Skip to main content

A Fast Deterministic Detection of Small Pattern Graphs in Graphs Without Large Cliques

  • Conference paper
  • First Online:
WALCOM: Algorithms and Computation (WALCOM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10167))

Included in the following conference series:

Abstract

We show that for several pattern graphs on four vertices (e.g., \(C_4\)), their induced copies in host graphs with n vertices and no clique on \(k+1\) vertices can be deterministically detected in time \(\tilde{O}(n^{\omega }k^{\mu }+n^2k^2),\) where \(\tilde{O}(f)\) stands for \(O(f (\log f)^c )\) for some constant c,  and \(\mu \approx 0.46530\). The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., \(K_4\)), in host graphs with n vertices and no independent set on \(k+1\) vertices.

By using the concept of Ramsey numbers, we can extend our method for induced subgraph isomorphism to include larger pattern graphs having a set of independent vertices with the same neighborhood and n-vertex host graphs without cliques on \(k+1\) vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Alon, N., Dao, P., Hajirasouliha, I., Hormozdiari, F., Sahinalp, S.C.: Biomolecular network motif counting and discovery by color coding. Bioinformatics (ISMB 2008) 24(13), 241–249 (2008)

    Article  Google Scholar 

  2. Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14(4), 926–934 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chung, F.R.K., Grinstead, C.M.: A survey of bounds for classical ramsey numbers. J. Graph Theory 7, 25–37 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Eisenbrand, F., Grandoni, F.: On the complexity of fixed parameter clique and dominating set. Theoret. Comput. Sci. 326, 57–67 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eschen, E.M., Hoàng, C.T., Spinrad, J., Sritharan, R.: On graphs without a C4 or a diamond. Discret. Appl. Math. 159(7), 581–587 (2011)

    Article  MATH  Google Scholar 

  6. Floderus, P., Kowaluk, M., Lingas, A., Lundell, E.-M.: Detecting and counting small pattern graphs. SIAM J. Discret. Math. 29(3), 1322–1339 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Floderus, P., Kowaluk, M., Lingas, A., Lundell, E.-M.: Induced subgraph isomorphism: are some patterns substantially easier than others? Theoret. Comput. Sci. 605, 119–128 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garey, M.R., Johnson, D.S.: Computers and Intractability - A Guide to the Theory of NP-Completeness. Bell Laboratories, Murray Hill (1979)

    MATH  Google Scholar 

  9. Gąsieniec, L., Kowaluk, M., Lingas, A.: Faster multi-witnesses for Boolean matrix product. Inf. Process. Lett. 109, 242–247 (2009)

    Article  MATH  Google Scholar 

  10. Hoàng, C.T., Kaminski, M., Sawada, J., Sritharan, R.: Finding and listing induced paths and cycles. Discret. Appl. Math. 161(4–5), 633–641 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM J. Comput. 7, 413–423 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kloks, T., Kratsch, D., Müller, H.: Finding and counting small induced subgraphs efficiently. Inf. Process. Lett. 74(3–4), 115–121 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kowaluk, M., Lingas, A., Lundell, E.-M.: Counting and detecting small subgraphs via equations and matrix multiplication. SIAM J. Discret. Math. 27(2), 892–909 (2013)

    Article  MATH  Google Scholar 

  14. Kuramochi, M., Karypis, G.: Finding frequent patterns in a large sparse graph. Data Min. Knowl. Disc. 11, 243–271 (2005)

    Article  MathSciNet  Google Scholar 

  15. Le Gall, F.: Faster algorithms for rectangular matrix multiplication. In: Proceedings of 53rd Symposium on Foundations of Computer Science (FOCS), pp. 514–523 (2012)

    Google Scholar 

  16. Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of 39th International Symposium on Symbolic and Algebraic Computation, pp. 296–303 (2014)

    Google Scholar 

  17. Nes̆etr̆il, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Math. Univ. Carol. 26(2), 415–419 (1985)

    MathSciNet  MATH  Google Scholar 

  18. Olariu, S.: Paw-free graphs. Inf. Process. Lett. 28, 53–54 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schank, T., Wagner, D.: Finding, counting and listing all triangles in large graphs, an experimental study. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 606–609. Springer, Heidelberg (2005). doi:10.1007/11427186_54

    Chapter  Google Scholar 

  20. Wolinski, C., Kuchcinski, K., Raffin, E.: Automatic design of application-specific reconfigurable processor extensions with UPaK synthesis kernel. ACM Trans. Des. Autom. Electron. Syst. 15(1), 1–36 (2009)

    Article  Google Scholar 

  21. Vassilevska, V.: Efficient algorithms for path problems in weighted graphs. Ph.D. thesis, CMU, CMU-CS-08-147 (2008)

    Google Scholar 

  22. Williams, V.V., Wang, J.R., Williams, R., Yu, H.: Finding four-node subgraphs in triangle time. In: Proceedings of SODA, pp. 1671–1680 (2015)

    Google Scholar 

  23. Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of 44th Annual ACM Symposium on Theory of Computing (STOC), pp. 887–898 (2012)

    Google Scholar 

Download references

Acknowledgments

The research has been supported in part by the grant of polish National Science Center 2014/13/B/ST6/00770 and Swedish Research Council grant 621-2011-6179, respectively.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrzej Lingas .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Kowaluk, M., Lingas, A. (2017). A Fast Deterministic Detection of Small Pattern Graphs in Graphs Without Large Cliques. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53925-6_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53924-9

  • Online ISBN: 978-3-319-53925-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics