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General polytopes are not determined by their edge-graph (up to combinatorial equivalence). But I came accross the statement that 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 and their realizations, but I am not familiar with their literature.

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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

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