Tangles, tree-decompositions and grids in matroids

Authors:
Jim Geelen;Bert Gerards;Geoff Whittle
Affiliations:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada;Centrum Wiskunde & Informatica, Amsterdam, The Netherlands and Technische Universiteit Eindhoven, Eindhoven, The Netherlands;School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand
Venue:
Journal of Combinatorial Theory Series B
Year:
2009

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Abstract

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.