Irreducible Triangulations of Surfaces with Boundary

Abstract

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O(g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.

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Correspondence to Éric Colin de Verdière.

Additional information

This work was done during the first author’s internship at École normale supérieure. The internship was funded by the Agence Nationale de la Recherche under the Triangles project of the Programme blanc ANR-07-BLAN-0319.

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Boulch, A., Colin de Verdière, É. & Nakamoto, A. Irreducible Triangulations of Surfaces with Boundary. Graphs and Combinatorics 29, 1675–1688 (2013). https://doi.org/10.1007/s00373-012-1244-1

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Keywords

  • Topological graph theory
  • Surface
  • Triangulation
  • Irreducible triangulation
  • Homotopy

Mathematics Subject Classification (2000)

  • 05C10
  • 57M15
  • 57N05