Generous Roman domination stability in graphs

Let G=(V,E) be a simple graph and f a function defined from V to \{۰,۱,۲,۳\}. A vertex u with f(u)=۰ is called an undefended vertex with respect to f if it is not adjacent to a vertex v with f(v)\geq۲. The function f is called a generous Roman dominating function (GRD-function) if for every vertex with f(u)=۰ there exists at least a vertex v with f(v)\geq۲ adjacent to u such that the function f^{\prime}:V\rightarrow\{۰,۱,۲,۳\}, defined by f^{\prime}(u)=\alpha, f^{\prime}(v)=f(v)-\alpha where \alpha\in\{۱,۲\}, 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)\leq۲, and if further T has maximum degree \Delta\geq۳, then \mathrm{st}_{\gamma_{gR}}^{-}(T)\leq\Delta-۱.

کلیدواژه ها:

Generous Roman domination ، Generous Roman domination stability ، trees