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Generous Roman domination stability in graphs | ||
| Journal of Discrete Mathematics and Its Applications | ||
| دوره 11، شماره 1، خرداد 2026، صفحه 1-12 اصل مقاله (280.03 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) ≥ 2. 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) ≥ 2 adjacent to u such that the function f': V→{0,1,2,3}, defined by f'(u)=α, f'(v)=f(v)-α, where α∈{1,2}, and f'(w)=f(w) if w∈ 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 ϒgR(G). The ϒgR(G)-stability st_ϒgR(G) (resp. ϒgR--stability st_ϒgR-(G), ϒgR+-stability st_ϒgR+(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 ϒgR-stability for some special classes of graphs, and then we present some bounds on st_ϒgR(G). We also characterize graphs with large st_ϒgR(G). Moreover, we show that if T is a nontrivial tree, then st_{ϒgR}(T) ≤ 2, and if further T has maximum degree Δ ≥ 3, then st_{ϒgR-(T)} ≤ Δ-1. | ||
| کلیدواژه‌ها | ||
| Generous Roman domination؛ Generous Roman domination stability؛ trees | ||
| مراجع | ||
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