Untangling a Polygon

  • János Pach
  • Gábor Tardos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)


The following problem was raised by M. Watanabe. Let P be a self-intersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P, i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n)2/3) vertices. On the other hand, every polygon P can be untangled in at most n − Ω(√n) steps. Some related questions are also considered.


Planar Graph Random Permutation Simple Polygon Closed Polygon Black Edge 
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.


  1. [ES35]
    P. Erdõs and G. Szekeres, A combinatorial problem in geometry, Compositio Mathematica 2 (1935), 463–470.MathSciNetGoogle Scholar
  2. [PSS94]
    J. Pach, F. Shahrokhi, and M. Szegedy, Applications of the crossing number, Proc. 10th ACM Symposium on Computational Geometry, 1994, 198–202. Also in: Algorithmica 16 (1996), 111–117.MathSciNetCrossRefGoogle Scholar
  3. [PW98]
    J. Pach and R. Wenger, Embedding planar graphs with fixed vertex locations, in: Graph Drawing’ 98 (Sue Whitesides, ed.), Lecture Notes in Computer Science 1547, Springer-Verlag, Berlin, 1998, 263–274.CrossRefGoogle Scholar
  4. [SV94]
    O. Sýkora and I. Vrťo, On VLSI layouts of the star graph and related networks, Integration, The VLSI Journal 17 (1994), 83–93.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • János Pach
    • 1
  • Gábor Tardos
    • 1
  1. 1.Rényi Institute of MathematicsHungarian Academy of SciencesHungary

Personalised recommendations