Variants on the minimum rank problem: A survey II

TL;DR

This survey paper reviews recent advances in the minimum rank problem for graphs, including variants like positive semidefinite and zero forcing parameters, highlighting key developments since previous surveys.

Contribution

It provides a comprehensive overview of new results and variants in the minimum rank problem for graphs, updating the field since prior surveys.

Findings

Discussion of positive semidefinite minimum rank

Analysis of zero forcing parameters

Overview of minimum rank for patterns

Abstract

The minimum rank problem for a (simple) graph $G$ is to determine the smallest possible rank over all real symmetric matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. This paper surveys the many developments on the (standard) minimum rank problem and its variants since the survey paper \cite{FH}. In particular, positive semidefinite minimum rank, zero forcing parameters, and minimum rank problems for patterns are discussed.