Trees of tangles in abstract separation systems

TL;DR

This paper establishes canonical and non-canonical tree-of-tangles theorems for abstract separation systems, simplifying proofs and unifying results across graphs, matroids, and other systems with submodular order functions.

Contribution

It introduces a unified framework for tree-of-tangles theorems in abstract separation systems, extending and simplifying existing results.

Findings

Proves canonical and non-canonical tree-of-tangles theorems for submodular systems

Simplifies proofs of known theorems for graphs and matroids

Unifies various tree-of-tangles results under a common framework

Abstract

We prove canonical and non-canonical tree-of-tangles theorems for abstract separation systems that are merely structurally submodular. Our results imply all known tree-of-tangles theorems for graphs, matroids and abstract separation systems with submodular order functions, with greatly simplified and shortened proofs.