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Encoding Toroidal Triangulations

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Abstract

Poulalhon and Schaeffer introduced an elegant method to linearly encode a planar triangulation optimally. The method is based on performing a special depth-first search algorithm on a particular orientation of the triangulation: the minimal Schnyder wood. Recent progress toward generalizing Schnyder woods to higher genus enables us to generalize this method to the toroidal case. In the plane, the method leads to a bijection between planar triangulations and some particular trees. For the torus we obtain a similar bijection but with particular unicellular maps (maps with only one face).

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Acknowledgments

We thank Luca Castelli Aleardi, Nicolas Bonichon, Eric Fusy and Frédéric Meunier for fruitful discussions about this work. This work was supported by the Grant EGOS ANR-12-JS02-002-01 and the project-team GALOIS supported by LabEx PERSYVAL-Lab ANR-11-LABX-0025.

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Authors and Affiliations

  1. GIPSA-Lab, University Grenoble Alpes, Grenoble, France

    Vincent Despré

  2. CNRS, LIRMM, University Montpellier, Montpellier, France

    Daniel Gonçalves

  3. CNRS, G-SCOP, University Grenoble Alpes, Grenoble, France

    Benjamin Lévêque

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  1. Vincent Despré
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  2. Daniel Gonçalves
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  3. Benjamin Lévêque
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Correspondence to Benjamin Lévêque.

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Despré, V., Gonçalves, D. & Lévêque, B. Encoding Toroidal Triangulations. Discrete Comput Geom 57, 507–544 (2017). https://doi.org/10.1007/s00454-016-9832-0

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  • DOI: https://doi.org/10.1007/s00454-016-9832-0

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