Irreducible triangulations of surfaces with boundary

TL;DR

This paper studies irreducible triangulations of surfaces with boundary, proving that their vertex count is linearly bounded by genus and boundary components, extending known results from closed surfaces.

Contribution

It extends the bound on the number of vertices in irreducible triangulations to surfaces with boundary, providing a simpler, elementary proof.

Findings

Vertex count is O(g+b) for irreducible triangulations of surfaces with boundary.

The proof is elementary and simpler than previous methods for surfaces without boundary.

The result generalizes known bounds from closed to bounded surfaces.

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 boundaries 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.