The Parameterized Complexity of Graph Cyclability

  • Petr A. Golovach
  • Marcin Kamiński
  • Spyridon Maniatis
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)


The cyclability of a graph is the maximum integer k for which every k vertices lie on a cycle. The algorithmic version of the problem, given a graph G and a non-negative integer k, decide whether the cyclability of G is at least k, is NP-hard. We prove that this problem, parameterized by k, is coW[1]-hard. We give an FPT algorithm for planar graphs that runs in time \(2^{2^{O(k^2\log k)}}\cdot n^2\). Our algorithm is based on a series of graph theoretical results on cyclic linkages in planar graphs.


Planar Graph Parameterized Complexity Tree Decomposition Colored Vertex Graph Minor 
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.
    Adler, I., Dorn, F., Fomin, F.V., Sau, I., Thilikos, D.M.: Fast minor testing in planar graphs. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 97–109. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Adler, I., Kolliopoulos, S.G., Krause, P.K., Lokshtanov, D., Saurabh, S., Thilikos, D.: Tight bounds for linkages in planar graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 110–121. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Adler, I., Kolliopoulos, S.G., Thilikos, D.M.: Planar disjoint-paths completion. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 80–93. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Aldred, R.E., Bau, S., Holton, D.A., McKay, B.D.: Cycles through 23 vertices in 3-connected cubic planar graphs. Graphs and Combinatorics 15(4), 373–376 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dirac, G.A.: In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen. Math. Nachr. 22, 61–85 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefGoogle Scholar
  8. 8.
    Downey, R., Fellows, M.: Fixed-parameter tractability and completeness. III. Some structural aspects of the W hierarchy. In: Complexity Theory, pp. 191–225. Cambridge Univ. Press, Cambridge (1993)Google Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. In: 21st Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, MB, vol. 87, pp. 161–178 (1992)Google Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. I. Basic results. SIAM J. Comput. 24(4), 873–921 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science 141(1-2), 109–131 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Flandrin, E., Li, H., Marczyk, A., Woźniak, M.: A generalization of dirac’s theorem on cycles through K vertices in K-connected graphs. Discrete Mathematics 307(7), 878–884 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  14. 14.
    Fomin, F.V., Golovach, P.A., Thilikos, D.M.: Contraction obstructions for treewidth. J. Comb. Theory, Ser. B 101(5), 302–314 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and intractability. W. H. Freeman and Co. (1979) a guide to the theory of NP-completeness, A Series of Books in the Mathematical SciencesGoogle Scholar
  16. 16.
    Grötschel, M.: Hypohamiltonian facets of the symmetric travelling salesman polytope. Zeitschrift für Angewandte Mathematik und Mechanik 58, 469–471 (1977)Google Scholar
  17. 17.
    Gu, Q.-P., Tamaki, H.: Improved bounds on the planar branchwidth with respect to the largest grid minor size. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 85–96. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Kawarabayashi, K.I.: An improved algorithm for finding cycles through elements. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 374–384. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Kawarabayashi, K., Wollan, P.: A shorter proof of the graph minor algorithm: the unique linkage theorem. In: FOCS 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 771–780 (2008)Google Scholar
  20. 20.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Habilitation thesis (September 2002)Google Scholar
  21. 21.
    Perkovic, L., Reed, B.A.: An improved algorithm for finding tree decompositions of small width. Int. J. Found. Comput. Sci. 11(3), 365–371 (2000)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Plummer, M., Győri, E.: A nine vertex theorem for 3-connected claw-free graphs. Studia Scientiarum Mathematicarum Hungarica 38(1), 233–244 (2001)Google Scholar
  23. 23.
    Robertson, N., Seymour, P.: Graph Minors. XXII. irrelevant vertices in linkage problems. Journal of Combinatorial Theory, Series B 102(2), 530–563 (2012), CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Robertson, N., Seymour, P.D.: Graph Minors. X. Obstructions to Tree-decomposition. J. Combin. Theory Series B 52(2), 153–190 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Robertson, N., Seymour, P.D.: Graph Minors. XIII. The disjoint paths problem. J. Combin. Theory, Ser. B 63(1), 65–110 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Robertson, N., Seymour, P.D.: Graph Minors. XXI. Graphs with unique linkages. J. Combin. Theory Ser. 99(3), 583–616 (2009), CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Watkins, M., Mesner, D.: Cycles and connectivity in graphs. Canad. J. Math. 19, 1319–1328 (1967)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Marcin Kamiński
    • 2
  • Spyridon Maniatis
    • 3
  • Dimitrios M. Thilikos
    • 3
    • 4
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Institute of Computer ScienceUniversity of WarsawWarsawPoland
  3. 3.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  4. 4.AlGCo project-team, CNRS, LIRMMMontpellierFrance

Personalised recommendations