On the computational complexity of Roman{2}-domination in grid graphs

A Roman{2}-dominating function of a graph G = (V, E) is a function f : V ? {0, 1, 2} such that every vertex x ? V with f (x) = 0 either there exists at least one vertex y ? N(x) with f (y) = 2 or there are at least two vertices u, v ? N(x) with f (u) = f (v) = 1. The weight of a Roman{2}-dominating function f on G is defined to be the value of S-x?V f (x). The minimum weight of a Roman{2}-dominating function on G is called the Roman{2}-domination number of G. In this paper, we prove that the decision problem associated with Roman{2}-domination number is N P-complete even when restricted to subgraphs of grid graphs. Additionally, we answer an open question about the approximation hardness of Roman{2}-domination problem for bounded degree graphs.