Strong Menger Connectivity on the Class of Hypercube-like Networks.

Abstract

Motivated by some research works in networks with faults, we are interested in the connectivity issue of a network. Suppose that a network G has a set F of faulty vertices and let G−F is the resulting network with those vertices in F removed. We say that a k-regular graph G is strongly Menger connected if each pair u and v of G−F are connected by min{degf (u), degf (v)} vertex-disjoint fault-free paths in G−F, where degf (u) and degf (v) are the degree of u and v in G − F, respectively. In this paper, we consider an n-dimensional Hypercube-like networks HLn. We show that for each pair u and v in HLn−F, where F is a set of vertices, for |F| ≦ n−2, there are min{degf (u),degf (v)} vertex-disjoint fault- free paths connecting u to v, for n≧3.