Induced Subgraph Isomorphism: Are Some Patterns Substantially Easier Than Others?

  • Peter Floderus
  • Mirosław Kowaluk
  • Andrzej Lingas
  • Eva-Marta Lundell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


The complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph. For instance, the larger the maximum independent set of the pattern graph is the more efficient algorithms are known. The situation seems to be substantially different in the case of induced subgraph isomorphism for pattern graphs of fixed size. We present two results which provide evidence that no topology of an induced subgraph of fixed size can be easier to detect or count than an independent set of related size. We show that:
  • Any fixed pattern graph that has a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k. It follows in particular that an induced path on k vertices is not easier to detect than an independent set on ⌈k/2 ⌉ vertices, and that an induced even cycle on k vertices is not easier to detect than an independent set on k/2 vertices. In view of linear time upper bounds on induced paths of length three and four, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and induced complete split graphs.

  • For an arbitrary pattern graph H on k vertices with no isolated vertices, there is a simple subdivision of H, resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one, that is not easier to detect or count than an independent set on k vertices, respectively.

Finally, we show that the so called diamond, paw and C 4 are not easier to detect as induced subgraphs than an independent set on three vertices.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chen, Y., Flum, J.: On Parametrized Path and Chordless Path Problems. In: Proc. IEEE Conference on Computational Complexity, pp. 250–263 (2007)Google Scholar
  2. 2.
    Coppersmith, D.: Rectangular matrix multiplication revisited. Journal of Complexity 13, 42–49 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. J. of Symbolic Computation 9, 251–280 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A Linear Recognition Algorithm for Cographs. SIAM J. Comput. 14(4), 926–934 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parametrized Complexity. Springer, New York (1999)CrossRefGoogle Scholar
  6. 6.
    Eisenbrand, F., Grandoni, F.: On the complexity of fixed parameter clique and dominating set. Theoretical Computer Science 326, 57–67 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Eschen, E.M., Hoàng, C.T., Spinrad, J., Sritharan, R.: On graphs without a C4 or a diamond. Discrete Applied Mathematics 159(7), 581–587 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-completeness. W.H. Freeman and Company, New York (2003)Google Scholar
  9. 9.
    Hoàng, C.T., Kaminski, M., Sawada, J., Sritharan, R.: Finding and listing induced paths and cycles. Manuscript (2010)Google Scholar
  10. 10.
    Huang, X., Pan, V.Y.: Fast rectangular matrix multiplications and applications. Journal of Complexity 14(2), 257–299 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM Journal on Computing 7(4), 413–423 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kloks, T., Kratsch, D., Müller, H.: Finding and counting small induced subgraphs efficiently. Information Processing Letters 74(3-4), 115–121 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kowaluk, M., Lingas, A., Lundell, E.M.: Counting and detecting small subgraphs via equations and matrix multiplication. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1468–1476 (2011)Google Scholar
  14. 14.
    Nešetřil, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae 26(2), 415–419 (1985)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Olariu, S.: Paw-Free Graphs. Information Processing Letters 28, 53–54 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Papadimitriu, C.H., Yannakakis, M.: On limited nondeterminism and the complexity of the VC-dimension. Journal of Computer and System Sciences 53, 161–170 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Vassilevska, V.: Efficient Algorithms for Path Problems in Weighted Graphs. PhD thesis, CMU, CMU-CS-08-147 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peter Floderus
    • 1
  • Mirosław Kowaluk
    • 2
  • Andrzej Lingas
    • 3
  • Eva-Marta Lundell
    • 3
  1. 1.The Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Institute of InformaticsWarsaw UniversityWarsawPoland
  3. 3.Department of Computer ScienceLund UniversityLundSweden

Personalised recommendations