A vertex labelling of a graph G is a partial map f from the vertex-set V of G into a set L. The set L is known as the vertex label set. An edge labelling of a graph G is a partial map f from the edge-set E of G into a set L. The set L is known as the edge label set. A labelled graph is built by assigning labels successively to vertices or edges after the (unlabelled) graph has been constructed.

For the following operations, T may be interpreted as either the vertex-set or edge-set of some graph. The variable t may be interpreted as either a vertex or an edge.

Subsections

• Labelling a Graph
• Reading Labels
• Testing for Labels
• Deleting Labels

AssignLabel(t, l) : GrphEdge, . ->

Assigns the label l to the edge or vertex t

AssignLabel(G, i, l) : Grph, RngIntElt, . ->

Assign the label l to the ith vertex of G.

AssignLabel(G, i, j, l) : Grph, RngIntElt, RngIntElt, . ->

Assign the label l to the edge between the ith and jth vertices of G.

AssignLabels(S, L) : {@ GrphVert @}, SeqEnum ->

AssignLabels(S, L) : [GrphEdge], SeqEnum ->

AssignLabels(S, L) : {@ GrphEdge @}, SeqEnum ->

Assigns the labels in L to the corresponding edges or vertices in the sequence or indexed set S. If for some edge or vertex t the corresponding entry in L is not defined then any existing label of t is removed.

AssignLabels(G, S, L) : Grph, [RngIntElt], SeqEnum ->

Assign the labels in L to the vertices of G whose indices are the corresponding elements in S. If for some vertex v whose index is in S the corresponding label in L is not defined then any existing label of v is removed.

AssignLabels(G, S, L) : Grph, SeqEnum, SeqEnum ->

Assign the labels in L to the edges of G specified in S. The elements of S must be sets or sequences of two integers, depending on whether G is undirected or directed respectively. Each element of S will correspond to the edge between the ith and jth vertices of G, where i and j are the two integers in the element.

AssignLabels(T, L) : GrphVertSet, SeqEnum ->

AssignLabels(T, L) : GrphEdgeSet, SeqEnum ->

Assigns the labels in L to the corresponding edges or vertices in T If for some edge or vertex t the corresponding entry in L is not defined then any existing label of t is removed.

Label(t) : GrphEdge -> .

The label of the edge or vertex t. It is an error if t does not have a label defined.

The label of the ith vertex of G. It is an error if the vertex does not have a label defined.

The label of the edge between the ith and jth vertices of G. It is an error if the edge does not have a label defined.

Labels(S) : [GrphEdge] -> SeqEnum

The sequence L of labels of the edges or vertices in S. If an element of S has no label, then the corresponding entry in L is undefined.

Labels(T) : GrphVertSet -> SeqEnum

Labels(T) : GrphEdgeSet -> SeqEnum

The sequence L of labels of the edges or vertices in T. If an element of T has no label, then the corresponding entry in L is undefined.

The sequence L of labels of the vertices of G whose indices are in S. If a vertex whose index is in S has no label, then the corresponding entry in L is undefined.

VertexLabels(G) : Grph -> SeqEnum

The sequence L of labels of the vertices of G. If a vertex of G has no label, then the corresponding entry in L is undefined.

The sequence L of labels of the edges of G specified in S. The elements of S must be sets or sequences of two integers, depending on whether G is undirected or directed respectively. Each element of S will correspond to the edge between the ith and jth vertices of G, where i and j are the two integers in the element. If the edge has no label, then the corresponding element of L is undefined.

EdgeLabels(G) : Grph -> SeqEnum

The sequence L of labels of the edges of G. If an edge of G has no label, then the corresponding entry in L is undefined.

IsLabelled(t) : GrphEdge -> BoolElt

True if and only if the edge or vertex t has a label.

True if and only if the i-th vertex of G has a label.

True if and only if the edge between the ith and jth vertices of G has a label.

DeleteLabel(t) : GrphEdge ->

Removes the label of the edge or vertex t.

DeleteLabel(G, i) : Grph, RngIntElt ->

Removes the label of the ith vertex of G.

DeleteLabel(G, i, j) : Grph, RngIntElt, RngIntElt ->

Remove the label of the edge between the ith and jth vertices of G.

DeleteLabels(S) : [GrphEdge] ->

Remove the labels of the edges or vertices in S.

DeleteLabels(G, S) : Grph, [RngIntElt] ->

Remove the labels of the vertices of G whose indices are in S.

DeleteLabels(G, S) : Grph, SeqEnum ->

Remove the labels of the edges of G specified in S. The elements of S must be sets or sequences of two integers, depending on whether G is undirected or directed respectively. Each element of S will correspond to the edge between the ith and jth vertices of G, where i and j are the two integers in the element.

DeleteLabels(T) : GrphVertSet ->

DeleteLabels(T) : GrphEdgeSet ->

Remove the labels of the edges or vertices in T.

> K34, V, E := BipartiteGraph(3, 4);
> L := [ IsEven(Distance(V!1, v)) select "red" else "blue" : >                                        v in Vertices(K34) ];
> AssignLabels(Vertices(K34), L);
> VertexLabels(K34);                                          
[ red, red, red, blue, blue, blue, blue ]

> G<a,b> := FPGroup(Sym(4));
> I,m := Transversal(G, sub<G | 1>);
> S := Setseq(Generators(G));
> N := [{m(a*b) : b in S} : a in I];      
> graph := StandardGraph(Digraph<I | N>);
> AssignLabels(VertexSet(graph), IndexedSetToSequence(I));
> for i in [1..#I] do
>     AddEdges( graph, [[i, Index(I, m(I[i]*s))] : s in S], S);
> end for;

> &*EdgeLabels(graph, [[1,2],[2,5],[5,4],[4,1]]);
a^4
> G;
Finitely presented group G on 2 generators
Relations
    b^2 = Id(G)
    a^4 = Id(G)
    (a^-1 * b)^3 = Id(G)