Roman domination excellent graphs: trees

A Roman dominating function (RDF) on a graph $G=(V,E)$ is a labeling $f : V \rightarrow {0, 1, 2}$suchthateveryvertexwithlabel$0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V)={\mathrm{\Sigma }}_{v\in V}f(v)$. 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 $h_x(x)≠0$. 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.