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New Invariants for the Graph Isomorphism Problem

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Abstract

In this paper, we introduce a novel polynomial-time algorithm to compute graph invariants based on the idea of a modified random walk on graphs. Though not proved to be a full graph invariant yet, our method gives the right answer for the graph instances other well-known methods could not compute (such as special Fürer gadgets and point-line incidence graphs of finite projective planes of higher degrees).

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References

  1. L. Babai, P. Erdos, and S. M. Selkow, “Random graph isomorphism,” SIAM J. Comput., 9, No. 3, 628–635 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Babai, D. Yu. Grigoryev, and D. M. Mount, “Isomorphism of graphs with bounded eigenvalue multiplicity,” in: Proc. STOC (1982), pp. 310–324.

  3. R. B. Boppana, J. Hastad, and S. Zachos, “Does co-NP have short interactive proofs?” Inform. Process. Lett., 25, No. 2, 127–132 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. R. Buss, “Alogtime algorithms for tree isomorphism, comparison, and canonization,” in: Computational Logic and Proof Theory (Vienna, 1997), Lect. Notes Comput. Sci., 1289, Springer-Verlag, Berlin (1997), pp. 18–33.

  5. J.-Yi Cai, M. Fürer, and N. Immerman, “An optimal lower bound on the number of variables for graph identification,” Combinatorica, 12, No. 4, 389–410 (1992).

  6. I. S. Filotti and J. N. Mayer, “A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus,” in: Proc. STOC (1980), pp. 236–243.

  7. J. Hopcroft and R. Tarjan, “AV 2 algorithm for determining isomorphism of planar graphs,” Inf. Process. Lett., 1, 32–34 (1971).

  8. J. E. Hopcroft and J. K. Wong, “Linear time algorithm for isomorphism of planar graphs,” in: Proc. STOC (1974), pp. 310–324.

  9. S. Lindell, “A logspace algorithm for tree canonization,” in: Proc. 24 Annual ACM Symp. on Theory of Computing (1992), pp. 400–404.

  10. E. M. Luks, “Isomorphism of graphs of bounded valence can be tested in polynomial time,” J. Comput. System Sci., 25, No. 1, 42–65 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Luks, “Parallel algorithms for permutation groups and graph isomorphism,” in: Proc. 27 IEEE Symp. on Foundations of Computer Science (1986), pp. 292–302.

  12. R. Mathon, “A note on the graph isomorphism counting problem,” Inform. Process. Lett., 8, 131–132 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  13. B. McKay, The Nauty System, http://cs.anu.edu.au/~bdm/nauty.

  14. T. Miyazaki, “The complexity of McKay’s canonical labeling algorithm,” in: Groups and Computation, II (New Brunswick, NJ, 1995), DIMACS Ser. Discr. Math. Theor. Comput. Sci., 28, Am. Math. Soc., Providence, Rhode Island (1997), pp. 239–256.

  15. E. Moorhouse, “Projective planes of small order,” http://www.uwyo.edu/moorhouse/pub/planes.

  16. U. Sch¨oning, “Graph isomorphism is in the low hierarchy,” J. Comput. System Sci., 37, No. 3, 312–323 (1988).

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Correspondence to A. Gamkrelidze.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 97, Proceedings of the International Conference “Lie Groups, Differential Equations, and Geometry,” June 10–22, 2013, Batumi, Georgia, Part 2, 2015.

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Gamkrelidze, A., Varamashvili, L. & Hotz, G. New Invariants for the Graph Isomorphism Problem. J Math Sci 218, 754–761 (2016). https://doi.org/10.1007/s10958-016-3061-1

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  • DOI: https://doi.org/10.1007/s10958-016-3061-1

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