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)


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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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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