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

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.