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On Graph Isomorphism for Restricted Graph Classes

  • Johannes Köbler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic.

In this paper, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions.

Keywords

Planar Graph Color Class Graph Class Graph Isomorphism Direct Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agrawal, M., Saxena, N.: Automorphisms of finite rings and applications to complexity of problems. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 1–17. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Àlvarez, C., Jenner, B.: A very hard log-space counting class. Theoretical Computer Science 107(1), 3–30 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arvind, V., Köbler, J.: On hypergraph and graph isomorphism with bounded color classes. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 384–395. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Arvind, V., Kurur, P., Vijayaraghavan, T.: Bounded color multiplicity graph isomorphism is in the #L hierarchy. In: Proc. 20th Annual IEEE Conference on Computational Complexity, pp. 13–27. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  5. 5.
    Arvind, V., Torán, J.: Isomorphism testing: Pespective and open problems. Bulletin of the European Association of Theoretical Computer Science (BEATCS) 86 (2005)Google Scholar
  6. 6.
    Babai, L.: Moderately exponential bounds for graph isomorphism. In: Gecseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 34–50. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  7. 7.
    Babai, L.: Trading group theory for randomness. In: Proc. 17th ACM Symposium on Theory of Computing, pp. 421–429. ACM Press, New York (1985)Google Scholar
  8. 8.
    Babai, L.: A Las Vegas-NC algorithm for isomorphism of graphs with bounded multiplicity of eigenvalues. In: Proc. 27th IEEE Symposium on the Foundations of Computer Science, pp. 303–312. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  9. 9.
    Babai, L.: Automorphism groups, isomorphism, reconstruction. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 1447–1540. Elsevier, Amsterdam (1995)Google Scholar
  10. 10.
    Babai, L., Grigoryev, D., Mount, D.: Isomorphism of graphs with bounded eigenvalue multiplicity. In: Proc. 14th ACM Symposium on Theory of Computing, pp. 310–324. ACM Press, New York (1982)Google Scholar
  11. 11.
    Babai, L., Luks, E.: Canonical labeling of graphs. In: Proc. 15th ACM Symposium on Theory of Computing, pp. 171–183 (1983)Google Scholar
  12. 12.
    Babai, L., Luks, E., Seress, Á.: Permutation groups in NC. In: Proc. 19th ACM Symposium on Theory of Computing, pp. 409–420. ACM Press, New York (1987)Google Scholar
  13. 13.
    Balcázar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I. In: EATCS Monographs on Theoretical Computer Science, 2nd edn., Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Bodlaender, H.: Polynomial algorithm for graph isomorphism and chromatic index on partial k-trees. Journal of Algorithms 11, 631–643 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Booth, K.: Isomorphism testing for graphs, semigroups, and finite automata are polynomially equivalent problems. SIAM Journal on Computing 7(3), 273–279 (1978)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Boppana, R., Hastad, J., Zachos, S.: Does co-NP have short interactive proofs? Information Processing Letters 25(2), 27–32 (1987)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Buntrock, G., Damm, C., Hertrampf, U., Meinel, C.: Structure and importance of logspace-MOD classes. Mathematical Systems Theory 25, 223–237 (1992)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Buss, S.: Alogtime algorithms for tree isomorphism, comparison, and canonization. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) KGC 1997. LNCS, vol. 1289, pp. 18–33. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  19. 19.
    Cai, J., Fürer, M., Immerman, N.: An optimal lower bound for the number of variables for graph identification. Combinatorica 12, 389–410 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Information and Control 64, 2–22 (1985)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Evdokimov, S., Karpinski, M., Ponomarenko, I.: On a new high dimensional Weisfeiler-Lehman algorithm. Journal of Algebraic Combinatorics 10, 29–45 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Furst, M., Hopcroft, J., Luks, E.: Polynomial time algorithms for permutation groups. In: Proc. 21st IEEE Symposium on the Foundations of Computer Science, pp. 36–41. IEEE Computer Society Press, Los Alamitos (1980)Google Scholar
  23. 23.
    Grohe, M.: Fixed-points logics on planar graphs. In: Proceedings of the 13th Symposium on Logic in Computer Science, pp. 6–15 (1998)Google Scholar
  24. 24.
    Grohe, M.: Isomorphism testing for embeddable graphs through definability. In: Proc. 32th ACM Symposium on Theory of Computing, pp. 63–72 (2000)Google Scholar
  25. 25.
    Grohe, M., Mariño, J.: Definability and descriptive complexity on databases of bounded tree-width. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 70–82. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  26. 26.
    Grohe, M., Verbitsky, O.: Testing graph isomorphism in parallel by playing a game (2006) (manuscript)Google Scholar
  27. 27.
    Gurevich, Y.: From invariants to canonization. Bulletin of the European Association of Theoretical Computer Science (BEATCS) 63 (1997)Google Scholar
  28. 28.
    Hopcroft, J.E., Tarjan, R.E.: Isomorphism of planar graphs (working paper). In: Miller, R., Thatcher, J. (eds.) Complexity of computer computations, pp. 131–152. Plenum Press, New York (1972)CrossRefGoogle Scholar
  29. 29.
    Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. Journal of the ACM 21, 549–568, 620 (1974)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hopcroft, J.E., Wong, J.: Linear time algorithm for isomorphisms of planar graphs. In: Proc. 6th ACM Symposium on Theory of Computing, pp. 172–184 (1974)Google Scholar
  31. 31.
    Immerman, N.: Number of quantifiers is better than number of tape cells. Journal of Computer and System Sciences 22(3), 384–406 (1981)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Impagliazzo, R., Levin, L.A.: No better ways to generate hard NP-instances than picking uniformly at random. In: Proc. 31st IEEE Symposium on the Foundations of Computer Science, pp. 812–821. IEEE Computer Society Press, Los Alamitos (1990)Google Scholar
  33. 33.
    Filotti, J.M.I.S.: A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. In: Proc. 12th ACM Symposium on Theory of Computing, pp. 236–243. ACM Press, New York (1980)Google Scholar
  34. 34.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. Journal of Computer and System Sciences 66, 549–566 (2003)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem: Its Structural Complexity. In: Progress in Theoretical Computer Science, Birkhäuser, Boston (1993)Google Scholar
  36. 36.
    Lichtenstein, D.: Isomorphism for graphs embaddable on the projective plane. In: Proc. 12th ACM Symposium on Theory of Computing, pp. 218–224. ACM Press, New York (1980)Google Scholar
  37. 37.
    Lindell, S.: A logspace algorithm for tree canonization. In: Proc. 24th ACM Symposium on Theory of Computing, pp. 400–404. ACM Press, New York (1992)Google Scholar
  38. 38.
    Luks, E.: Isomorphism of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences 25, 42–65 (1982)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Luks, E.: Parallel algorithms for permutation groups and graph isomorphism. In: Proc. 27th IEEE Symposium on the Foundations of Computer Science, pp. 292–302. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  40. 40.
    Mathon, R.: A note on the graph isomorphism counting problem. Information Processing Letters 8, 131–132 (1979)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Miller, G.: Isomorphism testing for graphs of bounded genus. In: Proc. 12th ACM Symposium on Theory of Computing, pp. 225–235. ACM Press, New York (1980)Google Scholar
  42. 42.
    Miller, G., Reif, J.: Parallel tree contraction, part 2: Further applications. SIAM Journal on Computing 20, 1128–1147 (1991)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Ponomarenko, I.: The isomorphism problem for classes of graphs that are invariant with respect to contraction (Russian). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 174, 147–177 (1988)Google Scholar
  44. 44.
    Reingold, O.: Undirected st-connectivity in log-space. In: Proc. 37th ACM Symposium on Theory of Computing, pp. 376–385. ACM Press, New York (2005)Google Scholar
  45. 45.
    Schöning, U.: Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences 37, 312–323 (1988)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20, 865–877 (1991)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Torán, J.: On the hardness of graph isomorphism. SIAM Journal on Computing 33(5), 1093–1108 (2004)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Zemlyachenko, V.N.: Canonical numbering of trees (Russian). In: Proc. Seminar on Comb. Anal. at Moscow State University (1970)Google Scholar
  49. 49.
    Zemlyachenko, V.N., Konienko, N., Tyshkevich, R.I.: Graph isomorphism problem (Russian). In: The Theory of Computation I, Notes Sci. Sem. LOMI 118 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Johannes Köbler
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinGermany

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