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

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 $\text{calM}(G)$ of a graph $G$, given its lattice of integer flows $\text{calF}(G)$. The algorithm can then be applied to compute any other 2-isomorphism invariants (that is, matroid invariants) of $G$ from $\text{calF}(G)$. Our method is based on a result of Amini which describes the relationship between the geometry of the Voronoi cell of $\text{calF}(G)$ and the structure of $G$.