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On the Identification of Vertices and Edges Using Cycles

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2227))

Abstract

The subgraphs C 1, C 2, . . . , C k of a graph G are said to identify the vertices (resp. the edges) of G if the sets j : v ∈ C j (resp. j : e ∈ C j) are nonempty for all the vertices v (edges e) and no two are the same. We consider the problem of minimizing k when the subgraphs Ci are required to be cycles or closed walks. The motivation comes from maintaining multiprocessor systems, and we study the cases when G is the binary hypercube, or the two-dimensional p-ary space with respect to the Lee metric.

Research supported by the Academy of Finland under grant 44002.

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References

  1. U. Blass, I. Honkala, M. G. Karpovsky, S. Litsyn, Short dominating paths and cycles in binary hypercubes, Annals of Combinatorics, to appear.

    Google Scholar 

  2. U. Blass, I. Honkala, S. Litsyn, Bounds on identifying codes, Discrete Mathematics, to appear.

    Google Scholar 

  3. U. Blass, I. Honkala, S. Litsyn, On binary codes for identification, Journal of Combinatorial Designs 8 (2000), 151–156.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Chakrabarty, M. G. Karpovsky, L. B. Levitin, “Fault isolation and diagnosis in multiprocessor systems with point-to-point connections” Fault tolerant parallel and distributed systems, Kluwer (1998), pp. 285–301.

    Google Scholar 

  5. I. Charon, O. Hudry, A. Lobstein, Identifying codes with small radius in some infinite regular graphs, Electronic Journal of Combinatorics, submitted.

    Google Scholar 

  6. I. Charon, I. Honkala, O. Hudry, A. Lobstein, General bounds for identifying codes in some infinite regular graphs, Electronic Journal of Combinatorics, submitted.

    Google Scholar 

  7. I. Charon, I. Honkala, O. Hudry, A. Lobstein, The minimum density of an identifying code in the king lattice, Discrete Mathematics, submitted.

    Google Scholar 

  8. G. Cohen, I. Honkala, A. Lobstein, G. Zémor, On codes identifying vertices in the two-dimensional square lattice with diagonals, IEEE Transactions on Computers 50 (2001), 174–176.

    Article  Google Scholar 

  9. G. Cohen, I. Honkala, A. Lobstein, G. Zémor, Bounds for codes identifying vertices in the hexagonal grid, SIAM Journal on Discrete Mathematics 13 (2000), 492–504.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Cohen, I. Honkala, A. Lobstein, G. Zémor, “On identifying codes,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Proceedings of the DIMACS Workshop on Codes and Association Schemes, November 9–12, 1999, pp. 97–109.

    Google Scholar 

  11. I. Honkala, M. Karpovsky, S. Litsyn: Cycles for identifying vertices and edges in graphs, submitted.

    Google Scholar 

  12. I. Honkala, T. Laihonen, S. Ranto, On codes identifying sets of vertices in Hamming spaces, Designs, Codes and Cryptography, to appear.

    Google Scholar 

  13. I. Honkala, T. Laihonen, S. Ranto, On strongly identifying codes, Discrete Mathematics, to appear.

    Google Scholar 

  14. I. Honkala, A. Lobstein: On the density of identifying codes in the square lattice, submitted.

    Google Scholar 

  15. M. G. Karpovsky, K. Chakrabarty, L. B. Levitin, On a new class of codes for identifying vertices in graphs, IEEE Transactions on Information Theory 44 (1998), 599–611.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. G. Karpovsky, S. M. Chaudhry, L. B. Levitin, Detection and location of given sets of errors by nonbinary linear codes, preprint.

    Google Scholar 

  17. M. G. Karpovsky, L. B. Levitin, F. S. Vainstein, Diagnosis by signature analysis of test responses, IEEE Transactions on Computers 43 (1994), 1–12.

    Article  Google Scholar 

  18. M. G. Karpovsky, S. M. Chaudhry, L. B. Levitin, “Multiple signature analysis: a framework for built-in self-diagnostic,” Proceedings of the 22nd International Symposium on Fault-Tolerant Computing, July 8–10, 1992, Boston, Massachusetts (1992), pp. 112–119.

    Google Scholar 

  19. T. Laihonen, Sequences of optimal identifying codes, IEEE Transactions on Information Theory, submitted.

    Google Scholar 

  20. T. Laihonen, Optimal codes for strong identification, European Journal of Combinatorics, submitted.

    Google Scholar 

  21. T. Laihonen, S. Ranto, Families of optimal codes for strong identification, Discrete Applied Mathematics, to appear.

    Google Scholar 

  22. S. Ranto, I. Honkala, T. Laihonen, Two families of optimal identifying codes in binary Hamming spaces, IEEE Transactions on Information Theory, submitted.

    Google Scholar 

  23. Rosen, K. H. (ed.), Handbook of discrete and combinatorial mathematics, CRC Press, Boca Raton, 2000.

    MATH  Google Scholar 

  24. L. Zakrevski, M. G. Karpovsky, “Fault-tolerant message routing in computer networks,” Proc. Int. Conf. on Parallel and Distributed Processing Techniques and Applications (1999), pp. 2279–2287.

    Google Scholar 

  25. L. Zakrevski, M. G. Karpovsky, “Fault-tolerant message routing for multiprocessors,” Parallel and Distributed Processing J. Rolim (ed.), Springer (1998), pp. 714–731.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Iiro Honkala

  2. College of Engineering, Boston University, 02215, Boston, MA, USA

    Mark G. Karpovsky

  3. Department of Electrical Engineering-Systems, Tel-Aviv University, 69978, Ramat-Aviv, Israel

    Simon Litsyn

Authors
  1. Iiro Honkala
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  2. Mark G. Karpovsky
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  3. Simon Litsyn
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Editor information

Editors and Affiliations

  1. Department of Mathematics, RMIT University, GPO Box 2476V, 3001, Melbourne, Australia

    Serdar Boztaş

  2. Department of Computing, Macquarie University, 2109, NSW, Australia

    Igor E. Shparlinski

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© 2001 Springer-Verlag Berlin Heidelberg

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Honkala, I., Karpovsky, M.G., Litsyn, S. (2001). On the Identification of Vertices and Edges Using Cycles. In: Boztaş, S., Shparlinski, I.E. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2001. Lecture Notes in Computer Science, vol 2227. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45624-4_32

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  • DOI: https://doi.org/10.1007/3-540-45624-4_32

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42911-1

  • Online ISBN: 978-3-540-45624-7

  • eBook Packages: Springer Book Archive

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