Fixed-parameter tractability of Graph Isomorphism in graphs with an excluded minor

TL;DR

This paper demonstrates that Graph Isomorphism and Canonization problems are fixed-parameter tractable for graphs excluding a fixed minor, using a canonical decomposition approach based on graph connectivity and automorphism analysis.

Contribution

It introduces a fixed-parameter tractable algorithm for Graph Isomorphism in H-minor-free graphs, with a novel canonical decomposition technique.

Findings

Algorithm runs in f(H)·n^{O(1)} time for H-minor-free graphs

Graphs can be decomposed into unbreakable parts with bounded automorphisms

Unbreakable H-minor-free graphs can be canonically decomposed into parts with few automorphisms and bounded treewidth

Abstract

We prove that Graph Isomorphism and Canonization in graphs excluding a fixed graph $H$ as a minor can be solved by an algorithm working in time $f(H)\cdot n^{O(1)}$, where $f$ is some function. In other words, we show that these problems are fixed-parameter tractable when parameterized by the size of the excluded minor, with the caveat that the bound on the running time is not necessarily computable. The underlying approach is based on decomposing the graph in a canonical way into unbreakable (intuitively, well-connected) parts, which essentially provides a reduction to the case where the given $H$-minor-free graph is unbreakable itself. This is complemented by an analysis of unbreakable $H$-minor-free graphs, performed in a second subordinate manuscript, which reveals that every such graph can be canonically decomposed into a part that admits few automorphisms and a part that has bounded treewidth.