×

Paired bondage in trees. (English) Zbl 1161.05022

Summary: Let \(G=(V,E)\) be a graph with \(\delta(G) \geq1\). A set \(D \subseteq V\) is a paired dominating set if \(D\) is dominating, and the induced subgraph left angle \(\langle D\rangle\) contains a perfect matching. The paired domination number of \(G\), denoted by \(\gamma_p(G)\), is the minimum cardinality of a paired dominating set of \(G\). The paired bondage number, denoted by \(b_p(G)\), is the minimum cardinality among all sets of edges \(E'\subseteq E\) such that \(\delta(G-E') \geq 1\) and \(\gamma_p(G-E') > \gamma_p(G)\). We say that \(G\) is a \(\gamma_p\)-strongly stable graph if, for all \(E'\subseteq E\), either \(\gamma_p(G-E')= \gamma_p(G)\) or \(\delta(G-E')=0\). We discuss the basic properties of paired bondage and give a constructive characterization of \(\gamma_p\)-strongly stable trees.

MSC:

05C05 Trees
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Favaron, O.; Henning, M. A., Paired-domination in claw-free cubic graphs, Graphs Combin., 20, 447-456 (2004) · Zbl 1054.05074
[2] Fink, J. F.; Jacobson, M. S.; Kinch, L. F.; Roberts, J., The bondage number of a graph, Discrete Math., 86, 47-57 (1990) · Zbl 0745.05056
[3] Haynes, T. W.; Hedetniemi, S. T.; Slater, P. J., Fundamentals of Domination in Graphs (1998), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0890.05002
[4] Haynes, T. W.; Hedetniemi, S. T.; Slater, P. J., Domination in Graphs. Advanced Topics (1998), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0883.00011
[5] Haynes, T. W.; Slater, P. J., Paired-domination in graphs, Networks, 32, 199-206 (1998) · Zbl 0997.05074
[6] Haynes, T. W.; Slater, P. J., Paired-domination and the paired-domatic number, Congr. Numer., 109, 65-72 (1995) · Zbl 0904.05052
[7] Hartnell, B. L.; Rall, D. F., A characterization of trees in which no edge is essential to the domination number, Ars Combin., 33, 65-76 (1992) · Zbl 0767.05078
[8] Kang, L.; Yuan, J., Bondage number of planar graphs, Discrete Math., 222, 191-198 (2000) · Zbl 0961.05055
[9] Liu, H.; Sun, L., The bondage and connectivity of a graph, Discrete Math., 263, 289-293 (2003) · Zbl 1014.05050
[10] Proffitt, K. E.; Haynes, T. W.; Slater, P. J., Paired-domination in grid graphs, Congr. Numer., 150, 161-172 (2001) · Zbl 0988.05067
[11] Qiao, H.; Kang, L.; Cardei, M.; Ding-Zhu, Paired-domination of trees, J. Global Optim., 25, 43-54 (2003) · Zbl 1013.05055
[12] Shan, E.; Kang, L.; Henning, M. A., A characterization of trees with equal total domination and paired-domination numbers, Australas. J. Combin., 30, 31-39 (2004) · Zbl 1054.05081
[13] Wu, Y.; Fan, Q., The bondage number of four domination parameters for trees, J. Math., Wuchan Univ., 24, 267-270 (2004) · Zbl 1051.05070
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.