Closure property of contraction-depth of matroids

Contraction⁎-depth is a 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 coloops only): the contraction⁎-depth of M plus one is equal to the minimum contraction-depth of a matroid containing M as a restriction.