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(n3) 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.Comment: 22 pages, 5 figure