Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

We give an isomorphism test that runs in time n^{polylog(h)} on all n-vertex graphs excluding some h-vertex graph as a topological subgraph. Previous results state that isomorphism for such graphs can be tested in time n^{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^{polylog(d)} (FOCS 2018) and for graphs of Hadwiger number h running in time n^{polylog(h)} (FOCS 2020).