Document Open Access Logo

Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

Authors Adrian Dumitrescu , János Pach

PDF

File

Thumbnail PDF
PDF
LIPIcs.GD.2024.9.pdf
  • Filesize: 0.63 MB
  • 10 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Convexity
  • complete geometric Graph
  • crossing Family
  • plane Subgraph

Metrics

Abstract

A complete geometric graph consists of a set P of n points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant c < 1, such that every complete geometric graph on n points can be partitioned into at most cn plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set P is dense, which means that the ratio between the maximum and the minimum distances in P is of the order of Θ(√n).

Cite As Get BibTex

Adrian Dumitrescu and János Pach. Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 9:1-9:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.9

Author Details

Adrian Dumitrescu
  • Algoresearch L.L.C., Milwaukee, WI, USA
János Pach
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • EPFL, Lausanne, Switzerland

Funding

Work on this paper was supported by ERC Advanced Grant no. 882971 "GeoScape." Work by the second named author was partially supported by NKFIH (National Research, Development and Innovation Office) grant K-131529.

Acknowledgements

We thank the anonymous reviewers for carefully reading the manuscript.

References

  1. Oswin Aichholzer, Jan Kyncl, Manfred Scheucher, Birgit Vogtenhuber, and Pavel Valtr. On crossing-families in planar point sets. Comput. Geom., 107:101899, 2022. URL: https://doi.org/10.1016/J.COMGEO.2022.101899.
  2. Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber. Edge partitions of complete geometric graphs. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 6:1-6:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.SOCG.2022.6.
  3. Jin Akiyama and Mikio Kano. Path factors of a graph. In Frank Harary and John S. Maybee, editors, Graph and Applications, Proceedings of the First Colorado Symposium on Graph Theory, pages 1-21. Wiley, New York, 1985. Google Scholar
  4. Gabriela Araujo, Adrian Dumitrescu, Ferran Hurtado, Marc Noy, and Jorge Urrutia. On the chromatic number of some geometric type kneser graphs. Comput. Geom., 32(1):59-69, 2005. URL: https://doi.org/10.1016/J.COMGEO.2004.10.003.
  5. Boris Aronov, Paul Erdös, Wayne Goddard, Daniel J. Kleitman, Michael Klugerman, János Pach, and Leonard J. Schulman. Crossing families. Comb., 14(2):127-134, 1994. URL: https://doi.org/10.1007/BF01215345.
  6. Imre Bárány and Maria Prodromou. On maximal convex lattice polygons inscribed in a plane convex set. Israel Journal of Mathematics, 154(1):337-360, 2006. Google Scholar
  7. Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, and David R. Wood. Partitions of complete geometric graphs into plane trees. Comput. Geom., 34(2):116-125, 2006. URL: https://doi.org/10.1016/J.COMGEO.2005.08.006.
  8. Peter Brass, William O. J. Moser, and János Pach. Research Problems in Discrete Geometry. Springer, 2005. Google Scholar
  9. Michael B. Dillencourt, David Eppstein, and Daniel S. Hirschberg. Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5-17, 2000. URL: https://doi.org/10.7155/JGAA.00023.
  10. Herbert Edelsbrunner, Pavel Valtr, and Emo Welzl. Cutting dense point sets in half. Discret. Comput. Geom., 17(3):243-255, 1997. URL: https://doi.org/10.1007/PL00009291.
  11. Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio mathematica, 2:463-470, 1935. Google Scholar
  12. Paul Erdős and George Szekeres. On some extremum problems in elementary geometry. Ann. Univ. Sci. Budapest. Eötvös Sect. Math, 3(4):53-62, 1960. Google Scholar
  13. William S. Evans and Noushin Saeedi. On problems related to crossing families. CoRR, abs/1906.00191, 2019. URL: https://arxiv.org/abs/1906.00191.
  14. Hillel Furstenberg and Yitzhak Katznelson. An ergodic Szemerédi theorem for commuting transformations. Journal d’Analyse Mathématique, 34(1):275-291, 1978. Google Scholar
  15. István Kovács and Géza Tóth. Dense point sets with many halving lines. Discret. Comput. Geom., 64(3):965-984, 2020. URL: https://doi.org/10.1007/S00454-019-00080-3.
  16. Jirí Matousek. Lectures on Discrete Geometry, volume 212 of Graduate texts in mathematics. Springer, 2002. Google Scholar
  17. Jaroslav Nešetřil. Ramsey theory. In Ron Graham, Martin Grötschel, and László Lovász, editors, Handbook of Combinatorics, Vol. II, chapter 25, pages 1331-1403. Elsevier, Amsterdam, first edition, 1995. Google Scholar
  18. Johannes Obenaus and Joachim Orthaber. Edge partitions of complete geometric graphs (part 1). CoRR, abs/2108.05159, 2021. URL: https://arxiv.org/abs/2108.05159.
  19. János Pach, Natan Rubin, and Gábor Tardos. Planar point sets determine many pairwise crossing segments. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 1158-1166. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316328.
  20. János Pach, Morteza Saghafian, and Patrick Schnider. Decomposition of geometric graphs into star-forests. In Michael A. Bekos and Markus Chimani, editors, Graph Drawing and Network Visualization - 31st International Symposium, GD 2023, Isola delle Femmine, Palermo, Italy, September 20-22, 2023, Revised Selected Papers, Part I, volume 14465 of Lecture Notes in Computer Science, pages 339-346. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49272-3_23.
  21. János Pach and József Solymosi. Halving lines and perfect cross-matchings. Contemporary Mathematics, 223:245-250, 1999. Google Scholar
  22. DHJ Polymath. A new proof of the density Hales-Jewett theorem. Annals of Mathematics, pages 1283-1327, 2012. Google Scholar
  23. Andrew Suk. On the Erdős-Szekeres convex polygon problem. Journal of the American Mathematical Society, 30:1047-1053, 2017. URL: https://doi.org/10.1090/jams/869.
  24. Endre Szemerédi. On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27(1):199-245, 1975. Google Scholar
  25. László Fejes Tóth, Gábor Fejes Tóth, and Włodzimierz Kuperberg. Lagerungen. Springer, 2023. Google Scholar
  26. Pavel Valtr. Convex independent sets and 7-holes in restricted planar point sets. Discret. Comput. Geom., 7:135-152, 1992. URL: https://doi.org/10.1007/BF02187831.
  27. Pavel Valtr. Planar point sets with bounded ratios of distances. Dissertation, Freie Univ., 1994. Google Scholar
  28. Pavel Valtr. Lines, line-point incidences and crossing families in dense sets. Comb., 16(2):269-294, 1996. URL: https://doi.org/10.1007/BF01844852.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact