On 2-rainbow domination and Roman domination in graphs

A 2-rainbow dominating function of a graph G is a function g that assigns to each vertex a set of colors chosen from the set {1, 2} so that for each vertex with g(v) = ∅ we have ⋃ u∈N(v) g(u) = {1, 2}. The minimum of g(V (G)) = ∑ v∈V (G) |g(v)| over all such functions is called the 2-rainbow domination number γ2r(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The minimum of f(V (G)) = ∑ u∈V (G) f(u) over all such functions is called the Roman domination number γR(G). We first prove that γR(G)/γr2(G) ≤ 3/2 for every graph G and we improve this ratio for all trees. Then we present some bounds for the 2-rainbow domination number in graphs. In particular, we give an upper bound on the 2rainbow domination number for every tree of order at least three in terms of the number of vertices, stems and leaves of the tree. ∗ This research was supported by “Programmes Nationaux de Recherche: Code 8/u09/510”. † Also at: School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran. This research was in part supported by a grant from IPM (No. 91050016). 86 MUSTAPHA CHELLALI AND NADER JAFARI RAD