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Holes in Convex and Simple Drawings

Authors Helena Bergold , Joachim Orthaber , Manfred Scheucher , Felix Schröder

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Computational geometry
Keywords
  • simple topological graph
  • convexity hierarchy
  • k-gon
  • k-hole
  • empty k-cycle
  • Erdős-Szekeres theorem
  • Empty Hexagon theorem
  • planar point set
  • pseudolinear drawing

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Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős-Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of k-holes for simple drawings and study their existence with respect to the convexity hierarchy. We present a family of simple drawings without 4-holes and prove a generalization of Gerken’s empty hexagon theorem for convex drawings. A crucial intermediate step will be the structural investigation of pseudolinear subdrawings in convex drawings.

Cite As Get BibTex

Helena Bergold, Joachim Orthaber, Manfred Scheucher, and Felix Schröder. Holes in Convex and Simple Drawings. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 5:1-5:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.5

Author Details

Helena Bergold
  • Institute of Computer Science, Freie Universität Berlin, Germany
  • TUM School of Computation, Information and Technology, Technische Universität München, Germany
Joachim Orthaber
  • Institute of Software Technology, Graz University of Technology, Austria
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany
Felix Schröder
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  • Institute of Mathematics, Technische Universität Berlin, Germany

Funding

  • Bergold, Helena: Supported by DFG-Research Training Group "Facets of Complexity" (DFG-GRK 2434).
  • Orthaber, Joachim: Supported by the Austrian Science Fund (FWF) grant W1230.
  • Scheucher, Manfred: Supported by DFG Grant SCHE 2214/1-1.
  • Schröder, Felix: Supported by the GAČR Grant no. 23-04949X.

References

  1. Oswin Aichholzer, Ruy Fabila-Monroy, Hernán González-Aguilar, Thomas Hackl, Marco A. Heredia, Clemens Huemer, Jorge Urrutia, Pavel Valtr, and Birgit Vogtenhuber. On k-gons and k-holes in point sets. Computational Geometry, 48(7):528-537, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.12.007.
  2. Oswin Aichholzer, Thomas Hackl, Clemens Huemer, Ferran Hurtado, and Birgit Vogtenhuber. Large bichromatic point sets admit empty monochromatic 4-gons. SIAM Journal on Discrete Mathematics, 23(4), 2010. URL: https://doi.org/10.1137/090767947.
  3. Oswin Aichholzer, Thomas Hackl, Alexander Pilz, Pedro Ramos, Vera Sacristán, and Birgit Vogtenhuber. Empty triangles in good drawings of the complete graph. Graphs and Combinatorics, 31:335-345, 2015. URL: https://doi.org/10.1007/s00373-015-1550-5.
  4. Alan Arroyo, Dan McQuillan, R. Bruce Richter, and Gelasio Salazar. Levi’s Lemma, pseudolinear drawings of K_n, and empty triangles. Journal of Graph Theory, 87(4):443-459, 2018. URL: https://doi.org/10.1002/jgt.22167.
  5. Alan Arroyo, Dan McQuillan, R. Bruce Richter, and Gelasio Salazar. Convex drawings of the complete graph: topology meets geometry. Ars Mathematica Contemporanea, 22(3), 2022. URL: https://doi.org/10.26493/1855-3974.2134.ac9.
  6. Alan Arroyo, R. Bruce Richter, and Matthew Sunohara. Extending drawings of complete graphs into arrangements of pseudocircles. SIAM Jorunal on Discrete Mathematics, 35(2):1050-1076, 2021. URL: https://doi.org/10.1137/20M1313234.
  7. Martin Balko, Radoslav Fulek, and Jan Kynčl. Crossing Numbers and Combinatorial Characterization of Monotone Drawings of K_n. Discrete & Computational Geometry, 53(1):107-143, 2015. URL: https://doi.org/10.1007/s00454-014-9644-z.
  8. Imre Bárány and Zoltán Füredi. Empty simplices in Euclidean space. Canadian Mathematical Bulletin, 30(4):436-445, 1987. URL: https://doi.org/10.4153/cmb-1987-064-1.
  9. Helena Bergold, Stefan Felsner, Meghana M. Reddy, Joachim Orthaber, and Manfred Scheucher. Plane Hamiltonian cycles in convex drawings. In Proceedings of the 40th International Symposium on Computational Geometry (SoCG 2024), pages 18:1-18:16, 2024. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.18.
  10. Helena Bergold, Stefan Felsner, Manfred Scheucher, Felix Schröder, and Raphael Steiner. Topological Drawings meet Classical Theorems from Convex Geometry. Discrete & Computational Geometry, 2022. URL: https://doi.org/10.1007/s00454-022-00408-6.
  11. Helena Bergold, Joachim Orthaber, Manfred Scheucher, and Felix Schröder. Holes in convex and simple drawings. arXiv, 2024. URL: https://arxiv.org/abs/2409.01723.
  12. Peter Brass, William O. J. Moser, and János Pach. Research Problems in Discrete Geometry. Springer, 2005. URL: https://doi.org/10.1007/0-387-29929-7.
  13. Olivier Devillers, Ferran Hurtado, Gyula Károlyi, and Carlos Seara. Chromatic variants of the Erdős-Szekeres theorem on points in convex position. Computational Geometry, 26(3):193-208, 2003. URL: https://doi.org/10/bb9h93.
  14. Paul Erdős. Some problems on elementary geometry. Australian Mathematical Society, 2:2-3, 1978. URL: https://users.renyi.hu/~p_erdos/1978-44.pdf.
  15. Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935. URL: http://www.renyi.hu/~p_erdos/1935-01.pdf.
  16. Alfredo García, Alexander Pilz, and Javier Tejel. On plane subgraphs of complete topological drawings. Ars Mathematica Contemporanea, 20(1):69-87, 2021. URL: https://doi.org/10.26493/1855-3974.2226.e93.
  17. Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Empty triangles in generalized twisted drawings of K_n. In Graph Drawing and Network Visualization, volume 13764 of LNCS, pages 40-48. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22203-0_4.
  18. Tobias Gerken. Empty Convex Hexagons in Planar Point Sets. Discrete & Computational Geometry, 39(1):239-272, 2008. URL: https://doi.org/10.1007/s00454-007-9018-x.
  19. Heiko Harborth. Empty triangles in drawings of the complete graph. Discrete Mathematics, 191(1-3):109-111, 1998. URL: https://doi.org/10.1016/S0012-365X(98)00098-3.
  20. Marijn J. H. Heule and Manfred Scheucher. Happy ending: An empty hexagon in every set of 30 points. In Bernd Finkbeiner and Laura Kovács, editors, Proc. 30th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'24), volume 14570 of LNCS, pages 61-80. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-57246-3_5.
  21. Joseph D. Horton. Sets with no empty convex 7-gons. Canadian Mathematical Bulletin, 26:482-484, 1983. URL: https://doi.org/10.4153/CMB-1983-077-8.
  22. Jiří Matoušek. Lectures on Discrete Geometry. Springer, 2002. URL: https://doi.org/10.1007/978-1-4613-0039-7.
  23. Carlos M. Nicolás. The Empty Hexagon Theorem. Discrete & Computational Geometry, 38(2):389-397, 2007. URL: https://doi.org/10.1007/s00454-007-1343-6.
  24. János Pach, József Solymosi, and Géza Tóth. Unavoidable configurations in complete topological graphs. Discrete & Computational Geometry, 30(2):311-320, 2003. URL: https://doi.org/10.1007/s00454-003-0012-9.
  25. Nabil H. Rafla. The good drawings D_n of the complete graph K_n. PhD thesis, McGill University, Montreal, 1988. URL: https://escholarship.mcgill.ca/concern/theses/x346d4920.
  26. Gerhard Ringel. Extremal problems in the theory of graphs. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), volume 8590. Publ. House Czechoslovak Acad. Sci Prague, 1964. Google Scholar
  27. Manfred Scheucher. A SAT Attack on Erdős-Szekeres Numbers in ℝ^d and the Empty Hexagon Theorem. Computing in Geometry and Topology, 2(1):2:1-2:13, 2023. URL: https://doi.org/10.57717/cgt.v2i1.12.
  28. Andrew Suk and Ji Zeng. Unavoidable patterns in complete simple topological graphs. In Graph Drawing and Network Visualization, volume 13764 of LNCS, pages 3-15. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22203-0_1.
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