outer independent roman domination number of trees

A roman dominating function (rdf) on a graph g= (v,e) is a function f:v→{0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. an rdf f is called an outer independent roman dominating function (oirdf) if the set of vertices assigned a 0 under f is an independent set. the weight of an oirdf is the sum of its function values over all vertices, and the outer independent roman domination number γoir(g) is the minimum weight of an oirdf on g. in this paper, we show that if t is a tree of order n≥3 with s(t) support vertices, then γ_oir(t)≤min{5n/6,3n+s(t)/4}. moreover, we characterize the tress attaining each bound.