Some notes on the Roman domination number and Italian domination number in graphs
Maryam Hajibaba, Nader Jafari Rad · Journal of Physics Conference Series · 2017
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, Σ u ∈ N ( v ) f ( u ) ≥ 2. The weight of an Italian dominating function f is defined as w ( f ) = f ( V ) = Σ u ∈ 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 γ 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 ) = Σ v ∈ V f ( v ), and the minimum weight of a DRDF on G is the double Roman domination number of G , denoted by γ dR ( G ). In this paper we show that γ dR ( G )/2 ≤ γ I ( G ) ≤ 2 γ dR ( G )/3, and characterize all trees T with γ I ( T ) = 2 γ dR ( T )/3.