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Given: A planar undirected connected graph $G$ in which degree of every vertex is $2$ or more.

Fact: A planar graph can have multiple planar embeddings.

Question: Give an efficient algorithm to find the set $S$ of face cycles corresponding to any one of the planar embeddings. We do not care about from which specific embedding $S$ is made from.

For example, a graph $G$ has the following two planar embeddings:

Two Embeddings

The above illustration shows two of may be several other planar embeddings of the same graph.

S for Embedding1: $\{ CDFC, ABDCA, ACEA, BDEB, ABEA, DEGD \}$

S for Embedding2: $\{ CFDC, CDGEC, DEGD, EBDE, ECAE, EABE \}$

S for EmbeddingN: $\{ ... \}$

We want our algorithm to return any one of the above sets.

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    $\begingroup$ de Fraysseix, H.; Pach, J; and Pollack, R. "Small Sets Supporting Fáry Embeddings of Planar Graphs." Proc. of the 20th Symposium on the Theory of Computing. ACM, pp. 426-433, 1988. $\endgroup$
    – user16034
    Jul 17, 2023 at 10:18

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