# The k-Rainbow Bondage Number of a Digraph

• Volume: 35, Issue: 2, page 261-270
• ISSN: 2083-5892

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## Abstract

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Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) 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) = Ø 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 kRDF f is the value w(f) = ∑v∈V |f(v)|. The k-rainbow domination number of a digraph D, denoted by γrk(D), is the minimum weight of a kRDF of D. The k-rainbow bondage number brk(D) of a digraph D with maximum in-degree at least two, is the minimum cardinality of all sets A′ ⊆ A for which γrk(D−A′) > γrk(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.

## How to cite

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Jafar Amjadi, et al. "The k-Rainbow Bondage Number of a Digraph." Discussiones Mathematicae Graph Theory 35.2 (2015): 261-270. <http://eudml.org/doc/271087>.

abstract = {Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) 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) = Ø 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 kRDF f is the value w(f) = ∑v∈V |f(v)|. The k-rainbow domination number of a digraph D, denoted by γrk(D), is the minimum weight of a kRDF of D. The k-rainbow bondage number brk(D) of a digraph D with maximum in-degree at least two, is the minimum cardinality of all sets A′ ⊆ A for which γrk(D−A′) > γrk(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.},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {k-rainbow dominating function; k-rainbow domination number; k-rainbow bondage number; digraph; -rainbow dominating function; -rainbow domination number; -rainbow bondage number},
language = {eng},
number = {2},
pages = {261-270},
title = {The k-Rainbow Bondage Number of a Digraph},
url = {http://eudml.org/doc/271087},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Seyed Mahmoud Sheikholeslami
AU - Lutz Volkmann
TI - The k-Rainbow Bondage Number of a Digraph
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 2
SP - 261
EP - 270
AB - Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) 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) = Ø 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 kRDF f is the value w(f) = ∑v∈V |f(v)|. The k-rainbow domination number of a digraph D, denoted by γrk(D), is the minimum weight of a kRDF of D. The k-rainbow bondage number brk(D) of a digraph D with maximum in-degree at least two, is the minimum cardinality of all sets A′ ⊆ A for which γrk(D−A′) > γrk(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.
LA - eng
KW - k-rainbow dominating function; k-rainbow domination number; k-rainbow bondage number; digraph; -rainbow dominating function; -rainbow domination number; -rainbow bondage number
UR - http://eudml.org/doc/271087
ER -

## References

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10. [10] S.M. Sheikholeslami and L. Volkmann, The k-rainbow domatic number of a graph, Discuss. Math. Graph Theory 32 (2012) 129-140. doi:10.7151/dmgt.1591[Crossref] Zbl1255.05139
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