on trees with equal roman domination and outer-independent roman domination numbers

A roman dominating function (rdf) 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 roman dominating function f is called an outer-independent roman dominating function (oirdf) on g if the set {v∈v∣f(v)=0} is independent. the (outer-independent) roman domination number γr(g) (γoir(g)) is the minimum weight of an rdf (oirdf) on g. clearly for any graph g, γr(g)≤γoir(g). in this paper, we provide a constructive characterization of trees t with γr(t)=γoir(t).