Double Roman domination stability in graphs

A double Roman dominating function (DRDF) on a graph $G$ is a function $f:V\to \{0,1,2,3\}$ having the property that if $f\left(v\right)=0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f\left(w\right)=3$, and if $f\left(v\right)=1$, then the vertex $v$ must have at least one neighbor $w$ with $f\left(w\right)\ge 2$. The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number ${\gamma }_{dR}\left(G\right)$ is the minimum weight of a DRDF on $G$. The ${\gamma }_{dR}$-stability (${\gamma }_{dR}^{-}$-stability, ${\gamma }_{dR}^{+}$-stability) of $G$, denoted by $s{t}_{{\gamma }_{dR}}\left(G\right)$ ($s{t}_{{\gamma }_{dR}}^{-}\left(G\right)$, $s{t}_{{\gamma }_{dR}}^{+}\left(G\right)$), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the double Roman domination number. In this paper, we determine the exact values on the ${\gamma }_{dR}$-stability of some special classes of graphs, and present some bounds on $s{t}_{{\gamma }_{dR}}\left(G\right)$. In addition, for a tree $T$ with maximum degree $\Delta$, we show that $s{t}_{{\gamma }_{dR}}\left(T\right)=1$ and $s{t}_{{\gamma }_{dR}}^{-}\left(T\right)\le \Delta$, and characterize the trees that achieve the upper bound.