By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. their face lattices are isomorphic.


I know that two polytopes can have isomorphic graphs while being non-equivalent, e.g. neighborly polytopes. However, all examples I know of are polytopes of different dimension. So I wonder:



> 
> **Question:** Can there be two non-equivalent polytopes of the same dimension with the same graph?
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Especially, are all $k$-neighborly polytopes of the same dimension equivalent?



Can two non-equivalent polytopes of same dimension have the same graph?
There are many non-equivalent neighborly polytopes, already in dimension $d=4$. See for example


* [Arnau Padrol. *"Many Neighborly Polytopes and Oriented Matroids"*](https://link.springer.com/article/10.1007/s00454-013-9544-7)


