Note: 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 @c"R(G) of G is the minimum of @?"v"@?"V"("G")f(v) over such functions. A Roman dominating function of G of weight @c"R(G) is called a @c"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 @c"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, @c"R(G)+@c(G)2@?n, where @c(G) is the domination number of G. Also we prove that for any @c"R-function f=(V"0,V"1,V"2) of a connected graph G of order n>=3, |V"0|>=n5+1, |V"1|@?4n5-2 and |V"2|@?2n5.