Reconstructibility of matroid polytopes

TL;DR

This paper investigates the reconstructibility of matroid polytopes from their graphs and dual graphs, providing counterexamples, positive results, and an efficient algorithm for reconstructing vertices.

Contribution

It introduces the concept of class reconstructibility, shows that matroid polytopes are not reconstructible from their graphs but are class reconstructible, and provides a polynomial-time algorithm for vertex reconstruction.

Findings

Matroid polytopes are not reconstructible from their graphs.

Matroid polytopes are class reconstructible from their graphs.

An $O(n^3)$ algorithm for reconstructing vertices from the graph.

Abstract

We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present a $O(n^3)$ algorithm that computes the vertices of a matroid polytope from its $n$-vertex graph. Moreover, our proof includes a characterisation of all matroids with isomorphic basis exchange graphs.