Restricted Connectivity for Some Interconnection Networks

Abstract

A vertex-cut $S$ of a connected graph $G$ is called a restricted vertex-cut if no vertex $u$ of the graph $G$ satisfies $N_G(u)\subseteq S$. The restricted connectivity $\kappa '(G)$ of the graph $G$ is the cardinality of a minimum restricted vertex-cut of $G$; this is a more refined index than the connectivity parameter $\kappa (G)$. In this paper, we prove that $\kappa '(K_m\times G_2)=2k_2+m-2$, where $G_2$ is a $k_2\ (\ge 2)$-regular and maximally connected graph with girth $g(G_2)\ge 4$; and $\kappa '(G_1\times G_2)=2k_1+2k_2-2$, where $G_i$ is a $k_i\ (\ge 2)$-regular and maximally connected graph with girth $g(G_i)\ge 4$ for $1\le i\le 2$. Furthermore, we determine the restricted connectivity of a class of minimal Abelian Cayley graphs.