Independent Double Roman Domination Stability in Graph

Abstract

An independent double Roman dominating function (IDRD-function) on a graph $G$ is a function $f :V(G)\to \{0, 1, 2, 3\}$ having the property that (i) if $f(v) = 0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w) = 3$, and if $f(v) = 1$, then the vertex $v$ must have at least one neighbor $w$ with $f(w) \ge2$, and (ii) the subgraph induced by the vertices with positive weight under $f$ is edgeless. The weight of an IDRD-function is the sum of its function values over all vertices, and the independent double Roman domination number (IDRD-number) $i_{dR}(G)$ is the minimum weight of an IDRD-function on $G$. The $i_{dR}$-stability ($i^-_{dR}$-stability, $i^+_{dR}$-stability) of $G$, denoted by ${\rm st}_{i_{dR}}(G)$ (${\rm st}^-_{i_{dR}}(G)$, ${\rm st}^+_{i_{dR}}(G)$), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the independent double Roman domination number. In this paper, we first determine the exact values on the $i_{dR}$-stability of some special classes of graphs, and then present some bounds on ${\rm st}_{i_{dR}}(G)$. In addition, for a tree $T$ with maximum degree $\Delta$, we show that ${\rm st}_{i_{dR}}(T)=1$ and ${\rm st}^-_{i_{dR}}(T)\le \Delta$, and characterize the trees that achieve the upper bound.