On the (2,2)-domination number of trees

You Lu; Xinmin Hou; Jun-Ming Xu

Discussiones Mathematicae Graph Theory (2010)

  • Volume: 30, Issue: 2, page 185-199
  • ISSN: 2083-5892

Abstract

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Let γ(G) and γ 2 , 2 ( G ) denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that ( 2 ( γ ( T ) + 1 ) ) / 3 ≤ γ 2 , 2 ( T ) ≤ 2 γ ( T ) . Moreover, we characterize all the trees achieving the equalities.

How to cite

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You Lu, Xinmin Hou, and Jun-Ming Xu. "On the (2,2)-domination number of trees." Discussiones Mathematicae Graph Theory 30.2 (2010): 185-199. <http://eudml.org/doc/271068>.

@article{YouLu2010,
abstract = {Let γ(G) and $γ_\{2,2\}(G)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $(2(γ(T)+1))/3 ≤ γ_\{2,2\}(T) ≤ 2γ(T)$. Moreover, we characterize all the trees achieving the equalities.},
author = {You Lu, Xinmin Hou, Jun-Ming Xu},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination number; total domination number; (2,2)-domination number; -domination number},
language = {eng},
number = {2},
pages = {185-199},
title = {On the (2,2)-domination number of trees},
url = {http://eudml.org/doc/271068},
volume = {30},
year = {2010},
}

TY - JOUR
AU - You Lu
AU - Xinmin Hou
AU - Jun-Ming Xu
TI - On the (2,2)-domination number of trees
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 2
SP - 185
EP - 199
AB - Let γ(G) and $γ_{2,2}(G)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $(2(γ(T)+1))/3 ≤ γ_{2,2}(T) ≤ 2γ(T)$. Moreover, we characterize all the trees achieving the equalities.
LA - eng
KW - domination number; total domination number; (2,2)-domination number; -domination number
UR - http://eudml.org/doc/271068
ER -

References

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  1. [1] T.J. Bean, M.A. Henning and H.C. Swart, On the integrity of distance domination in graphs, Australas. J. Combin. 10 (1994) 29-43. Zbl0815.05036
  2. [2] G. Chartrant and L. Lesniak, Graphs & Digraphs (third ed., Chapman & Hall, London, 1996). 
  3. [3] M. Fischermann and L. Volkmann, A remark on a conjecture for the (k,p)-domination number, Utilitas Math. 67 (2005) 223-227. Zbl1077.05074
  4. [4] M.A. Henning, Trees with large total domination number, Utilitas Math. 60 (2001) 99-106. Zbl1011.05045
  5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (New York, Marcel Deliker, 1998). Zbl0890.05002
  6. [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (New York, Marcel Deliker, 1998). Zbl0883.00011
  7. [7] T. Korneffel, D. Meierling and L. Volkmann, A remark on the (2,2)-domination number Discuss. Math. Graph Theory 28 (2008) 361-366, doi: 10.7151/dmgt.1411. Zbl1156.05044

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