Editing Planar graph (section)

===Dual graph===
[[Image:dual graphs.svg|thumb|100px|A planar graph and its [[Dual graph|dual]]]]

Given an embedding {{mvar|G}} of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the ''[[dual graph]]'' {{mvar|G*}} as follows: we choose one vertex in each face of {{mvar|G}} (including the outer face) and for each edge {{mvar|e}} in {{mvar|G}} we introduce a new edge in {{mvar|G*}} connecting the two vertices in {{mvar|G*}} corresponding to the two faces in {{mvar|G}} that meet at {{mvar|e}}. Furthermore, this edge is drawn so that it crosses {{mvar|e}} exactly once and that no other edge of {{mvar|G}} or {{mvar|G*}} is intersected. Then {{mvar|G*}} is again the embedding of a (not necessarily simple) planar graph; it has as many edges as {{mvar|G}}, as many vertices as {{mvar|G}} has faces and as many faces as {{mvar|G}} has vertices. The term "dual" is justified by the fact that {{math|1=''G''** = ''G''}}; here the equality is the equivalence of embeddings on the [[sphere]]. If {{mvar|G}} is the planar graph corresponding to a convex polyhedron, then {{mvar|G*}} 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.