All pairs suffice

Abstract

A P-set of a symmetric matrix $A$ is a set $\alpha$ of indices such that the nullity of the matrix obtained from $A$ by removing rows and columns indexed by $\alpha$ is $|\alpha|$ more than that 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.