Branch-depth is minor closure of contraction-deletion-depth
<|reference_start|>Branch-depth is minor closure of contraction-deletion-depth: The notion of branch-depth for matroids was introduced by DeVos, Kwon and Oum as the matroid analogue of the tree-depth of graphs. The contraction-deletion-depth, another tree-depth like parameter of matroids, is the number of recursive steps needed to decompose a matroid by contractions and deletions to single elements. Any matroid with contraction-deletion-depth at most d has branch-depth at most d. However, the two notions are not functionally equivalent as contraction-deletion-depth of matroids with branch-depth two can be arbitrarily large.   We show that the two notions are functionally equivalent for representable matroids when minor closures are considered. Namely, an F-representable matroid has small branch-depth if and only if it is a minor of an F-representable matroid with small contraction-deletion-depth. This implies that any class of F-representable matroids has bounded branch-depth if and only if it is a subclass of the minor closure of a class of F-representable matroids with bounded contraction-deletion-depth.<|reference_end|>
@article{briański2024branch-depth,
    title={Branch-depth is minor closure of contraction-deletion-depth},
    author={Marcin Bria'nski and Daniel Kr'av{l} and Krist'yna Pek'arkov'a},
    journal={arXiv preprint arXiv:2402.16215},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.16215},
    primaryClass={math.CO cs.DM}
}