Geodesic
The k-Rainbow Bondage Number of a Digraph
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 261-270.

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Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (k RDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set 1, 2, . . ., k such that for any vertex v ∈ V with f(v) = ∅ the condition ⋃_ u ∈ N^−(v) f(u) = 1, 2, . . ., k is fulfilled, where N^− (v) is the set of in-neighbors of v. The weight of a k RDF f is the value ω (f) = ∑_v ∈ V |f(v)|. The k-rainbow domination number of a digraph D, denoted by γ_rk (D), is the minimum weight of a k RDF of D. The k-rainbow bondage number b_rk (D) of a digraph D with maximum in-degree at least two, is the minimum cardinality of all sets A^'⊆ A for which γ_rk (D−A^' ) gt; γ_rk (D). In this paper, we establish some bounds for the k-rainbow bondage number and determine the k-rainbow bondage number of several classes of digraphs.
Keywords: k-rainbow dominating function, k-rainbow domination number, k-rainbow bondage number, digraph 
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Amjadi, Jafar; Mohammadi, Negar; Sheikholeslami, Seyed Mahmoud; Volkmann, Lutz. The k-Rainbow Bondage Number of a Digraph. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 261-270. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a5/

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