  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).
