Advertisement

From Invariants to Canonization in Parallel

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

Abstract

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.

Keywords

Rotation System Colored Graph Input Graph Closed Walk Canonization Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Copyright information

© 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

Personalised recommendations