Nested sequences of principal minors and potential stability

Sufficient conditions are determined for an n-by-n zero-nonzero pattern to allow a nested sequence of nonzero principal minors. A method is then given to sign such a pattern so that it allows a nested sequence of k-by-k principal minors with sign (−1)k for 1 k n. It is a classical result that a sign pattern with such a sequence is potentially stable; thus classes of potentially stable sign patterns can be identified. A necessary and sufficient condition for a certain tree sign-pattern class to be potentially stable is also given.