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.
Structural submodularity and tangles in abstract separation systems
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abstract
We prove a tangle-tree theorem and a tangle duality theorem for abstract separation systems $\vec S$ that are submodular in the structural sense that, for every pair of oriented separations, $\vec S$ contains either their meet or their join defined in some universe $\vec U$ of separations containing $\vec S$. This holds, and is widely used, if $\vec U$ comes with a submodular order function and $\vec S$ consists of all its separations up to some fixed order. Our result is that for the proofs of these two theorems, which are central to abstract tangle theory, it suffices to assume the above structural consequence for $\vec S$, and no order function is needed.
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math.CO 1years
2023 1verdicts
UNVERDICTED 1representative citing papers
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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.