The kk-rainbow bondage number of a graph

For a positive integer kk, a kk-rainbow dominating function   (kRDF) of a graph GG is a function ff from the vertex set V(G)V(G) to the set of all subsets of the set {1,2,…,k}{1,2,…,k} such that for any vertex v∈V(G)v∈V(G) with f(v)=0̸f(v)=0̸ the condition ⋃u∈N(v)f(u)={1,2,…,k}⋃u∈N(v)f(u)={1,2,…,k} is fulfilled, where N(v)N(v) is the neighborhood of vv. The weight   of a kRDF ff is the value ω(f)=∑v∈V|f(v)|ω(f)=∑v∈V|f(v)|. The kk-rainbow domination number   of a graph GG, denoted by γrk(G)γrk(G), is the minimum weight of a kRDF of G. The 1-rainbow domination is the same as the ordinary domination. The kk-rainbow bondage number brk(G)brk(G) of a graph GG with maximum degree at least two is the minimum cardinality of all sets E′⊆E(G)E′⊆E(G) for which γrk(G−E′)>γrk(G)γrk(G−E′)>γrk(G). Note that br1(G)br1(G) is the classical bondage number b(G)b(G). In this paper, we initiate the study of the kk-rainbow bondage number in graphs and we present some bounds for brk(G)brk(G). In addition, we determine the 2-rainbow bondage number of some classes of graphs.