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Disjoint paths in the enhanced hypercube with a faulty subgraph

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Abstract

Problems about embedding of disjoint paths in interconnection networks have received much attention in recent years. A connected graph G is strong Menger connected if there are \(\min \{d_G(u), d_G(v)\}\) internally disjoint paths joining any two distinct vertices u and v in G. The enhanced hypercube \(Q_{n,k}\) is an important variant of the hypercube \(Q_n\) that retains many desirable properties of the hypercube. In order to study its fault tolerance, we consider the problem of embedding internally disjoint paths in an enhanced hypercube when part of the network is faulty. We show that the subgraph obtained from the enhanced hypercube \(Q_{n,k}\) \((2\le k\le n)\) by deleting the vertices of a faulty subnetwork \(Q_s\) \((1\le s\le n-1)\) or \(Q_{s,k}\) \((k\le s\le n-1)\) is strong Menger connected.

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References

  1. Benini, L., Micheli, G.D.: Networks on chips: a new SoC paradigm. Computer 35(1), 70–78 (2002)

    Article  Google Scholar 

  2. Chen, Y.C., Chen, M.H., Tan, J.J.M.: Maximally local connectivity and connected components of augmented cubes. Inf. Sci. 273, 387–392 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, Q., Li, P., Xu, M.: Conditional (edge-) fault-tolerant strong Menger (edge) connectivity of folded hypercubes. Theoret. Comput. Sci. 728, 1–8 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. El-Amawy, A., Latifi, S.: Properties and performance of folded hypercubes. IEEE Trans. Parallel Distrib. Syst. 2(1), 31–42 (1991)

    Article  Google Scholar 

  5. Ghasemi, M.: Some results about the reliability of folded hypercubes. Bull. Malays. Math. Sci. Soc. 44(2), 1093–1099 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gu, M.M., Chang, J.M., Hao, R.X.: Strong Menger connectedness of augmented k-ary n-cubes. Comput. J. 64(5), 812–825 (2021)

    Article  MathSciNet  Google Scholar 

  7. Guo, J., Lu, M.: Edge-fault-tolerant strong Menger edge connectivity of bubble-sort star graphs. Discret. Appl. Math. 297, 109–119 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guo, L., Qin, C., Xu, L.: Subgraph fault tolerance of distance optimally edge connected hypercubes and folded hypercubes. J. Parallel Distrib. Comput. 138, 190–198 (2020)

    Article  Google Scholar 

  9. Lai, C.N.: Optimal node-disjoint paths in folded hypercubes. J. Parallel Distrib. Comput. 147, 100–107 (2021)

    Article  Google Scholar 

  10. Lee, C.W., Hsieh, S.Y., Yang, S.S.: \({\rm R}_{3}\)-connectivity of folded hypercubes. Discret. Appl. Math. 285, 261–273 (2020)

    Article  MATH  Google Scholar 

  11. Lin, C.K., Zhang, L., Fan, J., Wang, D.: Structure connectivity and substructure connectivity of hypercubes. Theoret. Comput. Sci. 634, 97–107 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ma, M., Yu, J.: Edge-disjoint paths in faulty augmented cubes. Discret. Appl. Math. 294, 108–114 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ma, M., West, D., Xu, J.M.: The vulnerability of the diameter of the enhanced hypercubes. Theoret. Comput. Sci. 694, 60–65 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Menger, K.: Zur allgemeinen kurventheorie. Fund. Math. 10, 96–115 (1927)

    Article  MATH  Google Scholar 

  15. Oh, E., Chen, J.: On strong Menger-connectivity of star graphs. Discret. Appl. Math. 129, 499–511 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Qiao, Y., Yang, W.: Edge disjoint paths in hypercubes and folded hypercubes with conditional faults. Appl. Math. Comput. 294, 96–101 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Sabir, E., Mamut, A., Vumar, E.: The extra connectivity of the enhanced hypercubes. Theoret. Comput. Sci. 799, 22–31 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tzeng, N.F., Wei, S.: Enhanced hypercubes. IEEE Trans. Comput. 40(3), 284–294 (1991)

    Article  Google Scholar 

  19. Wang, D.: Diagnosability of hypercubes and enhanced hypercubes under the comparison diagnosis model. IEEE Trans. Comput. 48(12), 1369–1374 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, Y., Lin, C.K., Zhou, S., Tian, T.: Subgraph-based strong Menger connectivity of hypercube and exchanged hypercube. Int. J. Found. Comput. Sci. 32(3), 305–330 (2021)

    Article  MathSciNet  Google Scholar 

  21. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, Englewood Cliffs (2001)

    Google Scholar 

  22. Xu, J.M.: Combinatorial Theory in Networks. Science Press, Beijing (2013)

    Google Scholar 

  23. Xu, M., Li, P.: Edge-fault-tolerant strong Menger edge connectivity on regular graphs. Theoret. Comput. Sci. 847, 39–48 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, L., Zhou, S., Liu, J., Yin, S.: Reliability measure of multiprocessor system based on enhanced hypercubes. Discret. Appl. Math. 289, 125–138 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, S., Yang, W.: Conditional connectivity of folded hypercubes. Discret. Appl. Math. 257, 388–392 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions. This work is partially supported by Shandong Provincial Natural Science Foundation (No. ZR2021MF012) and National Natural Science Foundation of China (No. 62076039).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, Shandong, China

    Meijie Ma & Chaoming Guo

  2. School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China

    Xiang-Jun Li

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  1. Meijie Ma
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  2. Chaoming Guo
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Correspondence to Meijie Ma.

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Ma, M., Guo, C. & Li, XJ. Disjoint paths in the enhanced hypercube with a faulty subgraph. J. Appl. Math. Comput. 69, 1343–1354 (2023). https://doi.org/10.1007/s12190-022-01794-z

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  • DOI: https://doi.org/10.1007/s12190-022-01794-z

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