The Rank Partition and the Covering Number of the Elements of the Dual Matroid

We present results that characterize the covering number and the rank partition of the dual of a matroid \(M\) using properties of \(M\). We prove, in particular, that the elements of covering number \(2\) in \(M^*\) are the elements of the closure of the maximal \(2\)-transversals of \(M\).

From the results presented it can be seen that every matroid \(M\) is a weak map image of a transversal matroid with the same rank partition.