Obstructions for bounded branch-depth in matroids

Abstract

© 2021 J. Pascal Gollin, Kevin Hendrey, Dillon Mayhew, and Sang-il Oum cb Licensed under a Creative Commons Attribution License (CC-BY).DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un,2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.