Geodesic
Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 71-93.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: domination, semitotal domination, trees 
@article{DMGT_2016_36_1_a5,
     author = {Henning, Michael A. and Marcon, Alister J.},
     title = {Vertices {Contained} {In} {All} {Or} {In} {No} {Minimum} {Semitotal} {Dominating} {Set} {Of} {A} {Tree}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {71--93},
     publisher = {mathdoc},
     volume = {36},
     number = {1},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a5/}
}
TY  - JOUR
AU  - Henning, Michael A.
AU  - Marcon, Alister J.
TI  - Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2016
SP  - 71
EP  - 93
VL  - 36
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a5/
LA  - en
ID  - DMGT_2016_36_1_a5
ER  - 
%0 Journal Article
%A Henning, Michael A.
%A Marcon, Alister J.
%T Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree
%J Discussiones Mathematicae. Graph Theory
%D 2016
%P 71-93
%V 36
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a5/
%G en
%F DMGT_2016_36_1_a5
Henning, Michael A.; Marcon, Alister J. Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 71-93. http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a5/

[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

[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

[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

[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

[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