Roman {2}-Domination Problem in Graphs

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