Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

Abstract

We give an isomorphism test that runs in time \(n^{\operatorname{polylog}(h)} \) 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 \(n^{\operatorname{polylog}(n)} \) (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).