Journal of Applied Mathematics

Bondage Numbers of C4 Bundles over a Cycle Cn

Moo Young Sohn, Fu-Tao Hu, and Jaeun Lee

Full-text: Open access

Abstract

Graph bundles generalize the notion of covering graphs and graph products. Graph bundles have been applied in computer architecture and communication networks. The bondage number is an important parameter for measuring the vulnerability and stability of the network domination under link failure. The bondage number b(G) of a graph G is the minimum number of edges whose removal enlarges the domination number. In this paper, we show that the bondage number of every C4 bundles over a cycle Cn  (n4) is equal to 4.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 520251, 5 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808292

Digital Object Identifier
doi:10.1155/2013/520251

Mathematical Reviews number (MathSciNet)
MR3142571

Zentralblatt MATH identifier
06950726

Citation

Sohn, Moo Young; Hu, Fu-Tao; Lee, Jaeun. Bondage Numbers of ${C}_{4}$ Bundles over a Cycle ${C}_{n}$. J. Appl. Math. 2013 (2013), Article ID 520251, 5 pages. doi:10.1155/2013/520251. https://projecteuclid.org/euclid.jam/1394808292


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References

  • J.-M. Xu, Theory and Application of Graphs, vol. 10 of Network Theory and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
  • T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, NY, USA, 1998.
  • T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Eds., Domination in Graphs: Advanced Topics, vol. 209, Marcel Dekker, New York, NY, USA, 1998.
  • J. F. Fink, M. S. Jacobson, L. F. Kinch, and J. Roberts, “The bondage number of a graph,” Discrete Mathematics, vol. 86, no. 1–3, pp. 47–57, 1990.
  • F.-T. Hu and J.-M. Xu, “On the complexity of the bondage and reinforcement problems,” Journal of Complexity, vol. 28, no. 2, pp. 192–201, 2012.
  • J.-M. Xu, “On bondage numbers of graphs: a survey with some comments,” International Journal of Combinatorics, vol. 2013, Article ID 595210, 34 pages, 2013.
  • F.-T. Hu and J.-M. Xu, “Bondage number of mesh networks,” Frontiers of Mathematics in China, vol. 7, no. 5, pp. 813–826, 2012.
  • J. E. Dunbar, T. W. Haynes, U. Teschner, and L. Volkmann, “Bondage, insensitivity, and reinforcement,” in Domination in Graphs: Advanced Topics, T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Eds., vol. 209 of Monographs and Textbooks in Pure and Applied Mathematics, pp. 471–489, Marcel Dekker, New York, NY, USA, 1998.
  • M. Y. Sohn, X.-D. Yuan, and H. S. Jeong, “The bondage number of ${C}_{3}\times {C}_{n}$,” Journal of the Korean Mathematical Society, vol. 44, no. 6, pp. 1213–1231, 2007.
  • L. Kang, M. Y. Sohn, and H. K. Kim, “Bondage number of the discrete torus ${C}_{n}\times {C}_{4}$,” Discrete Mathematics, vol. 303, no. 1–3, pp. 80–86, 2005.
  • J. Huang and J.-M. Xu, “The bondage numbers and efficient dominations of vertex-transitive graphs,” Discrete Mathematics, vol. 308, no. 4, pp. 571–582, 2008.
  • C. J. Xiang, X. D. Yuan, and M. Y. Sohn, “Domination and bondage number of ${C}_{5}\times {C}_{n}^{\ast\,\!}$,” Ars Combinatoria, vol. 97, pp. 299–310, 2010.
  • M. Y. Sohn and J. X. Cao, “Bondage number of the graph bundle ${C}_{n}{{\square}}^{\phi }{C}_{5}$ having reflection voltage assignment,” Unpublished manuscipt.
  • J. H. Kwak and J. Lee, “Isomorphism classes of graph bundles,” Canadian Journal of Mathematics, vol. 42, no. 4, pp. 747–761, 1990.
  • T. Pisanski and J. Vrabec, “Graph čommentComment on ref. [14?]: Please provide more information for these references [14,15?].bundles,” Unpublished manuscipt, 1982.
  • V. G. Vizing, “The cartesian product of graphs,” Vyčisl Sistemy, vol. 9, pp. 30–43, 1963.
  • B. Brešar, P. Dorbec, W. Goddard et al., “Vizing's conjecture: a survey and recent results,” Journal of Graph Theory, vol. 69, no. 1, pp. 46–76, 2012.
  • B. Zmazek and J. Žerovnik, “On domination numbers of graph bundles,” Journal of Applied Mathematics & Computing, vol. 22, no. 1-2, pp. 39–48, 2006.
  • S. Klavžar and N. Seifter, “Dominating Cartesian products of cycles,” Discrete Applied Mathematics, vol. 59, no. 2, pp. 129–136, 1995. \endinput