Hypergraph characterization of split matroids

TL;DR

This paper studies split matroids through hypergraph characterizations, introduces elementary split matroids, and explores their properties, including closure under various operations and a forbidden minor characterization, with applications to binary split matroids.

Contribution

It introduces elementary split matroids, provides a hypergraph characterization, and establishes their closure properties and forbidden minor characterization.

Findings

Elementary split matroids are characterized by hypergraphs.

The class is closed under duality, minors, and truncation.

Complete list of binary split matroids is provided.

Abstract

We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being closed under duality and taking minors. Furthermore, these matroids proved to be useful in giving exact asymptotic bounds for the dimension of the Dressian, and also implied new results on the rays of the tropical Grassmannians. In the present paper, we introduce the notion of elementary split matroids, a subclass of split matroids that contains all connected split matroids. We give a hypergraph characterization of elementary split matroids in terms of independent sets, and show that the proposed class is closed not only under duality and taking minors but also truncation. We further show that, in contrast to split matroids, the proposed class can be characterized by a single forbidden minor. As an application, we provide a complete list of binary split matroids.