Unions and Products of Graphs

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

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

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) : GrphDir, GrphDir -> GrphDir

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) : GrphDir, GrphDir -> GrphDir

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) : GrphDir, GrphDir -> GrphDir

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

(a)

u_1 = v_1 and u_2 adj v_2, or

(b)

u_2 = v_2 and u_1 adj v_1.

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

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

(a)

u_1 adj v_1, or

(b)

u_1 = v_1 and u_2 adj v_2.

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

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.

G ^ n : GrphUnd, RngIntElt -> GrphUnd

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.