Super connectivity of Kronecker product of complete bipartite graphs and complete graphs

Abstract

Let G1 and G2 be two graphs. The Kronecker product G1×G2 has vertex set V(G1×G2)=V(G1)×V(G2) and edge set E(G1×G2)={(u1,v1)(u2,v2):u1u2∈E(G1) and v1v2∈E(G2)}. In this paper we determine that the super-connectivity of Km,r×Kn for n≥3 is (n−2)(m+r). That is, for n≥3, m≥1 and r≥1, at least (n−2)(m+r) vertices need to be removed to get a disconnected graph that contains no isolated vertices. We also determine that the super-connectivity of Km×Kn is mn−4, where n≥m≥2 and n≥3. We generalize our result by establishing the h-extra-connectivity of Km,r×Kn for n≥3, where 1≤h≤m+r−1. More precisely we show that the smallest number of vertices that need to be removed from Km,r×Kn so that the resulting graph is disconnected and each component has more than h vertices is (n−2)(m+r).