Skip to main content
SpringerLink
Log in

Fault tolerance of hypercubes and folded hypercubes

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

Let \(G = (V,E)\) be a connected graph. The conditional edge connectivity \(\lambda _\delta ^k(G)\) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects \(G\) and each component of \(G - F\) has \(\delta \ge k\). We assume that \(F \subseteq E\) is an edge set, \(F\) is called edge extra-cut, if \(G - F\) is not connected and each component of \(G - F\) has more than \(k\) vertices. The edge extraconnectivity \(\lambda _\mathrm{e}^k(G)\) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bondy JA, Murty USR (1976) Graph theory and its application. Academic Press, New York

    Google Scholar 

  2. Cheng E, Lesniak L, Lipman MJ, Lipták L (2009) Conditional matching preclusion sets. Inf Sci 179:1092–1101

    Article  MATH  Google Scholar 

  3. Duh DR, Chen GH, Fang JF (1995) Algorithms and properties of a new two-level network with folded hypercubes as basic modules. IEEE Trans Parallel Distrib Syst 6(7):714–723

    Article  Google Scholar 

  4. Esfahanian AH, Hakimi SL (1988) On computing a conditional edge-connectivity of a graph. Inf Process Lett 27:195–199

    Article  MATH  MathSciNet  Google Scholar 

  5. El-Amawy A, Latifi S (1991) Properties and performance of folded hypercubes. IEEE Trans Parallel Distrib Syst 2:31–42

    Article  Google Scholar 

  6. F\(\acute{a}\)brega J, Fiol MA (1996) On the extraconnectivity of graphs. Discret Math 155:49–57

    Google Scholar 

  7. Guo L, Guo X (2009) Connectivity of the Mycielskian of a digraph. Appl Math Lett 22:1622–1625

    Article  MATH  MathSciNet  Google Scholar 

  8. Guo L, Qin C, Guo X (2010) Super connectivity of Kronecker products of graphs. Inf Process Lett 110:659–661

    Article  MATH  MathSciNet  Google Scholar 

  9. Guo L, Liu R, Guo X (2012) Super connectivity and super edge connectivity of the Mycielskian of a graph. Graph Comb 28:143–147

    Article  Google Scholar 

  10. Harary F (1983) Conditional connectivity. Networks 13:346–357

    Article  Google Scholar 

  11. Liaw SC, Chang GJ (1998) Generalized diameters and Rabin numbers of networks. J Combin Optim 2:371–384

    Article  MathSciNet  Google Scholar 

  12. Lai CN, Chen GH, Duh DR (2002) Constructing one-to-many disjoint paths in folded hypercubes. IEEE Trans Comput 51(1):33–45

    Article  MathSciNet  Google Scholar 

  13. Latifi S, Hedge M, Naraghi-Pour M (1994) Conditional connectivity measures for large multiprocessor systems. IEEE Trans Comput 43(2):218–222

    Article  Google Scholar 

  14. Lin M-S, Chang M-S, Chen D-J (1999) Efficient algorithms for reliability analysis of distributed computing systems. Inf Sci 117:89–106

    Article  MathSciNet  Google Scholar 

  15. Meng JX, Ji YH (2002) On a kind of restricted edge connectivty of graphs. Discret Appl Math 117:183–193

    Article  MATH  MathSciNet  Google Scholar 

  16. Sim\(\acute{o}\) E, Yebra JLA (1997) The vulnerability of the diameter of folded n-cubes, Discrete Math 174:317–322

    Google Scholar 

  17. Valafar H, Arabnia HR, Williams G (2004) Distributed global optimization and its development on the multiring network. Int J Neural Parallel Sci Comput 12(4):465–490

    MATH  MathSciNet  Google Scholar 

  18. Wang D (2001) Embedding Hamiltonian cycles into folded hypercubes with faulty links. J Parallel Distrib Comput 61:545–564

    Article  MATH  Google Scholar 

  19. Xu JM (2000) Restricted edge connectivity of vertex transitive graphs. Chin Ann Math Ser A 21:605–608

    MATH  Google Scholar 

  20. Zhu Q, Xu JM (2006) On restricted edge connectivity and extra edge connectivity of hypercubes and foled hypercubes. J Univ Sci Technol China 36(3):246–253

    Google Scholar 

  21. Zhu Q, Xu JM, Hou X, Xu M (2007) On reliability of the foled hypercubes. Inf Sci 177:1782–1788

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhang Z, Yuan JJ (2007) Degree conditions for retricted edge connectivity and isoperimetric-edge-connectivity to be optimal. Discret Math. 307:293–298

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The project is supported by NSFC (No. 11301440, 11171279), Xiamen University of Technology (YKJ12030R), the Foundation to the Educational Committee of Fujian (JA13240, JA13025, JA13034), the Natural Science Foundation of Fujian Province (2013J05006), Natural Sciences Foundation of Guangxi Province (No. 2012GXNSFBA053005, 2011GXNSFA018144). We would like to thank the referees for kind help and valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Xiamen University of Technology, Xiamen , 361024, Fujian, People’s Republic of China

    Litao Guo

  2. School of Mathematical Sciences, Xiamen University, Xiamen , 361005, Fujian, People’s Republic of China

    Litao Guo & Xiaofeng Guo

Authors
  1. Litao Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofeng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Litao Guo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, L., Guo, X. Fault tolerance of hypercubes and folded hypercubes. J Supercomput 68, 1235–1240 (2014). https://doi.org/10.1007/s11227-013-1078-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-013-1078-5

Keywords

Search

Navigation