Upper Total Domination in Claw-Free Cubic Graphs

Abstract

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some other vertex in S. A total dominating set S is minimal if no proper subset of S is a total dominating set of G. The upper total domination number, $\Gamma _t(G)$, of G is the maximum cardinality of a minimal total dominating set of G. A claw-free graph is a graph that does not contain a claw $K_{1,3}$ as an induced subgraph. It is known, or can be readily deduced, that if $G \ne K_4$ is a connected claw-free cubic graph of order n, then $\frac{1}{3}n \le \alpha (G) \le \frac{2}{5}n$, and $\frac{1}{3}n \le \Gamma (G) \le \frac{1}{2}n$, and these bounds are tight, where $\alpha (G)$ and $\Gamma (G)$ denote the independence number and upper domination number, respectively, of G. In this paper, we prove that if G is a connected claw-free cubic graph of order n, then $\frac{4}{9}n \le \Gamma _t(G) \le \frac{3}{5}n$.