Tangles, trees, and flowers

TL;DR

This paper introduces a method to construct tree decompositions that encapsulate all significant $k$-separations related to a robust tangle in matroids or graphs, enhancing understanding of their connectivity structure.

Contribution

It provides a new tree decomposition technique that displays all non-trivial $k$-separations associated with a robust tangle in matroids and graphs.

Findings

Tree decompositions display all relevant $k$-separations.

Applicable to matroids and graphs with robust tangles.

Improves understanding of connectivity structures.

Abstract

A tangle of order $k$ in a matroid or graph may be thought of as a "$k$-connected component". For a tangle of order $k$ in a matroid or graph that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or graph that displays, up to a certain natural equivalence, all of the $k$-separations of the matroid or graph that are non-trivial with respect to the tangle.