Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings

TL;DR

This paper characterizes bichromatic point sets with a unique non-crossing perfect matching, providing geometric descriptions and an efficient algorithm to identify such sets in the plane.

Contribution

It introduces a characterization of point sets with exactly one non-crossing matching and presents an O(nlogn) algorithm to verify this property.

Findings

Characterization of bichromatic sets with unique non-crossing matchings

Geometric descriptions of such point sets

Efficient O(nlogn) algorithm for verification

Abstract

Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing matching. We give several geometric descriptions of such sets, and find an O(nlogn) algorithm that checks whether a given bichromatic set has this property.