Some constraints for the graphs of convex polytopes are well known:
In 3 dimensions, the graph must be planar and 3-connected (actually a complete characterization, which likely cannot be extended to higher dimensions).
In $k$ dimensions, it must be $k$-connected.
What are other constraints?
I feel that there should be many more such constraints, but I cannot find any reference, so any hint or pointer would be really appreciated.
Such constraints define the realizability of a graph as a $d$-dimensional polytope. Some such constraints can be found in Grünbaum, Convex Polytopes, 2003, Chapter 11.