Discrete & Computational Geometry

, Volume 42, Issue 4, pp 542–569 | Cite as

Untangling a Planar Graph

  • Xavier Goaoc
  • Jan Kratochvíl
  • Yoshio Okamoto
  • Chan-Su Shin
  • Andreas Spillner
  • Alexander Wolff
Open Access
Article

Abstract

A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem.

Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least \(\sqrt{((\log n)-1)/\log\log n}\) vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least \(\sqrt{n/2}\) vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δG of G with \(\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1\) and an n-vertex outerplanar graph H and a drawing δH of H with \(\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1\) . Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

Keywords

Graph drawing Straight-line drawing Planarity NP-hardness Hardness of approximation Moving vertices Untangling Point-set embeddability 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Xavier Goaoc
    • 1
  • Jan Kratochvíl
    • 2
  • Yoshio Okamoto
    • 3
  • Chan-Su Shin
    • 4
  • Andreas Spillner
    • 5
  • Alexander Wolff
    • 6
  1. 1.LORIA–INRIA Grand EstNancyFrance
  2. 2.Department of Applied Mathematics and Institute of Theoretical Computer ScienceCharles UniversityPragueCzech Republic
  3. 3.Graduate School of Information Science and EngineeringTokyo Institute of TechnologyTokyoJapan
  4. 4.School of Electronics and Information EngineeringHankuk University of Foreign StudiesYonginKorea
  5. 5.School of Computing SciencesUniversity of East AngliaNorwichUK
  6. 6.Faculteit Wiskunde en InformaticaTechnische Universiteit EindhovenEindhovenThe Netherlands

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