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Lattice path matroids: the excluded minors

Joseph E. Bonin

Abstract

A lattice path matroid is a transversal matroid for which some collection of incomparable intervals in some linear order on the ground set is a presentation. We characterize the minor-closed class of lattice path matroids by its excluded minors.

Lattice path matroids: the excluded minors

Abstract

A lattice path matroid is a transversal matroid for which some collection of incomparable intervals in some linear order on the ground set is a presentation. We characterize the minor-closed class of lattice path matroids by its excluded minors.

Paper Structure

This paper contains 10 sections, 44 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Connected and cyclic flats
  4. Lattice path matroids
  5. Parallel connections
  6. The Excluded Minors of Lattice Path Matroids
  7. Parallel connections
  8. Connectivity
  9. Fundamental flats
  10. The last step

Key Result

Lemma 2.1

A loopless matroid is determined by its collection of nontrivial connected flats and their ranks.

Figures (2)

  • Figure 1: The excluded minors of lattice path matroids. Those in the top row are in infinite families of excluded minors.
  • Figure 2: Two minors that arise in the proof of Lemma \ref{['lem:lpmpc2']}.

Theorems & Definitions (60)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • proof
  • Lemma 2.9
  • ...and 50 more