Forbidden minor characterization of polytope skeletons

Say that a graph is "$$d$$-dimensional" if it is the node-disjoint union of $$1$$-skeletons of closed convex polytopes in $$d$$ dimensions, or a subgraph thereof. So the $$2$$-dimensional graphs are exactly those that are the disjoint union of cycles and paths, the $$3$$-dimensional graphs are exactly the planar graphs, and so on. (Is there a standard name for this?)

Clearly for any fixed $$d$$ the $$d$$-dimensional graphs are minor-closed, so by the graph minor theorem they can be characterized by finitely many forbidden minors. Is it known what the forbidden minor sets are for general $$d$$, or even for any $$d > 3$$? Alternately, is it known that the forbidden minor sets need to be unreasonably large (like how the torus needs $$>1000$$ forbidden minors)?

• Shall the $3$-dimensional graphs be $3$-connected? – Bullet51 Apr 12 at 7:29
• @Bullet51: No, since we are allowed to take subgraphs. – Ilya Bogdanov Apr 12 at 10:07
• I'm wondering whether there's any obstruction at all: one can take the skeleton of the 3d cubic lattice and embed any graph in it. – Bullet51 Apr 12 at 15:24