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.

翻译：对于拟阵的分支深度概念由DeVos、Kwon和Oum引入，作为图树深度的拟阵类比。收缩-删除深度是拟阵的另一个类树深度参数，指通过收缩和删除操作将拟阵分解为单个元素所需的递归步数。任何收缩-删除深度不超过d的拟阵，其分支深度至多为d。然而，这两个概念并非功能等价，因为分支深度为2的拟阵其收缩-删除深度可以任意大。我们证明，在考虑子式闭包时，对于可表示拟阵，这两个概念是功能等价的。具体而言，一个F-可表示拟阵具有小分支深度当且仅当它是某个具有小收缩-删除深度的F-可表示拟阵的子式。这意味着，任何F-可表示拟阵类具有有界分支深度当且仅当它包含于某个具有有界收缩-删除深度的F-可表示拟阵类的子式闭包中。