An Upper Bound for the Total Restrained Domination Number of Graphs

Let G be a graph with vertex set V. A set ${D \subseteq V}$ is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in ${V \setminus D}$ has a neighbor in ${V \setminus D}$ . The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γ _tr (G). In this paper, we prove that if G is a connected graph of order n ≥ 4 and minimum degree at least two, then ${\gamma_{tr}(G) \leq n-\sqrt3]{n \over 4}}$ .