On $\{k\}$-Roman graphs: complexity of recognition and the case of split graphs

For a positive integer $k$, a $\{k\}$-Roman dominating function of a graph $G=(V,E)$ is a function $f\text{\hspace{0.17em}⁣}:V\to \{0,1,\dots ,k\}$ satisfying $f(N(v))\ge k$ for each vertex $v\in V$ with $f(v)=0$. Every graph $G$ satisfies ${\gamma }_{\{Rk\}}(G)\le k\gamma (G)$, where ${\gamma }_{\{Rk\}}(G)$ denotes the minimum weight of a $\{k\}$-Roman dominating function of $G$ and $\gamma (G)$ is the domination number of $G$. In this work we study graphs for which the equality is reached, called \emph{$\{k\}$-Roman graphs}. This extends the concept of $\{k\}$-Roman trees studied by Wang et al. in 2021 to general graphs. We prove that for every $k\ge 3$, the problem of recognizing $\{k\}$-Roman graphs is NP-hard, even when restricted to split graphs. We provide partial answers to the question of which split graphs are $\{2\}$-Roman: we characterize $\{2\}$-Roman split graphs that can be decomposed with respect to the split join operation into two smaller split graphs and classify the $\{k\}$-Roman property within two specific families of split graphs that are prime with respect to the split join operation: suns and their complements.