Computing the (k-)monopoly number of direct product of graphs
Let G = (V,E) be a simple graph without isolated vertices and minimum degree ?(G), and let k ? {1-??(G)/2? ,..., ?(G)/2c?} be an integer. Given a set M ? V, a vertex v of G is said to be k-controlled by M if ?M(v)? ?(v)/2 + k where ?M(v) represents the quantity of neighbors v has in M and ?(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.