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For a positive integer k, a k-rainbow dominating function 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, ...,k such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃u ∈ N(v)f(u) = 1,2, ...,k is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set $f₁,f₂,...,f_d$ of k-rainbow dominating functions on G with the property that ${\sum }_{i=1}^d\left|f_i\left(v\right)\right|\le k$ for each v ∈ V(G), is called a k-rainbow dominating family (of...