Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

Abstract

We give an isomorphism test that runs in time npolylog(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 npolylog(n) (Babai, STOC 2016) and nf(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 npolylog(d) (FOCS 2018) and for graphs of Hadwiger number h running in time npolylog(h) (FOCS 2020).