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A Schnyder-Type Drawing Algorithm for 5-Connected Triangulations

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Graph Drawing and Network Visualization (GD 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14466))

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Abstract

We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5 regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing.

OB was partially supported by NSF Grant DMS-2154242. EF was partially supported by the project ANR19-CE48-011-01 (COMBINÉ), and the project ANR-20-CE48-0018 (3DMaps).

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Notes

  1. 1.

    If we use an affine transformation to have \(v_1\), \(v_2\) and \(v_5\) placed at (0, 0), (0, 1) and (1, 0) respectively, then we get a drawing with vertex-coordinates in \(\mathbb {Q}(\sqrt{5})\).

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Author information

Authors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, USA

    Olivier Bernardi & Shizhe Liang

  2. CNRS/LIGM, Université Gustave Eiffel, Champs-sur-Marne, France

    Éric Fusy

Authors
  1. Olivier Bernardi
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  2. Éric Fusy
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  3. Shizhe Liang
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Correspondence to Éric Fusy .

Editor information

Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Osnabrück University, Osnabrück, Germany

    Markus Chimani

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Bernardi, O., Fusy, É., Liang, S. (2023). A Schnyder-Type Drawing Algorithm for 5-Connected Triangulations. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14466. Springer, Cham. https://doi.org/10.1007/978-3-031-49275-4_8

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  • DOI: https://doi.org/10.1007/978-3-031-49275-4_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49274-7

  • Online ISBN: 978-3-031-49275-4

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