Bounds on the restrained Roman domination number of a graph

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating} function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 induce a subgraph with no isolated vertex.} The weight of a restrained Roman dominating function is the value $omega(f)=sum_{uin V(G)} f(u)$. The minimum weight of a restrained Roman dominating function of $G$ is called the { em restrained Roman domination number} of $G$ and denoted by $gamma_{rR}(G)$. In this paper we establish some sharp bounds for this parameter.