that kiss. The circle packing theorem, first proved by Paul Koebe in 1936, states that a graph is planar if and only if it is a coin graph. This result provides an easy proof of Fáry's theorem, that every simple planar graph can be embedded in the plane in such a way that its edges are straight line segments that do not cross each other. If one places each vertex of the graph at the center of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do not cross any of the other edges. Planar graph density The meshedness coefficient or density of a planar graph, or network, is the ratio of the number of bounded faces (the same as the circuit rank of the graph, by Mac Lane's planarity criterion) by its maximal possible values for a graph with vertices: The density obeys , with for a completely sparse planar graph (a tree), and for a completely dense (maximal) planar graph. Dual graph Given an embedding of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the dual graph as follows: we choose one vertex in each face of (including the outer face) and for each edge in we introduce a new edge in connecting the two vertices in corresponding to the two faces in that meet at . Furthermore, this edge is drawn so that it crosses exactly once and that no other edge of or is intersected. Then is again the embedding of a (not necessarily simple) planar graph; it has as many edges as , as many vertices as has faces and as many faces as has vertices. The term "dual" is justified by the fact that ; here the equality is the equivalence of embeddings on the sphere. If is the planar graph corresponding to a convex polyhedron, then is the planar graph corresponding to the dual polyhedron. Duals are useful because many properties of the dual graph are related in simple ways to properties of the original graph, enabling results to be proven about graphs by examining their dual graphs. While the dual constructed for a particular embedding is unique (up to isomorphism), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-homeomorphic) embeddings. Families of planar graphs Maximal planar graphs A simple graph is called maximal planar if