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Abstract
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Let k ≥ 1 be an integer, and let G be a graph. A k-rainbow dominating function (or a k-RDF)
of G is a function f from the vertex set V(G) to the family of all subsets of {1, 2, . . . , k} such that for every v ∈ V(G) with f (v) = ∅, the condition u∈NG(v) f (u) = {1, 2, . . . , k} is fulfilled, where NG(v) is the open neighborhood of v. The weight of a k-RDF f of G is the
valueω(f ) = v∈V(G) | f (v) |. The k-rainbow domination number of G, denoted by γrk(G), is
the minimum weight of a k-RDF of G. The 1-rainbow domination is the same as the classical
domination.
The k-rainbow reinforcement number of G, denoted by rrk(G), is the minimum number
of edges that must be added to G in order to decrease the k-rainbow domination number.
In this paper, we study the k-rainbow reinforcement number of graphs to compare γrk and
γrk′ for k ̸= k′, and present some sharp bounds concerning the invariant
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