Edge-Removal and Non-Crossing Perfect Matchings
Aviv Sheyn, Ran J. Tessler
TL;DR
This work analyzes how many edges can be removed from a complete geometric graph on $2n$ points while still guaranteeing a non-crossing perfect matching. It proves a tight upper bound of $h(G)=2n-2$ in general, and shows that when the hull boundary contains at most $n+1$ points one can remove $n$ edges and preserve a matching, using structural lemmas and induction. In the random setting, the authors demonstrate that with high probability one can remove $n+\Theta(n/\log n)$ edges and still retain a non-crossing perfect matching, by decomposing the plane into convex regions and applying convex-set probabilistic bounds. The methods combine obstruction-structure analysis (trees with hull-boundary vertices), Ham Sandwich partitions, and random convex-set theory to derive sharp thresholds for edge-removal robustness of non-crossing matchings, with implications for geometric computation and probabilistic geometry.
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 $n + Θ(n/log (n))$ 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.