We give an isomorphism test for graphs of Euler genus $g$ running in time $2^{O(g^4 \log g)}n^{O(1)}$. Our algorithm provides the first explicit upper bound on the dependence on $g$ for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time $f(g)n$ for some function $f$ (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude $K_{3,h}$ as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, we introduce $(t,k)$-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm. This concept may be of independent interest.

翻译：我们提出一种针对欧拉亏格为$g$的图的同构测试算法，运行时间为$2^{O(g^4 \log g)}n^{O(1)}$。该算法首次给出了以输入图欧拉亏格为参数的FPT同构测试中依赖$g$的显式上界。先前唯一的FPT算法运行时间为$f(g)n$（其中$f$为某个函数，Kawarabayashi 2015）。实际上，我们的算法甚至适用于仅禁止$K_{3,h}$作为子式的输入图。对于此类图，此前没有已知的FPT同构测试。该算法基于群论、组合学和图论方法的巧妙结合。特别地，我们引入了$(t,k)$-WL有界图概念，它为将群论技术与标准Weisfeiler-Leman算法结合提供了有力工具。这一概念可能具有独立的研究价值。