Koebe–Andreev–Thurston circle packing theorem

Statement

If $G$ is a finite connected planar graph, then a circle packing in the plane whose tangency graph is isomorphic to $G$ exists. If $G$ has a fixed planar embedding, the circle packing can be chosen so that the cyclic order of tangencies around each circle corresponds to the order of edges around each vertex in the embedding. If $G$ is maximal planar, this packing is unique, up to reflections in lines and Möbius transformations.