We consider a simple graph G = (V, E) without isolated vertices and of minimum degree δ(G). Let k be an integer number such that k ∈ {1 − ⌈δ(G)/2⌉,..., ⌊(G)/2⌋}. A vertex υ of G is said to be k-controlled by a set M ⊂ V, if $\delta _M(v)\geq \frac{\delta (v)}{2}+k$ where δM(υ) represents the number of neighbors υ has in M and δ(υ) the degree of υ....

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