Outer-independent k-rainbow domination.

An outer-independent k-rainbow dominating function of a graph G is a function f from V (G) to the set of all subsets of { 1 , 2 , ... , k } such that both the following hold: (i) { 1 , ... , k } = ⋃ u ∈ N (v) f (u) whenever v is a vertex with f (v) = ∅ , and (ii) the set of all v ∈ V (G) with f (v) = ∅ is independent. The outer-independent k-rainbow domination number of G is the invariant γ o i r k (G) , which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by an outer-independent k-rainbow dominating function. In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it. [ABSTRACT FROM AUTHOR]