Obstructions and dualities for matroid depth parameters

TL;DR

This paper explores structural properties of matroid depth parameters, establishing bounds on obstructions for contraction*-, contraction-, and deletion-depth, and introduces a dual notion called deletion*--depth for finite field representable matroids.

Contribution

It proves bounds on obstructions for contraction*-depth in certain matroids and extends these results to related depth notions and their duals.

Findings

Obstructions for contraction*-depth are bounded in size.

Results extend to contraction- and deletion-depth.

Introduces and analyzes deletion*-depth as a dual concept.

Abstract

Contraction$^*$-depth is considered to be one of the analogues of graph tree-depth in the matroid setting. In this paper, we investigate structural properties of contraction$^*$-depth of matroids representable over finite fields and rationals. In particular, we prove that the obstructions for contraction$^*$-depth for these classes of matroids are bounded in size. From this we derive analogous results for related notions of contraction-depth and deletion-depth. Moreover, we define a dual notion to contraction$^*$-depth, named deletion$^*$-depth, for $\mathbb{F}$-representable matroids, and by duality extend our results from contraction$^*$-depth to this notion.