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Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

Daniel Neuen

TL;DR

The paper resolves Graph Isomorphism for graphs excluding a fixed $h$-vertex topological subgraph by combining a Weisfeiler-Leman–driven initial-set extraction with a $t$-CR refinement framework and powerful group-theoretic tools. The core innovation is a 3-WL based construction of an isomorphism-invariant initial set $X$ whose $t$-closure $D$ yields a decomposition into $t$-CR-bounded parts, enabling polylogarithmic-in-$h$ isomorphism testing on the parts and efficient assembly of global isomorphisms. The approach unifies and extends prior polylogarithmic-exponent GI results for graphs of bounded degree and Hadwiger-number, while also revealing a refined structure of the automorphism groups in these sparse classes through tree decompositions with bounded adhesion. This work advances the practical and theoretical understanding of GI on sparse graph classes and opens avenues for extending these techniques to moderately dense graph families.

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).

Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

TL;DR

The paper resolves Graph Isomorphism for graphs excluding a fixed -vertex topological subgraph by combining a Weisfeiler-Leman–driven initial-set extraction with a -CR refinement framework and powerful group-theoretic tools. The core innovation is a 3-WL based construction of an isomorphism-invariant initial set whose -closure yields a decomposition into -CR-bounded parts, enabling polylogarithmic-in- isomorphism testing on the parts and efficient assembly of global isomorphisms. The approach unifies and extends prior polylogarithmic-exponent GI results for graphs of bounded degree and Hadwiger-number, while also revealing a refined structure of the automorphism groups in these sparse classes through tree decompositions with bounded adhesion. This work advances the practical and theoretical understanding of GI on sparse graph classes and opens avenues for extending these techniques to moderately dense graph families.

Abstract

We give an isomorphism test that runs in time on all -vertex graphs excluding some -vertex vertex graph as a topological subgraph. Previous results state that isomorphism for such graphs can be tested in time (Babai, STOC 2016) and for some function (Grohe and Marx, SIAM J. Comp., 2015). Our result also unifies and extends previous isomorphism tests for graphs of maximum degree running in time (SIAM J. Comp., 2023) and for graphs of Hadwiger number running in time (SIAM J. Comp., 2023).

Paper Structure

This paper contains 24 sections, 40 theorems, 138 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Topological Subgraphs
  5. Weisfeiler-Leman Algorithm
  6. Allowing Color Refinement to Split Small Color Classes
  7. Finding the Initial Set
  8. Basic Tools
  9. The Closure Graph
  10. Interaction between Closure Sets
  11. Constructing Disjoint Paths
  12. Paths of Odd Length
  13. Paths of Even Length
  14. Group-Theoretic Techniques for Isomorphism Testing
  15. Basics
  16. ...and 9 more sections

Key Result

Theorem 1.1

The Graph Isomorphism Problem for graphs excluding some $h$-vertex graph as a topological subgraph can be solved in time $n^{\operatorname{polylog}(h)}$.

Figures (3)

  • Figure 1: Visualization of a graph $G$ and the sequence of colorings described in Definition \ref{['def:t-cr-bounded']} for $t = 2$. The coloring $\chi_4$ is discrete, so $G$ is $2$-CR-bounded.
  • Figure 2: Visualization of the interaction between the closure sets.
  • Figure 3: Visualization of the graph decomposition.

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: BollobasT98KomlosS96
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: GroheNW23
  • Theorem 4.1
  • Corollary 4.2
  • proof
  • ...and 89 more