TITLE: Visualizing polyhedra from their 1-skeletons
QUESTION [7 upvotes]: 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.]

REPLY [2 votes]: As has been noted one can draw diagrams of convex 3-dimensional polyhedra as graphs in the plane. Such diagrams are plane and 3-connected (Steinitz's Theorem) and are essentially unique but can be very different in visual appearance because one has the freedom to choose the infinite face. So if the polyhedron has a k-gon face one can draw a plane representation (with straight line edges and convex regions except for the infinite face) with a k-gon as the infinite face.