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A Canonical Tree Decomposition for Chirotopes

Authors Mathilde Bouvel, Valentin Feray, Xavier Goaoc, Florent Koechlin

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Author Details

Mathilde Bouvel
  • Université de Lorraine, CNRS, INRIA, LORIA, F-54000 Nancy, France
Valentin Feray
  • Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
Xavier Goaoc
  • Université de Lorraine, CNRS, INRIA, LORIA, F-54000 Nancy, France
Florent Koechlin
  • Université Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93340 Villetaneuse, France

Funding

  • Koechlin, Florent: This work was done while FK was a postdoctoral fellow in LORIA, funded by ANR ASPAG (ANR-17-CE40-0017), IUF and INRIA.

Acknowledgements

The authors thank Emo Welzl for discussion leading to footnote 2 and the anonymous referees for helpful comments.

Cite AsGet BibTex

Mathilde Bouvel, Valentin Feray, Xavier Goaoc, and Florent Koechlin. A Canonical Tree Decomposition for Chirotopes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.23

Abstract

We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets, and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
Keywords
  • Order type
  • modular decomposition
  • counting triangulations
  • mutually avoiding point sets
  • generating functions
  • rewriting systems

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