Flip Distance to a Non-crossing Perfect Matching

A perfect straight-line matching M on a finite set P of points in the plane is a set of segments such that each point in P is an endpoint of exactly one segment. M is non-crossing if no two segments in M cross each other. Given a perfect straight-line matching M with at least one crossing, we can remove this crossing by a flip operation. The flip operation removes two crossing segments on a point set Q and adds two non-crossing segments to attain a new perfect matching M ′. It is well known that after a finite number of flips, a non-crossing matching is attained and no further flip is possible. However, prior to this work, no non-trivial upper bound on the number of flips was known. If g(n) (resp. k(n)) is the maximum length of the longest (resp. shortest) sequence of flips starting from any matching of size n, we show that g(n) = O(n) and g(n) = Ω(n) (resp. k(n) = O(n) and k(n) = Ω(n)). Van Leeuwen and Schoone showed with the same argument and the same definition of flip how to transform a Hamilton cycle to a non-crossing Hamilton cycle on a set of n points within O(n) flips [17]. Therefore, we do not consider the main result (our upper bound on g(n)) as a new contribution, because the used technique is exactly the same. We want to use these proceedings to draw attention again on this old problem and hope to stimulate research that will close the gap between the upper and lower bound.