Hardness and Algorithmic Results for Roman \{3\}-Domination

A Roman $\{3\}$-dominating function on a graph $G = (V, E)$ is a function $f: V \rightarrow \{0, 1, 2, 3\}$ such that for each vertex $u \in V$, if $f(u) = 0$ then $\sum_{v \in N(u)} f(v) \geq 3$ and if $f(u) = 1$ then $\sum_{v \in N(u)} f(v) \geq 2$. The weight of a Roman $\{3\}$-dominating function $f$ is $\sum_{u \in V} f(u)$. The objective of \rtd{} is to compute a Roman $\{3\}$-dominating function of minimum weight. The problem is known to be NP-complete on chordal graphs, star-convex bipartite graphs and comb-convex bipartite graphs. In this paper, we study the complexity of \rtd{} and show that the problem is NP-complete on split graphs. In addition, we prove that the problem is W[2]-hard parameterized by weight. On the positive front, we present a polynomial-time algorithm for block graphs, thereby resolving an open question posed by Chaudhary and Pradhan [Discrete Applied Mathematics, 2024].