If the connected planar graph G has no edges, it is an isolated vertex and V+F-E=1+1-0=2. Otherwise, choose any edgeThis is the proof used by van Lint and Wilson. Other variants are possible -- ife. Ifeconnects two vertices, contract it, reducing V and E by one. Otherwise, it is a Jordan curve and separates two faces; remove it and reduce F and E by one. In either case the result follows by induction.

Proofs of Euler's Formula.

From the Geometry Junkyard,
computational
and recreational geometry pointers.

David Eppstein,
Theory Group,
ICS,
UC Irvine.

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