Edge-Removal and Non-Crossing Perfect Matchings

TL;DR

This paper investigates the maximum number of edges that can be removed from a complete geometric graph while still ensuring a non-crossing perfect matching exists, providing bounds for special and random point configurations.

Contribution

It establishes new bounds on edge removal in geometric graphs that guarantee the existence of non-crossing perfect matchings, including for convex hulls and random points.

Findings

In convex hulls with at most n+1 points, n edges can be removed.

For random points, with high probability, up to n + Θ(n/log n) edges can be removed.

Upper bounds are discussed for eliminating all non-crossing perfect matchings.

Abstract

We study the following problem - How many arbitrary edges can be removed from a complete geometric graph with 2n vertices such that the resulting graph always contains a perfect non-crossing matching? We first address the case where the boundary of the convex hull of the original graph contains at most $n + 1$ points. In this case we show that n edges can be removed, one more than the general case. In the second part we establish a lower bound for the case where the $2n$ points are randomly chosen. We prove that with probability which tends to 1, one can remove any $n + \Theta(n/log (n))$ edges but the residual graph will still contain a non-crossing perfect matching. We also discuss the upper bound for the number of arbitrary edges one must remove in order to eliminate all the non-crossing perfect matchings.