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MSOL Restricted Contractibility to Planar Graphs

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Part of the Lecture Notes in Computer Science book series (LNTCS,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).

Keywords

  • Planar Graph
  • Hexagonal Grid
  • Incidence Graph
  • Pendant Edge
  • Topological Embedding

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.

The first author acknowledges support of Special focus on Algorithmic Foundations of the Internet, NSF grant #CNS-0721113 and mgvis.com http://mgvis.com . The latter three authors acknowledge support of ESF Eurogiga project GraDR as GAČR GIG/11/E023.

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Abello, J., Klavík, P., Kratochvíl, J., Vyskočil, T. (2012). MSOL Restricted Contractibility to Planar Graphs. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_19

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  • DOI: https://doi.org/10.1007/978-3-642-33293-7_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33292-0

  • Online ISBN: 978-3-642-33293-7

  • eBook Packages: Computer ScienceComputer Science (R0)