Connected Matchings
Oswin Aichholzer, Sergio Cabello, Viola M\'esz\'aros, Patrick, Schnider, Jan Soukup

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 points in the plane in general position has a straight-line matching with at least edges whose segments form a connected set, and such a matching can be computed in 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 edges. We also consider a colored version, where each edge of the matching should connect points with different colors.
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Taxonomy
TopicsAdvanced Graph Theory Research
