Double Roman domination subdivision number in graphs

For a graph G = (V, E), a double Roman dominating function is a function f : V(G) -> {0, 1, 2, 3} having the property that if f(v) = 0, then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f(w) = 3, and if f(v) = 1, then vertex v must have at least one neighbor with f(w) >= 2. The weight of a double Roman dominating function f is the value w(f) = Sigma(v is an element of V(G)) f(v). The double Roman domination number of a graph G, denoted by gamma(dR)(G), equals the minimum weight of a double Roman dominating function on G. The double Roman domination subdivision number sd(gamma dR) (G) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double Roman domination number. In this paper, we first show that the decision problem associated with sd(gamma dR) (G) is NP-hard and then establish upper bounds on the double Roman domination subdivision number for arbitrary graphs.