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Vertices contained in all or in no minimum semitotal dominating set of a tree. (English) Zbl 1329.05232
Summary: Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, \(\gamma(G)\), and the total domination number, \(\gamma_t(G)\). A set \(S\) of vertices in a graph \(G\) is a semitotal dominating set of \(G\) if it is a dominating set of \(G\) and every vertex in \(S\) is within distance 2 of another vertex of \(S\). The semitotal domination number, \(\gamma_{t2}(G)\), is the minimum cardinality of a semitotal dominating set of \(G\). We observe that \(\gamma(G) \leq \gamma_{t2}(G) \leq \gamma_t(G)\). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.

MSC:
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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[1] M. Blidia, M. Chellali and S. Khelifi, Vertices belonging to all or no minimum double dominating sets in trees, AKCE Int. J. Graphs. Comb. 2 (2005) 1-9. · Zbl 1076.05058
[2] 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. 260 (2003) 37-44. doi:10.1016/S0012-365X(02)00447-8 · Zbl 1013.05054
[3] W. Goddard, M.A. Henning and C.A. McPillan, Semitotal domination in graphs, Util. Math. 94 (2014) 67-81. · Zbl 1300.05220
[4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998). · Zbl 0890.05002
[5] M.A. Henning, Recent results on total domination in graphs: A survey, Discrete Math. 309 (2009) 32-63. doi:10.1016/j.disc.2007.12.044 · Zbl 1219.05121
[6] M.A. Henning and A.J. Marcon, On matching and semitotal domination in graphs, Discrete Math. 324 (2014) 13-18. doi:10.1016/j.disc.2014.01.021 · Zbl 1284.05196
[7] M.A. Henning and A.J. Marcon, Semitotal domination in graphs: Partition and algorithmic results, Util. Math., to appear. · Zbl 1284.05196
[8] M.A. Henning and M.D. Plummer, Vertices contained in all or in no minimum paired-dominating set of a tree, J. Comb. Optim. 10 (2005) 283-294. doi:10.1007/s10878-005-4107-3 · Zbl 1122.05071
[9] M.A. Henning and A. Yeo, Total domination in graphs (Springer Monographs in Mathematics, 2013). · Zbl 1408.05002
[10] C.M. Mynhardt, Vertices contained in every minimum dominating set of a tree, J. Graph Theory 31 (1999) 163-177. doi:10.1002/(SICI)1097-0118(199907)31:3〈163::AID-JGT2〉3.0.CO;2-T
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