  Circulant contraction minors play a key role for characterizing ideal
circular matrices in terms of minimally non ideal structures. In this article
we prove necessary and sufficient conditions for a circular matrix $A$ to have
circulant contraction minors in terms of circuits in a digraph associated with
$A$. In the particular case when $A$ itself is a circulant matrix, our result
provides an alternative characterization to the one previously known from the
literature.
