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.
Profiles of separations: in graphs, matroids and beyond
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abstract
We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications. These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem.
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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.