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

Michael A. Henning; Alister J. Marcon

Discussiones Mathematicae Graph Theory (2016)

  • Volume: 36, Issue: 1, page 71-93
  • ISSN: 2083-5892

Abstract

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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.

How to cite

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Michael A. Henning, and Alister J. Marcon. "Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree." Discussiones Mathematicae Graph Theory 36.1 (2016): 71-93. <http://eudml.org/doc/276976>.

@article{MichaelA2016,
abstract = {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.},
author = {Michael A. Henning, Alister J. Marcon},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination; semitotal domination; trees},
language = {eng},
number = {1},
pages = {71-93},
title = {Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree},
url = {http://eudml.org/doc/276976},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Michael A. Henning
AU - Alister J. Marcon
TI - Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree
JO - Discussiones Mathematicae Graph Theory
PY - 2016
VL - 36
IS - 1
SP - 71
EP - 93
AB - 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.
LA - eng
KW - domination; semitotal domination; trees
UR - http://eudml.org/doc/276976
ER -

References

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  1. [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. Zbl1076.05058
  2. [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] Zbl1013.05054
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  4. [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998). Zbl0890.05002
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  6. [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] Zbl1284.05196
  7. [7] M.A. Henning and A.J. Marcon, Semitotal domination in graphs: Partition and algorithmic results, Util. Math., to appear. Zbl1284.05196
  8. [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] Zbl1122.05071
  9. [9] M.A. Henning and A. Yeo, Total domination in graphs (Springer Monographs in Mathematics, 2013). 
  10. [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] Zbl0931.05063

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