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A new minor-closed class of transversal matroids

Gerry Toft

TL;DR

This work characterises when a single-element contraction $M/e$ of a transversal matroid $M$ remains transversal by linking it to the structure of minimal $(e,\mathcal{A})$-presenting graphs; in particular, $M/e$ is transversal iff these graphs are trees, and it provides an explicit presentation of $M/e$ when this holds. Building on this, the authors define path-circular matroids $M(\mathcal{P})$ from path-circular collections on a graph, and prove that contracting a path yields another path-circular matroid, establishing that this new class is closed under minors. Path-circular matroids generalise bicircular and multi-path matroids, unifying these minor-closed families within a single framework. The results supply structural tools for analyzing minors of transversal matroids and introduce a robust new minor-closed transversal class with potential for broader applications in matroid theory.

Abstract

We provide a characterisation of when a single-element contraction of a transversal matroid is itself transversal. Using this characterisation, we define a new class of transversal matroids closed under minors, which we call path-circular matroids. Path-circular matroids generalise both of the well-known classes of bicircular matroids and multi-path matroids.

A new minor-closed class of transversal matroids

TL;DR

This work characterises when a single-element contraction of a transversal matroid remains transversal by linking it to the structure of minimal -presenting graphs; in particular, is transversal iff these graphs are trees, and it provides an explicit presentation of when this holds. Building on this, the authors define path-circular matroids from path-circular collections on a graph, and prove that contracting a path yields another path-circular matroid, establishing that this new class is closed under minors. Path-circular matroids generalise bicircular and multi-path matroids, unifying these minor-closed families within a single framework. The results supply structural tools for analyzing minors of transversal matroids and introduce a robust new minor-closed transversal class with potential for broader applications in matroid theory.

Abstract

We provide a characterisation of when a single-element contraction of a transversal matroid is itself transversal. Using this characterisation, we define a new class of transversal matroids closed under minors, which we call path-circular matroids. Path-circular matroids generalise both of the well-known classes of bicircular matroids and multi-path matroids.

Paper Structure

This paper contains 8 sections, 23 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Basic Results
  3. Transversal Matroids
  4. Cyclic Flats
  5. Presentations
  6. Mason's $\alpha$-Criterion
  7. Single-Element Contractions
  8. Path-Circular Matroids

Key Result

Theorem 1.1

Let $M$ be a transversal matroid with presentation $\mathcal{A}$ such that $|\mathcal{A}| = r(M)$, and let $e \in E(M)$. Let $G$ be a minimal $(e, \mathcal{A})$-presenting graph. Then $M / e$ is transversal if and only if $G$ is a tree.

Figures (3)

  • Figure 1: Minimal $(e, \mathcal{A}_i)$-presenting graphs.
  • Figure 2: A multi-path matroid with ground set given by the black vertices (A), and an isomorphic path-circular matroid (B).
  • Figure 3: A path-circular matroid (A), and the contraction of the dashed path (B). The new vertices are shown in grey.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 28 more