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A solution to a conjecture on the generalized connectivity of graphs

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Abstract

The generalized k-connectivity \(\kappa _k(G)\) of a graph G was introduced by Chartrand et al. in (Bull Bombay Math Colloq 2:1–6, 1984), which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we consider the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. We give a confirmative solution to a conjecture raised by S. Li in Ph.D. Thesis (2012). We also give a polynomial-time algorithm to a problem of generalized k-edge-connectivity.

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References

  • Bondy JA, Murty USR (2008) Graph theory, GTM 244. Springer, Berlin

    Book  MATH  Google Scholar 

  • Chartrand G, Kappor S, Lesniak L, Lick D (1984) Generalized connectivity in graphs. Bull Bombay Math Colloq 2:1–6

    Google Scholar 

  • Chartrand G, Okamoto F, Zhang P (2010) Rainbow trees in graphs and generalized connectivity. Networks 55(4):360–367

    MathSciNet  MATH  Google Scholar 

  • Grötschel M (1997) The Steiner tree packing problem in \(VLSI\) design. Math Program 78:265–281

    Article  MathSciNet  MATH  Google Scholar 

  • Grötschel M, Martin A, Weismantel R (1996) Packing Steiner trees: a cutting plane algorithm and commputational results. Math Program 72:125–145

    Article  MATH  Google Scholar 

  • Jain K, Mahdian M, Salavatipour M (2003) Packing Steiner trees. In: Proceedings of the 14th \(ACM\) symposium on discterte algorithms, Baltimore, pp. 266–274

  • Li S (2012) Some topics on generalized connectivity of graphs. Ph.D. Thesis, Nankai University

  • Li S, Li X (2012) Note on the hardness of generalized connectivity. J Comb Optim 24:389–396

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Mao Y (2014) The generalied \(3\)-connectivity of lexigraphical product graphs. Discret Math Theor Comput Sci 16(1):339–354

    Google Scholar 

  • Li X, Mao Y (2015a) On extremal graphs with at most \(\ell \) internally disjoint Steiner trees connecting any \(n-1\) vertices. Graphs Comb

  • Li X, Mao Y (2015b) Nordhaus–Gaddum-type results for the generalized edge-connectivity of graphs. Discret Appl Math 185:102–112

  • Li X, Mao Y (2015c) The minimal size of a graph with given generalized 3-edge-connectivity. Ars Comb 118:63–72

  • Li S, Li X, Zhou W (2010) Sharp bounds for the generalized connectivity \(\kappa _3(G)\). Discret Math 310:2147–2163

    Article  MATH  Google Scholar 

  • Li H, Li X, Sun Y (2012a) The generalied 3-connectivity of Cartesian product graphs. Discret Math Theor Comput Sci 14(1):43–54

    Article  MATH  Google Scholar 

  • Li S, Li W, Li X (2012b) The generalized connectivity of complete bipartite graphs. Ars Comb 104:65–79

  • Li S, Li W, Li X (2014a) The generalized connectivity of complete equipartition \(3\)-partite graphs. Bull. Malays Math Sci Soc 37(1):103–121

  • Li X, Mao Y, Sun Y (2014b) On the generalized (edge-)connectivity of graphs. Australas J Comb 58(2):304–319

  • Okamoto F, Zhang P (2010) The tree connectivity of regular complete bipartite graphs. J Comb Math Comb Comput 74:279–293

    MathSciNet  MATH  Google Scholar 

  • Sherwani NA (1999) Algorithms for VLSI physical design automation, 3rd edn. Kluwer Academic Publishers, London

    MATH  Google Scholar 

Download references

Acknowledgments

Supported by NSFC (Nos. 11071130, 11161037, 11501223, 11501271), the “973” program, the Scientific Research Funds of Huaqiao University (No. 14BS311), and the Science Found of Qinghai Province (No. 2014-ZJ-907).

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Authors and Affiliations

  1. School of Mathematics Science, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Lily Chen

  2. Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin, 300071, People’s Republic of China

    Lily Chen, Xueliang Li, Mengmeng Liu & Yaping Mao

  3. Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, People’s Republic of China

    Mengmeng Liu

  4. Department of Mathematics, Qinghai Normal University, Xining, 810008, Qinghai, People’s Republic of China

    Yaping Mao

Authors
  1. Lily Chen
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  2. Xueliang Li
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  3. Mengmeng Liu
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  4. Yaping Mao
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Correspondence to Lily Chen.

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Chen, L., Li, X., Liu, M. et al. A solution to a conjecture on the generalized connectivity of graphs. J Comb Optim 33, 275–282 (2017). https://doi.org/10.1007/s10878-015-9955-x

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  • DOI: https://doi.org/10.1007/s10878-015-9955-x

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