Total restrained domination in unicyclic graphs

Let G = (V, E) be a graph. A set S subset of 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 gamma(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 gamma(tr)(U) >= inverted right perpendicularn/2inverted left perpendicular and provide a characterization of graphs achieving this bound.