Consider a polygonization of the plane by convex polygons of a given minimal size that meet edge-to-edge and vertex-to-vertex.

^{What's the “official” name of such a polygonization?}

Such polygonizations of the plane induce infinite graphs.

How can such abstract graphs be characterized?

Somehow like this: “A graph is induced by a polygonization of the plane iff it is infinite, planar, 3-vertex-connected, and *P*.” (The question asks for property *P*, since infinite, planar and 3-vertex-connected those graphs obviously are.)

Is it true, that the graphs that are induced by a polygonization of the

sphereare exactly the polyhedral graphs which in turn are exactly the finite planar 3-vertex-connected graphs?

Finally I want to know:

Can the graphs be characterized that are induced by a polygonization of

anysurface?

^{For the record: I asked this question at MSE before but it didn't earn a lot of interest.}