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 $f(v) =2$‎. ‎A {\em restrained Roman dominating}‎ ‎function $f$ is a 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 $\omega(f)=\sum_{u\in V(G)} f(u)$‎. ‎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.‎