Connected Matchings

TL;DR

The paper proves that for any set of points in the plane, there exists a connected straight-line matching with a guaranteed minimum number of edges, and provides an efficient algorithm to find such matchings.

Contribution

It introduces a new lower bound for connected matchings in planar point sets and presents an $O(n \,\log n)$ algorithm to compute them, also exploring a colored variant.

Findings

Established a lower bound of (5n+1)/27 edges for connected matchings.

Provided an $O(n \log n)$ algorithm to find such matchings.

Showed that some point sets have at most ceil((n-1)/3) edges in connected matchings.

Abstract

We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, and such a matching can be computed in $O(n \log n)$ time. As an upper bound, we show that for some planar point sets in general position the largest matching whose segments form a connected set has $\lceil \frac{n-1}{3}\rceil$ edges. We also consider a colored version, where each edge of the matching should connect points with different colors.