We show that several reducibility notions coincide when applied to the Graph Isomorphism (GI) problem. In particular we show that if a set is many-one logspace reducible to GI, then it is in fact many-one AC 0 reducible to GI. For the case of Turing reducibilities we show that for any k ≥ 0 an NC k + 1 reduction to GI can be transformed into an AC k reduction to the same problem.


Computational complexity reducibilities graph isomorphism 


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  1. 1.
    Agrawal, M., Allender, E., Rudich, S.: Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem. JCSS 57, 17–143 (1998)MathSciNetGoogle Scholar
  2. 2.
    Álvarez, C., Balcázar, J.L., Jenner, B.: Adaptive Logspace Reducibilities and Parallel Time. Math. Systems Theory 28, 117–140 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC1. Journal of Computer and System Sciences 41, 274–306 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Information and Control 64(1), 2–22 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hoffmann, C.M. (ed.): Group-Theoretic Algorithms and Graph Isomorphism. LNCS, vol. 136. Springer, Heidelberg (1982)zbMATHGoogle Scholar
  6. 6.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. Journal of Computer and System Sciences 66, 549–566 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Köbler, J., Schöning, U., Torán, J.: Graph Isomorphism: its Structural Complexity, Birkhäuser, Boston (1992)Google Scholar
  8. 8.
    Ogihara, M.: Equivalence of NCk and ACk − 1 closures of NP and other classes. Information and Computation 120(1), 55–58 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ruzzo, W.: On uniform circuit complexity. Journal of Computer and System Sciences 22, 365–383 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Selman, A.: Promise problems complete for complexity classes. Information and Computation 78, 87–98 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Torán, J.: On the hardness of Graph Isomorphism. SIAM Journal on Computing 33(5), 1093–1108 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Wilson, C.B.: Decomposing NC and AC. SIAM Journal on Computing 19(2), 384–396 (1990)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jacobo Torán
    • 1
  1. 1.Institut für Theoretische Informatik, Universität Ulm, D-89069 UlmGermany

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