For a positive integer k , a {k}-Roman dominating function of a graph G = (V,E) is a function f: V —> {0,1,... ,k} satisfying f(N(v)) ≥ k for each vertex v ε V with f(v) = 0. Every graph G satisfes γ { Rk } (G) ≤ kγ(G) , where γ { Rk } ( G ) denotes the minimum weight of a { k }-Roman dominating function of G and γ(G) is the domination number of G . In this work we study graphs for which the equality is reached, called {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 ≥ 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.