All pairs suffice for a P-set

A P-set of a symmetric matrix A is a set α of indices such that the nullity of the matrix obtained from A by removing rows and columns indexed by α   is |α||α| more than the nullity of A. It is known that each subset of a P-set is a P-set. It is also known that a set of indices such that each singleton subset is a P-set need not be a P-set. This note shows that if all pairs of vertices of a set with at least two elements are P-sets, then the set is a P-set.