TITLE: Abstract characterization of polygonizations
QUESTION [5 upvotes]: 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
  sphere are 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 any surface?

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

REPLY [2 votes]: For the OP's claim re *infinite*polyhedral graphs, the answer is yes, this is true, and a proof is in my paper:
Rivin, Igor. "Combinatorial optimization in geometry." Advances in Applied Mathematics 31.1 (2003): 242-271.
Basically, you can construct a circle packing with any prescribed (three-connected) combinatorics. What you lose when you go from finite to infinite is uniqueness, in a spectacular way: it should be true that one can get the carrier of the packing to be any Jordan domain.