Suppose we have a connected plane graph $G$, together with a rotation system, i.e. for each vertex $v$ in $G$ we have a cyclic permutation $\pi_v$ of the edges adjacent to $v$, which describes the order in which the edges appear around $v$, lets say in clockwise order. Does this rotation system already describe $G$ up to homeomorphism ?
I would be thankful for any answers or helpful references.
The short answer to my question would be no. For a counterexample look at the following plane graphs:
They are embeddings of the same planar graph, and their rotation systems agree. However their exterior face is different, thus there can not be any homeomorphism of the plane, sendeng one to the other. But we may notice that they are topologically equivalent.