On the Roman domination number of a graph

A Roman dominating function of a graph G is a labeling f : V ( G ) ⟶ { 0 , 1 , 2 } such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γ R ( G ) of G is the minimum of ∑ v ∈ V ( G ) f ( v ) over such functions. A Roman dominating function of G of weight γ R ( G ) is called a γ R ( G ) -function. A Roman dominating function f : V ⟶ { 0 , 1 , 2 } can be represented by the ordered partition ( V 0 , V 1 , V 2 ) of V , where V i = { v ∈ V ∣ f ( v ) = i } . Cockayne et al. [E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi, S.T. Hedetniemi, On Roman domination in graphs, Discrete Math. 278 (2004) 11–22] posed the following question: What can we say about the minimum and maximum values of | V 0 | , | V 1 | , | V 2 | for a γ R -function f = ( V 0 , V 1 , V 2 ) of a graph G ? In this paper we first show that for any connected graph G of order n ≥ 3 , γ R ( G ) + γ ( G ) 2 ≤ n , where γ ( G ) is the domination number of G . Also we prove that for any γ R -function f = ( V 0 , V 1 , V 2 ) of a connected graph G of order n ≥ 3 , | V 0 | ≥ n 5 + 1 , | V 1 | ≤ 4 n 5 − 2 and | V 2 | ≤ 2 n 5 .