×

Captive domination in graphs. (English) Zbl 1458.05183

Summary: In this paper, a new definition of graph domination called “Captive Domination” is introduced. The proper subset of the vertices of a graph \(G\) is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which doesn’t belong to the dominating set. The inverse captive domination is also introduced. The lower and upper bounds for the number of edges of the graph are presented by using the captive domination number. Moreover, the lower and upper bounds for the captive domination number are found by using the number of vertices. The condition when the total domination and captive domination number are equal to two is discussed and obtained results. The captive domination in complement graphs is discussed. Finally, the captive dominating set of paths and cycles are determined.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abdlhusein, M. A. and Al-Harere, M. N., Pitchfork Domination and It’s Inverse for Corona and Join Operations in Graphs, Proceedings of International Mathematical SciencesI(2) (2019) 51-55.
[2] Al-Harere, M. N. and Abdlhusein, M. A., Pitchfork domination in graphs, Discrete Mathematics, Algorithms and Applications12(2) (2020) 2050025. · Zbl 1456.05122
[3] Al-Harere, M. N. and Breesam, A. T., Further results on bi-domination in graphs, AIP Conf. Proc.2096 (2019) 020013.
[4] Al-Harere, M. N. and Khuda Bakhash, P. A., Tadpole domination in graphs, Baghdad Sci. J.15 (2018) 466-471.
[5] Cockayne, E. J., Dawes, R. M., Hedetniemi, S. T., Total domination in graphs, Networks: Int. J.10(3) (1980) 211-219. · Zbl 0447.05039
[6] Chellali, M., Haynes, T. W., Hedetniemi, S. T. and Rae, A. M., [1,2]-Set in graphs, Discrete Appl. Math.161(18) (2013) 2885-2893. · Zbl 1287.05098
[7] Harary, F., Graph Theory (Addison-Wesley, Reading Mass, 1969). · Zbl 0182.57702
[8] Haynes, T. W., Hedetniemi, S. T. and Slater, P. J., Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). · Zbl 0890.05002
[9] Haynes, T. W., Hedetniemi, S. T. and Slater, P. J., Domination in Graphs-Advanced Topics (Marcel Dekker Inc., 1998). · Zbl 0883.00011
[10] Jabour, A. A. and Omran, A. A., Domination in discrete topology graph, AIP Conf. Proc.2183 (2019) 030006.
[11] Khodkar, A., Samadi, B. and Golmohammadi, H. R., \((k, k^\prime, k^{\prime \prime})\)-domination in graphs, J. Combin. Math. Combin.Comput.98 (2016) 343-349. · Zbl 1360.05124
[12] Natarajan, C., Ayyaswamy, S. K. and Sathiamoorthy, G., A note on hop domination number of some special families of graphs, Int. J. Pure Appl. Math.119(12) (2018) 14165-14171.
[13] Omran, A. A. and Hamed Oda, H., Hn domination in graphs, Baghdad Sci. J.16(1) (2019) 242-247.
[14] Omran, A. A. and Rajihy, Y., Some properties of frame domination in graphs, J. Eng. Appl. Sci.12 (2017) 8882-8885.
[15] Omran, A. A. and Al Hwaeer, H. J., Modern roman domination in graphs, Basrah Journal of Science36(1) (2018) 45-54.
[16] Ore, O., Theory of Graphs (American Mathematical Society, Providence, R.I., 1962). · Zbl 0105.35401
[17] Saradha, N. and Swaminathan, V., Connected equitable co-independent domination of a graph, Int. J. Pure Appl. Math.101(5) (2015) 721-726.
[18] Yang, X. and Wu, B., [1, 2]-Domination in graphs, Discrete Appl. Math.175 (2014) 79-86. · Zbl 1298.05261
[19] Zhang, X., Shao, Z. and Yang, H., The \([a,b]\)-domination and \([a,b]\)-total domination of graphs, J. Math. Res.9(3) (2017) 38-45.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.