Tangle-Tree Duality: In Graphs, Matroids and Beyond

Abstract

We apply a recent tangle-tree duality theorem in abstract separation systems to derive tangle-tree-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets.Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a tangle-type duality theorem for tree-width.Our results can also be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width. * This is an extended version of [9] available only in preprint form.

Structural submodularity and tangles in abstract separation systems

“…This paper is, in a sense, the capstone of a comprehensive project [2,3,4,5,8,9,10,11,13,12,17,18] whose aim has been to utilize the idea of tangles familiar from Robertson and Seymour's graph minors project as a way of capturing clusters in other contexts, such as image analysis [14], genetics [6], or the social sciences [7]. The idea is to use tangles, which in graphs are certain consistent ways of orienting their low-order separations, as an indirect way of capturing 'fuzzy' clusters -ones that cannot easily be described by simply listing their elements -by instead orienting all those low-order separations towards them.…”

“…In order to prove these theorems, or to apply them to concrete cases of abstract separation systems, e.g. as in [12,14], one so far still needed a further ingredient of graph tangles: a submodular order function on the separation system considered. Our aim in this paper is to show that one can do without this: we shall prove that a structural consequence of the existence of a submodular order function, a consequence that can be expressed in terms of abstract separation systems, can replace the assumption that such a function exists in the proofs of the above two theorems.…”

“…The abstract tangles in Theorem 3 are not the only F-tangles for which such a statement holds. In [12] the tangle-tree duality theorem of [13] is used to prove such a statement for a broad class of F-tangles, albeit under a stronger assumption: one needs there that → S is not just structurally submodular, as is our assumption here throughout our paper, but that U has a submodular order function and → S is the set separations up to some fixed order (and therefore, in particular, submodular).…”

“…Our result is that for the proofs of these two theorems, which are central to abstract tangle theory, it suffices to assume the above structural consequence for → S , and no order function is needed.Abstract separation systems were first introduced in [8]; see there for a gentle formal introduction and any terminology we forgot to define below. Motivation for why they are interesting can be found in the introductory sections of [11,13,12] and in [14]. In what follows we provide a self-contained account of just the definitions and basic facts about abstract separation systems that we need in this paper.A separation system ( → S , ≤, * ) is a partially ordered set with an order-reversing involution * :S is → s * , which we usually denote by ← s .…”

“…Abstract separation systems were first introduced in [8]; see there for a gentle formal introduction and any terminology we forgot to define below. Motivation for why they are interesting can be found in the introductory sections of [11,13,12] and in [14]. In what follows we provide a self-contained account of just the definitions and basic facts about abstract separation systems that we need in this paper.…”