Highly unbreakable graph with a fixed excluded minor are almost rigid

TL;DR

This paper demonstrates that graphs excluding a fixed minor and satisfying unbreakability conditions are nearly rigid, enabling a partition into a bounded treewidth part and a small labeling family, crucial for graph isomorphism algorithms.

Contribution

It proves that such graphs can be almost rigidly partitioned, advancing understanding of their structure and aiding fixed-parameter algorithms for Graph Isomorphism.

Findings

Graphs excluding a fixed minor are almost rigid under unbreakability conditions.

Vertices can be partitioned into a bounded treewidth part and a small labeling family.

This structural result is key for fixed-parameter algorithms for Graph Isomorphism.

Abstract

A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$ excludes a fixed complete graph $K_h$ as a minor and satisfies certain unbreakability guarantees, then $G$ is almost rigid in the following sense: the vertices of $G$ can be partitioned in an isomorphism-invariant way into a part inducing a graph of bounded treewidth and a part that admits a small isomorphism-invariant family of labelings. This result is the key ingredient in the fixed-parameter algorithm for Graph Isomorphism parameterized by the Hadwiger number of the graph, which is presented in a companion paper.