All pairs suffice

TL;DR

This paper proves that if every pair of vertices in a set is a P-set for a symmetric matrix, then the entire set is also a P-set, simplifying the identification of P-sets.

Contribution

It establishes that a set with all pairs as P-sets must itself be a P-set, providing a new criterion for P-set characterization.

Findings

Sets with all pairs as P-sets are themselves P-sets.

Simplifies the process of identifying P-sets in symmetric matrices.

Provides theoretical insight into the structure of P-sets.

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.