On the Computation of Schrijver's Kernels

Abstract

The geometry of a graph $G$ embedded on a closed oriented surface $S$ can be probed by counting the intersections of $G$ with closed curves on $S$. Of special interest is the map $c \mapsto \mu_G(c)$ counting the minimum number of intersections between $G$ and any curve freely homotopic to a given curve $c$. Schrijver [On the uniqueness of kernels, 1992] calls $G$ a kernel if for any proper graph minor $H$ of $G$ we have $\mu_H < \mu_G$. Hence, $G$ admits a minor $H$ which is a kernel and such that $\mu_G = \mu_H$. We show how to compute such a minor kernel of $G$ in $O(n^3 \log n)$ time where $n$ is the number of edges of $G$, and $g\ge 2$ is the genus of $S$. Our algorithm leverages a tight bound on the size of minimal bigons in a system of closed curves. It also relies on several subroutines of independent interest including the computation of the area enclosed by a curve and a test of simplicity for the lift of a curve in the universal covering of $S$. As a consequence of our minor kernel algorithm and a recent result of Dubois [Making multicurves cross minimally on surfaces, 2024], after a preprocessing that takes $O(n^3 \log n)$ time and $O(n)$ space, we are able to compute $\mu_G(c)$ in $O(g (n + \ell) \log(n + \ell))$ time given any closed walk $c$ with $\ell$ edges. The state-of-the-art algorithm by Colin de Verdi\`ere and Erickson [Tightening non-simple paths and cycles on surfaces, 2010] would avoid constructing a kernel but would lead to a computation of $\mu_G(c)$ in $O(g n \ell \log(n \ell))$ time (with a preprocessing that takes $O(gn\log n)$ time and $O(gn)$ space). Another consequence of the computation of minor kernels is the ability to decide in polynomial time whether two graph minors $H$ and $H'$ of $G$ satisfy $\mu_H = \mu_{H'}$.