Total Domination in Graphs with Given Girth
- 100 Downloads
A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ t (G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g, then γ t (G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs.
KeywordsGraphs Hypergraphs Total domination number Transversals Girth
AMS subject classification05C70
Unable to display preview. Download preview PDF.
- 1.Archdeacon, D., Ellis-Monaghan, J., Fischer, D., Froncek, D., Lam, P.C.B., Seager, S., Wei, B., Yuster, R.: Some remarks on domination. J. Graph Theory 46, 207–210 (2004)Google Scholar
- 2.Chvátal, V., McDiarmid, C.: Small transversals in hypergraphs. Combinatorica 12, 19–26 (1992)Google Scholar
- 3.Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks 10, 211–219 (1980)Google Scholar
- 4.Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.) Fundamentals of Domination in Graphs. Marcel Dekker, Inc. New York (1998)Google Scholar
- 5.Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.) Domination in Graphs: Advanced Topics. Marcel Dekker, Inc. New York (1998)Google Scholar
- 6.Haynes, T.W., Henning, M.A.: Upper bounds on the total domination number. To appear in Ars Combin Google Scholar
- 7.Henning, M.A.: Graphs with large total domination number. J. Graph Theory 35(1), 21–45 (2000)Google Scholar
- 8.Löwenstein, C., Rautenbach, D.: Domination in graphs with minimum degree at least two and large girth. To appear in Graphs Combin Google Scholar
- 9.Thomassé, S., Yeo, A.: Total domination of graphs and small transversals of hypergraphs. Combinatorica 27, 473–487 (2007)Google Scholar
- 10.Tuza, Z.: Covering all cliques of a graph. Discrete Math. 86, 117–126 (1990)Google Scholar