Total restrained domination in claw-free graphs with minimum degree at least two

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 γ t r ( G ) , is the smallest cardinality of a total restrained dominating set of G . We will show that if G is claw-free, connected, has minimum degree at least two and G is not one of nine exceptional graphs, then γ t r ( G ) ≤ 4 n 7 .