Testing Graph Isomorphism in Parallel by Playing a Game

  • Martin Grohe
  • Oleg Verbitsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)


Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in \({\rm TC^1}\subseteq\mbox{\rm NC\ensuremath{^{2}}}\). If no counting quantifiers are needed, then Graph Isomorphism for C is even in AC1. The definability conditions can be checked by designing a winning strategy for suitable Ehrenfeucht-Fraïssé games with a logarithmic number of rounds. The parallel isomorphism algorithm this approach yields is a simple combinatorial algorithm known as the Weisfeiler-Lehman (WL) algorithm.

Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC1, answering an open question from [9]. Furthermore, we obtain an AC1 algorithm for testing isomorphism of rotation systems (combinatorial specifications of graph embeddings). The AC1 upper bound was known before, but the fact that this bound can be achieved by the simple WL algorithm is new. Combined with other known results, it also yields a new AC1 isomorphism algorithm for planar graphs.


Planar Graph Rotation System Order Logic Input Graph Tree Decomposition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babai, L.: Moderately exponential bound for graph isomorphism. In: Gecseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 34–50. Springer, Heidelberg (1981)Google Scholar
  2. 2.
    Babai, L.: Automorphism groups, isomorphism, reconstruction. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, Ch. 27, pp. 1447–1540. Elsevier Publ., Amsterdam (1995)Google Scholar
  3. 3.
    Babai, L., Grigoryev, D., Yu., M.D.M.: Isomorphism of graphs with bounded eigenvalue multiplicity. In: Proc. of the 14th ACM Symp. on Theory of Computing, pp. 310–324 (1982)Google Scholar
  4. 4.
    Babai, L., Luks, E.M.: Canonical labeling of graphs. In: Proc. of the 15th ACM Symposium on Theory of Computing, pp. 171–183 (1983)Google Scholar
  5. 5.
    Bodlaender, H.L.: Polynomial algorithms for Graph Isomorphism and Chromatic Index on partial k-trees. J. Algorithms 11, 631–643 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bohman, T., Frieze, A.: Łuczak, T., Pikhurko, O., Smyth, C., Spencer, J., Verbitsky, O.: The first order definability of trees and sparse random graphs. E-print (2005),
  7. 7.
    Boppana, R.B., Stad, H.J., Zachos, S.: Does co-NP have short interactive proofs? Inf. Process. Lett. 25, 127–132 (1987)MATHCrossRefGoogle Scholar
  8. 8.
    Cai, J.-Y., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 389–410 (1992)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chandrasekharan, N.: Isomorphism testing of k-trees is in NC, for fixed k. Inf. Process. Lett. 34, 283–287 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Del Greco, J.G., Sekharan, C.N., Sridhar, R.: Fast parallel reordering and isomorphism testing of k-trees. Algorithmica 32, 61–72 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Evdokimov, S., Karpinski, M., Ponomarenko, I.: On a new high dimensional Weisfeiler-Lehman algorithm. J. Algebraic Combinatorics 10, 29–45 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Evdokimov, S., Ponomarenko, I.: On highly closed cellular algebras and highly closed isomorphism. In: Electronic J. Combinatorics, vol. 6 (1999) #R18Google Scholar
  13. 13.
    Filotti, I.S., Mayer, J.N.: A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. In: Proc. of the 12th ACM Symp. on Theory of Computing, pp. 236–243 (1980)Google Scholar
  14. 14.
    Grohe, M.: Fixed-point logics on planar graphs. In: Proc. of the Ann. Conf. on Logic in Computer Science, pp. 6–15 (1998)Google Scholar
  15. 15.
    Grohe, M.: Isomorphism testing for embeddable graphs through definability. In: Proc. of the 32nd ACM Ann. Symp. on Theory of Computing, pp. 63–72 (2000)Google Scholar
  16. 16.
    Grohe, M., Marino, 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
  17. 17.
    Grohe, M., Verbitsky, O.: Testing graph isomorphism in parallel by playing a game. E-print (2006),
  18. 18.
    Hopcroft, J.E., Tarjan, R.E.: Isomorphism of planar graphs (working paper). In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of computer computations, pp. 131–152. Plenum Press, New York (1972)Google Scholar
  19. 19.
    Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs. In: Proc. of the 6th ACM Symp. on Theory of Computing, pp. 172–184 (1974)Google Scholar
  20. 20.
    Immerman, N.: Descriptive complexity. Springer, Heidelberg (1999)MATHGoogle Scholar
  21. 21.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness Results for Graph Isomorphism. J. Comput. Syst. Sci. 66, 549–566 (2003)MATHCrossRefGoogle Scholar
  22. 22.
    Karp, R.M., Ramachandran, V.: Parallel algorithms for shared-memory machines. In: van Leeuwen, J. (ed.) Algorithms and complexity. Handbook of theoretical computer science, pp. 869–941. Elsevier Publ., Amsterdam (1990)Google Scholar
  23. 23.
    Kim, J.-H., Pikhurko, O., Spencer, J., Verbitsky, O.: How complex are random graphs in first order logic? In: Random Structures and Algorithms, vol. 26, pp. 119–145 (2005)Google Scholar
  24. 24.
    Lindell, S.: A logspace algorithm for tree canonization. In: Proc. of the 24th Ann. ACM Symp. on Theory of Computing, pp. 400–404 (1992)Google Scholar
  25. 25.
    Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25 (1982) 42–65Google Scholar
  26. 26.
    Miller, G.L.: Isomorphism testing for graphs of bounded genus. In: Proc. of the 12th ACM Symp. on Theory of Computing, pp. 225–235 (1980)Google Scholar
  27. 27.
    Miller, G.L., Reif, J.H.: Parallel tree contraction. Part 2: further applications. SIAM J. Comput. 20, 1128–1147 Google Scholar
  28. 28.
    Mohar, B., Thomassen, C.: Graphs on surfaces. The John Hopkins University Press (2001)Google Scholar
  29. 29.
    Pikhurko, O., Spencer, J., Verbitsky, O.: Succinct definitions in first order graph theory. Annals of Pure and Applied Logic 139, 74–109 (2006)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Pikhurko, O., Veith, H., Verbitsky, O.: First order definability of graphs: tight bounds on quantifier rank. Discrete Applied Mathematics (to appear)Google Scholar
  31. 31.
    Ponomarenko, I.N.: The isomorphism problem for classes of graphs that are invariant with respect to contraction. Computational Complexity Theory 3. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 174, 147–177 (1988)Google Scholar
  32. 32.
    Ramachandran, V., Reif, J.: Planarity testing in parallel. J.Comput.Syst.Sci. 49, 517–561 (1994)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree-width. J.Algorithms 7, 309–322 (1986)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Schöning, U.: Graph isomorphism is in the low hierarchy. J. Comput.Syst.Sci. 37, 312–323 (1988)MATHCrossRefGoogle Scholar
  35. 35.
    Spencer, J.: The strange logic of random graphs. Springer, Heidelberg (2001)MATHGoogle Scholar
  36. 36.
    Torán, J.: On the hardness of graph isomorphism. SIAM J.Comput. 33, 1093–1108 (2004)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Verbitsky, O.: The first order definability of graphs with separators via the Ehrenfeucht game. Theor.Comput.Sci. 343, 158–176 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Grohe
    • 1
  • Oleg Verbitsky
    • 1
  1. 1.Institut für InformatikHumboldt Universität zu BerlinBerlinGermany

Personalised recommendations