Total restrained domination in graphs of diameter 2 or 3

Abstract For a given connected graph G=(V,E), a set D tr ⊆ V ( G ) is a total restrained dominating set if it is a dominating set and both 〈D t r 〉 and 〈V(G)−D t r 〉 do not contain isolated vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by γ t r (G). In this paper, we continue the study of total restrained domination number of graphs. We first give some results on total restrained domination number of graphs. And then, we characterize all graphs G of order n for which (1) γ t r (G)=n, (2) γ(G)=1 and γ t r (G)=3, and (3) γ t r (G)=2. Furthermore, we give some bounds on total restrained domination number of graphs with diameter 3. Finally, we present some bounds for total restrained domination number of some planar graphs with diameter 2 and γ-set of cardinality 2.