Cyclic-Vertex Connectivity Of Cartesian Product Graphs

Let G be a graph with vertex set V(G)and edge set E(G).A cyclic vertex-cut of a graph G is a vertex set S such that G-S is disconnected and at least two of its components contain cycles.If G has a cyclic vertex-cut,then it is said to be cyclically separable.For a cyclically separable graph G,the cyclic vertex-connectivity κc(G)is defined as the cardinality of a minimum cyclic vertex-cut.our main results are summarized as follows:Let Gi be a ki(≥2)-regular and maximally connected graph with g(Gi)≥ 5 for i = 1,2.In this paper,we mainly prove that κc(Km□G2)= 3k2 + m-3 for m ≥ 3 and κc(G1□G2)=4k1 + 4k2-8.In addition,we state sufficient conditions to guarantee κc(K2□G2)= 2κ(G2).We also show that κc(C3□Cn1□Cn2□…Cnk)= 6k and κc(Cn1□Cn2…Cnk)=8A-8,where Cni is a cycle with ni≥ 4 for i = 1,2,…,k.