We prove a tree-of-tangles theorem and a tangle-tree duality theorem for abstract separation systems → S that are submodular in the structural sense that, for every pair of oriented separations, → S contains either their meet or their join defined in some universe U of separations containing → S . This holds, and is widely used, if U comes with a submodular order function and → S consists of all its separations up to some fixed order. 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 . An (unoriented) sepas separation systems. For separation universes, therefore, submodularity is used with the narrower meaning of being endowed with a submodular order function [8].

A k-block in a graph G is a maximal set of at least k vertices no two of which can be separated in G by deleting fewer than k vertices. The block number β(G) of G is the maximum integer k for which G contains a k-block.We prove a structure theorem for graphs without a (k+1)-block, showing that every such graph has a tree-decomposition in which every torso has at most k vertices of degree 2k 2 or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most 2k 2 .We also study k-blocks in graphs from classes of graphs G that exclude some fixed graph as a topological minor, and prove that every G ∈ G satisfies β(G) ≤ c 3 |G| for some constant c = c(G).Moreover, we show that every graph of tree-width at least 2k 2 has a minor containing a k-block. This bound is best possible up to a multiplicative constant.

We prove that, for all ℓ and s, every graph of sufficiently large treewidth contains either a complete bipartite graph Ks,s or a chordless cycle of length greater than ℓ.

Diestel and Müller showed that the connected tree-width of a graph G, i. e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of G and the largest length of a geodesic cycle in G. We improve their bound to one that is of correct order of magnitude. Finally, we construct a graph whose connected tree-width exceeds the connected order of any of its brambles. This disproves a conjecture by Diestel and Müller asserting an analogue of tree-width duality.

No abstract