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

Abstract

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 ∈ V, with f(v) = 0, Σu∈N(v) f(u) ≥ 2. The weight of an Italian dominating function f is defined as w(f) = f(V ) = Σu∈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 γI(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) = Σv∈V f(v), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γdR(G). In this paper we show that γdR(G)/2 ≤ γI(G) ≤ 2γdR(G)/3, and characterize all trees T with γI(T) = 2γdR(T)/3.