graph theory – Minor prohibited characterization of polytopic skeletons

Say that a graph is "$$d$$"dimensional" if it is the node-disjoint union of $$1$$-bones of convex polytopes closed in $$d$$ dimensions, or a sub-graph of them. Therefore the $$2$$Three-dimensional graphs are exactly those which are the disjoint union of cycles and paths, the $$3$$The three-dimensional graphs are exactly the planar graphs, and so on. (Is there a standard name for this?)

Clearly for any fixed $$d$$ the $$d$$Three-dimensional graphs are in the minority. Thus, they can be characterized by the minimal number of minors prohibited. Do we know what sets of minors are banned for the general public? $$d$$, or even for everything $$d> 3$$? Alternatively, do we know that sets of prohibited minors must be excessively bulky (as is the case for the torus)? $$> 1000$$ minors banned)?