A single theorem showing that any efficient k-tangle-distinguishing tree-decomposition of a graph can be refined so each part is either too small for a k-tangle or minimal while containing one.
Duality theorems for blocks and tangles in graphs
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abstract
We prove a duality theorem applicable to a a wide range of specialisations, as well as to some generalisations, of tangles in graphs. It generalises the classical tangle duality theorem of Robertson and Seymour, which says that every graph either has a large-order tangle or a certain low-width tree-decomposition witnessing that it cannot have such a tangle. Our result also yields duality theorems for profiles and for $k$-blocks. This solves a problem studied, but not solved, by Diestel and Oum and answers an earlier question of Carmesin, Diestel, Hamann and Hundertmark.
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math.CO 1years
2023 1verdicts
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Optimal trees of tangles: refining the essential parts
A single theorem showing that any efficient k-tangle-distinguishing tree-decomposition of a graph can be refined so each part is either too small for a k-tangle or minimal while containing one.