Flip Distance to a Non-crossing Perfect Matching

Abstract

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 $M^{\prime }$. 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 $g(n)=O(n^3)$ and $g(n)=\Omega (n^2)$ (resp.~$k(n)=O(n^2)$ and $k(n)=\Omega (n)$).

References in corpus (3)

• Compatible Geometric Matchings
• Flip Graphs of Degree-Bounded (Pseudo-)Triangulations
• Flips in combinatorial pointed pseudo-triangulations with face degree at most four

Cited by in corpus (1)

• Flip Distance to some Plane Configurations