Abstarct: 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 value ω(f) = ∑_(u∈V(G))▒|f(v)| . The k-rainbow domination number of G, denoted by γ_rk (G),is the minimum weight of a k-RDF of G. In this thesis, we focus on the reinforcement concept for the domination. The reinforcement number of a graph G with γ_((G)) ⩾ 2, denoted by r(G), is the minimum number of edges that must be added to G in order to decrease the domination number. Now we extend the reinforcement number to the rainbow domination. Let k ⩾ 1 be an integer. For a graph G, a subset F of E((G) ) ̅the k-rainbow reinforcement set (or a k - RRS) of G if γ_rk(G + F) < γ_rk(G). The k-rainbow reinforcement number of a graph G with γ_rk(G) ⩾ k +1, denoted by γ_rk(G),is the minimum size of a k - RRS of G.