  For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0,
1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a
neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is
the sum $f (V) = \cup_{v\in V} f (v)$, and the minimum weight of a Roman
dominating function on $G$ is the Roman domination number $\gamma_R(G)$ of $G$.
The Roman bondage number $b_R(G)$ of $G$ is the minimum cardinality of all sets
$F \subseteq E$ for which $\gamma_R(G - F) > \gamma_R(G)$. A graph $G$ is in
the class $\mathcal{R}_{UVR}$ if the Roman domination number remains unchanged
when a vertex is deleted. In this paper we obtain tight upper bounds for
$\gamma_R(G)$ and $b_R(G)$ provided a graph $G$ is in $\mathcal{R}_{UVR}$. We
present necessary and sufficient conditions for a tree to be in the class
$\mathcal{R}_{UV R}$. We give a constructive characterization of
$\mathcal{R}_{UVR}$-trees using labellings.
