The characterization of perfect Roman domination stable trees

A \emph{perfect Roman dominating function} (PRDF) on a graph $G=(V,E)$ is a function $f:V\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to exactly one vertex $v$ for which $f(v)=2$. The weight of a PRDF is the value $w(f)={\sum }_{u\in V}f(u)$. The minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman domination number ${\gamma }_R^p(G)$} of $G$. A graph $G$ is perfect Roman domination domination stable if the perfect Roman domination number of $G$ remains unchanged under the removal of any vertex. In this paper, we characterize all trees that are perfect Roman domination stable.