Bounds on total domination in claw-free cubic graphs

Abstract

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 ${\gamma }_{\mathrm{t}}(G)$. If $G$ does not contain $K_{1,3}$ as an induced subgraph, then $G$ is said to be claw-free. It is shown in [D. Archdeacon, J. Ellis-Monaghan, D. Fischer, D. Froncek, P.C.B. Lam, S. Seager, B. Wei, R. Yuster, 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 ${\gamma }_{\mathrm{t}}(G)⩽n/2$. Two 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 ${\gamma }_{\mathrm{t}}(G)⩽5n/11$.