Tools that are able to take an arbitrary polyhedral graph as input and draw the corresponding polyhedron perspectively will most surely rely on an abstract representation of the graph, e.g. by its adjacency matrix. From this abstract representation - presumably - they will also draw the embedded version of the graph (without edges crossing).
I am interested in the underlying algorithms and/or heuristics of
drawing the embedded graph from the adjacency matrix
drawing the polyhedron from the adjacency matrix
drawing the polyhedron from the embedded graph
drawing the embedded graph from the polyhedron
I am asking for references.
Computer programs will most certainly deal with (1) and (2) while humans typically have to solve problems (3) and (4).
I guess that experts have some mental techniques to visualize a polyhedron from looking at its 1-skeleton.
Can these techniques be described, made explicit, and taught?
[Side question: If anyone could give me a visualization of the hexahedral graph 5, I would be thankful.]