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Comment by jurjenhaitsma | Hacker News Reader

This is pretty choice - reminds me of untangling a circuit layout... Never really looked into it that much, but I'd imagine there must be a mathematical technique that can be applied. My first instinct has always been to put the point with the most connections in the centre, then map out from that.

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Take the vertex whose vertices cross the biggest number of other vertices and move it. Repeat until nothing crosses. I remember this problem in maths ages ago, there is a way to know if you can't untangle something- look up the water gas electricity problem

A graph is nonplanar if and only if it has a subgraph that looks like either (1) five vertices all connected to one another or (2) two sets of three vertices, with every vertex in one set connected to every vertex in the other.

"Looks like" means: can be turned into by repeatedly replacing -o- (a vertex with two edges coming out of it) with -- (a single edge between the vertex's two old neighbours).

See http://en.wikipedia.org/wiki/Planar_graph for details.

Good luck. The problem may be NP complete in general.

Well, for the cases on planarity.net, it seems O(n) to me. Or maybe something like O(n log n)

It's O(n) in the general case, so certainly also for the special cases on planarity.net.

Could you give me a source or the name of the algorithm, that allows this running time?

P.S. I found a O(n) running time for planarity testing - but not embedding.

The Wikipedia article on planarity testing cites, for instance, this paper: http://jgaa.info/accepted/2004/BoyerMyrvold2004.8.3.pdf which describes a not-insanely-complicated algorithm that constructs planar embeddings (or proves they don't exist) in linear time.

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