### Motivation

There are at least three *interpretations* of an abstract polyhedral (= planar 3-vertex-connected) graph:

the

**1-skeleton of a convex polyhedron**(when embedded into $\mathbb{R}^3$)a

**polygonization of the sphere**(when embedded into the sphere $\mathbb{S}^2$)a

**polygonization of a polygon**– for each of its faces (when embedded into the plane $\mathbb{R}^2$)

In any case there are many geometric *realizations*:

of a polyhedron

of a polygonization of the sphere

of a polygon and its polygonizations

### Question

I'd like to understand in an abstract setting:

What do these interpretations and realizations have to do with each other?

### Example

For many (but not all) polyhedral graphs there is a realization as a convex polyhedron that is inscribable into the sphere. A central projection of this polyhedron back onto the sphere induces a polygonization of the sphere.

Taken for granted is Steinitz' theorem. The question is *not* about this.

**EDIT**: For completeness' sake I should mention embeddings of a polyhedral graph into:

the hyperbolic space $\mathbb{H}^2$

the 3-dimensional sphere $\mathbb{S}^3$

the hyperbolic space $\mathbb{H}^3$