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.

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.