Linear Time Algorithm for Computing a Small Biclique in Graphs without Long Induced Paths

  • Aistis Atminas
  • Vadim V. Lozin
  • Igor Razgon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an additional parameter s and assuming that G does not have induced (i.e. chordless) paths of length s. We prove that under this parameterization the problem becomes fixed-parameter linear. The main tool in our proof is a Ramsey-type theorem stating that a graph with a long (not necessarily induced) path contains either a long induced path or a large biclique.


Bipartite Graph Common Neighbor Linear Time Algorithm Pigeonhole Principle Ramsey Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arbib, C., Mosca, R.: Polynomial algorithms for special cases of the balanced complete bipartite subgraph problem. J. Combin. Math. Combin. Comput. 39, 3–22 (1999)MathSciNetGoogle Scholar
  2. 2.
    Bascó, G., Tuza, Z.: A characterization of graphs without long induced paths. J. of Graph Theory 14, 455–464 (1990)zbMATHCrossRefGoogle Scholar
  3. 3.
    Binkele-Raible, D., Fernau, H., Gaspers, S., Liedloff, M.: Exact exponential-time algorithms for finding bicliques. Inf. Process. Lett. 111(2), 64–67 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Broersma, H., Golovach, P.A., Paulusma, D., Song, J.: Updating the complexity status of coloring graphs without a fixed induced linear forest. Theor. Comput. Sci. 414(1), 9–19 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bulatov, A.A., Marx, D.: Constraint Satisfaction Parameterized by Solution Size. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 424–436. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Chen, Y., Thurley, M., Weyer, M.: Understanding the Complexity of Induced Subgraph Isomorphisms. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 587–596. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 1–16. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    Demaine, E., Gutin, G.Z., Marx, D., Stege, U.: 07281 open problems – structure theory and FPT algorithmcs for graphs, digraphs and hypergraphs. In: Demaine, E., Gutin, G.Z., Marx, D., Stege, U. (eds.) Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs, Dagstuhl, Germany. Dagstuhl Seminar Proceedings, vol. (07281), Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2007)Google Scholar
  10. 10.
    Diestel, R.: Graph Theory, 3rd edn. Springer (2005)Google Scholar
  11. 11.
    Dong, J.: Some results on graphs without long induced paths. J. of Graph Theory 22, 23–28 (1996)zbMATHCrossRefGoogle Scholar
  12. 12.
    Feige, U., Kogan, S.: Hardness of approximation of the balanced complete bipartite subgraph problem. Technical Report MCS04-04, Weizmann Institute of Science (2004)Google Scholar
  13. 13.
    Fellows, M., Gaspers, S., Rosamond, F.: Multivariate complexity theory. In: Blum, E.K., Aho, A.V. (eds.) Computer Science: The Hardware, Software and Heart of It, pp. 269–293. Springer (2011)Google Scholar
  14. 14.
    Fellows, M.R., Langston, M.A.: On search, decision and the efficiency of polynomial-time algorithms (extended abstract). In: STOC, pp. 501–512 (1989)Google Scholar
  15. 15.
    Golovach, P.A., Paulusma, D., Song, J.: Coloring Graphs without Short Cycles and Long Induced Paths. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 193–204. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Johnson, D.S.: The NP-completeness column: An ongoing guide. J. Algorithms 8(3), 438–448 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Korobitsyn, D.: On the complexity of determining the domination number in monogenic classes of graphs. Diskretnaya Matematika 2(3), 90–96 (1990) (in Russian)zbMATHGoogle Scholar
  18. 18.
    Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of Coloring Graphs without Forbidden Induced Subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Kühn, D., Osthus, D.: Induced subdivisions in K\(_{\mbox{s, s}}\)-free graphs of large average degree. Combinatorica 24(2), 287–304 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Le, V.B., Randerath, B., Schiermeyer, I.: On the complexity of 4-coloring graphs without long induced paths. Theor. Comput. Sci. 389(1-2), 330–335 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lozin, V.V., Mosca, R.: Maximum independent sets in subclasses of P\(_{\mbox{5}}\)-free graphs. Inf. Process. Lett. 109(6), 319–324 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lozin, V.V., Rautenbach, D.: Some results on graphs without long induced paths. Inf. Process. Lett. 88(4), 167–171 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Murphy, O.J.: Computing independent sets in graphs with large girth. Discrete Applied Mathematics 35(2), 167–170 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Randerath, B., Schiermeyer, I.: 3-colorability in P for P\(_{\mbox{6}}\)-free graphs. Discrete Applied Mathematics 136(2-3), 299–313 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Woeginger, G.J., Sgall, J.: The complexity of coloring graphs without long induced paths. Acta Cybernetica 15(1), 107–117 (2001)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aistis Atminas
    • 1
  • Vadim V. Lozin
    • 1
  • Igor Razgon
    • 2
  1. 1.DIMAP and Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of Computer ScienceUniversity of LeicesterLeicesterUK

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