Another (less known) characterization of planar graphs is Schnyder's theorem, which characterizes planar graphs according to order dimension. That is, a graph is planar if and only if its incidence poset has order dimension at most 3.

Also, I would be remiss to not mention the beautiful (strong) Hanani-Tutte theorem:

A graph is planar if and only if it has a drawing in the plane such that every two non-adjacent edges cross an even number of times.

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Another (less known) characterization of planar graphs is Schnyder's theorem, which characterizes planar graphs according to order dimension. That is, a graph is planar if and only if its incidence poset has order dimension at most 3.

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Another (less known) characterization of planar graphs is Schnyder's theorem

http://en.wikipedia.org/wiki/Schnyder's_theorem, which characterizes planar graphs according to order dimension. That is, a graph is planar if and only if its incidence poset has order dimension at most 3.

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