Point sets and functions inducing tangles of set separations

TL;DR

This paper characterizes tangles induced by data points or functions in broader discrete structures, providing two characterizations based on coverage and a new duality concept for oriented set separations.

Contribution

It introduces two characterizations of tangles induced by points or functions, extending the concept beyond graphs and matroids with a novel duality notion.

Findings

Two characterizations of point-induced tangles are provided.

A new duality concept for oriented set separations is introduced.

The work broadens the understanding of tangles in discrete structures.

Abstract

Tangles, as introduced by Robertson and Seymour, were designed as an indirect way of capturing clusters in graphs and matroids. They have since been shown to capture clusters in much broader discrete structures too. But not all tangles are induced by a set of points, let alone a cluster. We characterise those that are: the tangles that are induced by a subset of or function on the set of data points whose connectivity structure they are meant to capture. We offer two such characterisations. The first is in terms of how many small sides of a tangle's separations it takes to cover the ground set. The second uses a new notion of duality for oriented set separations that becomes possible if these are no longer required to be separations of graph or matroids.