Article

Discrete & Computational Geometry

, Volume 28, Issue 4, pp 585-592

First online:

Untangling a Polygon

  • PachAffiliated withCourant Institute, New York University, New York, NY 10012, USA pach@cims.nyu.edu
  • , TardosAffiliated withCourant Institute, New York University, New York, NY 10012, USA pach@cims.nyu.edu

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Abstract

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
$$n-\Omega(\sqrt{n})$$
steps. Some related questions are also considered.