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# 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 (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.

Combinatorial Dual Graph, Dual Graph, Uniquely Embeddable Graph

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## References

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.

## Referenced on Wolfram|Alpha

Geometric Dual Graph

## Cite this as:

Weisstein, Eric W. "Geometric Dual Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeometricDualGraph.html