Roman {2}-domination in Graphs and Graph Products

For a graph G = (V, E) of order n, a Roman {2}-dominating function f : V → {0, 1, 2} has the property that for every vertex v ∈ V with f(v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1 under f. In this paper, we classify all graphs with Roman {2}-domination number belonging to the set {2, 3, 4, n − 2, n − 1, n}. Furthermore, we obtain some results about Roman {2}-domination number of some graph operations.