Hello everyone
I'm writing a problem for a contest in my college, so I need generate a planar graph. I need to make sure the graph is planar. Is there any trick to generate such graphs?
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Kuratowski's theorem gives a sufficient and necessary condition for a graph being planar. I'm unsure though how could you check that.
There is a O(n^2) algorithm for detecting if a graph is minor of an other graph. But probably its not something easy to implement for a contest !
There is Gamma-algorithm, try to google it, cuz I found only russian tutorial version
Well, when I had the problem like this, I used random dots and two treaps. You can connect dots as in a Cartesian tree, then invert priority, and connect dots again. So you have the planar graph in O(N) time.
If you want more edges, I guess it can be done from geometry easily.