Connectivity of Strong Products of Graphs

Definition(s): Let G = (V,E) be a graph. A set S ⊆ V is called separating in G if G− S is not connected. The connectivity of G, written κ(G), is the minimum size of a set S, such that G−S is not connected or has only one vertex. A separating set in G with cardinality κ(G) is called a κ-set in G. Let G1 = (V1, E1) and G2 = (V2, E2) be a graphs. Strong product G1 G2 of graphs G1 and G2 is the graph with V (G1 G2) = V1 × V2, where vertices (x1, x2) and (y1, y2) are adjacent if one of the following occurs • x1 = y1 and x2y2 ∈ E2, • x2 = y2 and x1y1 ∈ E1, • x1y1 ∈ E1 and x2y2 ∈ E2. If a set I ⊂ V1 × V2 is of the form I = S1 × V2 or I = V1 × S2, where Si is a separating set in Gi = (Vi, Ei) for i = 1, 2, then we call it an I-set in G1 G2. Let S1 and S2 be arbitrary separating sets in G1 and G2, respectively, and let A1, . . . , Ak be the connected components of G1 − S1 and B1, . . . , B` the connected components of G2 − S2. Then for any i ≤ k and j ≤ ` L = (S1 ×Bi) ∪ (S1 × S2) ∪ (Aj × S2) is called an L-set in G1 G2. Clearly, I-sets and L-sets are separating in G1 G2.