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When I asked the question "cycles and faces in planar graphs", I learned that the numbers of vertices in the faces are not unique, if the planar graph is only $2$-connected.

My question now is : How can I find all possible embeddings (in particular, all possible sizes for the faces) for a given $2$-connected planar graph ?

Is it enough to know the cycles, or do I need more information ?

And finally, is there a straightfort method to construct an arbitary embedding ?

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