Generous Roman Domination Subdivision Number in Graph

Abstract

Let $G = (V, E)$ be a simple graph, and let $f: V \to \{0, 1, 2, 3\}$ be a function. A vertex \( u \) is considered an undefended vertex with respect to \( f \) if \( f(u) = 0 \) and there is no adjacent vertex \( v \) satisfying \( f(v) \geq 2 \).  A function \( f \) is termed a generous Roman dominating function (GRD-function) if, for every vertex \( u \) with \( f(u) = 0 \), there exists at least one adjacent vertex \( v \) such that \( f(v) \geq 2 \) and the modified function \( f': V \to \{0,1,2,3\} \), defined as
\(   f'(u) = \alpha, \quad f'(v) = f(v) - \alpha,\) where \( \alpha \in \{1,2\} \), and
\(
f'(w) = f(w) \quad \text{for all } w \in V \setminus \{u, v\},
\) ensures that no vertex remains undefended. The weight of a GRD-function \( f \) is defined as
\(
f(V) = \sum_{u \in V} f(u).
\) The smallest possible weight of a GRD-function on \( G \) is known as the generous Roman domination number of \( G \), denoted by \( \gamma_{gR}(G) \). The generous Roman domination subdivision number, represented as \( \mathrm{sd}_{\gamma_{gR}}(G) \), is the minimum number of edges that must be subdivided (where each edge in \( G \) can be subdivided at most once) to increase the generous Roman domination number. In this paper, we establish upper bounds on the generous Roman domination subdivision number. Furthermore, we determine the exact value of this parameter for certain families of graphs, including paths, cycles, and ladders. Further, we present several sufficient conditions for a graph \( G \) to have a small value of \( sd_{\gamma_{gR}}(G) \).
\end{abstract}
\keywords{generous Roman domination, generous Roman domination subdivision number}