Geometric Dual Graph

Given a planar graph , its geometric dual  is constructed by placing a vertex in each region of  (including the exterior region) and, if two regions have an edge  in common, joining the corresponding vertices by an edge  crossing only . 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 (Harary 1994, p. 115), and so may simply be called "the" dual graph.

REFERENCES

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 113-115, 1994.