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

Addendum to Matroid Tree-Width

Hliněn´y and Whittle have shown that the traditional treewidth notion of a graph can be defined without an explicit reference to vertices, and that it can be naturally extended to all matroids. Unfortunately their original paper Matroid tree-width, European J. Combin. 27 (2006), 1117–1128, as pointed out by Isolde Adler in 2007, contained some incorrect arguments. It is the purpose of this addendum to correct the affected proofs. (All the theorems and results of the original paper remain valid.