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 (1932) showed that the geometric dual graph and combinatorial dual graph are equivalent for planar graphs (Fleischner
1973; Harary 1994, p. 115), and so may simply be called "the" dual
graph.

Fleischner, H. "The Uniquely Embeddable Planar Graphs." Disc. Math.4, 347-358, 1973.Harary, F. Graph
Theory. Reading, MA: Addison-Wesley, pp. 113-115, 1994.Whitney,
H. "Congruent Graphs and the Connectivity of Graphs." Amer. J. Math.54,
150-168, 1932.