A Roman \3\-dominating function on a graph G = (V, E) is a function f: V \0, 1, 2, 3\ such that for each vertex u V, if f (u) = 0 then ₕ ₍ (ₔ) f (v) 3 and if f (u) = 1 then ₕ ₍ (ₔ) f (v) 2. The weight of a Roman \3\-dominating function f is ₔ ₕ f (u). The objective of 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 and show that the problem is NP-complete on split graphs. In addition, we prove that the problem is W2-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.
Sangam Balchandar Reddy (Sun,) studied this question.
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