On the size of k-irreducible triangulations
This provides a fundamental result in combinatorial topology, with implications for algorithms in computational geometry and graph theory.
The paper tackles the problem of bounding the size of k-irreducible triangulations on surfaces of genus g, proving an optimal bound of O(k^2g) triangles, which improves the previous best bound of k^{O(k)} g^2.
A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of a surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].