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Crossing and Non-Crossing Families

Authors Todor Antić , Martin Balko , Birgit Vogtenhuber

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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatorics
Keywords
  • crossing family
  • non-crossing family
  • geometric graph

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Abstract

For a finite set P of points in the plane in general position, a crossing family of size k in P is a collection of k line segments with endpoints in P that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of n points in the plane in general position. It is widely believed that this size should be linear in n.
Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets P that do not contain a non-crossing family of size m, which is a collection of 4 disjoint subsets P₁, P₂, P₃, and P₄ of P, each containing m points of P, such that for every choice of 4 points p_i ∈ P_i, the set {p₁,p₂,p₃,p₄} is such that p₄ is in the interior of the triangle formed by p₁,p₂,p₃. We prove that, for every m ∈ ℕ, each set P of n points in the plane in general position contains either a crossing family of size n/2^{O(√{log{m}})} or a non-crossing family of size m, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time O(nm^{1+o(1)}). We also prove that a crossing family of size Ω(n/m) or a non-crossing family of size m in P can be found in expected time O(n).

Cite As Get BibTex

Todor Antić, Martin Balko, and Birgit Vogtenhuber. Crossing and Non-Crossing Families. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.19

Author Details

Todor Antić
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Birgit Vogtenhuber
  • Institute of Algorithms and Theory, Graz University of Technology, Austria

Funding

  • Antić, Todor: supported by grant no. 23-04949X of the Czech Science Foundation (GAČR).
  • Balko, Martin: supported by grant no. 23-04949X of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE 24/SCI/008).

Acknowledgements

We would like to thank the organizers of the the 19th European Research Week on Geometric Graphs (GGWeek 2024) where this research was initiated. We would also like to thank Joachim Orthaber, Bettina Speckmann, and Viola Mészáros for interesting discussions about the problem during the early stages of the research. We also thank the anonymous reviewers for their valuable comments and suggestions, which helped improve the quality of this work.

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