On k-restricted connectivity of direct product of graphs

Abstract

Let [Formula: see text] be a graph. A vertex-cut [Formula: see text] of [Formula: see text] is said to be k-restricted if every component of [Formula: see text] has at least [Formula: see text] vertices, and cyclic if [Formula: see text] has at least two components which contain a cycle. The minimum cardinality over all [Formula: see text]-restricted vertex-cuts of [Formula: see text] is called the k-restricted connectivity of [Formula: see text] and is denoted by [Formula: see text]. Also the minimum cardinality over all cyclic vertex-cuts of [Formula: see text] is called the cyclic connectivity of graph [Formula: see text] and is denoted by [Formula: see text]. In this paper, the [Formula: see text]-restricted connectivity and cyclic connectivity of the direct product of two graphs [Formula: see text] and [Formula: see text] is obtained for some [Formula: see text], where [Formula: see text] is a complete graph, and [Formula: see text] is a complete graph, a complete bipartite graph or a cycle.