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Let $P$ be the space of all possible planar graphs.

Fact: A planar graph may have multiple duals based on its embedding.

Let $D_p$ be the set of all possible duals of a planar graph $p\in P$

Let $S$ be the space given by union of $D_p$ for all $p \in P$

Is $S = P ?$

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If the dual of the dual of a planar graph is isomorphic to the original graph, then yes, it must be. Let $G$ be a planar graph and $G^*$ its dual. Then, since $G^*$ is a planar graph, then $G^{**}$ is its dual. Therefore, any graph $G$ that is planar or the dual of a planar belongs in both $S$ and $P$:

If $G \in P$, then $G^* \in S$. But clearly $S \subseteq P$, so $G^* \in P$. Then also $G^{**} \in S$, but since $G \simeq G^{**}$, $G \in S$ as well, hence $P \subseteq S$.

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