General polytopes are [not determined by their edge-graph](https://mathoverflow.net/questions/309826/can-two-non-equivalent-polytopes-of-same-dimension-have-the-same-graph/380096#380096) (up to combinatorial equivalence). But I came accross the statement that [zonotopes](https://en.wikipedia.org/wiki/Zonohedron#Zonotopes) are determined in this way.



> 
> **Question:** Is this true? And where is this proven?
> 
> 
> 


I suppose that this is somehow proven in the language of [*oriented matroids*](https://en.wikipedia.org/wiki/Oriented_matroid) and their realizations, but I am not familiar with their literature.



Are zonotopes determined by their edge-graph?
Yes, the face lattice of a zonotope is determined by its graph. This is Theorem 6.14 of Bjorner, A., Edelman, P. H., and Ziegler, G. M. (1990). Hyperplane arrangements with a lattice of regions. Discrete Comput. Geom., 5(3):263–288.


The result uses the relation between hyperplane arrangements and zonotopes.


The other result on the reconstruction of zonotopes that I am aware of is that cubical zonotopes are reconstructed from their dual graph. This is in the paper E. Babson, L. Finschi, and K. Fukuda, Cocircuit graphs and ef- ficient orientation reconstruction in oriented matroids., Eur. J. Comb. 22, no. 5 (2001), pp. 587–600.


Hope this helps.


Regards,
Guillermo


