Roman domination excellent graphs: trees

A Roman dominating function (RDF) on a graph G=(V,E) is a labeling f:V→{0,1,2} such that every vertex with label 0 has a neighbor with label 2. The weight of f is the value f(V)=Σv∈Vf(v) The Roman domination number, γR(G), of G is the minimum weight of an RDF on G. An RDF of minimum weight is called a γR-function. A graph G is said to be γR-excellent if for each vertex x∈V there is a γR-function hx on G with hx(x)≠0. We present a constructive characterization of γR-excellent trees using labelings. A graph G is said to be in class UVR if γ(G−v)=γ(G) for each v∈V, where γ(G) is the domination number of G. We show that each tree in UVR is γR-excellent.