The k-rainbow reinforcement numbers in graphs

For a positive integer k, a k-rainbow dominatingfunction (kRDF) of a graph G is a function f from the vertexset V (G) to the set of all subsets of the set {1, 2, . . . , k} suchthat for any vertex v ϵ V (G) with f(v) = ø∅ the conditionUuϵN(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is theneighborhood of v. The weight of a kRDF f is the valueω(f) = Σ(vϵV) |f(v)|. The k-rainbow domination number of agraph G, denoted by γrk(G), is the minimum weight of a kRDFof G. The 1-rainbow domination is the same as the ordinarydomination. The k-rainbow reinforcement number r(rk)(G) of agraph G is the minimum number of edges that must be added toG in order to decrease the k-rainbow domination number. Here,we study the k-rainbow reinforcement number in graphs and wedetermine the 2-rainbow reinforcement number of some classesof graphs.