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Tangles, tree-decompositions and grids in matroids. (English) Zbl 1229.05070
Summary: A tangle in a matroid is an obstruction to small branch-width. In particular, the maximum order of a tangle is equal to the branch-width. We prove that: (i) there is a tree-decomposition of a matroid that “displays” all of the maximal tangles, and (ii) when \(M\) is representable over a finite field, each tangle of sufficiently large order “dominates” a large grid-minor. This extends results of Robertson and Seymour concerning Graph Minors.

05B35 Combinatorial aspects of matroids and geometric lattices
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C83 Graph minors
Full Text: DOI
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