Total restrained domination in unicyclic graphs

Abstract

Let G = (V, E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V 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 minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then γ_tr(U) ≥ [n/2], and provide a characterization of graphs achieving this bound.