Abstract
The power domination number, \(\gamma _\mathrm{{P}}(G)\), is the minimum cardinality of a power dominating set. In this paper, we study the power domination number of some graph products. A general upper bound for \(\gamma _\mathrm{{P}}(G\, \Box \,H)\) is obtained. We determine some sharp upper bounds for \(\gamma _\mathrm{{P}}(G\, \Box \,H)\) and \(\gamma _\mathrm{{P}}(G\times H)\), where the graph H has a universal vertex. We characterize the graphs G and H of order at least four for which \(\gamma _\mathrm{{P}}(G\, \Box \,H)=1\). The generalized power domination number of the lexicographic product is also obtained.
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Acknowledgments
The first author is supported by Maulana Azad National Fellowship (F1- 17.1/2012-13/MANF-2012-13-CHR-KER-15793) of the University Grants Commission, India.
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Varghese, S., Vijayakumar, A. (2016). On the Power Domination Number of Graph Products. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_31
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