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Lattice Path Matroids: Structural Properties

Joseph E. Bonin, Anna de Mier

Abstract

This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.

Lattice Path Matroids: Structural Properties

Abstract

This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.

Paper Structure

This paper contains 15 sections, 54 theorems, 15 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Basic Structural Properties of Lattice Path Matroids
  4. Minors, Duals, and Direct Sums
  5. Connectivity and Spanning Circuits
  6. Circuits and Connected Flats
  7. Generalized Catalan Matroids
  8. Fundamental Flats and a Characterization of Lattice Path Matroids
  9. Lattice Path Matroids as Transversal Matroids
  10. Maximal and Minimal Presentations
  11. Recognizing Lattice Path Matroids
  12. A Geometric Description of Lattice Path Matroids
  13. Relation to Other Classes of Transversal Matroids
  14. Higher Connectivity
  15. Notch Matroids and their Excluded Minors

Key Result

Lemma 1.1

Let $\mathcal{A}=(D_1,D_2,\ldots,D_k)$ be a presentation of a rank-$r$ transversal matroid $M$. If some basis of $M$ is a transversal of $(D_{i_1},D_{i_2},\ldots,D_{i_r})$, with $i_1<i_2<\cdots<i_r$, then $(D_{i_1},D_{i_2},\ldots,D_{i_r})$ is also a presentation of $M$.

Figures (15)

  • Figure 1: A lattice path presentation and geometric representation of a lattice path matroid.
  • Figure 2: Presentations of two lattice path matroids and their direct sum.
  • Figure 3: Presentations of a lattice path matroid and its dual.
  • Figure 4: The lattice path interpretation of the shortening of intervals that yields a presentation of a single-element deletion.
  • Figure 5: The shaded regions show the presentations of $M|F_i$, $M|G_j$, and $M|(F_i\cap G_j)$.
  • ...and 10 more figures

Theorems & Definitions (88)

  • Lemma 1.1
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • ...and 78 more