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Topological Drawings Meet Classical Theorems from Convex Geometry

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Abstract

In this article we discuss classic theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph \(K_n\), any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. As our main result, we present a generalization of Kirchberger’s theorem that is of purely combinatorial nature. It turned out that this classic theorem also applies to “generalized signotopes”—a combinatorial generalization of simple topological drawings, which we introduce and investigate in the course of this article. As indicated by their name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines. We also present a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory’s theorem in the plane and discuss further classic theorems from Convex Geometry in the context of simple topological drawings.

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Notes

  1. The authors of [8] use the term face-convex instead of cell-convex.

  2. Arrangements supporting a drawing of \(K_n\) are also known as pseudoconfigurations of points and can be considered as oriented matroids of rank 3 (cf. [17, Chapt. 5.3]). For a formal definition of arrangements of pseudolines, we refer the interested reader to [19] or [17, Chapt. 6].

References

  1. Ábrego, B., Aichholzer, O., Fernández-Merchant, S., Hackl, T., Pammer, J., Pilz, A., Ramos, P., Salazar, G., Vogtenhuber, B.: All good drawings of small complete graphs. In: 31th European Workshop on Computational Geometry (Ljubljana 2015). Book of Abstracts, pp. 57–60. http://eurocg15.fri.uni-lj.si/pub/eurocg15-book-of-abstracts.pdf

  2. Aichholzer, O., Hackl, T., Pilz, A., Salazar, G., Vogtenhuber, B.: Deciding monotonicity of good drawings of the complete graph. In: 16th Spanish Meeting on Computational Geometry (Barcelona 2015), pp. 33–36 (2015)

  3. Alon, N., Kleitman, D.J.: Piercing convex sets and the Hadwiger–Debrunner \((p, q)\)-problem. Adv. Math. 96(1), 103–112 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arocha, J.L., Bárány, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discrete Comput. Geom. 42(2), 142–154 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arroyo, A., Bensmail, J., Richter, R.B.: Extending drawings of graphs to arrangements of pseudolines. In: 36th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 164, # 9. Leibniz-Zent. Inform., Wadern (2020)

  6. Arroyo, A., McQuillan, D., Richter, R.B., Salazar, G.: Drawings of \(K_n\) with the same rotation scheme are the same up to triangle-flips (Gioan’s theorem). Aust. J. Comb. 67, 131–144 (2017)

    MATH  Google Scholar 

  7. Arroyo, A., McQuillan, D., Richter, R.B., Salazar, G.: Levi’s lemma, pseudolinear drawings of \(K_n\), and empty triangles. J. Graph Theory 87(4), 443–459 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arroyo, A., McQuillan, D., Richter, R.B., Salazar, G.: Convex drawings of the complete graph: topology meets geometry. Ars Mathematica Contemporanea (2021). https://doi.org/10.26493/1855-3974.2134.ac9

  9. Arroyo, A., Richter, R.B., Sunohara, M.: Extending drawings of complete graphs into arrangements of pseudocircles. SIAM J. Discrete Math. 35(2), 1050–1076 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bachem, A., Wanka, A.: Separation theorems for oriented matroids. Discrete Math. 70(3), 303–310 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bachem, A., Wanka, A.: Euclidean intersection properties. J. Comb. Theory Ser. B 47(1), 10–19 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Balko, M., Fulek, R., Kynčl, J.: Crossing numbers and combinatorial characterization of monotone drawings of \(K_n\). Discrete Comput. Geom. 53(1), 107–143 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2–3), 141–152 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bárány, I., Shlosman, S.B., Szücs, A.: On a topological generalization of a theorem of Tverberg. J. Lond. Math. Soc. 23(1), 158–164 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bárány, I., Soberón, P.: Tverberg’s theorem is 50 years old: a survey. Bull. Am. Math. Soc. 55(4), 459–492 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002)

  17. Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and Its Applications, vol. 46. Cambridge University Press, Cambridge (1999)

  18. Chung, F.R.K., Graham, R.L., Frankl, P., Shearer, J.B.: Some intersection theorems for ordered sets and graphs. J. Comb. Theory Ser. A 43(1), 23–37 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Felsner, S., Goodman, J.E.: Pseudoline arrangements. In: Handbook of Discrete and Computational Geometry, 3rd ed., pp. 125–157 (chapter 5). CRC Press, Boca Raton (2018)

  20. Felsner, S., Weil, H.: Sweeps, arrangements and signotopes. Discrete Appl. Math. 109(1–2), 67–94 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Frick, F., Soberón, P.: The topological Tverberg problem beyond prime powers (2020). arXiv:2005.05251

  22. Gioan, E.: Complete graph drawings up to triangle mutations. Discrete Comput. Geom. 67(4), 985–1022 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Goaoc, X., Paták, P., Patáková, Z., Tancer, M., Wagner, U.: Bounding Helly numbers via Betti numbers. In: A Journey Through Discrete Mathematics, pp. 407–447. Springer, Cham (2017)

  24. Goodman, J.E., Pollack, R.: Helly-type theorems for pseudoline arrangements in \({ P}^{2}\). J. Comb. Theory Ser. A 32(1), 1–19 (1982)

    Article  MATH  Google Scholar 

  25. Guy, R.K.: Crossing numbers of graphs. In: Graph Theory and Applications (Kalamazoo 1972). Lecture Notes in Mathematics, vol. 303, pp. 111–124. Springer, Berlin (1972)

  26. Helly, E.: Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatsh. Math. Phys. 37(1), 281–302 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  27. Holmsen, A.F.: The intersection of a matroid and an oriented matroid. Adv. Math. 290, 1–14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Holmsen, A.F., Pach, J., Tverberg, H.: Points surrounding the origin. Combinatorica 28(6), 633–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kalai, G.: Colorful Caratheodory revisited (2009). http://gilkalai.wordpress.com/2009/03/15/colorful-caratheodory-revisited

  30. Kalai, G., Meshulam, R.: A topological colorful Helly theorem. Adv. Math. 191(2), 305–311 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Keller, Ch., Smorodinsky, Sh., Tardos, G.: Improved bounds on the Hadwiger–Debrunner numbers. Isr. J. Math. 225(2), 925–945 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kirchberger, P.: Über Tchebychefsche Annäherungsmethoden. Math. Ann. 57(4), 509–540 (1903)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kynčl, J.: Improved enumeration of simple topological graphs. Discrete Comput. Geom. 50(3), 727–770 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kynčl, J.: Simple realizability of complete abstract topological graphs simplified. Discrete Comput. Geom. 64(1), 1–27 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Özaydin, M.: Equivariant maps for the symmetric group (1987). https://minds.wisconsin.edu/handle/1793/63829

  36. Pach, J., Tóth, G.: How many ways can one draw a graph? Combinatorica 26(5), 559–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pammer, J.: Rotation Systems and Good Drawings. MSc thesis, Graz University of Technology (2014). http://diglib.tugraz.at/rotation-systems-and-good-drawings-2014

  38. Patáková, Z.: Bounding radon number via Betti numbers. In: 36th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 164, # 61. Leibniz-Zent. Inform., Wadern (2020)

  39. Roudneff, J.-P.: Tverberg-type theorems for pseudoconfigurations of points in the plane. Eur. J. Comb. 9(2), 189–198 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Helena Bergold

  2. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Stefan Felsner, Manfred Scheucher, Felix Schröder & Raphael Steiner

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  1. Helena Bergold
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  2. Stefan Felsner
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Correspondence to Helena Bergold.

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We thank Alan Arroyo, Emo Welzl, Heiko Harborth, and Geza Tóth for inspiring discussions and the reviewers for helpful comments. A special thanks goes to Patrick Schnider for his simplification of the construction in the proof of Proposition 3. R. Steiner and H. Bergold were funded by DFG-GRK 2434. S. Felsner and M. Scheucher were supported by the DFG Grant FE 340/12-1. M. Scheucher was supported by the DFG Grant SCHE 2214/1-1 and by the internal research funding “Post-Doc-Funding” from Technische Universität Berlin.

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Bergold, H., Felsner, S., Scheucher, M. et al. Topological Drawings Meet Classical Theorems from Convex Geometry. Discrete Comput Geom 70, 1121–1143 (2023). https://doi.org/10.1007/s00454-022-00408-6

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