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Structure fault tolerance of balanced hypercubes

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Abstract

Let H be a connected subgraph of a given graph G. The H-structure connectivity of G is the cardinality of a minimal set \({\mathcal {F}}\) of subgraphs of G such that every element in \({\mathcal {F}}\) is isomorphic to H, and the removal of all the elements of \({\mathcal {F}}\) will disconnect G. The H-substructure connectivity of graph G is the cardinality of a minimal set \({\mathcal {F}}'\) of subgraphs of G such that every element in \({\mathcal {F}}'\) is isomorphic to a connected subgraph of H, and the removal of all the elements of \({\mathcal {F}}'\) will disconnect G. The two parameters were proposed by Lin et al. in (Theor Comput Sci 634:97–107, 2016), where no restrictions on \({\mathcal {F}}\) and \({\mathcal {F}}'\). In Lü and Wu (Bull Malays Math Sci Soc 43(3):2659–2672, 2020), the authors imposed some restrictions on \({\mathcal {F}}\) (resp. \({\mathcal {F}}'\)) for the n-dimensional balanced hypercube \(\text {BH}_n\) and requires that two elements in \({\mathcal {F}}\) (resp. \({\mathcal {F}}'\)) cannot share a vertex. Under such restrictions, they determined the (restricted) H-structure and (restricted) H-substructure connectivity of \(\text {BH}_n\) for \(H\in \{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}\). In this paper, we follow (2016) for the definitions of the two parameters and determine the H-structure and H-substructure connectivity of \(\text {BH}_n\) for \(H\in \{K_{1,t},P_k,C_4\}\), where \(K_{1,t}\) is the star on \(t+1\) vertices with \(1\le t\le 2n\) and \(P_k\) is a path of length k with \(1\le k\le 7\). Some of our main results show that the H-structure connectivity (resp. H-substructure connectivity) of \(\text {BH}_n\) is equal to the restricted H-structure connectivity (resp. restricted H-substructure connectivity) of \(\text {BH}_n\) for \(H\in \{K_{1,1},K_{1,2},C_4\}\), but the \(K_{1,3}\)-structure connectivity (resp. \(K_{1,3}\)-substructure connectivity) of \(\text {BH}_n\) is not equal to the restricted \(K_{1,3}\)-structure connectivity (resp. restricted \(K_{1,3}\)-substructure connectivity) of \(\text {BH}_n\) unless \(n=\lceil 2n/3\rceil\).

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References

  1. Lin CK, Zhang L, Fan J et al (2016) Structure connectivity and substructure connectivity of hypercubes. Theor Comput Sci 634:97–107

    Article  MathSciNet  Google Scholar 

  2. Lü H, Wu T (2020) Structure and substructure connectivity of balanced hypercubes. Bull Malays Math Sci Soc 43(3):2659–2672

    Article  MathSciNet  Google Scholar 

  3. Harary F (1983) Conditional connectivity. Networks 13(3):347–357

    Article  MathSciNet  Google Scholar 

  4. Yang Y, Wang S (2012) Conditional connectivity of star graph networks under embedding restriction. Inf Sci 199:187–192

    Article  MathSciNet  Google Scholar 

  5. Wang X, Fan J, Zhou J et al (2016) The restricted \(h\)-connectivity of the data center network DCell. Discrete Appl Math 203:144–157

    Article  MathSciNet  Google Scholar 

  6. Lü H (2017) On extra connectivity and extra edge-connectivity of balanced hypercubes. Int J Comput Math 94(4):813–820

    Article  MathSciNet  Google Scholar 

  7. Li P, Xu M (2018) Fault-tolerant strong Menger (edge) connectivity and 3-extra edge-connectivity of balanced hypercubes. Theor Comput Sci 707:56–68

    Article  MathSciNet  Google Scholar 

  8. You L, Fan J, Han Y (2018) Super spanning connectivity on WK-recursive networks. Theor Comput Sci 713:42–55

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  10. Fàbrega J, Fiol MA (1994) Extraconnectivity of graphs with large girth. Discrete Math 127(1–3):163–170

    Article  MathSciNet  Google Scholar 

  11. Fàbrega J, Fiol MA (1996) On the extraconnectivity of graphs. Discrete Math 155(1–3):49–57

    Article  MathSciNet  Google Scholar 

  12. Mane SA (2018) Structure connectivity of hypercubes. Akce Int J Graph Combinator 15(1):49–52

    Article  MathSciNet  Google Scholar 

  13. Sabir E, Meng J (2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor Comput Sci 711:44–45

    Article  MathSciNet  Google Scholar 

  14. Yang Y (2019) Characterization of minimum structure- and substructure- cuts of hypercubes. Comput J 62:1313–1321

    Article  MathSciNet  Google Scholar 

  15. Li D, Hu X, Liu H (2019) Structure connectivity and substructure connectivity of twisted hypercubes. Theor Comput Sci 796:169–179

    Article  MathSciNet  Google Scholar 

  16. Wang G, Lin CK, Cheng B et al (2019) Structure fault-tolerance of generalized hypercube. Comput J 62(10):1463–1476

    MathSciNet  Google Scholar 

  17. Pan Z, Cheng D (2020) Structure connectivity and substructure connectivity of the crossed cube. Theor Comput Sci 824:67–80

    Article  MathSciNet  Google Scholar 

  18. Li M, Zhang S, Li R et al (2019) Structure fault tolerance of \(k\)-ary \(n\)-cube networks. Theor Comput Sci 795:213–218

    Article  MathSciNet  Google Scholar 

  19. Lv Y, Fan J, Hsu DF et al (2018) Structure connectivity and substructure connectivity of \(k\)-ary \(n\)-cube networks. Inf Sci 433–434:115–124

    Article  MathSciNet  Google Scholar 

  20. Li C, Lin S, Li S (2020) Structure connectivity and substructure connectivity of star graphs. Discrete Appl Math 284:472–480

    Article  MathSciNet  Google Scholar 

  21. Cao J, Shi M, Feng L (2016) On the edge-hyper-hamiltonian laceability of balanced hypercubes. Discuss Math Graph T 36(4):805–817

    Article  MathSciNet  Google Scholar 

  22. Cheng D (2018) Cycles embedding in balanced hypercubes with faulty edges and vertices. Discrete Appl Math 238:56–69

    Article  MathSciNet  Google Scholar 

  23. Cheng D (2019) Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes. Discrete Appl Math 262:56–71

    Article  MathSciNet  Google Scholar 

  24. Lü H, Wu T (2019) Edge-disjoint Hamiltonian cycles of balanced hypercubes. Inform Process Lett 144:25–30

    Article  MathSciNet  Google Scholar 

  25. Lü H, Wang F (2019) Hamiltonian paths passing through prescribed edges in balanced hypercubes. Theor Comput Sci 761:23–33

    Article  MathSciNet  Google Scholar 

  26. Xu M, Hu XD, Xu JM (2007) Edge-pancyclicity and hamiltonian laceability of the balanced hypercubes. Appl Math Comput 189(2):1393–1401

    MathSciNet  MATH  Google Scholar 

  27. Yang MC (2013) Conditional diagnosability of balanced hypercubes under the MM* model. J Supercomput 65:1264–1278

    Article  Google Scholar 

  28. Yang Y, Zhang L (2019) Fault-tolerant-prescribed hamiltonian laceability of balanced hypercubes. Inform Process Lett 145:11–15

    Article  MathSciNet  Google Scholar 

  29. Zhou JX, Kwak JH, Feng YQ et al (2017) Automorphism group of the balanced hypercubes. Ars Math Contemp 12:145–154

    Article  MathSciNet  Google Scholar 

  30. Lü H, Li X, Zhang H (2012) Matching preclusion for balanced hypercubes. Theor Comput Sci 465:10–20

    Article  MathSciNet  Google Scholar 

  31. Huang K, Wu J (1996) Area efficient layout of balanced hypercubes. Int J High Speed Electron Syst 6(4):631–646

    Article  Google Scholar 

  32. Wu J, Huang K (1997) The balanced hypercube: a cube-based system for fault-tolerant applications. IEEE Trans Comput 46(4):484–490

    Article  MathSciNet  Google Scholar 

  33. Zhou JX, Wu ZL, Yang SC et al (2015) Symmetric property and reliability of balanced hypercube. IEEE Trans Comput 64(3):876–881

    Article  MathSciNet  Google Scholar 

  34. Bondy JA, Murty USR (2018) Graph theory. Springer, New York

    MATH  Google Scholar 

  35. Yang MC (2012) Super connectivity of balanced hypercubes. Appl Math Comput 219(3):970–975

    MathSciNet  MATH  Google Scholar 

  36. Yang DW, Feng YQ, Lee J (2018) On extra connectivity and extra edge-connectivity of balanced hypercubes. Appl Math Comput 320:464–473

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their heartfelt thanks to the reviewers for their comments and suggestions which are very helpful to improve the quality of this paper.

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Correspondence to Yuxing Yang.

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The third author of this paper is partly supported by Research Project of Shanxi Scholarship Council of China (No. 2020-122).

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Yang, Y., Li, X. & Li, J. Structure fault tolerance of balanced hypercubes. J Supercomput 77, 3885–3898 (2021). https://doi.org/10.1007/s11227-020-03419-3

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