Geodesic
Roman {2}-Domination Problem in Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 641-660.

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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) = ∑v∈V f(v), and the minimum weight of an R2DF on G is the Roman 2-domination number γR2(G). An R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman 2-domination number iR2(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 γR2(G) is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of iR2(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 γR2(G) in any block graph G, which is a generalization of trees.
Keywords: Roman {2}-domination, domination, algorithms 
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Chen, Hangdi; Lu, Changhong. Roman {2}-Domination Problem in Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 641-660. http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a18/

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