Subsections

• Converting between Graphs and Digraphs

Union(G, H) : GrphUnd, GrphUnd -> GrphUnd

G join H : GrphDir, GrphDir -> GrphDir

G join H : GrphUnd, GrphUnd -> GrphUnd

Given graphs G and H with disjoint vertex sets V(G) and V(H), respectively, construct their union, i.e. the graph with vertex set V(G) union V(H), and edge set E(G) union E(H).

EdgeUnion(G, H) : GrphUnd, GrphUnd -> GrphUnd

Given graphs G and H having the same number of vertices, construct their edge union K. This construction identifies the i-th vertex of G with the i-th vertex of H for all i. The edge union has the same vertex set as G (and hence as H) and vertices u and v of K are adjacent if and only if either u and v are adjacent in G or u and v are adjacent in H.

CompleteUnion(G, H) : GrphUnd, GrphUnd -> GrphUnd

Given graphs G and H with disjoint vertex sets V(G) and V(H), respectively, construct the complete union of G and H. This graph consists of the union of G and H (Union(G, H)), together with edges uv, for all u in V(G) and all v in V(H).

CartesianProduct(G, H) : GrphUnd, GrphUnd -> GrphUnd

Given graphs G and H with disjoint vertex sets V(G) and V(H), respectively, form the product K = G x H of G and H. The product has vertex set V(G) x V(H). Two vertices u = (u_1, u_2) and v = (v_1, v_2) of K are adjacent when either

• u_1 = v_1 and u_2 adj v_2, or
• u_2 = v_2 and u_1 adj v_1.

LexProduct(G, H) : GrphUnd, GrphUnd -> GrphUnd

Given graphs G and H with disjoint vertex sets V(G) and V(H), respectively, form the lexicographic product K of G and H. The lexicographic product has vertex set V(G) x V(H). Two vertices u = (u_1, u_2) and v = (v_1, v_2) of K are adjacent when either

• u_1 adj v_1, or
• u_1 = v_1 and u_2 adj v_2.

TensorProduct(G, H) : GrphUnd, GrphUnd -> GrphUnd

Given graphs G and H with disjoint vertex sets V(G) and V(H), respectively, form the tensor product K of G and H. This graph has vertex set V(G) x V(H). Two vertices u = (u_1, u_2) and v = (v_1, v_2) of K are adjacent when u_1 adj v_1 and u_2 adj v_2.

Given a graph G and a positive integer n, construct the n-th power K of G. This graph has the same vertex set as G, and vertices u and v of K are adjacent if and only if the distance between u and v in G is less than or equal to n.

Given a graph G, produce a digraph D whose vertex set is the same as that of G and whose edge set consists of the edges of G, each given an arbitary direction. The edges of D are always directed from the lower numbered vertex to the higher numbered vertex. Thus, if G contains the edge { i, j }, then D will have the edge [i, j] if i < j, otherwise the edge [j, i].

The underlying graph G of the digraph D. Thus, G has the same vertex set as D, and two vertices u and v are adjacent in G if and only if, in D, there is either an edge directed from u to v or from v to u.

The underlying digraph D of the graph G. Thus, D has the same vertex set as G, and if vertices u and v are adjacent in G then in D, there will be both an edge directed from u to v and an edge directed from v to u.