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)→{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 ω(f)=∑u∈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 γrr(g). in this paper we establish some sharp bounds for this parameter.