Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree

• Autorzy: Michael A. Henning , Alister J. Marcon
• Języki publikacji: EN

Abstrakty

EN

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, γ(G), and the total domination number, γt(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, γt2(G), is the minimum cardinality of a semitotal dominating set of G. We observe that γ(G) ≤ γt2(G) ≤ γt(G). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.

Słowa kluczowe

EN

domination; semitotal domination; trees

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2016
• Tom: 36
• Numer: 1
• Strony: 71-93

Daty

wydano

2016-02-01

otrzymano

2014-12-18

poprawiono

2015-04-13

zaakceptowano

2015-04-13

online

2016-01-19

Twórcy

autor

Michael A. Henning

• Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006, South Africa

autor

Alister J. Marcon

• Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006, South Africa

Bibliografia

• [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.
• [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[Crossref]
• [3] W. Goddard, M.A. Henning and C.A. McPillan, Semitotal domination in graphs, Util. Math. 94 (2014) 67–81.
• [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998).
• [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[Crossref][WoS]
• [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[Crossref]
• [7] M.A. Henning and A.J. Marcon, Semitotal domination in graphs: Partition and algorithmic results, Util. Math., to appear.
• [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[Crossref]
• [9] M.A. Henning and A. Yeo, Total domination in graphs (Springer Monographs in Mathematics, 2013).
• [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[Crossref]

Identyfikatory

DOI

10.7151/dmgt.1844