Abstract
A subset D of the vertex set V(G) of a graph G is called a [1, k]-dominating set if every vertex from \(V-D\) is adjacent to at least one vertex and at most k vertices of D. A [1, k]-dominating set with minimum number of vertices is called a \(\gamma _{[1,k]}(G)\)-set, and the number of its vertices is called the [1, k]-domination number of G and is denoted by \(\gamma _{[1,k]}(G)\). In this paper, we express the computation of the [1, k]-domination number of lexicographic products \(G\circ H\) as an optimization problem over certain partitions of V(G). Nonetheless, in special cases explicit formulas are possible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bishnu, A., Dutta, K., Gosh, A., Pual, S.: \([1, j]\)-set problem in graphs. Discrete Math. 339, 2215–2525 (2016)
Chellali, M., Favaron, O., Hansberg, A., Volkmann, L.: \(k\)-domination and \(k\)-independence in graphs: a survey. Graphs Combin. 28, 1–55 (2012)
Chellali, M., Favaron, O., Haynes, T.W., Hedetniemi, S.T., McRae, A.: Independent \([1, k]\)-sets in graphs. Australas. J. Combin. 59, 144–156 (2014)
Chellali, M., Haynes, T.W., Hedetniemi, S.T., McRae, A.: \([1,2]\)-sets in graphs. Discrete Appl. Math. 161, 2885–2893 (2013)
Dejter, I.J.: Quasiperfect domination in triangular lattices. Discuss. Math. Graph Theory 29, 179–198 (2009)
Etesami, O., Ghareghani, N., Habib, M., Hooshmandasl, M., Naserasr, R., Sharifani, P.: When an optimal dominating set with given constraints exists. Theor. Comput. Sci. 780, 54–65 (2019)
Fink, J.F., Jacobson, M.S.: \(n\)-domination, \(n\)-dependence and forbidden subgraphs. In: Alavi, Y., Chartrand, G., Lick D.R., Wall, C.E. (eds.) Graph Theory with Applications to Algorithms and Computer Science, pp. 301–311. Wiley, New York (1985)
Goharshady, A.K., Hooshmandasl, M.R., Meybodi, M.A.: \([1,2]\)-sets and \([1,2]\)-total sets in trees with algorithms. Discrete Appl. Math. 198, 136–146 (2016)
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs. CRC Press, Boca Raton (2011)
Klavžar, S., Peterin, I., Yero, I.G.: Graphs that are simultaneously efficient open domination and efficient closed domination graphs. Discrete Appl. Math. 217, 613–621 (2017)
Kuziak, D., Peterin, I., Yero, I.G.: Efficient open domination in graph products. Discrete Math. Theor. Comput. Sci. 16, 105–120 (2014)
Sampathkumar, E.: \((1, k)\)-domination in graph. J. Math. Phys. Sci. 22, 613–619 (1988)
Taylor, D.T.: Perfect \(r\)-codes in lexicographic products of graphs. ARS Combin. 93, 215–223 (2009)
Yang, X., Wu, B.: \([1,2]\)-domination in graphs. Discrete Appl. Math. 175, 79–86 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Xueliang Li.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Iztok Peterin was partially supported by Slovenian research agency under the Grants P1-0297, J1-1693 and J1-9109.
Rights and permissions
About this article
Cite this article
Ghareghani, N., Peterin, I. & Sharifani, P. [1, k]-Domination Number of Lexicographic Products of Graphs. Bull. Malays. Math. Sci. Soc. 44, 375–392 (2021). https://doi.org/10.1007/s40840-020-00957-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-00957-0