On a conjecture involving a bound for the total restrained domination number of a graph

Abstract

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 in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γtr(G), is the smallest cardinality of a total restrained dominating set of G. In Koh et al. (2013), it is proved that if G is a graph of order n≥4 and δ(G)≥2, then γtr(G)≤n−n43. It is further conjectured that this bound can be improved to γtr(G)≤n−θ(n). In this paper we show that if G is a graph with no C3 components and δ(G)≥2, then γtr(G)≤n−n2.