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Remarks on the bondage number of planar graphs

Miranca Fischermann, Dieter Rautenbach, Lutz Volkmann
2003 Discrete Mathematics  
The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number (G) of G.  ...  Yuan proved b(G) 6 8 for every connected planar graph G, and therefore, they conÿrmed the conjecture for (G) ¿ 7.  ...  Acknowledgements We are grateful to one of the anonymous referees for his useful suggestions and detailed hints.  ... 
doi:10.1016/s0012-365x(02)00449-1 fatcat:pqdym7kvb5gvdpq2nzojgjzczi

The complexity of the bondage problem in planar graphs [article]

Valentin Bouquet
2022 arXiv   pre-print
The bondage number b(G) of a graph G is the smallest cardinality of a set of edges A⊆ E(G), such that γ(G-A)=γ(G)+1.  ...  In this paper, we show that 1-Bondage is 𝖭𝖯-hard, even for the class of 3-regular planar graphs, the class of subcubic claw-free graphs, and the class of bipartite planar graphs of maximum degree 3,  ...  Acknowledgements: The author express its gratitude to François Delbot, Christophe Picouleau and Stéphane Rovedakis for carefully reading the manuscript, and for providing helpful comments.  ... 
arXiv:2107.11216v2 fatcat:zi46cqb3evdtfmcxylov3yxlnq

On Bondage Numbers of Graphs: A Survey with Some Comments

Jun-Ming Xu
2013 International Journal of Combinatorics  
The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of .  ...  This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers.  ...  The author would like to express his gratitude to the anonymous referees for their kind suggestions and comments on the original paper, to Professor Sheikholeslami and Professor Jafari Rad for sending  ... 
doi:10.1155/2013/595210 fatcat:6zg7jz7i5jenzc2e2izfdy2txq

The bondage number of graphs on topological surfaces and Teschner's conjecture

Andrei Gagarin, Vadim Zverovich
2013 Discrete Mathematics  
We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable  ...  The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number.  ...  Acknowledgement The authors are grateful to the anonymous referees for their valuable comments and remarks.  ... 
doi:10.1016/j.disc.2012.12.018 fatcat:y4foeftg6bgxlaq7ivjihhm53q

On Bondage Numbers of Graphs -- a survey with some comments [article]

Jun-Ming Xu
2012 arXiv   pre-print
The bondage number of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number of G.  ...  This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.  ...  We conclude this subsection with a question on bondage numbers of planar graphs.  ... 
arXiv:1204.4010v1 fatcat:px2cuvbi6fhl5ci6hpoz5fyh3y

Upper bounds for the bondage number of graphs on topological surfaces

Andrei Gagarin, Vadim Zverovich
2013 Discrete Mathematics  
The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G.  ...  We show that, for a graph G having the maximum vertex degree Δ(G) and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, b(G)<{Δ(G)+h+2, Δ(G)+k+1}.  ...  Acknowledgement The authors are grateful to the anonymous referees for their comments and suggestions which led to an improved presentation of the results.  ... 
doi:10.1016/j.disc.2011.10.018 fatcat:76vjg673nzazzjvuinodpitx6q

Upper bounds on the bondage number of a graph

Vladimir Dimitrov Samodivkin
2018 Electronic Journal of Graph Theory and Applications  
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number.  ...  We also present upper bounds for the bondage number of graphs in terms of the girth, domination number and Euler characteristic.  ...  We propose a new type of upper bound on the bondage number of a graph.  ... 
doi:10.5614/ejgta.2018.6.1.1 fatcat:6ohhgnnxb5gs7hblypg4j2xomu

On The Double Roman bondage numbers of Graphs [article]

N. Jafari Rad, H.R. Maimani, M. Momeni, F. Rahimi Mahid
2019 arXiv   pre-print
The weight of a DRDF f is the sum f (V) =∑_u∈ V f (u). The minimum weight of a DRDF on a graph G is the double Roman domination number of G and is denoted by γ_dR(G).  ...  We determine the double roman bondage number in several families of graphs, and present several bounds for the double roman bondage number.  ...  Bauer, Harary, Nieminen and Suffel [4] introduced the concept of bondage number in graphs.  ... 
arXiv:1905.06724v1 fatcat:grd6kxuuw5aelmxtp344q4vgyy

Note on the bondage number of graphs on topological surfaces [article]

Vladimir Samodivkin
2012 arXiv   pre-print
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number.  ...  In this paper we present new upper bounds for b(G) in terms of girth, order and Euler characteristic.  ...  Recently, the following results on bondage number of graphs on surfaces were obtained. Theorem 2 (Gagarin and Zverovich [4] ).  ... 
arXiv:1208.6203v1 fatcat:odm3isiwnrfa7g5kdkjtt5nkai

Page 8090 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews  
Aachen) ; Volkmann, Lutz (D-AACH-M2; Aachen) Remarks on the bondage number of planar graphs.  ...  Summary: “The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number 2  ... 

The bondage number of chordal graphs [article]

Valentin Bouquet
2022 arXiv   pre-print
The bondage number b(G) of a graph G is the smallest cardinality of a set edges A⊆ E(G) such that γ(G-A)=γ(G)+1. A chordal graph is a graph with no induced cycle of length four or more.  ...  In this paper, we prove that the bondage number of a chordal graph G is at most the order of its maximum clique, that is, b(G)≤ω(G). We show that this bound is best possible.  ...  Acknowledgements: The author would like to thank Christophe Picouleau, Stéphane Rovedakis and François Delbot for providing helpful comments.  ... 
arXiv:2203.09256v1 fatcat:umsvmzcb6vcdzaz4qzklr72c7q

Contents

2003 Discrete Mathematics  
Volkmann Remarks on the bondage number of planar graphs 57 Y. Guan and K.L. Williams Profile minimization on triangulated triangles 69 T.W. Haynes, M.A. Henning and P.J.  ...  Selvaraju Decompositions of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs 137 S. Wang and J. Hao The extreme set condition of a graph 151 J. Wu, J.  ... 
doi:10.1016/s0012-365x(02)00779-3 fatcat:c6ruhvugjzazdi7o6mlx4drqrq

On the Roman Bondage Number of Graphs on surfaces [article]

Vladimir Samodivkin
2014 arXiv   pre-print
The Roman bondage number b_R(G) is the cardinality of a smallest set of edges whose removal from G results in a graph with Roman domination number not equal to γ_R(G).  ...  We also show that the Roman bondage number of every graph which admits a 2-cell embedding on a surface with non negative Euler characteristic does not exceed 15.  ...  [1] gave upper bounds on the Roman bondage number of planar graphs.  ... 
arXiv:1407.0367v1 fatcat:4qcnxv6jcfaltmlkej3o4rek7u

Author index to volume 260

2003 Discrete Mathematics  
Pyatkin, On 4-chromatic edge-critical regular graphs of high connectivity , M., D. Rautenbach and L. Volkmann, Remarks on the bondage number of planar graphs (1-3) 57-67 Fischermann, M., D.  ...  Kriger, On some questions of Razpet regarding binomial coefficients (Note) (1-3) 177-182 Berman, A. and X.-D. Zhang, Bipartite density of cubic graphs (1-3) 27-35 Bohman, T. and R.R.  ... 
doi:10.1016/s0012-365x(02)00785-9 fatcat:xjvlplv3ezalvfsvh7xhhrcl2y

New upper bounds for the bondage number of a graph in terms of its maximum degree and Euler characteristic [article]

Jia Huang, Jian Shen
2020 arXiv   pre-print
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number.  ...  We also establish further improved upper bounds for b(G) when the girth, order, or size of the graph G is large compared with its Euler characteristic χ.  ...  If G is a planar graph then we have b(G) ≤ ∆(G) + 1. The best upper bound known so far for the bondage number of a planar graph is the following. Theorem 1.5 (Kang and Yuan [11] ).  ... 
arXiv:2002.00765v1 fatcat:sm57gycvlzaozo6dksmi7nsqia
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