Roman domination excellent graphs: trees

A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = \Sigma_{v\in V} f(v)$. The Roman domination number, $\gamma_R(G)$, of $G$ is the minimum weight of an RDF on $G$. An RDF of minimum weight is called a $\gamma_R$-function. A graph G is said to be $\gamma_R$-excellent if for each vertex $x \in V$ there is a $\gamma_R$-function $h_x$ on $G$ with $h_x(x) \not = 0$. We present a constructive characterization of $\gamma_R$-excellent trees using labelings. A graph $G$ is said to be in class $UVR$ if $\gamma(G-v) = \gamma (G)$ for each $v \in V$, where $\gamma(G)$ is the domination number of $G$. We show that each tree in $UVR$ is $\gamma_R$-excellent.