The k-Rainbow Bondage Number of a Digraph

• Autorzy: Jafar Amjadi , Negar Mohammadi , Seyed Mahmoud Sheikholeslami , Lutz Volkmann
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

k-rainbow dominating function; k-rainbow domination number; k-rainbow bondage number; digraph

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2015
• Tom: 35
• Numer: 2
• Strony: 261-270

Daty

wydano

2015-05-01

otrzymano

2014-02-10

poprawiono

2014-06-12

zaakceptowano

2014-06-12

online

2015-04-18

Twórcy

autor

Jafar Amjadi

• Department of Mathematics Azarbaijan Shahid Madani University Tabriz, I.R. Iran

autor

Negar Mohammadi

• Department of Mathematics Azarbaijan Shahid Madani University Tabriz, I.R. Iran

autor

Seyed Mahmoud Sheikholeslami

• Department of Mathematics Azarbaijan Shahid Madani University Tabriz, I.R. Iran

autor

Lutz Volkmann

• Lehrstuhl II für Mathematik RWTH Aachen University 52056 Aachen, Germany

Bibliografia

• [1] J. Amjadi, A. Bahremandpour, S.M. Sheikholeslami and L. Volkmann, The rainbow domination number of a digraph, Kragujevac J. Math. 37 (2013) 257-268.
• [2] B. Brešar, M.A. Henning and D.F. Rall, Rainbow domination in graphs, Taiwanese J. Math. 12 (2008) 213-225.
• [3] B. Brešar and T.K. Šumenjak, On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 2394-2400. doi:10.1016/j.dam.2007.07.018[WoS][Crossref]
• [4] G.J. Chang, J. Wu and X. Zhu, Rainbow domination on trees, Discrete Appl. Math. 158 (2010) 8-12. doi:10.1016/j.dam.2009.08.010[Crossref]
• [5] Ch. Tong, X. Lin, Y. Yang and M.Luo, 2-rainbow domination of generalized Petersen graphs P(n, 2), Discrete Appl. Math. 157 (2009) 1932-1937. doi:10.1016/j.dam.2009.01.020[Crossref]
• [6] N. Dehgardi, S.M. Sheikholeslami and L. Volkmann, The k-rainbow bondage number of a graph, Discrete Appl. Math. 174 (2014) 133-139. doi:10.1016/j.dam.2014.05.006[Crossref][WoS]
• [7] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, The bondage number of a graph, Discrete Math. 86 (1990) 47-57. doi:10.1016/0012-365X(90)90348-L[Crossref]
• [8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998).
• [9] D. Meierling, S.M. Sheikholeslami and L. Volkmann, Nordhaus-Gaddum bounds on the k-rainbow domatic number of a graph, Appl. Math. Lett. 24 (2011) 1758-1761. doi:10.1016/j.aml.2011.04.046[Crossref][WoS]
• [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]
• [11] D.B. West, Introduction to Graph Theory (Prentice-Hall, Inc., 2000).
• [12] Y. Wu and N. Jafari Rad, Bounds on the 2-rainbow domination number of graphs, Graphs Combin. 29 (2013) 1125-1133. doi:10.1007/s00373-012-1158-y[Crossref]
• [13] G. Xu, 2-rainbow domination in generalized Petersen graphs P(n, 3), Discrete Appl. Math. 157 (2009) 2570-2573. doi:10.1016/j.dam.2009.03.016 [Crossref]

Identyfikatory

DOI

10.7151/dmgt.1797