On the monopolies of lexicographic product graphs: bounds and closed formulas

We consider a simple graph G = (V, E) without isolated vertices and of minimum degree delta(G). Let k be an integer number such that k is an element of {1 - inverted right perpendicular delta(G)/2inverted left perpendicular,..., left perpendicular delta(G)/2right perpendicular}. A vertex v of G is said to be k-controlled by a set M subset of V, if delta(M) (v) >= + delta(v)/2 + k where delta(M)(v) represents the number of neighbors v has in M and delta(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 in G is the k-monopoly number of G. In this article we study the k-monopolies of the lexicographic product of graphs. Specifically we obtain several relationships between the k-monopoly number of this product graph and the k-monopoly numbers and/or order of its factors. Moreover, we bound (or compute the exact value) of the k-monopoly number of several families of lexicographic product graphs.