Contracting a Single Element in a Transversal Matroid

TL;DR

This paper presents a polynomial-time algorithm to determine if contracting a single element in a transversal matroid results in another transversal matroid, and provides a transversal representation if it does.

Contribution

It introduces the first efficient method to decide single-element contraction preservation of transversality in matroids and offers a transversal representation when applicable.

Findings

Polynomial-time algorithm for single-element contraction check

Algorithm provides transversal representation when contraction preserves transversality

Discussion on potential extension to dual matroids

Abstract

It is well known that the class of transversal matroids is not closed under contraction or duality. In particular, after contracting a set of elements from a transversal matroid, the resulting matroid may or may not be transversal, and the computational complexity of determining whether it is transversal is not well understood. As a step toward resolving this, we provide a polynomial-time algorithm for determining whether a single-element contraction of a transversal matroid is transversal. In the case that the single-element contraction is transversal, the algorithm also provides a transversal representation. We then discuss possible applications of our techniques toward finding a polynomial-time algorithm to determine if the dual of a transversal matroid is transversal.