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Let $G$ a planar graph and $e$ an edge of $G$. Describe a way (method) to give a planar embedding of the graph such that $e$ is in the outer face of the graph.

Any help?

Thank you!

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1 Answer

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Draw the graph on a sphere. Declare a point next to the chosen edge to be the north pole and project the sphere stereographically to a plane.

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Can you explain a little more your idea? – passenger Feb 10 '12 at 21:01
Draw your graph on the sphere. Your edge $e$ divides two faces. Choose one of these two faces. Now imagine the sphere made of very elastic rubben, make a small hole to the chosen face and blow into it. Enlarge the hole and flatten the balloon (the sphere) to adhere to the plane. The face you chosed is now the outer face of the graph. – Mario Stefanutti Feb 11 '12 at 18:41
It would help, passenger, if you would indicate just exactly what part(s) of Henning's answer require more explanation. – Gerry Myerson Feb 12 '12 at 2:15

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