the restrained k-rainbow reinforcement numbers in graphs

for a positive integer k, a restrained k-rainbow dominating function (rkrdf) of a graph g is a function f from the vertex set v (g) to the set of all subsets of the set {1, 2, . . . , k} such that ∪ for any vertex v ϵ v (g) with f(v) = ∅ the conditions u_uϵn(v) f(u) = {1, 2, . . . , k} and |n(v) ç {u ϵ v (g) | f(u) = ∅}| ³ 1 are fulfilled, where n(v) is the open neighborhood of v. the weight of an rkrdf f is the value w(f) =σ_vϵv (g) |f(v)|. the restrained k-rainbow domination number of a graph g, denoted by _γrrk(g), is the minimum weight of an rkrdf of g. the restrained krainbow reinforcement number r_rrk(g) of a graph g is the minimum number of edges that must be added to g in order to decrease the restrained k-rainbow domination number. in this paper, we initiate the study of restrained k-rainbow reinforcement number in graphs and we present some sharp bounds on r_rrk(g). in particular, we determine the restrained 2-rainbow reinforcement number of some classes of graphs.