Tangles, tree-decompositions and grids in matroids

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.

[1]  Bert Gerards,et al.  Excluding a planar graph from GF(q)-representable matroids , 2007, J. Comb. Theory, Ser. B.

[2]  J. S. Dharmatilake.  A min-max theorem using matroid separations , 1996 .

[3]  W. T. Tutte.  Menger's theorem for matroids , 1965 .

[4]  Bert Gerards,et al.  On the excluded minors for the matroids of branch-width k , 2003, J. Comb. Theory, Ser. B.

[5]  B. Mohar,et al.  Graph Minors , 2009 .

[6]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[7]  Paul D. Seymour,et al.  Graph Minors. XVI. Excluding a non-planar graph , 2003, J. Comb. Theory, Ser. B.

[8]  Paul D. Seymour,et al.  Graph minors. X. Obstructions to tree-decomposition , 1991, J. Comb. Theory, Ser. B.

[9]  James G. Oxley,et al.  Matroid theory , 1992 .

[10]  Bert Gerards,et al.  Obstructions to branch-decomposition of matroids , 2006, J. Comb. Theory, Ser. B.