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 $\mu_H = \mu_{H'}$.

翻译：通过计数图$G$与封闭曲面$S$上闭曲线的交点数，可以探究嵌入在定向闭曲面$S$上的图$G$的几何性质。特别令人关注的是映射$c \mapsto \mu_G(c)$，它计算$G$与任意同伦于给定曲线$c$的曲线之间的最小交点数。Schrijver [On the uniqueness of kernels, 1992] 称$G$为核，如果对于$G$的任何真图子式$H$，都有$\mu_H < \mu_G$。因此，$G$存在一个作为核的子式$H$，且满足$\mu_G = \mu_H$。我们展示了如何在$O(n^3 \log n)$时间内计算$G$的这样一个子式核，其中$n$是$G$的边数，$g\ge 2$是$S$的亏格。我们的算法利用了闭曲线系统中最小二边形尺寸的紧界。它还依赖于几个具有独立意义的子程序，包括计算曲线所围区域的面积，以及在$S$的万有覆叠中测试曲线提升的简单性。作为我们子式核算法以及Dubois近期结果 [Making multicurves cross minimally on surfaces, 2024] 的推论，在经过$O(n^3 \log n)$时间和$O(n)$空间的预处理后，我们能够在$O(g (n + \ell) \log(n + \ell))$时间内计算任意具有$\ell$条边的闭游走$c$对应的$\mu_G(c)$。Colin de Verdi\`ere和Erickson [Tightening non-simple paths and cycles on surfaces, 2010] 的最先进算法虽然避免了构造核，但会导致在$O(g n \ell \log(n \ell))$时间内计算$\mu_G(c)$（其预处理需要$O(gn\log n)$时间和$O(gn)$空间）。计算子式核的另一个推论是能够在多项式时间内判定$G$的两个图子式$H$和$H'$是否满足$\mu_H = \mu_{H'}$。