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Is it possible to distinguish planar and non-planar graphs (networks as a matter of fact) by flows? That is, is there a flow criterion for a graph being planar or not?

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I doubt that. A graph is planar iff it doesn not contain a subdivision of $K_5$ or $K_{3,3}$. I don't see how flow enters there easily. –  Hagen von Eitzen Jan 25 '13 at 10:03

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