# Tangle-tree duality: in graphs, matroids and beyond

## TL;DR

This paper extends duality theorems for tangles to various width parameters in graphs and matroids, and introduces a duality theorem for clusters in large data sets, unifying and simplifying classical results.

## Contribution

It generalizes tangle duality to new parameters and data clustering, providing a unified framework for classical graph minor dualities and new applications.

## Key findings

- New duality theorems for tree-width, path-width, and tree-decompositions.
- Carving width is dual to edge-tangles.
- Unified proofs of classical duality theorems in graph minor theory.

## Abstract

We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets. Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width. Our results can be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width.