Lattice of Integer Flows and the Poset of Strongly Connected Orientations for Regular Matroids

TL;DR

This paper generalizes Amini's 2010 result relating the geometry of the Voronoi polytope of integer flow lattices to the poset of strongly connected orientations, extending it from graphs to regular matroids.

Contribution

It extends Amini's theorem from graphs to regular matroids, explicitly incorporating matroid duality to unify the results for integer flows and cuts.

Findings

Generalization of Amini's theorem to regular matroids

Explicit duality between integer flows and cuts in this context

Unified description of face posets and orientation posets for regular matroids

Abstract

A 2010 result of Amini provides a way to extract information about the structure of the graph from the geometry of the Voronoi polytope of the lattice of integer flows (which determines the graph up to two-isomorphism). Specifically, Amini shows that the face poset of the Voronoi polytope is isomorphic to the poset of strongly connected orientations of subgraphs. This answers a question raised by Caporaso and Viviani, and Amini also proves a dual result for integer cuts. In this paper we generalise Amini's result to regular matroids; in this context the theorem for integer cuts becomes a direct consequence of the theorem for integer flows, by making duality explicit as matroid duality.