How much cutting is needed to simplify the topology of a surface?

I will discuss bounds for several instances of this question that appeared in a joint work with Eric Colin de Verdière and Arnaud de Mesmay: for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces.

These questions, come from the computational topology and topological graph theory literature, and much of the work that we did involved setting a simple way to translate analogous results from Riemannian systolic geometry. I will give proofs of this translation.

Then I will talk about the separator theorem of Lipton and Tarjan that provides a bound on the Cheeger constant of any planar graph. It was in generalising this theorem to surfaces that computational topologists discovered discrete systolic geometry.