We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications.These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem. arXiv:1110.6207v5 [math.CO] 17 Apr 20171 The order of a separation {A, B} of a graph is the number |A ∩ B|. Separations of order k are k-separations; separations of order < k are (< k)-separations.2 Indeed, in Section 2 we shall introduce more abstract 'k-profiles', not necessarily induced by k-blocks, of which k-tangles -those of order k, in the terminology of [20] -are a prime example. See [1,2] for more on the relationship between profiles, blocks and tangles.

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph.As an application, we show that the k-blocks -the maximal vertex sets that cannot be separated by at most k vertices -of a graph G live in distinct parts of a suitable tree-decomposition of G of adhesion at most k, whose decomposition tree is invariant under the automorphisms of G. This extends recent work of Dunwoody and Krön and, like theirs, generalizes a similar theorem of Tutte for k = 2.Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all k simultaneously, all the k-blocks of a finite graph.

We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject to the requirement that they commute with graph isomorphisms. In particular, all the decompositions constructed are invariant under the automorphisms of the graph.

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 fewer than k other vertices. The block number β(G) of G is the largest integer k such that G has a k-block.We investigate how β interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a k-block, or which find all its k-blocks.The connectivity invariant β(G) has a dual width invariant, the blockwidth bw(G) of G. Our algorithms imply the duality theorem β = bw: a graph has a block-decomposition of width and adhesion < k if and only if it contains no k-block.

In Part I of this series we described three algorithms that construct canonical tree-decompositions of graphs which distinguish all their k-blocks and tangles of order k. We now establish lower bounds on the number of parts in these decompositions that contain such a block or tangle, and determine conditions under which such parts contain nothing but a k-block.