Constant Congestion Brambles
Meike Hatzel, Pawel Komosa, Marcin Pilipczuk, Manuel Sorge

TL;DR
This paper refines bounds on the size and structure of brambles in graphs with given treewidth, showing linear size for certain bounds and tight exponential upper bounds for others.
Contribution
It improves existing bounds by proving that brambles of certain orders can have linear size and provides tight exponential bounds for brambles with higher order.
Findings
Every graph with treewidth at least k has a bramble of order rac{1}{2} with congestion 2.
A tight exponential upper bound is established for brambles of order rac{1}{2}+elta in graphs with large treewidth.
The size of such brambles can be exponential in rac{1}{2}+elta of the treewidth.
Abstract
A bramble in an undirected graph is a family of connected subgraphs of such that for every two subgraphs and in the bramble either or there is an edge of with one endpoint in and the second endpoint in . The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph equals one plus the treewidth of . However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree -vertex expander a bramble of order requires size exponential in for any fixed . On the other hand, the combination of results of Grohe and Marx…
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