Restrained domination in cubic graphs

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γr(G), is the smallest cardinality of a restrained dominating set of G. A graph G is said to be cubic if every vertex has degree three. In this paper, we study restrained domination in cubic graphs. We show that if G is a cubic graph of order n, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma_{r}(G)\geq \frac{n}{4}$\end{document} , and characterize the extremal graphs achieving this lower bound. Furthermore, we show that if G is a cubic graph of order n, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma _{r}(G)\leq \frac{5n}{11}.$\end{document} Lastly, we show that if G is a claw-free cubic graph, then γr(G)=γ(G).