A Bramble like Witness for Large Branch-Width

TL;DR

This paper introduces a new framework for understanding large branch-width in graphs, using a witness similar to brambles, and applies it to profiles, a generalization of tangles and blocks.

Contribution

It develops a natural witness concept for large branch-width based on graph separations, extending the theory with applications to profiles.

Findings

Defined a bramble-like witness for large branch-width

Extended the framework to profiles, unifying tangles and blocks

Provided new insights into graph separation structures

Abstract

Using the new framework of graph separations developed recently by Diestel and Oum, we show that there is a natural witness to large branch-width which can be defined in a similar way to a bramble. We also apply our methods to a more recent object of study in the area, a common generalization of tangles and blocks, called profiles.