Computing the (k-)monopoly number of direct product of graphs

Abstract

Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ(G), and let k e {1 - 1δ(G)/2],…, [δ(G)/2J} be an integer. Given a set M c V, a vertex v of G is said to be k-controlled by M if (FORMULA) represents the quantity of neighbors v has in M and δ(v) the degree of v. The set M is called a k-monopoly if it k-controls every vertex v of G. The minimum cardinality of any k-monopoly is the k-monopoly number of G. In this article we study the k-monopoly number of direct product graphs. Specifically we obtain tight lower and upper bounds for the k-monopoly number of direct product graphs in terms of the k-monopoly numbers of its factors. Moreover, we compute the exact value for the k-monopoly number of several families of direct product graphs.