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On the Size of k-Irreducible Triangulations

Authors Vincent Delecroix , Oscar Fontaine , Arnaud de Mesmay

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LIPIcs.SoCG.2026.38.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
  • Theory of computation → Computational geometry
Keywords
  • surface
  • irreducible triangulation
  • system of curves
  • minimal position
  • systolic geometry

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Abstract

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has O(k²g) triangles, which is optimal. This is an improvement over the previous best bound k^O(k) g² of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].

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Vincent Delecroix, Oscar Fontaine, and Arnaud de Mesmay. On the Size of k-Irreducible Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.38

Author Details

Vincent Delecroix
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France
Oscar Fontaine
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France
Arnaud de Mesmay
  • LIGM Laboratoire Informatique Gaspard Monge Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France

Funding

This work was funded by the ANR-SNF project SUGAR (ANR-25-CE40-0416, SNF 200021E_238147).
  • Delecroix, Vincent: Funded by ANR MOST (ANR-23-CE40-0020) and ANR CarteEtPlus (ANR-23-CE48-0018).

Acknowledgements

We are grateful to anonymous reviewers for helpful remarks and suggestions.

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