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All pairs suffice

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 $|α|$ 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.