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Total k-domination in Cartesian product graphs

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Abstract

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a total k-dominating set if every vertex \(v\in V\) has at least k neighbors in S. The total k-domination number \(\gamma _{kt}(G)\) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of the Cartesian product of two graphs, and we investigate the relationship between the total k-domination number of the Cartesian product graph with respect to the total k-domination number in the factors of the product. We also study the total k-domination number in certain particular cases of Cartesian products of graphs and determine the exact values of this parameter.

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Acknowledgements

This Research was partly supported by Plan Nacional I+D+I Grants MTM2015-70531-P and MTM 2013–46374-P (Spain), Junta de Andalucía FQM-260 (Spain) and CONACYT (FOMIX-CONACYT-UAGro-249818) (Mexico).

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Authors and Affiliations

  1. Department of Economics, Quantitative Methods and Economic History, Universidad Pablo de Olavide, Carretera de Utrera Km. 1, 41013, Seville, Spain

    S. Bermudo

  2. Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico

    J. L. Sanchéz & J. M. Sigarreta

  3. Faculty of Computer Science and Mathematics, Holguín University, 80100, Holguín, Cuba

    J. L. Sanchéz

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  1. S. Bermudo
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  2. J. L. Sanchéz
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  3. J. M. Sigarreta
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Correspondence to S. Bermudo.

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Bermudo, S., Sanchéz, J.L. & Sigarreta, J.M. Total k-domination in Cartesian product graphs. Period Math Hung 75, 255–267 (2017). https://doi.org/10.1007/s10998-017-0191-2

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