Profiles of separations: in graphs, matroids and beyond

TL;DR

The paper introduces a unified tree-decomposition approach that distinguishes all tangles in graphs and matroids, with applications in clustering, image segmentation, and deriving the Gomory-Hu theorem.

Contribution

It presents a general decomposition theorem for combinatorial structures that unifies and extends existing results, including a new method for cluster analysis and image segmentation.

Findings

All tangles in graphs and matroids can be distinguished by a single automorphism-invariant tree-decomposition.

The theorem applies to broader combinatorial structures, enabling new applications.

It provides a new perspective on the Gomory-Hu theorem via edge-tangles.

Abstract

We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications. These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem.