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( h ) (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 ) (SIAM J. Comp., 2023) and for graphs of Hadwiger number h running in time n polylog( h ) (SIAM J. Comp., 2023).