The support and vertex/edge decorations of the original graphs are not retained in the graph resulting from applying any of the union functions below.
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) ∪V(H), and edge-set E(G) ∪E(H).
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.
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).
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 = (u1, u2) and v = (v1, v2) of K are adjacent when either
- (a)
- u1 = v1 and u2 adj v2, or
- (b)
- u2 = v2 and u1 adj v1.
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 = (u1, u2) and v = (v1, v2) of K are adjacent when either
- (a)
- u1 adj v1, or
- (b)
- u1 = v1 and u2 adj v2.
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 = (u1, u2) and v = (v1, v2) of K are adjacent when u1 adj v1 and u2 adj v2.
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.