The k-rainbow bondage number of a graph

Abstract 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 ) = 0 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 r k ( 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 r k ( G ) of a graph G with maximum degree at least two is the minimum cardinality of all sets E ′ ⊆ E ( G ) for which γ r k ( G − E ′ ) > γ r k ( G ) . Note that b r 1 ( 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 r k ( G ) . In addition, we determine the 2-rainbow bondage number of some classes of graphs.