Multi-Path Matroids

Abstract

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.

Figures (10)

• Figure 1: A lattice path presentation and geometric representation of a lattice path matroid.
• Figure 2: The $3$-whirl $\mathcal{W}^3$ as a multi-path matroid.
• Figure 3: A multi-path matroid that has multiple minimal presentations.
• Figure 4: The diagram $D(\mathcal{W}^3,1)$ with the labels on the North steps.
• Figure 5: The dual of the $3$-whirl $\mathcal{W}^3$ via flipping diagrams about the line $y=x$.

Theorems & Definitions (37)

• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• Lemma 3.4
• proof