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On the Complexity of Counting the Number of Vertices Moved by Graph Automorphisms

  • Antoni Lozano
  • Vijay Raghavan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1530)

Abstract

We consider the problem of deciding whether a given graph G has an automorphism which moves at least k vertices (where k is a function of ∣ V(G) ∣), a question originally posed by Lubiw (1981). Here we show that this problem is equivalent to the one of deciding whether a graph has a nontrivial automorphism, for k ∈ O(logn / loglogn ).

It is commonly believed that deciding isomorphism between two graphs is strictly harder than deciding whether a graph has a nontrivial automorphism. Indeed, we show that an isomorphism oracle would improve the above result slightly–using such an oracle, one can decide whether there is an automorphism which moves at least k′ vertices, where k′ ∈ O(logn).

If P \(\ne\) NP and Graph Isomorphism is not NP-complete, the above results are fairly tight, since it is known that deciding if there is an automorphism which moves at least n ε vertices, for any fixed ε ∈ (0, 1) , is NP-complete. In other words, a substantial improvement of our result would settle some fundamental open problems about Graph Isomorphism.

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References

  1. 1.
    Agrawal, M., Arvind, V.: A note on decision versus search for graph isomorphism. Information and Computation 131(2), 179–189 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arvind, V., Beigel, R., Lozano, A.: The complexity of modular graph automorphism. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 172–182. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Blum, M., Kannan, S.: Designing programs that check their work. Journal of the ACM 43, 269–291 (1995)CrossRefGoogle Scholar
  4. 4.
    Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1991)Google Scholar
  5. 5.
    Hoffmann, C.: Group-Theoretic Algorithms and Graph Isomorphism. In: Hoffmann, C.M. (ed.) Group-Theoretic Algorithms and Graph Isomorphism. LNCS, vol. 136. Springer, Heidelberg (1982)Google Scholar
  6. 6.
    Köbler, J., Schöning, U., Torán, J.: The graph isomorphism problem: its structural complexity. Birkhäuser, Boston (1993)Google Scholar
  7. 7.
    Ladner, R.: On the structure of polynomial-time reducibilities. Journal of the Assoc. Comput. Mach. 22, 155–171 (1975)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Lozano, A., Torán, J.: On the nonuniform complexity of the graph isomorphism problem. In: Proceedings of the 7th Structure in Complexity Theory Conference, pp. 118–129 (1992)Google Scholar
  9. 9.
    Lubiw, A.: Some NP-complete problems similar to graph isomorphism. SIAM Journal of Computing 10(1), 11–21 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Luks, E.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. System Sci. 25, 42–65 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mathon, R.: A note on the graph isomorphism counting problem. Information Processing Letters 8, 131–132 (1979)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Antoni Lozano
    • 1
  • Vijay Raghavan
    • 2
  1. 1.Department LSIUPC Jordi GironaBarcelonaE.U.
  2. 2.CS Dept.Vanderbilt UniversityNashvilleUSA

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