Given a planar graph G, its geometric dual G^* is constructed by placing a vertex in each region of G (including the exterior region) and, if two regions have an edge x in common, joining the corresponding vertices by an edge X^* crossing only x. The result is always a planar pseudograph. However, an abstract graph with more than one embedding on the sphere can give rise to more than one dual. Whitney showed that the geometric dual graph and combinatorial dual graph are equivalent for planar graphs, and so may simply be called "the" dual graph.

combinatorial dual graph | dual graph | uniquely embeddable graph