Understanding the Complexity of Induced Subgraph Isomorphisms

  • Yijia Chen
  • Marc Thurley
  • Mark Weyer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


We study left-hand side restrictions of the induced subgraph isomorphism problem: Fixing a class Open image in new window , for given graphs G Open image in new window and arbitrary H we ask for induced subgraphs of H isomorphic to G.

For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant.

We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if Open image in new window contains no arbitrarily large graphs.

All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs.

Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if Open image in new window there are classes Open image in new window such that the induced subgraph isomorphism problem on Open image in new window is neither in Open image in new window nor Open image in new window -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.


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  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bodirski, M., Grohe, M.: Non-dichotomies in Constraint Satisfaction Complexity. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008, Track B) (to appear, 2008)Google Scholar
  3. 3.
    Chen, Y., Flum, J.: On parameterized path and chordless path problems. In: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, pp. 250–263. IEEE Computer Society, Los Alamitos (2007)Google Scholar
  4. 4.
    Dalmau, V., Jonsson, P.: The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci. 329(1-3), 315–323 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming, CP 2002, pp. 310–326 (2002)Google Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput. 24(4), 873–921 (1995)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Flum, J., Grohe, M.: Fixed-parameter tractability, definability, and model-checking. SIAM J. Comput. 31(1), 113–145 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)MATHGoogle Scholar
  10. 10.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)Google Scholar
  11. 11.
    Ladner, R.E.: On the structure of polynomial time reducibility. J. ACM 22(1), 155–171 (1975)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    McCartin, C.: Parameterized counting problems. Ann. Pure Appl. Logic 138(1-3), 147–182 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yijia Chen
    • 1
  • Marc Thurley
    • 2
  • Mark Weyer
    • 2
  1. 1.BASICS, Department of Computer ScienceShanghai Jiaotong UniversityShanghaiChina
  2. 2.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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