Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

TL;DR

This paper presents a new isomorphism testing algorithm for graphs excluding certain small topological subgraphs, achieving faster runtimes than previous methods for specific graph classes.

Contribution

It introduces an isomorphism test with runtime $n^{ ext{polylog}(h)}$ for graphs excluding a fixed $h$-vertex topological subgraph, unifying and extending prior results.

Findings

Runs in time $n^{ ext{polylog}(h)}$ for excluded topological subgraphs

Unifies previous algorithms for bounded degree and Hadwiger number graphs

Improves efficiency over prior isomorphism tests for specific graph classes

Abstract

We give an isomorphism test that runs in time $n^{\operatorname{polylog}(h)}$ on all $n$-vertex graphs excluding some $h$-vertex 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).