## Irreducible Triangulations of Surfaces with Boundary

*A. Boulch, É. Colin de Verdière and A. Nakamoto*

Graphs and Combinatorics, Springer Verlag, 2012

### 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.