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.



Minimal combinatorial data needed to define a polytope
I don't know conditions of existence ($d$-connectivity is necessary), but the uniqueness was proved by Blind and Mani in 1987, see also the 1988 article "A simple way to tell a simple polytope from its graph" by Gil Kalai. Kalai's proof is also presented in Guenter Ziegler's book "Lectures on polytopes".


The simplicity assumption is important: for $d \ge 4$ there are different combinatorial classes of $1$-neighborly polytopes (polytopes whose $1$-skeleton is a complete graph).


