MSOL Restricted Contractibility to Planar Graphs

  • James Abello
  • Pavel Klavík
  • Jan Kratochvíl
  • Tomáš Vyskočil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7535)

Abstract

We study the computational complexity of graph planarization via edge contraction. The problem Contract asks whether there exists a set S of at most k edges that when contracted produces a planar graph. We give an FPT algorithm in time \(\mathcal{O}(n^2 f(k))\) which solves a more general problem P-RestrictedContract in which S has to satisfy in addition a fixed inclusion-closed MSOL formula P.

For different formulas P we get different problems. As a specific example, we study the ℓ-subgraph contractability problem in which edges of a set S are required to form disjoint connected subgraphs of size at most ℓ. This problem can be solved in time \(\mathcal{O}(n^2 f'(k,l))\) using the general algorithm. We also show that for ℓ ≥ 2 the problem is NP-complete. And it remains NP-complete when generalized for a fixed genus (instead of planar graphs).

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References

  1. 1.
    Asano, T., Hirata, T.: Edge-Contraction Problems. Journal Comput. System Sci. 26, 197–208 (1983)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small tree-width. SIAM Journal on Computing 25, 1305–1317 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Courcelle, B.: The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. ITA 26, 257–286 (1992)MathSciNetMATHGoogle Scholar
  4. 4.
    van’t Hof, P., Kamiński, M., Paulusma, D., Szeider, S., Thilikos, D.M.: On graph contractions and induced minors. Discrete Applied Mathematics 160, 799–809 (2012)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Golovach, P.A., van’t Hof, P., Paulusma, P.: Obtaining Planarity by Contracting Few Edges. CoRR, abs/1204.5113 (2012), http://arxiv.org/abs/1204.5113
  6. 6.
    Abello, J., Klavík, P., Kratochvíl, J., Vyskočil, T.: Matching and ℓ-Subgraph Contractibility to Planar Graphs. CoRR, abs/1204.6070 (2012), http://arxiv.org/abs/1204.6070
  7. 7.
    Fellows, M.R., Kratochvíl, J., Middendorf, M., Pfeiffer, F.: The Complexity of Induced Minors and Related Problems. Algorithmica 13(3), 266–282 (1995)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Alg. Discr. Meth. 4(3), 312–316 (1983)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Grohe, M.: Computing crossing numbers in quadratic time. J. Comput. Syst. Sci. 68(2), 285–302 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hopcroft, J.E., Tarjan, R.E.: Efficient Planarity Testing. J. ACM (JACM) 21(4), 549–568 (1974)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Heggernes, P., van’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. In: Proc. FSTTCS 2011, pp. 217–228 (2011)Google Scholar
  12. 12.
    Marx, D., Schlotter, I.: Obtaining a Planar Graph by Vertex Deletion. Algorithmica 62, 807–822 (2012)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bojan, M., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University PressGoogle Scholar
  14. 14.
    Perkovic̀, L., Reed, B.: An improved algorithm for finding tree decompositions of small width. International Journal of Foundations of Computer Science 11(3), 365–371 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B 63, 65–110 (1995)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • James Abello
    • 1
  • Pavel Klavík
    • 2
  • Jan Kratochvíl
    • 2
  • Tomáš Vyskočil
    • 2
  1. 1.DIMACS Center for Discrete Mathematics and Theorethical Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

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