Dual graph

for each plane region, and an edge for each edge joining two neighboring regions (see diagram).

= Properties =

The dual of a plane graph is always a plane graph.

Since dualizing graphs changes regions into vertices, vertices into regions, and edges into edges, it follows that if G′ is the dual of G then G is the dual of G′

Dual graphs are not unique, in the sense that a same graph can have non-isomorphic dual graphs (since the dual graph depends on a particular plane embedding). In the picture, G′ and G″ are not isomorphic because G′ has a vertex with order 7 (the outer region) but G″ doesn t (see diagram).

Because of the dualism, any result involving counting regions and vertices can be dualized exchanging them. Moreover, dualism is deeply related with the symmetric roll of faces and vertices of Euler characteristic for polyhedron.