On Efficiently Roman Dominatable Graphs

A \((2,2)\) packing on a graph \(G\) is a function \(f: V(G) \to \{0, 1, 2\}\) with \(f(N[v]) \leq 2\) for all \(v \in V(G)\). For a function \(f: V(G) \to \{0,1,2\}\), the Roman influence of \(f\), denoted by \(I_R(f)\), is defined to be \(I_R(f) = (|V_1|+|V_2|) + \sum_{v\in V_2} deg(v)\). The efficient Roman domination number of \(G\), denoted by \(F_R(G)\), is defined to be the maximum of \(I_R(f)\) such that \(f\) is a \((2,2)\)-packing. That is, \(F_R(G) = \text{max}\{I_R(f): f \text{ is a }(2,2)-\emph{packing}\}\). A \((2,2)\)-packing \(F_R(G)\) with \(F_R(G) = I_R(f)\) is called an \(F_R(G)\)-function. A graph \(G\) is said to be efficiently Roman dominatable if \(F_R(G) = n\), and when \(F_R(G) = n\), an \(F_R(G)\)-function is called an efficient Roman dominating function. In this paper, we focus our study on certain graphs which are efficiently Roman dominatable. We characterize the class of \(2 \times m\) and \(3 \times m\) grid graphs, trees, unicyclic graphs, and split graphs which are efficiently Roman dominatable.