# Bounding the Openk-Monopoly Number of Strong Product Graphs

• Volume: 38, Issue: 1, page 287-299
• ISSN: 2083-5892

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## Abstract

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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 [...] δM(v)≥δG(v)2+k ${\delta }_{M}\left(v\right)\ge \frac{{\delta }_{G}\left(v\right)}{2}+k$ , 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.

## How to cite

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Dorota Kuziak, Iztok Peterin, and Ismael G. Yero. "Bounding the Openk-Monopoly Number of Strong Product Graphs." Discussiones Mathematicae Graph Theory 38.1 (2018): 287-299. <http://eudml.org/doc/288387>.

@article{DorotaKuziak2018,
abstract = {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 [...] δM(v)≥δG(v)2+k $\delta _M (v) \ge \{\{\delta _G (v)\} \over 2\} + k$ , 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.},
author = {Dorota Kuziak, Iztok Peterin, Ismael G. Yero},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {open monopolies; strong product graphs; alliances; domination},
language = {eng},
number = {1},
pages = {287-299},
title = {Bounding the Openk-Monopoly Number of Strong Product Graphs},
url = {http://eudml.org/doc/288387},
volume = {38},
year = {2018},
}

TY - JOUR
AU - Dorota Kuziak
AU - Iztok Peterin
AU - Ismael G. Yero
TI - Bounding the Openk-Monopoly Number of Strong Product Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2018
VL - 38
IS - 1
SP - 287
EP - 299
AB - 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 [...] δM(v)≥δG(v)2+k $\delta _M (v) \ge {{\delta _G (v)} \over 2} + k$ , 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.
LA - eng
KW - open monopolies; strong product graphs; alliances; domination
UR - http://eudml.org/doc/288387
ER -

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