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