Some notes on the Roman domination number and Italian domination number in graphs

An Italian dominating function (or simply, IDF) on a graph G = (V, E) is a function f : V -> {0, 1, 2} that satisfies the property that for every vertex v is an element of V, with f (v) = 0, Sigma(u is an element of N(v)) f(u) >= 2. The weight of an Italian dominating function f is defined as w(f) = f (V) = Sigma(u is an element of V) f(u). The minimum weight among all of the Italian dominating functions on a graph G is called the Italian domination number of G, and is denoted by gamma I(G). A double Roman dominating function (or simply, DRDF) is a function f : V -> {0, 1, 2, 3} having the property that if f(v) = 0 for a vertex v, then v has at least two adjacent vertices assigned 2 under f or one adjacent vertex assigned 3 under f, and if f(v) = 1, then v has at least one neighbor with f (w) >= 2. The weight of a DRDF f is defined as the sum f (V) =Sigma(u is an element of V) f(u), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by gamma dR(G). In this paper we show that gamma dR(G)/2 <= gamma I(G) <= 2 gamma dR(G)/3, and characterize all trees T with gamma I(T) = 2 gamma dR(T)/3.