On the rainbow domination subdivision numbers in graphs

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃u∈N(v)f(u) = {1, 2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ω(f) =∑v∈V|f(v)|. The 2-rainbow domination number of a graph G, denoted by γr2(G), is the minimum weight of a 2RDF of G. The 2-rainbow domination subdivision number sdγr2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-rainbow domination number. It is conjectured that for any connected graph G of order n ≥ 3, sdγr2(G) ≤ γr2(G). In this paper, we first prove this conjecture for some classes of graphs and then we prove that for any connected graph G of order n ≥ 3, sdγr2(G) ≤ 2γr2(G) − 2.