Journal of Graph Algorithms and Applications

Special Issue on Selected Papers from the 15th International Conference and Workshops on Algorithms and Computation, WALCOM 2021

DOI: 10.7155/jgaa.00591

On Compatible Matchings

Oswin Aichholzer, Alan Arroyo, Zuzana Masárová, Irene Parada, Daniel Perz, Alexander Pilz, Josef Tkadlec, and Birgit Vogtenhuber

Vol. 26, no. 2, pp. 225-240, 2022. Regular paper.

Abstract A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled sets of $n$ points in convex position there exists a compatible matching with $\lfloor \sqrt {2n+1} -1\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has $O(n^{2/(\ell+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of labelings of these point sets such that any compatible matching consists only of a single edge.