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Optimal fault-tolerant Hamiltonicity of star graphs with conditional edge faults

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Abstract

The star graph is viewed as an attractive alternative to the hypercube. In this paper, we investigate the Hamiltonicity of an n-dimensional star graph. We show that for any n-dimensional star graph (n≥4) with at most 3n−10 faulty edges in which each node is incident with at least two fault-free edges, there exists a fault-free Hamiltonian cycle. Our result improves on the previously best known result for the case where the number of tolerable faulty edges is bounded by 2n−7. We also demonstrate that our result is optimal with respect to the worst case scenario, where every other node of a cycle of length 6 is incident with exactly n−3 faulty noncycle edges.

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References

  1. Akers SB, Horel D, Krishnamurthy B (1987) The star graph: an attractive alternative to the n-cube. In: Proceedings of the international conference on parallel processing, University Park, PA, 1987, pp 393–400

  2. Akers SB, Krishnamurthy B (1989) A group-theoretic model for symmetric interconnection networks. IEEE Trans Comput 38:555–566

    Article  MATH  MathSciNet  Google Scholar 

  3. Akl SG (1997) Parallel computation: models and methods. Prentice Hall, Englewood Cliffs

    Google Scholar 

  4. Ascheuer N (1995) Hamiltonian path problems in the on-line optimization of flexible manufacturing systems, Ph.D. Thesis, University of Technology, Berlin, Germany. Also available from ftp://ftp.zib.de/pub/zib-publications/reports/TR-96-03.ps

  5. Bagherzadeh N, Nassif N, Latifi S (1993) A routing and broadcasting scheme on faulty star graphs. IEEE Trans Comput 42:1398–1403

    Article  Google Scholar 

  6. Bermond JC (ed) (1992) Interconnection networks. Discrete Appl Math 37+38 (1992) special issue

  7. Bhuyan L, Agrawal DP (1984) Generalized hypercube and hyperbus structures for a computer network. IEEE Trans Comput C-33:323–333

    Article  Google Scholar 

  8. Day K, Tripathi AR (1994) A comparative study of topological properties of hypercubes and star graphs. IEEE Trans Parallel Distrib Syst 5:31–38

    Article  MathSciNet  Google Scholar 

  9. Fragopoulou P, Akl SG (1995) Optimal communication algorithms on star graphs using spanning tree constructions. J Parallel Distrib Comput 24:55–71

    Article  Google Scholar 

  10. Fu JS (2007) Conditional fault-tolerant Hamiltonicity of star graphs. Parallel Comput 33:488–496

    Article  MathSciNet  Google Scholar 

  11. Hsieh SY, Chen GH, Ho CW (2000) Hamiltonian laceability of star graphs. Networks 36:225–232

    Article  MATH  MathSciNet  Google Scholar 

  12. Hsieh SY, Chen GH, Ho CW (2001) Longest fault-free paths in star graphs with vertex faults. Theor Comput Sci 262:215–227

    Article  MATH  MathSciNet  Google Scholar 

  13. Hsieh SY, Chen GH, Ho CW (2001) Longest fault-free paths in star graphs with edge faults. IEEE Trans Comput 50:960–971

    Article  Google Scholar 

  14. Hsieh SY (2005) Embedding longest fault-free paths onto star graphs with more vertex faults. Theor Comput Sci 337:370–378

    Article  MATH  MathSciNet  Google Scholar 

  15. Hsu DF (1993) Interconnection networks and algorithms. Networks 23:211–450

    Article  Google Scholar 

  16. Jwo JS, Lakshmivarahan S, Dhall SK (1991) Embedding of cycles and grids in star graphs. J Circuits Syst Comput 1:43–74

    Article  Google Scholar 

  17. Leighton FT (1992) Introduction to parallel algorithms and architecture: arrays hypercubes. Morgan Kaufmann, San Mateo

    MATH  Google Scholar 

  18. Li TK, Tan JJM, Hsu LH (2004) Hyper Hamiltonian laceability on edge fault star graph. Inf Sci 165:59–71

    Article  MATH  MathSciNet  Google Scholar 

  19. Li TK (2005) Cycle embedding in star graphs with edge faults. Appl Math Comput 167:891–900

    Article  MATH  MathSciNet  Google Scholar 

  20. Lin CK, Huang HM, Hsu LH, Bau S (2005) Mutually independent Hamiltonian paths in star networks. Networks 46:110–117

    Article  MATH  MathSciNet  Google Scholar 

  21. Mendia VE, Sarkar D (1992) Optimal broadcasting on the star graph. IEEE Trans Parallel Distrib Syst 3:389–396

    Article  MathSciNet  Google Scholar 

  22. Miller Z, Pritikin D, Sudborough IH (1994) Near embeddings of hypercubes into Cayley graphs on the symmetrical group. IEEE Trans Comput 43:13–22

    Article  MathSciNet  Google Scholar 

  23. Ranka S, Wang JC, Yeh N (1993) Embedding meshes on the star graph. J Parallel Distrib Comput 19:131–135

    Article  MATH  MathSciNet  Google Scholar 

  24. Qiu K, Akl SG, Meijer H (1994) On some properties and algorithms for the star and pancake interconnection networks. J Parallel Distrib Comput 22:418–428

    Google Scholar 

  25. Tsai PY, Fu JS, Chen GH (2006) Hamiltonian-laceability in star graphs with conditional edge faults. In: Proceeding of the international computer symposium, Taipei, Taiwan, 2006, pp 144–152

  26. Tseng YC, Chang SH, Sheu JP (1997) Fault-tolerant ring embedding in a star graph with both link and node failures. IEEE Trans Parallel Distrib Syst 8:1185–1195

    Article  Google Scholar 

  27. Wang NC, Chu CP, Chen TS (2002) A dual-Hamiltonian-path-based multicasting strategy for wormhole-routed star graph interconnection networks. J Parallel Distrib Comput 62:1747–1762

    Article  MATH  Google Scholar 

  28. Wang NC, Yen CP, Chu CP (2005) Multicast communication in wormhole-routed symmetric networks with Hamiltonian cycle model. J Syst Arch 51:165–183

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Cheng Kung University, No. 1, University Road, Tainan, 70101, Taiwan

    Sun-Yuan Hsieh & Chang-De Wu

Authors
  1. Sun-Yuan Hsieh
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  2. Chang-De Wu
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Correspondence to Sun-Yuan Hsieh.

Additional information

An extended abstract of this paper under the title “Conditional Edge-Fault-Tolerant Hamiltonian Cycle Embedding of Star Graphs” was presented at the 13th International Conference on Parallel and Distributed Systems (ICPADS), 2007.

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Hsieh, SY., Wu, CD. Optimal fault-tolerant Hamiltonicity of star graphs with conditional edge faults. J Supercomput 49, 354–372 (2009). https://doi.org/10.1007/s11227-008-0242-9

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  • DOI: https://doi.org/10.1007/s11227-008-0242-9

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