FPT Algorithms for Domination in Biclique-Free Graphs

  • Jan Arne Telle
  • Yngve Villanger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)


A class of graphs is said to be biclique-free if there is an integer t such that no graph in the class contains K t,t as a subgraph. Large families of graph classes, such as any nowhere dense class of graphs or d-degenerate graphs, are biclique-free. We show that various domination problems are fixed-parameter tractable on biclique-free classes of graphs, when parameterizing by both solution size and t. In particular, the problems k -Dominating Set, Connected k -Dominating Set, Independent k -Dominating Set and Minimum Weight k -Dominating Set are shown to be FPT, when parameterized by t + k, on graphs not containing K t,t as a subgraph. With the exception of Connected k -Dominating Set all described algorithms are trivially linear in the size of the input graph.


Equivalence Class Planar Graph Input Graph Polynomial Kernel Graph Class 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abreu, M., Funk, M., Labbate, D., Napolitano, V.: A family of regular graphs of girth 5. Discrete Mathematics 308, 1810–1815 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alon, N., Gutner, S.: Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica 54, 544–556 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bollobás, B.: Extremal graph theory, Dover Books on Mathematics. Dover Publications (2004)Google Scholar
  5. 5.
    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
  6. 6.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Kernelization Hardness of Connectivity Problems in d-Degenerate Graphs. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 147–158. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Dawar, A., Kreutzer, S.: Domination problems in nowhere-dense classes. In: Kannan, R., Kumar, K.N. (eds.). FSTTCS LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, vol. 4, pp. 157–168 (2009)Google Scholar
  8. 8.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and h-minor-free graphs. J. ACM 52, 866–893 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through Colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I, LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  11. 11.
    Ellis, J.A., Fan, H., Fellows, M.R.: The dominating set problem is fixed parameter tractable for graphs of bounded genus. J. Algorithms 52, 152–168 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  14. 14.
    Golovach, P.A., Villanger, Y.: Parameterized Complexity for Domination Problems on Degenerate Graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 195–205. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)zbMATHCrossRefGoogle Scholar
  16. 16.
    Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289, 997–1008 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kratsch, S.: Co-nondeterminism in compositions: a kernelization lower bound for a ramsey-type problem. In: Rabani, Y. (ed.) SODA, pp. 114–122. SIAM (2012)Google Scholar
  18. 18.
    Misra, N., Philip, G., Raman, V., Saurabh, S., Sikdar, S.: FPT Algorithms for Connected Feedback Vertex Set. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 269–280. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Nesetril, J., de Mendez, P.O.: Structural properties of sparse graphs. Building bridges between Mathematics and Computer Science, vol. 19. Springer (2008)Google Scholar
  20. 20.
    Nesetril, J., de Mendez, P.O.: First order properties on nowhere dense structures. J. Symb. Log. 75, 868–887 (2010)zbMATHCrossRefGoogle Scholar
  21. 21.
    Nesetril, J., de Mendez, P.O.: On nowhere dense graphs. Eur. J. Comb. 32, 600–617 (2011)zbMATHCrossRefGoogle Scholar
  22. 22.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press (2006)Google Scholar
  23. 23.
    Philip, G., Raman, V., Sikdar, S.: Solving Dominating Set in Larger Classes of Graphs: FPT Algorithms and Polynomial Kernels. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 694–705. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  24. 24.
    Raman, V., Saurabh, S.: Short cycles make w -hard problems hard: Fpt algorithms for w -hard problems in graphs with no short cycles. Algorithmica 52, 203–225 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jan Arne Telle
    • 1
  • Yngve Villanger
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway

Personalised recommendations