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Independence of the Lower Bound for Geometric Triangulation Drawings on the Number of Chains (Poster Abstract)

Authors Vojtěch Vincíbr , Anežka Kasalová , Zuzana Reiffersová , Petr Filip

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • combinatorial triangulation
  • geometric triangulation
  • planar graph
  • d-tuple chain
  • switch subgraph

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Abstract

In 2026, Cruces, Huemer, and Lara established that a specific combinatorial triangulation can be drawn in at least Ω(3^{n/4}) ways on the double chain as geometric triangulation. We generalize their construction to a d-tuple chain configuration. By introducing the switch subgraph abstraction to count recursive drawing, we obtain the same asymptotic lower bounds Ω(3^{n/4}) ≈ Ω(1.316ⁿ) even when we increase the number of chains.

Cite As Get BibTex

Vojtěch Vincíbr, Anežka Kasalová, Zuzana Reiffersová, and Petr Filip. Independence of the Lower Bound for Geometric Triangulation Drawings on the Number of Chains (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 48:1-48:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.48

Author Details

Vojtěch Vincíbr
  • Charles University, Prague, Czech Republic
Anežka Kasalová
  • Charles University, Prague, Czech Republic
Zuzana Reiffersová
  • Charles University, Prague, Czech Republic
Petr Filip
  • Charles University, Prague, Czech Republic

Funding

This research was supported by grant no. 23-04949X of the Czech Science Foundation (GAČR).

Acknowledgements

We would like to thank Jan Kynčl for giving us helpful advice and introducing us to the problem.

References

  1. Belén Cruces, Clemens Huemer, and Dolores Lara. On the number of drawings of a combinatorial triangulation. Graphs Combin., 42(1):Paper No. 16, 11, 2026. URL: https://doi.org/10.1007/s00373-026-03010-2.
  2. Daniel Rutschmann and Manuel Wettstein. Chains, koch chains, and point sets with many triangulations. J. ACM, 70(3), May 2023. URL: https://doi.org/10.1145/3585535.
  3. Micha Sharir and Adam Sheffer. Counting triangulations of planar point sets. Electron. J. Combin., 18(1):Paper 70, 74, 2011. URL: https://doi.org/10.37236/557.
  4. Micha Sharir and Emo Welzl. Random triangulations of planar point sets. In Computational geometry (SCG'06), pages 273-281. ACM, New York, 2006. URL: https://doi.org/10.1145/1137856.1137898.
  5. William Thomas Tutte. A census of planar triangulations. Canadian J. Math., 14:21-38, 1962. URL: https://doi.org/10.4153/CJM-1962-002-9.
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