The contraction$^*$-depth is the matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted forest whose closure contains $G$ by proving the following for every matroid $M$ (except the trivial case when $M$ consists of loops and bridges only): the contraction$^*$-depth of $M$ plus one is equal to the minimum contraction-depth of a matroid containing $M$ as a restriction.

翻译：收缩$^*$-深度是与图的树深度相对应的拟阵深度参数。我们建立了经典图论结果的拟阵类比，该结果断言图$G$的树深度是其闭包包含$G$的有根森林的最小高度，通过证明对于每个拟阵$M$（除了$M$仅由环和桥组成的平凡情况）：$M$的收缩$^*$-深度加一等于包含$M$作为限制的拟阵的最小收缩深度。