Title: Tangle-tree duality in abstract separation systems,
Abstract: We prove a general width duality theorem for combinatorial structures with
well-defined notions of cohesion and separation. These might be graphs and
matroids, but can be much more general or quite different. The theorem asserts
a duality between the existence of high cohesiveness somewhere local and a
global overall tree structure.
We describe cohesive substructures in a unified way in the format of tangles:
as orientations of low-order separations satisfying certain consistency axioms.
These axioms can be expressed without reference to the underlying structure,
such as a graph or matroid, but just in terms of the poset of the separations
themselves. This makes it possible to identify tangles, and apply our
tangle-tree duality theorem, in very diverse settings.
Our result implies all the classical duality theorems for width parameters in
graph minor theory, such as path-width, tree-width, branch-width or rank-width.
It yields new, tangle-type, duality theorems for tree-width and path-width. It
implies the existence of width parameters dual to cohesive substructures such
as $k$-blocks, edge-tangles, or given subsets of tangles, for which no width
duality theorems were previously known.
Abstract separation systems can be found also in structures quite unlike
graphs and matroids. For example, our theorem can be applied to image analysis
by capturing the regions of an image as tangles of separations defined as
natural partitions of its set of pixels. It can be applied in big data contexts
by capturing clusters as tangles. It can be applied in the social sciences,
e.g. by capturing as tangles the few typical mindsets of individuals found by a
survey. It could also be applied in pure mathematics, e.g. to separations of
compact manifolds.