Upper Total Domination in Claw-Free Cubic Graphs

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, Γt(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _t(G)$$\end{document}, 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 K1,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{1,3}$$\end{document} as an induced subgraph. It is known, or can be readily deduced, that if G≠K4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \ne K_4$$\end{document} is a connected claw-free cubic graph of order n, then 13n≤α(G)≤25n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{3}n \le \alpha (G) \le \frac{2}{5}n$$\end{document}, and 13n≤Γ(G)≤12n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{3}n \le \Gamma (G) \le \frac{1}{2}n$$\end{document}, and these bounds are tight, where α(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (G)$$\end{document} and Γ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma (G)$$\end{document} 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 49n≤Γt(G)≤35n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{4}{9}n \le \Gamma _t(G) \le \frac{3}{5}n$$\end{document}.