On equality in an upper bound for the restrained and total domination numbers of a graph

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in S. The total domination number of a graph G without isolated vertices, denoted by γt(G), is the minimum cardinality of a total dominating set of G. Let δ and ∆ denote the minimum and maximum degrees, respectively, in G. If G is a graph of order n with δ ≥ 2, then it is shown that γr(G) ≤ n−∆, and we characterize ∗Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.