Restricted Connectivity for Some Interconnection Networks

Abstract

A vertex-cut $$S$$S of a connected graph $$G$$G is called a restricted vertex-cut if no vertex $$u$$u of the graph $$G$$G satisfies $$N_G(u)\subseteq S$$NG(u)⊆S. The restricted connectivity $$\kappa '(G)$$??(G) of the graph $$G$$G is the cardinality of a minimum restricted vertex-cut of $$G$$G; this is a more refined index than the connectivity parameter $$\kappa (G)$$?(G). In this paper, we prove that $$\kappa '(K_m\times G_2)=2k_2+m-2$$??(Km×G2)=2k2+m-2, where $$G_2$$G2 is a $$k_2\ (\ge 2)$$k2(?2)-regular and maximally connected graph with girth $$g(G_2)\ge 4$$g(G2)?4; and $$\kappa '(G_1\times G_2)=2k_1+2k_2-2$$??(G1×G2)=2k1+2k2-2, where $$G_i$$Gi is a $$k_i\ (\ge 2)$$ki(?2)-regular and maximally connected graph with girth $$g(G_i)\ge 4$$g(Gi)?4 for $$1\le i\le 2$$1≤i≤2. Furthermore, we determine the restricted connectivity of a class of minimal Abelian Cayley graphs.