between  [3Xdigraph1[103X  and
  [3Xdigraph2[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := ChainDigraph(3);;[127X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XHomomorphismsDigraphs(gr1, gr2);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 1 ] ), Transformation( [ 1, 3, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 1, 5, 4, 4, 5 ] ), Transformation( [ 2, 2, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1, 5, 4, 5 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XHomomorphismsDigraphsRepresentatives(gr1, CompleteDigraph(3));[127X[104X
    [4X[28X[ Transformation( [ 2, 1 ] ), Transformation( [ 2, 1, 2 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X7.3-4 DigraphMonomorphism[101X
  
  [33X[1;0Y[29X[2XDigraphMonomorphism[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0Y[10XDigraphMonomorphism[110X  returns  a single [13Xinjective[113X [2XDigraphHomomorphism[102X ([14X7.3-2[114X)
  between [3Xdigraph1[103X and [3Xdigraph2[103X if one exists, otherwise it returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := ChainDigraph(3);;[127X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphMonomorphism(gr1, gr1);[127X[104X
    [4X[28XIdentityTransformation[128X[104X
    [4X[25Xgap>[125X [27XDigraphMonomorphism(gr1, gr2);[127X[104X
    [4X[28XTransformation( [ 3, 1, 5, 4, 5 ] )[128X[104X
  [4X[32X[104X
  
  [1X7.3-5 MonomorphismsDigraphs[101X
  
  [33X[1;0Y[29X[2XMonomorphismsDigraphs[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMonomorphismsDigraphsRepresentatives[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0YThese  operations  behave  the  same  as  [2XHomomorphismsDigraphs[102X  ([14X7.3-3[114X) and
  [2XHomomorphismsDigraphsRepresentatives[102X   ([14X7.3-3[114X),   except  they  only  return
  [13Xinjective[113X homomorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := ChainDigraph(3);;[127X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[3, 5], [2], [3, 1], [], [4]]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XMonomorphismsDigraphs(gr1, gr2);[127X[104X
    [4X[28X[ Transformation( [ 1, 5, 4, 4, 5 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 5, 4, 5 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XMonomorphismsDigraphsRepresentatives(gr1, CompleteDigraph(3));[127X[104X
    [4X[28X[ Transformation( [ 2, 1 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X7.3-6 DigraphEpimorphism[101X
  
  [33X[1;0Y[29X[2XDigraphEpimorphism[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0Y[10XDigraphEpimorphism[110X  returns  a single [13Xsurjective[113X [2XDigraphHomomorphism[102X ([14X7.3-2[114X)
  between [3Xdigraph1[103X and [3Xdigraph2[103X if one exists, otherwise it returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := DigraphReverse(ChainDigraph(4));[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphRemoveEdge(CompleteDigraph(3), [1, 2]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEpimorphism(gr2, gr1);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphEpimorphism(gr1, gr2);[127X[104X
    [4X[28XTransformation( [ 3, 1, 2, 3 ] )[128X[104X
  [4X[32X[104X
  
  [1X7.3-7 EpimorphismsDigraphs[101X
  
  [33X[1;0Y[29X[2XEpimorphismsDigraphs[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XEpimorphismsDigraphsRepresentatives[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0YThese  operations  behave  the  same  as  [2XHomomorphismsDigraphs[102X  ([14X7.3-3[114X) and
  [2XHomomorphismsDigraphsRepresentatives[102X   ([14X7.3-3[114X),   except  they  only  return
  [13Xsurjective[113X homomorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := DigraphReverse(ChainDigraph(4));[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphSymmetricClosure(CycleDigraph(3));[127X[104X
    [4X[28X<immutable symmetric digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XEpimorphismsDigraphsRepresentatives(gr1, gr2);[127X[104X
    [4X[28X[ Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 1, 2, 3 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XEpimorphismsDigraphs(gr1, gr2);[127X[104X
    [4X[28X[ Transformation( [ 1, 2, 1, 3 ] ), Transformation( [ 1, 2, 3, 1 ] ), [128X[104X
    [4X[28X  Transformation( [ 1, 2, 3, 2 ] ), Transformation( [ 1, 3, 1, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 3, 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 2, 1, 3 ] ), Transformation( [ 3, 2, 3, 1 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X7.3-8 DigraphEmbedding[101X
  
  [33X[1;0Y[29X[2XDigraphEmbedding[102X( [3Xdigraph1[103X, [3Xdigraph2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0YAn  embedding  of  a  digraph  [3Xdigraph1[103X  into  another digraph [3Xdigraph2[103X is a
  [2XDigraphMonomorphism[102X   ([14X7.3-4[114X)  from  [3Xdigraph1[103X  to  [3Xdigraph2[103X  which  has  the
  additional property that a pair of vertices [10X[i, j][110X which have no edge [10Xi -> j[110X
  in  [3Xdigraph1[103X are mapped to a pair of vertices [10X[a, b][110X which have no edge [10Xa->b[110X
  in [3Xdigraph2[103X.[133X
  
  [33X[0;0YIn  other  words,  an  embedding  [10Xt[110X  is  an isomorphism from [3Xdigraph1[103X to the
  [2XInducedSubdigraph[102X ([14X3.3-3[114X) of [3Xdigraph2[103X on the image of [10Xt[110X.[133X
  
  [33X[0;0Y[10XDigraphEmbedding[110X  returns  a  single  embedding  if one exists, otherwise it
  returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := ChainDigraph(3);[127X[104X
    [4X[28X<immutable chain digraph with 3 vertices>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEmbedding(gr, CompleteDigraph(4));[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphEmbedding(gr, Digraph([[3], [1, 4], [1], [3]]));[127X[104X
    [4X[28XTransformation( [ 2, 4, 3, 4 ] )[128X[104X
  [4X[32X[104X
  
  [1X7.3-9 EmbeddingsDigraphs[101X
  
  [33X[1;0Y[29X[2XEmbeddingsDigraphs[102X( [3XD1[103X, [3XD2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XEmbeddingsDigraphsRepresentatives[102X( [3XD1[103X, [3XD2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0YThese  operations  behave  the  same  as  [2XHomomorphismsDigraphs[102X  ([14X7.3-3[114X) and
  [2XHomomorphismsDigraphsRepresentatives[102X   ([14X7.3-3[114X),   except  they  only  return
  embeddings of [3XD1[103X into [3XD2[103X.[133X
  
  [33X[0;0YSee also [2XIsDigraphEmbedding[102X ([14X7.3-11[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD1 := NullDigraph(2);[127X[104X
    [4X[28X<immutable empty digraph with 2 vertices>[128X[104X
    [4X[25Xgap>[125X [27XD2 := CycleDigraph(5);[127X[104X
    [4X[28X<immutable cycle digraph with 5 vertices>[128X[104X
    [4X[25Xgap>[125X [27XEmbeddingsDigraphsRepresentatives(D1, D2);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XEmbeddingsDigraphs(D1, D2);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 4, 4, 5, 1 ] ), [128X[104X
    [4X[28X  Transformation( [ 2, 5, 4, 5, 1 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 1, 5, 1, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 3, 5, 5, 1, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 4, 1, 1, 2, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 4, 2, 1, 2, 3 ] ), [128X[104X
    [4X[28X  Transformation( [ 5, 2, 2, 3, 4 ] ), [128X[104X
    [4X[28X  Transformation( [ 5, 3, 2, 3, 4 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X7.3-10 IsDigraphHomomorphism[101X
  
  [33X[1;0Y[29X[2XIsDigraphHomomorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphHomomorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X, [3Xcol1[103X, [3Xcol2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphEpimorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphEpimorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X, [3Xcol1[103X, [3Xcol2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphMonomorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphMonomorphism[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X, [3Xcol1[103X, [3Xcol2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphEndomorphism[102X( [3Xdigraph[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphEndomorphism[102X( [3Xdigraph[103X, [3Xx[103X, [3Xcol[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0Y[10XIsDigraphHomomorphism[110X returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is
  a homomorphism from the digraph [3Xsrc[103X to the digraph [3Xran[103X.[133X
  
  [33X[0;0Y[10XIsDigraphEpimorphism[110X  returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is
  a surjective homomorphism from the digraph [3Xsrc[103X to the digraph [3Xran[103X.[133X
  
  [33X[0;0Y[10XIsDigraphMonomorphism[110X returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is
  an injective homomorphism from the digraph [3Xsrc[103X to the digraph [3Xran[103X.[133X
  
  [33X[0;0Y[10XIsDigraphEndomorphism[110X returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is
  an endomorphism of the digraph [3Xdigraph[103X.[133X
  
  [33X[0;0YA  permutation or transformation [3Xx[103X is a [13Xhomomorphism[113X from a digraph [3Xsrc[103X to a
  digraph [3Xran[103X if the following hold:[133X
  
  [30X    [33X[0;6Y[10X[u  ^  [3Xx[103X[10X,  v ^ [3Xx[103X[10X][110X is an edge of [3Xran[103X whenever [10X[u, v][110X is an edge of [3Xsrc[103X;
        and[133X
  
  [30X    [33X[0;6Y[3Xx[103X fixes every [10Xi[110X which is not a vertex of [3Xsrc[103X.[133X
  
  [33X[0;0YSee also [2XGeneratorsOfEndomorphismMonoid[102X ([14X7.3-13[114X).[133X
  
  [33X[0;0YIf  [3Xcol1[103X  and  [3Xcol2[103X,  or  [3Xcol[103X,  are  given,  then they must represent vertex
  colourings;  see  [2XAutomorphismGroup[102X  ([14X7.2-5[114X)  for details of the permissible
  values  for  these  argument.  The  homomorphism  must  then  also  have the
  property:[133X
  
  [30X    [33X[0;6Y[10Xcol[i]  =  col[i  ^  x][110X  for all vertices [10Xi[110X of [3Xdigraph[103X, in the case of
        [10XIsDigraphEndomorphism[110X.[133X
  
  [30X    [33X[0;6Y[10Xcol1[i]  =  col2[i ^ x][110X for all vertices [10Xi[110X of [3Xsrc[103X, in the cases of the
        other operations.[133X
  
  [33X[0;0YSee also [2XDigraphsRespectsColouring[102X ([14X7.3-12[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsrc := Digraph([[1], [1, 2], [1, 3]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27Xran := Digraph([[1], [1, 2]]);[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(src, ran, Transformation([2, 1, 2]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(src, ran, Transformation([3, 3, 3]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(src, src, Transformation([3, 3, 3]));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),[127X[104X
    [4X[25X>[125X [27X                         [1, 2, 2], [1, 2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),[127X[104X
    [4X[25X>[125X [27X                         [2, 1, 1], [1, 2]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEndomorphism(src, Transformation([3, 3, 3]));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 2]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEpimorphism(src, ran, Transformation([3, 3, 3]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphMonomorphism(src, ran, Transformation([1, 2, 2]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEpimorphism(src, ran, Transformation([1, 2, 2]));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphMonomorphism(ran, src, ());[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.3-11 IsDigraphEmbedding[101X
  
  [33X[1;0Y[29X[2XIsDigraphEmbedding[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDigraphEmbedding[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X, [3Xcol1[103X, [3Xcol2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0Y[10XIsDigraphEmbedding[110X  returns [9Xtrue[109X if the permutation or transformation [3Xx[103X is a
  embedding  of  the  digraph  [3Xsrc[103X  into the digraph [3Xran[103X, while respecting the
  colourings [3Xcol1[103X and [3Xcol2[103X if given.[133X
  
  [33X[0;0YA  permutation  or  transformation  [3Xx[103X is a [13Xembedding[113X of a digraph [3Xsrc[103X into a
  digraph [3Xran[103X if [3Xx[103X is a monomorphism from [3Xsrc[103X to [3Xran[103X and the inverse of [3Xx[103X is a
  monomorphism  from  the  subdigraph of [3Xran[103X induced by the image of [3Xx[103X to [3Xsrc[103X.
  See also [2XIsDigraphHomomorphism[102X ([14X7.3-10[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsrc := Digraph([[1], [1, 2]]);[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xran := Digraph([[1], [1, 2], [1, 3]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphMonomorphism(src, ran, ());[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEmbedding(src, ran, ());[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEmbedding(src, ran, (), [2, 1], [2, 1, 1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xran := Digraph([[1, 2], [1, 2], [1, 3]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphMonomorphism(src, ran, IdentityTransformation);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEmbedding(src, ran, IdentityTransformation);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X7.3-12 DigraphsRespectsColouring[101X
  
  [33X[1;0Y[29X[2XDigraphsRespectsColouring[102X( [3Xsrc[103X, [3Xran[103X, [3Xx[103X, [3Xcol1[103X, [3Xcol2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThe   operation   [10XDigraphsRespectsColouring[110X  verifies  whether  or  not  the
  permutation or transformation [3Xx[103X respects the vertex colourings [3Xcol1[103X and [3Xcol2[103X
  of  the  digraphs  [3Xsrc[103X and [3Xrange[103X. That is, [10XDigraphsRespectsColouring[110X returns
  [9Xtrue[109X if and only if for all vertices [10Xi[110X of [3Xsrc[103X, [10Xcol1[i] = col2[i ^ x][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsrc := Digraph([[1], [1, 2]]);[127X[104X
    [4X[28X<immutable digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xran := Digraph([[1], [1, 2], [1, 3]]);[127X[104X
    [4X[28X<immutable digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [1, 2, 1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [2, 1, 1]);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X7.3-13 GeneratorsOfEndomorphismMonoid[101X
  
  [33X[1;0Y[29X[2XGeneratorsOfEndomorphismMonoid[102X( [3Xdigraph[103X[, [3Xcolors[103X][, [3Xlimit[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XGeneratorsOfEndomorphismMonoidAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of transformations.[133X
  
  [33X[0;0YAn  endomorphism  of  [3Xdigraph[103X  is a homomorphism [2XDigraphHomomorphism[102X ([14X7.3-2[114X)
  from  [3Xdigraph[103X  back to itself. [10XGeneratorsOfEndomorphismMonoid[110X, called with a
  single   argument,   returns   a  generating  set  for  the  monoid  of  all
  endomorphisms  of [3Xdigraph[103X. If [3Xdigraph[103X belongs to [2XIsImmutableDigraph[102X ([14X3.1-3[114X),
  then  the  value of [10XGeneratorsOfEndomorphismMonoid[110X will not be recomputed on
  future calls.[133X
  
  [33X[0;0YIf  the  [3Xcolors[103X  argument  is specified, then [10XGeneratorsOfEndomorphismMonoid[110X
  will  return  a generating set for the monoid of endomorphisms which respect
  the given colouring. The colouring [3Xcolors[103X can be in one of two forms:[133X
  
  [30X    [33X[0;6YA list of positive integers of size the number of vertices of [3Xdigraph[103X,
        where [3Xcolors[103X[10X[i][110X is the colour of vertex [10Xi[110X.[133X
  
  [30X    [33X[0;6YA  list  of  lists, such that [3Xcolors[103X[10X[i][110X is a list of all vertices with
        colour [10Xi[110X.[133X
  
  [33X[0;0YIf the [3Xlimit[103X argument is specified, then it will return only the first [3Xlimit[103X
  homomorphisms, where [3Xlimit[103X must be a positive integer or [10Xinfinity[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(List([1 .. 3], x -> [1 .. 3]));;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfEndomorphismMonoid(gr);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), [128X[104X
    [4X[28X  IdentityTransformation, Transformation( [ 1, 2, 1 ] ), [128X[104X
    [4X[28X  Transformation( [ 1, 2, 2 ] ), Transformation( [ 1, 1, 2 ] ), [128X[104X
    [4X[28X  Transformation( [ 1, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfEndomorphismMonoid(gr, 3);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), [128X[104X
    [4X[28X  IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27Xgr := CompleteDigraph(3);;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfEndomorphismMonoid(gr);[127X[104X
    [4X[28X[ Transformation( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ), [128X[104X
    [4X[28X  IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfEndomorphismMonoid(gr, [1, 2, 2]);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfEndomorphismMonoid(gr, [[1], [2, 3]]);[127X[104X
    [4X[28X[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ][128X[104X
  [4X[32X[104X
  
  [1X7.3-14 DigraphColouring[101X
  
  [33X[1;0Y[29X[2XDigraphColouring[102X( [3Xdigraph[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0YA [13Xproper colouring[113X of a digraph is a labelling of its vertices in such a way
  that  adjacent  vertices  have  different  labels. A [13Xproper [10Xn[110X-colouring[113X is a
  proper  colouring  that  uses  exactly  [10Xn[110X  colours.  Equivalently,  a proper
  ([10Xn[110X-)colouring of a digraph can be defined to be a [2XDigraphEpimorphism[102X ([14X7.3-6[114X)
  from   a   digraph   onto  the  complete  digraph  (with  [10Xn[110X  vertices);  see
  [2XCompleteDigraph[102X  ([14X3.5-11[114X).  Note  that a digraph with loops ([2XDigraphHasLoops[102X
  ([14X6.2-1[114X)) does not have a proper [10Xn[110X-colouring for any value [10Xn[110X.[133X
  
  [33X[0;0YIf   [3Xdigraph[103X   is   a   digraph  and  [3Xn[103X  is  a  non-negative  integer,  then
  [10XDigraphColouring([3Xdigraph[103X[10X,  [3Xn[103X[10X)[110X  returns  an epimorphism from [3Xdigraph[103X onto the
  complete digraph with [3Xn[103X vertices if one exists, else it returns [9Xfail[109X.[133X
  
  [33X[0;0YSee also [2XDigraphGreedyColouring[102X ([14X7.3-15[114X) and[133X
  
  [33X[0;0YNote that a digraph with at least two vertices has a 2-colouring if and only
  if it is bipartite, see [2XIsBipartiteDigraph[102X ([14X6.2-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphColouring(CompleteDigraph(5), 4);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphColouring(ChainDigraph(10), 1);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XD := ChainDigraph(10);;[127X[104X
    [4X[25Xgap>[125X [27Xt := DigraphColouring(D, 2);[127X[104X
    [4X[28XTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphColouring(D, t); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphGreedyColouring(D);[127X[104X
    [4X[28XTransformation( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )[128X[104X
  [4X[32X[104X
  
  [1X7.3-15 DigraphGreedyColouring[101X
  
  [33X[1;0Y[29X[2XDigraphGreedyColouring[102X( [3Xdigraph[103X, [3Xorder[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphGreedyColouring[102X( [3Xdigraph[103X, [3Xfunc[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphGreedyColouring[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0YA [13Xproper colouring[113X of a digraph is a labelling of its vertices in such a way
  that adjacent vertices have different labels. Note that a digraph with loops
  ([2XDigraphHasLoops[102X ([14X6.2-1[114X)) does not have any proper colouring.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph and [3Xorder[103X is a dense list consisting of all of the
  vertices  of  [3Xdigraph[103X  (in  any  order),  then [10XDigraphGreedyColouring[110X uses a
  greedy algorithm with the specified order to obtain some proper colouring of
  [3Xdigraph[103X, which may not use the minimal number of colours.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is a digraph and [3Xfunc[103X is a function whose argument is a digraph,
  and  that  returns  a dense list [3Xorder[103X, then [10XDigraphGreedyColouring([3Xdigraph[103X[10X,
  [3Xfunc[103X[10X)[110X returns [10XDigraphGreedyColouring([3Xdigraph[103X[10X, [3Xfunc[103X[10X([3Xdigraph[103X[10X))[110X.[133X
  
  [33X[0;0YIf  the  optional  second  argument  (either  a  list or a function), is not
  specified, then [2XDigraphWelshPowellOrder[102X ([14X7.3-16[114X) is used by default.[133X
  
  [33X[0;0YSee also [2XDigraphColouring[102X ([14X7.3-14[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphGreedyColouring(ChainDigraph(10));[127X[104X
    [4X[28XTransformation( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XDigraphGreedyColouring(ChainDigraph(10), [1 .. 10]);[127X[104X
    [4X[28XTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )[128X[104X
  [4X[32X[104X
  
  [1X7.3-16 DigraphWelshPowellOrder[101X
  
  [33X[1;0Y[29X[2XDigraphWelshPowellOrder[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of the vertices.[133X
  
  [33X[0;0Y[10XDigraphWelshPowellOrder[110X returns a list of all of the vertices of the digraph
  [3Xdigraph[103X   ordered   according   to  the  sum  of  the  number  of  out-  and
  in-neighbours, from highest to lowest.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphWelshPowellOrder(Digraph([[4], [9], [9], [], [127X[104X
    [4X[25X>[125X [27X                                    [4, 6, 9], [1], [], [], [127X[104X
    [4X[25X>[125X [27X                                    [4, 5], [4, 5]]));[127X[104X
    [4X[28X[ 5, 9, 4, 1, 6, 10, 2, 3, 7, 8 ][128X[104X
  [4X[32X[104X
  
  [1X7.3-17 ChromaticNumber[101X
  
  [33X[1;0Y[29X[2XChromaticNumber[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA non-negative integer.[133X
  
  [33X[0;0YA [13Xproper colouring[113X of a digraph is a labelling of its vertices in such a way
  that adjacent vertices have different labels. Equivalently, a proper digraph
  colouring  can  be defined to be a [2XDigraphEpimorphism[102X ([14X7.3-6[114X) from a digraph
  onto a complete digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph without loops (see [2XDigraphHasLoops[102X ([14X6.2-1[114X), then
  [10XChromaticNumber[110X  returns the least non-negative integer [10Xn[110X such that there is
  a  proper colouring of [3Xdigraph[103X with [10Xn[110X colours. In other words, for a digraph
  with  at  least  one vertex, [10XChromaticNumber[110X returns the least number [10Xn[110X such
  that [10XDigraphColouring([3Xdigraph[103X[10X, n)[110X does not return [9Xfail[109X. See [2XDigraphColouring[102X
  ([14X7.3-14[114X).[133X
  
  [33X[0;0YIt  is  possible to select the algorithm to compute the chromatic number via
  the  use  of  value  options. The permitted algorithms and values to pass as
  options are:[133X
  
  [30X    [33X[0;6Y[10Xlawler[110X - Lawler's Algorithm [Law76][133X
  
  [30X    [33X[0;6Y[10Xbyskov[110X - Byskov's Algorithm [Bys02][133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(NullDigraph(10));[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(CompleteDigraph(10));[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(CompleteBipartiteDigraph(5, 5));[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(NullDigraph(0));[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XD := PetersenGraph(IsMutableDigraph);[127X[104X
    [4X[28X<mutable digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(D);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(CompleteDigraph(10) : lawler);[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27XChromaticNumber(CompleteDigraph(10) : byskov);[127X[104X
    [4X[28X10[128X[104X
  [4X[32X[104X
  
  [1X7.3-18 DigraphCore[101X
  
  [33X[1;0Y[29X[2XDigraphCore[102X( [3XD[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of positive integers.[133X
  
  [33X[0;0YIf [3XD[103X is a digraph, then [10XDigraphCore[110X returns a list of vertices corresponding
  to the [10Xcore[110X of [3XD[103X. In particular, the subdigraph of [3XD[103X induced by this list is
  isomorphic to the core of [3XD[103X.[133X
  
  [33X[0;0YThe  [13Xcore[113X  of  a  digraph  [10XD[110X  is  the  minimal  subdigraph [3XC[103X of [10XD[110X which is a
  homomorphic image of [10XD[110X. The core of a digraph is unique up to isomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphSymmetricClosure(CycleDigraph(8));[127X[104X
    [4X[28X<immutable symmetric digraph with 8 vertices, 16 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphCore(D);[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27XD := PetersenGraph();[127X[104X
    [4X[28X<immutable digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphCore(D);[127X[104X
    [4X[28X[ 1 .. 10 ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[3], [3], [4], [5], [2]]);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphCore(D);[127X[104X
    [4X[28X[ 2, 3, 4, 5 ][128X[104X
  [4X[32X[104X
  
  [1X7.3-19 LatticeDigraphEmbedding[101X
  
  [33X[1;0Y[29X[2XLatticeDigraphEmbedding[102X( [3XL1[103X, [3XL2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X.[133X
  
  [33X[0;0YIf  [3XL1[103X  and  [3XL2[103X are lattice digraphs ([2XIsLatticeDigraph[102X ([14X6.3-3[114X) returns [9Xtrue[109X,
  then  [10XLatticeDigraphEmbedding[110X returns a single [13Xinjective[113X [2XDigraphHomomorphism[102X
  ([14X7.3-2[114X)  between  [3XL1[103X  and  [3XL2[103X,  with  the  property  that  it  is  a [13Xlattice
  homomorphism[113X. If no such homomorphism exists, [9Xfail[109X is returned.[133X
  
  [33X[0;0YA  [13Xlattice  homomorphism[113X  is a digraph homomorphism which respects meets and
  joins  of  every  pair  of  vertices.  Note  that  every  injective  lattice
  homomorphism  [10Xmap[110X is an embedding, in the sense that the inverse of [10Xmap[110X is a
  lattice homomorphism also.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := DigraphReflexiveTransitiveClosure(ChainDigraph(5));[127X[104X
    [4X[28X<immutable preorder digraph with 5 vertices, 15 edges>[128X[104X
    [4X[25Xgap>[125X [27XL1 := DigraphReflexiveTransitiveClosure(ChainDigraph(5));[127X[104X
    [4X[28X<immutable preorder digraph with 5 vertices, 15 edges>[128X[104X
    [4X[25Xgap>[125X [27XL2 := DigraphReflexiveTransitiveClosure(ChainDigraph(6));[127X[104X
    [4X[28X<immutable preorder digraph with 6 vertices, 21 edges>[128X[104X
    [4X[25Xgap>[125X [27XLatticeDigraphEmbedding(L1, L2);[127X[104X
    [4X[28XIdentityTransformation[128X[104X
    [4X[25Xgap>[125X [27XLatticeDigraphEmbedding(L2, L1);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X7.3-20 IsLatticeHomomorphism[101X
  
  [33X[1;0Y[29X[2XIsLatticeHomomorphism[102X( [3XL1[103X, [3XL2[103X, [3Xmap[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsLatticeEpimorphism[102X( [3XL1[103X, [3XL2[103X, [3Xmap[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsLatticeEmbedding[102X( [3XL1[103X, [3XL2[103X, [3Xmap[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsLatticeMonomorphism[102X( [3XL1[103X, [3XL2[103X, [3Xmap[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsLatticeEndomorphism[102X( [3XL[103X, [3Xmap[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YEach    of    the    function    described    in    this   section   (except
  [10XIsLatticeEndomorphism[110X)   takes   a  pair  of  digraphs  [3XL1[103X  and  [3XL2[103X,  and  a
  transformation  [3Xmap[103X, returning [9Xtrue[109X if [3Xmap[103X is a [13Xlattice homomorphism[113X from [3XL1[103X
  to  [3XL2[103X,  and  [9Xfalse[109X  otherwise.  If [3XL1[103X or [3XL2[103X is not a lattice, then [9Xfalse[109X is
  returned.[133X
  
  [33X[0;0YA  transformation  or  permutation  [3Xmap[103X  is  a  [13Xlattice  homomorphism[113X if [3Xmap[103X
  respects  meets  and  joins of every pair of vertices, and [3Xmap[103X fixes every [10Xi[110X
  which is not a vertex of [3XL1[103X.[133X
  
  [33X[0;0Y[10XIsLatticeHomomorphism[110X  returns [9Xtrue[109X if the permutation or transformation [3Xmap[103X
  is a lattice homomorphism from the lattice digraph [3XL1[103X to the lattice digraph
  [3XL2[103X.[133X
  
  [33X[0;0Y[10XIsLatticeEpimorphism[110X  returns  [9Xtrue[109X if the permutation or transformation [3Xmap[103X
  is  a  surjective  lattice  homomorphism  from the lattice digraph [3XL1[103X to the
  lattice digraph [3XL2[103X.[133X
  
  [33X[0;0Y[10XIsLatticeEmbedding[110X  returns [9Xtrue[109X if the permutation or transformation [3Xmap[103X is
  an injective lattice homomorphism from the lattice digraph [3XL1[103X to the lattice
  digraph   [3XL2[103X.   The   function   [10XIsLatticeMonomorphism[110X   is   a  synonym  of
  [10XIsLatticeEmbedding[110X.[133X
  
  [33X[0;0Y[10XIsLatticeEndomorphism[110X  returns [9Xtrue[109X if the permutation or transformation [3Xmap[103X
  is an lattice endomorphism of the lattice digraph [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := Digraph([[2, 4], [3, 7], [6], [5, 7], [6], [], [6]]);[127X[104X
    [4X[28X<immutable digraph with 7 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphRemoveVertex(G, 7);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := DigraphReflexiveTransitiveClosure(G);[127X[104X
    [4X[28X<immutable preorder digraph with 7 vertices, 22 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphReflexiveTransitiveClosure(D);[127X[104X
    [4X[28X<immutable preorder digraph with 6 vertices, 17 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDigraphEmbedding(D, G, IdentityTransformation);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeHomomorphism(D, G, IdentityTransformation);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 3], [4], [4], []]);[127X[104X
    [4X[28X<immutable digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := Digraph([[2, 3], [4], [4], [5], []]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XD := DigraphReflexiveTransitiveClosure(D);[127X[104X
    [4X[28X<immutable preorder digraph with 4 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := DigraphReflexiveTransitiveClosure(G);[127X[104X
    [4X[28X<immutable preorder digraph with 5 vertices, 14 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeEmbedding(D, G, IdentityTransformation);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeMonomorphism(D, G, IdentityTransformation);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xf := Transformation([1, 2, 3, 4, 4]);[127X[104X
    [4X[28XTransformation( [ 1, 2, 3, 4, 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeEpimorphism(G, D, f);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeEndomorphism(D, (2, 3));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
