Except for a few simple cases (typically pyramids and prisms) I find it hard to visualize a polyhedron from its 1-skeleton embedded in the plane, e.g. the hexahedral graph 5, as can be seen here.

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.]*

thepolyhedron" suggests you might be forgetting that these graphs are realized by an infinite variety of combinatorially equivalent polyhedra. – Joseph O'Rourke Jan 21 '13 at 13:18