A construction of the graphic matroid from the lattice of integer flows

TL;DR

This paper presents an algorithm to reconstruct the graphic matroid of a graph from its lattice of integer flows, enabling computation of various graph invariants based on geometric properties.

Contribution

It introduces a novel algorithmic method to derive the graphic matroid from the lattice of integer flows, expanding tools for graph isomorphism analysis.

Findings

Algorithm successfully reconstructs the graphic matroid from the lattice of integer flows.

Enables computation of 2-isomorphism invariants of graphs from lattice data.

Utilizes geometric properties of Voronoi cells related to the lattice.

Abstract

The lattice of integer flows of a graph is known to determine the graph up to 2-isomorphism (work of Su--Wagner and Caporaso--Viviani). In this paper we give an algorithmic construction of the graphic matroid $\calM(G)$ of a graph $G$, given its lattice of integer flows $\calF(G)$. The algorithm can then be applied to compute any other 2-isomorphism invariants (that is, matroid invariants) of $G$ from $\calF(G)$. Our method is based on a result of Amini which describes the relationship between the geometry of the Voronoi cell of $\calF(G)$ and the structure of $G$.