On 2-rainbow domination and Roman domination in graphs

A Roman dominating function of a graph G is a function f : V → {0, 1, 2} such that every vertex with 0 has a neighbor with 2. The minimum of f (V (G)) = ∑ v∈V f (v) over all such functions is called the Roman domination number γR(G). A 2-rainbow dominating function of a graphG is a function g that assigns to each vertex a set of colors chosen from the set {1, 2}, for each vertex v ∈ V (G) such that g(v) = ∅, we have ⋃ u∈N(v) g(u) = {1, 2}. The 2-rainbow domination number γr2(G) is the minimum ofw(g) = ∑ v∈V |g(v)| over all such functions. We prove γr2(G) ≤ γR(G) and obtain sharp lower and upper bounds for γr2(G) + γr2(G). We also show that for any connected graph G of order n ≥ 3, γr2(G)+ γ (G) 2 ≤ n. Finally, we give a proof of the characterization of graphs with γR(G) = γ (G)+ k for 2 ≤ k ≤ γ (G). © 2010 Elsevier Ltd. All rights reserved.