Let D = (V,A) be a finite and simple digraph. A II-rainbow dominating function (2RDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V with f(v) = ∅ the condition ⋃ u∈N−(v) f(u) = {1, 2} is fulfilled, where N−(v) is the set of in-neighbors of v. The weight of a 2RDF f is the value ω(f) = ∑ v∈V |f(v)|. The 2-rainbow d...

We obtain new results on 2-rainbow domination number of generalized Petersen graphs P(5k,k). In some cases (for infinite families), exact values are established, and in all other lower upper bounds given. particular, it is shown that, for k>3, γr2(P(5k,k))=4k k≡2,8mod10, γr2(P(5k,k))=4k+1 k≡5,9mod10, 4k+1≤γr2(P(5k,k))≤4k+2 k≡1,6,7mod10, 4k+1≤γr2(P(5k,k))≤4k+3 k≡0,3,4mod10.

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any v ∈ V (G), f(v) = ∅ implies