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Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks

Authors Mikkel Abrahamsen , Bartosz Walczak

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • geometric intersection graph
  • convex disk
  • homothet
  • similarity

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Abstract

For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^{hom}(A) and G^{sim}(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^{hom}(A) = G^{hom}(B) if and only if A and B are affine equivalent, and G^{sim}(A) = G^{sim}(B) if and only if A and B are similar.

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Mikkel Abrahamsen and Bartosz Walczak. Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.2

Author Details

Mikkel Abrahamsen
  • BARC, University of Copenhagen, Denmark
Bartosz Walczak
  • Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

Funding

  • Abrahamsen, Mikkel: The author is supported by Starting Grant 1054-00032B from the Independent Research Fund Denmark under the Sapere Aude research career programme. BARC is supported by the VILLUM Foundation grant 16582.
  • Walczak, Bartosz: The author is partially supported by the National Science Center of Poland grant 2015/17/D/ST1/00585.

References

  1. Anders Aamand, Mikkel Abrahamsen, Jakob Bæk Tejs Knudsen, and Peter Michael Reichstein Rasmussen. Classifying convex bodies by their contact and intersection graphs. In 37th International Symposium on Computational Geometry (SoCG 2021), pages 3:1-3:16, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.3.
  2. Jochen Alber and Jiří Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. Journal of Algorithms, 52(2):134-151, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.10.001.
  3. Marthe Bonamy, Édouard Bonnet, Nicolas Bousquet, Pierre Charbit, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora, and Stéphan Thomassé. EPTAS and subexponential algorithm for maximum clique on disk and unit ball graphs. Journal of the ACM, 68(2):9:1-9:38, 2021. URL: https://doi.org/10.1145/3433160.
  4. Édouard Bonnet, Nicolas Grelier, and Nicolas Miltzow. Maximum clique in disk-like intersection graphs. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020), pages 17:1-17:18, 2020. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2020.17.
  5. Sergio Cabello and Miha Jejčič. Refining the hierarchies of classes of geometric intersection craphs. Electronic Journal of Combinatorics, 24(1):P1.33, 19 pp., 2017. URL: https://doi.org/10.37236/6040.
  6. Marco Caoduro and András Sebő. Packing, hitting and coloring squares, 2022. URL: https://arxiv.org/abs/2206.02185.
  7. Ioannis Caragiannis, Aleksei V. Fishkin, Christos Kaklamanis, and Evi Papaioannou. A tight bound for online colouring of disk graphs. Theoretical Computer Science, 384(2-3):152-160, 2007. URL: https://doi.org/10.1016/j.tcs.2007.04.025.
  8. Jean Cardinal, Stefan Felsner, Tillmann Miltzow, Casey Tompkins, and Birgit Vogtenhuber. Intersection graphs of rays and grounded segments. Journal of Graph Algorithms and Applications, 22(2):273-295, 2018. URL: https://doi.org/10.7155/jgaa.00470.
  9. Steven Chaplick, Stefan Felsner, Udo Hoffmann, and Veit Wiechert. Grid intersection graphs and order dimension. Order, 35(2):363-391, 2018. URL: https://doi.org/10.1007/s11083-017-9437-0.
  10. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1-3):165-177, 1990. URL: https://doi.org/10.1016/0012-365X(90)90358-O.
  11. Adrian Dumitrescu and Minghui Jiang. Piercing translates and homothets of a convex body. Algorithmica, 61:94-115, 2011. URL: https://doi.org/10.1007/s00453-010-9410-4.
  12. Mihály Fekete. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift, 17:228-249, 1923. URL: https://doi.org/10.1007/BF01504345.
  13. Matt Gibson and Imran A. Pirwani. Algorithms for dominating set in disk graphs: breaking the log n barrier. In 18th Annual European Symposium on Algorithms (ESA 2010), pages 243-254, 2010. URL: https://doi.org/10.1007/978-3-642-15775-2_21.
  14. Albert Gräf, Martin Stumpf, and Gerhard Weißenfels. On coloring unit disk graphs. Algorithmica, 20:277-293, 1998. URL: https://doi.org/10.1007/PL00009196.
  15. Svante Janson and Jan Kratochvíl. Thresholds for classes of intersection graphs. Discrete Mathematics, 108(1-3):307-326, 1992. URL: https://doi.org/10.1016/0012-365X(92)90684-8.
  16. Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic connectivity in disk graphs, 2021. URL: https://arxiv.org/abs/2106.14935.
  17. Seog-Jin Kim, Alexandr Kostochka, and Kittikorn Nakprasit. On the chromatic number of intersection graphs of convex sets in the plane. Electronic Journal of Combinatorics, 11:R52, 12 pp., 2004. URL: https://doi.org/10.37236/1805.
  18. Seog-Jin Kim, Kittikorn Nakprasit, Michael J. Pelsmajer, and Jozef Skokan. Transversal numbers of translates of a convex body. Discrete Mathematics, 306(18):2166-2173, 2006. URL: https://doi.org/10.1016/j.disc.2006.05.014.
  19. Paul Koebe. Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 88:141-164, 1936. Google Scholar
  20. Jan Kratochvíl and Jiří Matoušek. Intersection graphs of segments. Journal of Combinatorial Theory, Series B, 62(2):289-315, 1994. URL: https://doi.org/10.1006/jctb.1994.1071.
  21. Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. Journal of Combinatorial Theory, Series B, 103(1):114-143, 2013. URL: https://doi.org/10.1016/j.jctb.2012.09.004.
  22. Colin McDiarmid and Tobias Müller. The number of disk graphs. European Journal of Combinatorics, 35:413-431, 2014. URL: https://doi.org/10.1016/j.ejc.2013.06.037.
  23. Irina G. Perepelitsa. Bounds on the chromatic number of intersection graphs of sets in the plane. Discrete Mathematics, 262(1-3):221-227, 2003. URL: https://doi.org/10.1016/S0012-365X(02)00501-0.
  24. Oded Schramm. Combinatorically prescribed packings and applications to conformal and quasiconformal maps, 2007. URL: https://arxiv.org/abs/0709.0710.
  25. My T. Thai, Ning Zhang, Ravi Tiwari, and Xiaochun Xu. On approximation algorithms of k-connected m-dominating sets in disk graphs. Theoretical Computer Science, 385(1-3):49-59, 2007. URL: https://doi.org/10.1016/j.tcs.2007.05.025.
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