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I encountered many problems that can be formulated as graph problem.
It is in general NP-hard but sometimes the graph can be proved to be planar.
Hence, I am interested in learning these problems and the algorithms.
So far as I know:
Hope someone can complete this list.
In this compendium of NP-complete problems, under planar in the index there are a good number (~25) of entries. These entries typically link to problems where planar input admits a PTAS.