Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the adjacencies with 1-dimensional faces? I do not know if that is important, but each and every vertex has exactly d adjacent vertices, and I'd expect the polytope to be simple.

I should add some information. The data I gave above is represented by a graph of which I know 1) it's connected 2) every vertex has the same valence d. I may come up with further constraints but these are the first I can think of.