Bounds on total domination in claw-free cubic graphs

Odile Favaron, Michael A. Henning
2008 Discrete Mathematics  
A set S of vertices in a graph G is a total dominating set, denoted by TDS, of G if every vertex of G is adjacent to some vertex in S (other than itself). The minimum cardinality of a TDS of G is the total domination number of G, denoted by t (G). If G does not contain K 1,3 as an induced subgraph, then G is said to be claw-free. It is shown in [Some remarks on domination, J. Graph Theory 46 (2004) 207-210.] that if G is a graph of order n with minimum degree at least three, then t (G) n/2. Two
more » ... infinite families of connected cubic graphs with total domination number one-half their orders are constructed in [O. Favaron, M.A. Henning, C.M. Mynhardt, J. Puech, Total domination in graphs with minimum degree three, J. Graph Theory 34(1) (2000) 9-19.] which shows that this bound of n/2 is sharp. However, every graph in these two families, except for K 4 and a cubic graph of order eight, contains a claw. It is therefore a natural question to ask whether this upper bound of n/2 can be improved if we restrict G to be a connected cubic claw-free graph of order at least 10. In this paper, we answer this question in the affirmative. We prove that if G is a connected claw-free cubic graph of order n 10, then t (G) 5n/11. (O. Favaron), henning@ukzn.ac.za (M.A. Henning).
doi:10.1016/j.disc.2007.07.007 fatcat:3ayowy544veadgrxyxlfekmvsu