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).