Sufficient Conditions for Maximally Edge-Connected Hypergraphs

Journal of the Operations Research Society of China volume 9, pages119–129(2021)Cite this article

Abstract

The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected. A graph or hypergraph is called maximally edge-connected if the edge-connectivity equals its minimum degree. In this paper, we show that some classical sufficient conditions for graphs to be maximally edge-connected can be generalized to hypergraphs.

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References

  1. 1.

    Whitney, H.: Congruent graphs and the connectivity of a graph. Am. J. Math. 54, 150–168 (1932)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Dankelmann, P., Meierling, D.: Maximally edge-connected hypergraphs. Discrete Math. 339, 33–38 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hellwig, A., Volkmann, L.: Maximally edge-connected graphs and digraphs-a survey. Discrete Math. 308, 3265–3296 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bahmanian, M.A., Šajna, M.: Connection and separation in hypergraphs. Theory Appl. Graphs 2(2), 1–24 (2015)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Berg, A.R., Jackson, B., Jordán, T.: Edge splitting and connectivity augmentation in directed hypergraphs. Discrete Math. 273, 71–84 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bernáth, A., Grappe, R., Szigeti, Z.: Augmenting the edge-connectivity of a hypergraph by adding a multipartite graph. J. Graph Theory 72, 291–312 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dewar, M., Pike, D., Proos, J.: Connectivity in hypergraphs. Canad. Math. Bull. 61, 252–271 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Jami, N., Szigeti, Z.: Edge-connectivity of permutation hypergraphs. Discrete Math. 312, 2536–2539 (2012)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Király, Z., Cosh, B., Jackson, B.: Local edge-connectivity augmentation in hypergraphs is NP-complete. Discrete Appl. Math. 158, 723–727 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hellwig, A., Volkmann, L.: Sufficient conditions for graphs to be \(\lambda ^{\prime }\)-optimal, super-edge-connected and maximally edge-connected. J. Graph Theory 48, 228–246 (2005)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Berge, C.: Graphs and Hypergraphs, North-Holland Mathematical Library, vol. 6, 2nd edn. Springer, Amsterdam (1976)

    Google Scholar 

  12. 12.

    Bretto, A.: Hypergraph Theory: an introduction. Springer, Berlin (2013)

    Google Scholar 

  13. 13.

    Soneoka, T., Nakada, H., Imase, M.: Sufficient conditions for dense graphs to be maximally connected. Proc. IEEE Int Conf Circuit Syst 85, 811–814 (1985)

    MATH  Google Scholar 

  14. 14.

    Soneoka, T., Nakada, H., Imase, M., Peyrat, C.: Sufficient conditions for maximally connected dense graphs. Discrete Math. 63, 53–66 (1987)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Dankelmann, P., Volkmann, L.: New sufficient conditions for equality of minimum degree and edge-connectivity. Ars Combin. 40, 270–278 (1995)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Plesník, J., Znám, S.: On equality of edge-connectivity and minimum degree of a graph. Arch. Math. (Brno) 25, 19–25 (1989)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Lesniak, L.: Resluts on the edge-connectivity of graphs. Discrete Math. 8, 351–354 (1974)

    MathSciNet  Article  Google Scholar 

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Author information

Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Lin-Ken Tong

  2. School of Management, Shanghai University, Shanghai, 200444, China

    Er-Fang Shan

Authors
  1. Lin-Ken Tong
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  2. Er-Fang Shan
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Corresponding author

Correspondence to Er-Fang Shan.

Additional information

This research was partially supported by the National Natural Science Foundation of China (Nos. 11571222 and 11871329).

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Tong, LK., Shan, EF. Sufficient Conditions for Maximally Edge-Connected Hypergraphs. J. Oper. Res. Soc. China 9, 119–129 (2021). https://doi.org/10.1007/s40305-018-0224-4

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Keywords

  • Hypergraph
  • Edge-connectivity
  • Maximally edge-connected

Mathematics Subject Classification

  • 05C65
  • 05C70