@article{13969,
  abstract     = {Bundling crossings is a strategy which can enhance the readability
of graph drawings. In this paper we consider good drawings, i.e., we require that
any two edges have at most one common point which can be a common vertex or a
crossing. Our main result is that there is a polynomial-time algorithm to compute an
8-approximation of the bundled crossing number of a good drawing with no toothed
hole. In general the number of toothed holes has to be added to the 8-approximation.
In the special case of circular drawings the approximation factor is 8, this improves
upon the 10-approximation of Fink et al. [14]. Our approach also works with the same
approximation factor for families of pseudosegments, i.e., curves intersecting at most
once. We also show how to compute a 9/2-approximation when the intersection graph of
the pseudosegments is bipartite and has no toothed hole.},
  author       = {Arroyo Guevara, Alan M and Felsner, Stefan},
  issn         = {1526-1719},
  journal      = {Journal of Graph Algorithms and Applications},
  number       = {6},
  pages        = {433--457},
  publisher    = {Brown University},
  title        = {{Approximating the bundled crossing number}},
  doi          = {10.7155/jgaa.00629},
  volume       = {27},
  year         = {2023},
}

@article{11938,
  abstract     = {A matching is compatible to two or more labeled point sets of size n with labels {1, . . . , 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 ⌊√2n + 1 − 1⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ). As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(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.},
  author       = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
  issn         = {1526-1719},
  journal      = {Journal of Graph Algorithms and Applications},
  number       = {2},
  pages        = {225--240},
  publisher    = {Brown University},
  title        = {{On compatible matchings}},
  doi          = {10.7155/jgaa.00591},
  volume       = {26},
  year         = {2022},
}