Untangling a Polygon
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.
KeywordsPlanar Graph Random Permutation Simple Polygon Closed Polygon Black Edge