Roman {2}-Domination Problem in Graphs

Chen, Hangdi ; Lu, Changhong

Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 641-660

Résumé

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 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.

Détail

Keywords: Roman {2}-domination, domination, algorithms