The k-rainbow bondage number of a graph

摘要

For a positive integer k, a k-rainbow dominating function (kRDF) 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 ∈ V(G) with f (v) = ? the condition∪_(u∈N(v)) f (u) = {1, 2,..., k} is fulfilled, where N(v) ? is the neighborhood of v. The weight of a kRDF f is the value ω(f) = ∑_(v∈V) |f (v)|. The k-rainbow domination number of a graph G, denoted by γ_(rk)(G), is the minimum weight of a kRDF of G. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow bondage number b_(rk)(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ? E(G) for which γ_(rk)(G ? E′) > γ_(rk)(G). Note that b_(r1)(G) is the classical bondage number b(G). In this paper, we initiate the study of the k-rainbow bondage number in graphs and we present some bounds for b_(rk)(G). In addition, we determine the 2-rainbow bondage number of some classes of graphs.