On the Algorithmic Complexity of Roman {2}-Domination (Italian Domination)

A Roman { 2 } -dominating function (R2DF) f : V ⟶ { 0 , 1 , 2 } of a graph G = ( V , E ) has the property that for every vertex v ∈ V with f ( v ) = 0 either there is a vertex u ∈ N ( v ) with f ( u ) = 2 or there are two vertices x , y ∈ N ( v ) with f ( x ) = f ( y ) = 1 . The weight of f is the sum f ( V ) = ∑ v ∈ V f ( v ) . The minimum weight of an R2DF on G is the Roman { 2 } -domination number of G . In this paper, we first show that the associated decision problem for Roman { 2 } -domination is NP-complete even when restricted to planar graphs. Then, we give a linear algorithm that computes the Roman { 2 } -domination number of a given unicyclic graph.