Untangling a polygon

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((nlogn)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. © Springer-Verlag Berlin Heidelberg 2002.