Geodesic
Bounding the Open k-Monopoly Number of Strong Product Graphs
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 287-299

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Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ, and let k ∈{ 1 − δ // 2 , . . ., δ // 2 } be an integer. Given a set M ⊂ V, a vertex v of G is said to be k-controlled by M if δ_M (v) ≥δ G(v) 2+k, where δ_M(v) represents the number of neighbors of v in M and δ_G(v) the degree of v in G. A set M is called an open k-monopoly if every vertex v of G is k-controlled by M. The minimum cardinality of any open k-monopoly is the open k-monopoly number of G. In this article we study the open k-monopoly number of strong product graphs. We present general lower and upper bounds for the open k-monopoly number of strong product graphs. Moreover, we study in addition the open 0-monopolies of several specific families of strong product graphs.
Keywords: open monopolies, strong product graphs, alliances, domination 
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Kuziak, Dorota; Peterin, Iztok; Yero, Ismael G. Bounding the Open k-Monopoly Number of Strong Product Graphs. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 287-299. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a22/