Vertices contained in all or in no minimum paired-dominating set of a tree. (English) Zbl 1122.05071

Summary: A set \(S\) of vertices in a graph \(G\) is a paired-dominating set of \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\) and if the subgraph induced by \(S\) contains a perfect matching. We characterize the set of vertices of a tree that are contained in all, or in no, minimum paired-dominating sets of the tree.


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


[1] G. Chartrand and L. Lesniak, Graphs & Digraphs: Third Edition, Chapman & Hall, London, 1996.
[2] M. Chellali and T.W. Haynes, ”On paired and double domination in graphs,” Utilitas Math., vol. 67, pp. 161–171, 2005. · Zbl 1069.05058
[3] M. Chellali and T.W. Haynes, ”Trees with unique minimum paired-dominating sets,” Ars Combin., vol. 73, pp. 3–12, 2004. · Zbl 1082.05023
[4] E.J. Cockayne, M.A. Henning, and C.M. Mynhardt, ”Vertices contained in all or in no minimum total dominating set of a tree,” Discrete Math., vol. 260, pp. 37–44, 2003. · Zbl 1013.05054
[5] O. Favaron and M.A. Henning, ”Paired domination in claw-free cubic graphs,” Graphs and Combinatorics, vol. 20, pp. 447–456, 2004. · Zbl 1054.05074
[6] S. Fitzpatrick and B. Hartnell, ”Paired-domination,” Discuss. Math.-Graph Theory, vol. 18, pp. 63–72, 1998. · Zbl 0916.05061
[7] P.L. Hammer, P. Hansen, and B. Simeone, ”Vertices belonging to all or to no maximum stable sets of a graph,” SIAM J. Algebraic Discrete Methods, vol. 3, no. 2, pp. 511–522, 1982. · Zbl 0496.90056
[8] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998a. · Zbl 0890.05002
[9] T. W. Haynes, S.T. Hedetniemi, and P.J. Slater (eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998b. · Zbl 0883.00011
[10] T.W. Haynes and P.J. Slater, ”Paired-domination and the paired-domatic number,” Congr. Numer., vol. 109, pp. 65–72, 1995. · Zbl 0904.05052
[11] T.W. Haynes and P.J. Slater, ”Paired-domination in graphs,” Networks, vol. 32, pp. 199–206, 1998. · Zbl 0997.05074
[12] C.M. Mynhardt, ”Vertices contained in every minimum dominating set of a tree,” J. Graph Theory, vol. 31, no. 3, pp. 163–177, 1999. · Zbl 0931.05063
[13] K.E. Proffitt, T.W. Haynes, and P.J. Slater, ”Paired-domination in grid graphs,” Congr. Numer., vol. 150, pp. 161–172, 2001. · Zbl 0988.05067
[14] H. Qiao, L. Kang, M. Cardei, and Ding-Zhu, ”Paired-domination of trees,” J. Global Optimization, vol. 25, pp. 43–54, 2003. · Zbl 1013.05055
[15] E. Shan, L. Kang, and M.A. Henning, ”A characterization of trees with equal total domination and paired-domination numbers,” Australasian J. Combinatorics, vol. 30, pp. 33–10, 2004. · Zbl 1054.05081
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.