Tangled up in Blue (A Survey on Connectivity, Decompositions, and Tangles)

TL;DR

This survey explores the theory of connectivity in systems using symmetric submodular functions, focusing on duality, canonical decompositions, and algorithmic aspects of tangles and decompositions.

Contribution

It provides a comprehensive overview of the abstract connectivity theory, emphasizing duality between decompositions and tangles, and discusses algorithmic considerations.

Findings

Duality between branch decompositions and tangles

Canonical decompositions into maximal tangles

Algorithmic approaches to connectivity systems

Abstract

We survey an abstract theory of connectivity, based on symmetric submodular set functions. We start by developing Robertson and Seymour's fundamental duality between branch decompositions (related to the better-known tree decompositions) and so-called tangles, which may be viewed as highly connected regions in a connectivity system. We move on to studying canonical decompositions of connectivity systems into their maximal tangles. Last, but not least, we will discuss algorithmic aspect of the theory.