We give an isomorphism test that runs in time $n^{\operatorname{polylog}(h)}$ on all $n$-vertex graphs excluding some $h$-vertex vertex graph as a topological subgraph. Previous results state that isomorphism for such graphs can be tested in time $n^{\operatorname{polylog}(n)}$ (Babai, STOC 2016) and $n^{f(h)}$ for some function $f$ (Grohe and Marx, SIAM J. Comp., 2015). Our result also unifies and extends previous isomorphism tests for graphs of maximum degree $d$ running in time $n^{\operatorname{polylog}(d)}$ (SIAM J. Comp., 2023) and for graphs of Hadwiger number $h$ running in time $n^{\operatorname{polylog}(h)}$ (SIAM J. Comp., 2023).

翻译：我们提出了一种同构测试算法，该算法在所有排除某个$h$顶点图作为拓扑子图的$n$顶点图上运行时间为$n^{\operatorname{polylog}(h)}$。先前的研究表明，此类图的同构测试可在$n^{\operatorname{polylog}(n)}$时间内完成（Babai, STOC 2016），或对于某个函数$f$在$n^{f(h)}$时间内完成（Grohe and Marx, SIAM J. Comp., 2015）。我们的结果还统一并扩展了针对最大度数为$d$的图的同构测试（运行时间$n^{\operatorname{polylog}(d)}$，SIAM J. Comp., 2023）以及针对Hadwiger数为$h$的图的同构测试（运行时间$n^{\operatorname{polylog}(h)}$，SIAM J. Comp., 2023）。