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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aazami, A.: Domination in graphs with bounded propagation: algorithms, formulations and hardness results. J. Comb. Optim. 19(4), 429–456 (2010)
AIM Minimum Rank: Special Graphs Work Group: Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl. 428(7), 1628–1648 (2008)
Baldwin, T.L., Mili, L., Boisen Jr., M.B., Adapa, R.: Power system observability with minimal phasor measurement placement. IEEE Trans. Power Syst. 8(2), 707–715 (1993)
Barrera, R., Ferrero, D.: Power domination in cylinders, tori and generalized Petersen graphs. Networks 58(1), 43–49 (2011)
Brešar, B., Klavžar, S., Rall, D.F.: Dominating direct products of graphs. Discrete Math. 307(13), 1636–1642 (2007)
Chang, G.J., Dorbec, P., Montassier, M., Raspaud, A.: Generalized power domination of graphs. Discrete Appl. Math. 160(12), 1691–1698 (2012)
Dorbec, D., Henning, M.A., Löwenstein, C., Montassier, M., Raspaud, A.: Generalized power domination in regular graphs. SIAM J. Discrete Math. 27(3), 1559–1574 (2013)
Dorbec, D., Mollard, M., Klavžar, S., Špacapan, S.: Power domination in product graphs. SIAM J. Discrete Math. 22(2), 554–567 (2008)
Dorfling, M., Henning, M.A.: A note on power domination in grid graphs. Discrete Appl. Math. 154(6), 1023–1027 (2006)
Guo, J., Niedermeier, R., Raible, D.: Improved algorithms and complexity results for power domination in graphs. Algorithmica 52(2), 177–202 (2008)
Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T., Henning, M.A.: Domination in graphs applied to electric power networks. SIAM J. Discrete Math. 15(4), 519–529 (2002)
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs. CRC Press, Taylor and Francis, Boca Raton (2011)
Liao, C.-S., Lee, D.-T.: Power domination problem in graphs. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 818–828. Springer, Heidelberg (2005)
Varghese, S.: Studies on some generalizations of line graph and the power domination problem in graphs. Ph. D thesis, Cochin University of Science and Technology, Cochin, India (2011)
Zhao, M., Kang, L., Chang, G.J.: Power domination in graphs. Discrete Math. 306(15), 1812–1816 (2006)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-29221-2_31
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29220-5
Online ISBN: 978-3-319-29221-2
eBook Packages: Computer ScienceComputer Science (R0)