Roman {2}-Domination Problem in Graphs

For a graph G = (V, E), a Roman {2}-dominating function (R2DF) f: V → {0, 1, 2} has the property that for every vertex v ? V with f(v) = 0, either there exists a neighbor u ? N(v), with f(u) = 2, or at least two neighbors x, y ? N(v) having f(x) = f(y) = 1. The weight of an R2DF f is the sum f(V) = vV f(v), and the minimum weight of an R2DF on G is the Roman {2}-domination number γ{R2}(G). An R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman {2}-domination number i{R2}(G) is the minimum weight of an independent Roman {2}-dominating function on G. In this paper, we show that the decision problem associated with γ{R2}(G) is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of i{R2}(T) in any tree T, which answers an open problem raised by Rahmouni and Chellali [Independent Roman {2}-domination in graphs, Discrete Appl. Math. 236 (2018) 408-414]. Moreover, we present a linear time algorithm for computing the value of γ{R2}(G) in any block graph G, which is a generalization of trees.