From Invariants to Canonization in Parallel

  • Johannes Köbler
  • Oleg Verbitsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)


A function f of a graph is called a complete graph invariant if two given graphs G and H are isomorphic exactly when f(G) = f(H). If additionally, f(G) is a graph isomorphic to G, then f is called a canonical form for graphs. Gurevich [9] proves that any polynomial-time computable complete invariant can be transformed into a polynomial-time computable canonical form. We extend this equivalence to the polylogarithmic-time model of parallel computation for classes of graphs having either bounded rigidity index or small separators. In particular, our results apply to three representative classes of graphs embeddable into a fixed surface, namely, to 3-connected graphs admitting either a polyhedral or a large-edge-width embedding as well as to all embeddable 5-connected graphs. Another application covers graphs with treewidth bounded by a constant k. Since for the latter class of graphs a complete invariant is computable in NC, it follows that graphs of bounded treewidth have a canonical form (and even a canonical labeling) computable in NC.


Rotation System Colored Graph Input Graph Closed Walk Canonization Problem 
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  1. 1.
    Arvind, V., Das, B., Mukhopadhyay, P.: On isomorphism and canonization of tournaments and hypertournaments. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 449–459. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Arvind, V., Torán, J.: Isomorphism testing: Pespective and open problems. Bulletin of the European Association of Theoretical Computer Science 86, 66–84 (2005)Google Scholar
  3. 3.
    Babai, L., Luks, E.: Canonical labeling of graphs. In: Proc. 15th ACM Symposium on Theory of Computing, pp. 171–183. ACM Press, New York (1983)Google Scholar
  4. 4.
    Chlebus, B.S., Diks, K., Radzik, T.: Testing isomorphism of outerplanar graphs in parallel. In: Koubek, V., Janiga, L., Chytil, M.P. (eds.) MFCS 1988. LNCS, vol. 324, pp. 220–230. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  5. 5.
    Filotti, I.S., Mayer, J.N.: 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
  6. 6.
    Fijavž, G., Mohar, B.: Rigidity and separation indices of Paley graphs. Discrete Mathematics 289, 157–161 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Fijavž, G., Mohar, B.: Rigidity and separation indices of graphs in surfaces. A manuscript in preparation, cited in [6]Google Scholar
  8. 8.
    Grohe, M.: Isomorphism testing for embeddable graphs through definability. In: Proc. 32th ACM Symposium on Theory of Computing, pp. 63–72. ACM Press, New York (2000)Google Scholar
  9. 9.
    Gurevich, Y.: From invariants to canonization. Bulletin of the European Association of Theoretical Computer Science 63, 115–119 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grohe, M., Verbitsky, O.: Testing graph isomorphism in parallel by playing a game. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 3–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Immerman, N., Lander, E.: Describing graphs: a first order approach to graph canonization. In: Selman, A.L. (ed.) Complexity Theory Retrospective, pp. 59–81. Springer, Heidelberg (1990)Google Scholar
  12. 12.
    Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, Boston (1993)zbMATHGoogle Scholar
  13. 13.
    Lichtenstein, D.: Isomorphism for graphs embeddable on the projective plane. In: Proc. 12th ACM Symposium on Theory of Computing, pp. 218–224. ACM Press, New York (1980)Google Scholar
  14. 14.
    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
  15. 15.
    Miller, G.L.: 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
  16. 16.
    Miller, G.L.: Isomorphism of k-contractible graphs. A generalization of bounded valence and bounded genus. Information and Computation 56, 1–20 (1983)zbMATHGoogle Scholar
  17. 17.
    Miller, G.L., Reif, J.H.: Parallel tree contraction. Part 2: Further applications. SIAM Journal on Computing 20, 1128–1147 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Mohar, B., Robertson, N.: Flexibility of polyhedral embeddings of graphs in surfaces. J. Combin. Theory, Ser. B 83, 38–57 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Mohar, B., Thomassen, C.: Graphs on surfaces. The John Hopkins University Press (2001)Google Scholar
  20. 20.
    Ponomarenko, I.: The isomorphism problem for classes of graphs closed under contraction. In: Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (in Russian), vol. 174, pp. 147–177 (1988); English translation in: Journal of Mathematical Sciences 55, 1621–1643 (1991)Google Scholar
  21. 21.
    Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Thomassen, C.: Embeddings of graphs with no short noncontractible cycles. J. Combin. Theory, Ser. B 48, 155–177 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Whitney, H.: 2-isomorphic graphs. Amer. Math. J. 55, 245–254 (1933)CrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Johannes Köbler
    • 1
  • Oleg Verbitsky
    • 2
  1. 1.Institut für InformatikHumboldt Universität zu BerlinBerlinGermany
  2. 2.Institute for Applied Problems of Mechanics and MathematicsLvivUkraine

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