Small cycle finding in a planar graph?

My first instinct is to use a method similar to a wall following maze solver. Essentially, follow edges, and always take the rightmost edge out of a vertex. Any cycles you encounter with this method will be boundaries of a face.

You'll have to keep track of which edges you've traversed in which direction. Once you've traversed an edge in both directions, you've identified the faces it separates. Once all edges have been traversed in both directions, you'll have identified all faces by their boundaries.

A "crossing edge", as you call it, is generally known as a chord. Thus, your problem is to find all chordless cycles.

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