Let us be given a topological graph $G$
on the unit sphere in $\mathbb{R}^3$
whose edges are minor arcs of great circles.
We suppose that the graph is $3$-vertex-connected
and that a pair of edges may only share a vertex incident to both edges.

We say that a polyhedron $P$ is a lifting of $G$
iff
$G$ is obtained by radial projection of the edge-skeleton of $P$ onto the unit sphere.

My question is whether there is a simple criterion to check
if a given spherical graph possesses a lifting.
Moreover, I would like to compute that lifting (if it exists).
(Computationally, I am interested in some spherical graphs with about $10$ to $20$ vertices)

The analogue planar question is classical and well-understood.
That is, a planar graph, i.e., a graph with vertices in $\mathbb{R}^2$ and straight edges,
can be lifted into a polyhedron,
iff it is possible to find an equilibrium configuration of forces,
see
[here][1].


  [1]: http://https://en.wikipedia.org/wiki/Steinitz's_theorem#Lifting