κ -rainbow domination and some related problems

Let G = (V,E) be a simple graph and k ∈ N. A κ -rainbow dominating function of G is a function ∫ : V  2{1,...,k} such that for each v ∈ V with∫ (v) = ∅, ∪u ∈N(v) f(u) = {1, ..., k}. The minimum of Σ v∈V |f(v)| over all κ -rainbow dominating functions f of G is denoted by r κ (G). In this paper, we consider small k and study some problems related to κ -rainbow domination theory.