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Generous Roman domination stability in graphs | ||
| Journal of Discrete Mathematics and Its Applications | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 17 تیر 1404 اصل مقاله (281.21 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2025.11827.1119 | ||
| نویسندگان | ||
| Seyed Mahmoud Sheikholeslami* 1؛ Mustapha Chellali2؛ Mariyeh Kor1 | ||
| 1Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran | ||
| 2LAMDA-RO Laboratory, Department of Mathematics, University of Blida | ||
| تاریخ دریافت: 16 اسفند 1403، تاریخ بازنگری: 05 اردیبهشت 1404، تاریخ پذیرش: 05 اردیبهشت 1404 | ||
| چکیده | ||
| 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 $\mathrm{st}_{\gamma_{gR}}^{-}(T)\leq\Delta-1$. | ||
| کلیدواژه‌ها | ||
| Generous Roman domination؛ Generous Roman domination stability؛ trees | ||
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آمار تعداد مشاهده مقاله: 2 تعداد دریافت فایل اصل مقاله: 3 |
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