On the security number of the Cartesian product of graphs

Abstract

A secure set of a graph is, intuitively, a set that can refute any attack from the neighborhood to its subsets. Formally, it is defined as a set S⊆V(G) such that |N[X]∩S|≥|N[X]−S| for all X⊆S. Although finding a minimum secure set is a computationally intractable problem, the minimum size of secure sets, called the security number, is studied for some specific graphs. Especially, determining the security number of the Cartesian product of graphs is one of the developed directions in this area. In this paper, we present an upper bound on the security number of the Cartesian product of general graphs, which is tight for some sparse graphs. We then determine the security number of K3m□K3n, the Cartesian product of complete graphs K3m and K3n, as well as good lower and upper bounds on the security number of the Cartesian product of complete graphs with any number of vertices.