Every k-tangle in a graph is the lift of a k-tangle in a topological minor of size bounded by a function of k, reducing the vertex-set induction problem to bounded-size graphs.
Tree sets
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abstract
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and matroids etc. Unlike graph-theoretical or order trees, these _tree sets_ can provide a suitable formalization of tree structure also for infinite graphs, matroids, and set partitions. Order trees reappear as oriented tree sets. We show how each of the above structures defines a tree set, and which additional information, if any, is needed to reconstruct it from this tree set.
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2024 1verdicts
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On vertex sets inducing tangles
Every k-tangle in a graph is the lift of a k-tangle in a topological minor of size bounded by a function of k, reducing the vertex-set induction problem to bounded-size graphs.