Matrix Representations of Transversal Matroids

TL;DR

This paper characterizes matrices associated with maximal presentations of transversal matroids, provides an algorithm for their construction, and links the problem to tropical rank determination.

Contribution

It offers a complete characterization of matrices from maximal presentations of transversal matroids and connects the problem to tropical rank computation.

Findings

Characterization of matrices from unique maximal presentations for rank 2

Algorithm for constructing such matrices

Relation between maximal presentation matrices and tropical rank

Abstract

A transversal matroid $M$ of rank $r$ on $[n]$ can be associated to a family of binary matrices corresponding to different presentations of $M$. We describe those matrices which arise from unique maximal presentations of size $r$- giving a complete characterization when $r=2$-, give an algorithm for producing such a matrix, and show that determining whether a matrix corresponds to a maximal presentation is related to determining the tropical rank of arbitrary matrices.