Discussiones Mathematicae Graph Theory 30(2) (2010) 185-199
doi: 10.7151/dmgt.1486

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You Lu,  Xinmin Hou  and  Jun-Ming Xu

Department of Mathematics
University of Science and Technology of China
Hefei, Anhui, 230026, China
e-mail: xmhou@ustc.edu.cn


Let γ(G) and γ2,2(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] ≤ γ2,2(T) ≤ 2γ(T). Moreover, we characterize all the trees achieving the equalities.

Keywords: domination number, total domination number, (2,2)-domination number.

2010 Mathematics Subject Classification: 05C69.


[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.
[2] G. Chartrant and L. Lesniak, Graphs & Digraphs (third ed., Chapman & Hall, London, 1996).
[3] M. Fischermann and L. Volkmann, A remark on a conjecture for the (k,p)-domination number, Utilitas Math. 67 (2005) 223-227.
[4] M.A. Henning, Trees with large total domination number, Utilitas Math. 60 (2001) 99-106.
[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (New York, Marcel Deliker, 1998).
[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (New York, Marcel Deliker, 1998).
[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.

Received 19 September 2008
Revised 4 June 2009
Accepted 4 June 2009