Displaying tangles and non-sequential separations

TL;DR

This paper introduces a method to construct a tree structure that simultaneously distinguishes all maximal tangles in a graph or matroid and displays relevant non-trivial separations for robust tangles, enhancing understanding of their structure.

Contribution

It provides a unified tree-based framework to distinguish maximal tangles and display key separations in graphs and matroids, extending previous structural analysis tools.

Findings

Existence of a tree distinguishing all maximal tangles

Tree displays all non-trivial separations for robust tangles

Framework applies to any graph or matroid

Abstract

We show that, for any graph or matroid, there is a tree that simultaneously distinguishes its maximal tangles, and, for each maximal tangle $\mathcal{T}$ that satisfies an additional robustness condition, displays all of the non-trivial separations relative to $\mathcal{T}$, up to a natural equivalence.