Authors:
D. Kuziak, I. Peterin, I.G. Yero
Title:
Bounding the open k-monopoly number of strong product graphs
Source:
Discussiones Mathematicae Graph Theory
Received 26.11.2015, Revised 15.12.2016, Accepted 19.12.2016, doi: 10.7151/dmgt.2017

Abstract:
Let G=(V, E) be a simple graph without isolated vertices and minimum degree δ, and let k∈{1-\left⌈ δ/2\right⌉, ...,\left⌊ δ/2\right⌋} be an integer. Given a set M⊂ V, a vertex v of G is said to be k-controlled by M if δM(v)≥ \frac{δG(v)}{2}+k, where δM(v) represents the {number} of neighbors of v in M and δG(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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