Obtaining Planarity by Contracting Few Edges

  • Petr A. Golovach
  • Pim van ’t Hof
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NP-complete. We show that it is fixed-parameter tractable when parameterized by k.

Keywords

Planar Graph Chordal Graph Outer Face Interior 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abello, J., Klavík, P., Kratochvíl, J., Vyskočil, T.: Matching and ℓ-subgraph contractibility to planar graphs. Manuscript, arXiv:1204.6070 (2012)Google Scholar
  2. 2.
    Asano, T., Hirata, T.: Edge-contraction problems. Journal of Computer and System Sciences 26, 197–208 (1983)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bouchet, A.: Orientable and nonorientable genus of the complete bipartite graph. Journal of Combinatorial Theory, Series B 24, 24–33 (1978)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Golovach, P.A., Kamiński, M., Paulusma, D., Thilikos, D.M.: Increasing the Minimum Degree of a Graph by Contractions. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 67–79. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Gu, Q.-P., Tamaki, H.: Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n 1 + ε) time. Theoretical Computer Science 412, 4100–4109 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Heggernes, P., van ’t Hof, P., Lévêque, B., Paul, C.: Contracting chordal graphs and bipartite graphs to paths and trees. In: Proc. LAGOS 2011. ENDM, vol. 37, pp. 87–92 (2011)Google Scholar
  8. 8.
    Heggernes, P., van ’t Hof, P., Lévêque, B., Lokshtanov, D., Paul, C.: Contracting Graphs to Paths and Trees. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 55–66. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Heggernes, P., van ’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. Algorithmica (to appear), doi: 10.1007/s00453-012-9670-2Google Scholar
  10. 10.
    Kamiński, M., Paulusma, D., Thilikos, D.M.: Contractions of Planar Graphs in Polynomial Time. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 122–133. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Kamiński, M., Thilikos, D.M.: Contraction checking in graphs on surfaces. In: Proc. STACS, pp. 182–193 (2012)Google Scholar
  12. 12.
    Kawarabayashi, K.: Planarity allowing few error vertices in linear time. In: Proc. FOCS, pp. 639–648 (2009)Google Scholar
  13. 13.
    Kawarabayashi, K., Reed, B.A.: Computing crossing number in linear time. In: Proc. STOC, pp. 382–390 (2007)Google Scholar
  14. 14.
    Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. Manuscript, arXiv:1110.4765 (2012)Google Scholar
  15. 15.
    Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62, 807–822 (2012)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Robertson, N., Seymour, P.D.: Quickly excluding a planar graph. Journal of Combinatorial Theory, Series B 62, 323–348 (1994)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Robertson, N., Seymour, P.D.: Graph minors XIII: The disjoint paths problem. Journal of Combinatorial Theory, Series B 63, 65–110 (1995)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Robertson, N., Seymour, P.D.: Graph minors XX: Wagner’s conjecture. Journal of Combinatorial Theory, Series B 92, 325–357 (2004)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph minors XXII: Irrelevant vertices in linkage problems. Journal of Combinatorial Theory, Series B 102, 530–563 (2012)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Pim van ’t Hof
    • 1
  • Daniël Paulusma
    • 2
  1. 1.Department of InformaticsUniversity of BergenNorway
  2. 2.School of Engineering and Computing SciencesDurham UniversityUK

Personalised recommendations