Generous Roman domination stability in graphs

Sheikholeslami, Seyed Mahmoud; Chellali, Mustapha; Kor, Mariyeh

doi:10.22061/jdma.2025.11827.1119

Generous Roman domination stability in graphs

Document Type : Full Length Article

Authors

¹ Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran

² LAMDA-RO Laboratory, Department of Mathematics, University of Blida

10.22061/jdma.2025.11827.1119

Abstract

Let $G=(V,E)$ be a simple graph and $f$ a function defined from $V$ to $\{0,1,2,3\}.$ A vertex $u$ with $f(u)=0$ is called an undefended vertex with respect to $f$ if it is not adjacent to a vertex $v$ with $f(v)\geq2$. The function $f$ is called a generous Roman dominating function (GRD-function) if for every vertex with $f(u)=0$ there exists at least a vertex $v$ with $f(v)\geq2$ adjacent to $u$ such that the function $f^{\prime}:V\rightarrow\{0,1,2,3\}$, defined by $f^{\prime}(u)=\alpha$, $f^{\prime}(v)=f(v)-\alpha$ where $\alpha\in\{1,2\}$, and $f^{\prime}(w)=f(w)$ if $w\in V-\{u,v\}$ has no undefended vertex. The weight of a GRD-function $f$ is the sum of its function values over all vertices, and the minimum weight of a GRD-function on $G$ is the generous Roman domination number $\gamma_{gR}(G)$. The $\gamma_{gR}$-stability $\mathrm{st}_{\gamma_{gR}}(G)$ (resp. $\gamma_{gR}^{-}$-stability $\mathrm{st}_{\gamma_{gR}}^{-}(G)$, $\gamma_{gR}^{+}$-stability $\mathrm{st}_{\gamma_{gR}}^{+}(G)$) of $G$ is defined as the order of the smallest set of vertices whose removal changes (resp. decreases, increases) the generous Roman domination number. In this paper, we first determine the exact values of $\gamma_{gR}$-stability for some special classes of graphs, and then we present some bounds on $\mathrm{st}_{\gamma_{gR}}(G)$. We also characterize graphs with large $\mathrm{st}_{\gamma_{gR}}(G)$.
Moreover, we show that if $T$ is a nontrivial tree, then $\mathrm{st}_{\gamma_{gR}}(T)\leq2,$ and if further $T$ has maximum degree $\Delta\geq3$, then $\mathrm{st}_{\gamma_{gR}}^{-}(T)\leq\Delta-1$.

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