Bounds on the total restrained domination number of a graph

Let G = (V, E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V - S is adjacent to a vertex in V - S. The total restrained domination number of G, denoted by γ_tr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ_tr(G) ≤ n - δ - 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r - regular, where 4 ≤ r ≤ n - 3, then γ_tr(G) ≤ n - diam(G) - r + 1.