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arXiv · 1701.02651

Tangle-tree duality: in graphs, matroids and beyond

Reinhard Diestel·Sang-il Oum
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Abstract

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.

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Reinhard Diestel, Sang-il Oum. 2017-01-10. Tangle-tree duality: in graphs, matroids and beyond. https://doi.org/10.1007/s00493-019-3798-5

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