48 research outputs found
Tangle-tree duality in abstract separation systems
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 -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.Comment: We have expanded Section 2 on terminology for better readability,
adding explanatory text, examples, and figures. This paper replaces the first
half of our earlier paper arXiv:1406.379
Limit-closed Profiles
Tangle-tree theorems are an important tool in structural graph theory, and
abstract separation systems are a very general setting in which tangle-tree
theorems can still be formulated and proven. For infinite abstract separation
systems, so far tangle-tree theorems have only been shown for special cases of
separation systems, in particular when the separation system arises from a
(locally finite) infinite graph. We present a tangle-tree theorem for infinite
separation systems where we do not place restrictions on the separation system
itself but on the tangles to be arranged in a tree.Comment: 12 pages, 2 figure
Tangle-tree duality: in graphs, matroids and beyond
We apply a recent duality theorem for tangles in abstract separation systems
to derive tangle-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
duality theorem for tree-width.
Our results can 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.Comment: arXiv admin note: text overlap with arXiv:1406.379