Set-theoretic equality in double dual graph

Question

Someone in the math fandom on tumblr was explaining the concept of a dual graph, and he was asked a tough question:

Is the dual of the dual [equal to] the original graph always, or just isomorphic?

Now, in the usual definition of "dual graph" that I am familiar with is rather informal, as is the one the original user was using to explain the concept. But it seems to me that if we are thinking of a plane graph as an abstract graph together with an embedding, it would be really hard to make a definition of the dual such that $G$ was literally, set-theoretically equal to $G^{**}$.

I'd be surprised if this question has any independent interest, but I was wondering if anyone knows a construction, or what sort of tools we might be able to use to construct an impossibility argument.

Answer 1

I believe what you are interested in is called a band decomposition -- here the vertices and faces of the graph are both sets of discs on the surface of a sphere, and the operation of taking the dual really is just changing which set of discs you call vertices, and so is in fact a set-theoretic involution (for connected graphs).