When is an ordering of edges in a graph a planar embedding?

Question

Is there a criterion which decides whether a given rotation system for a graph determines a planar embedding? That is, a lemma of the form:

A graph G = (V, E) is planar if and only if there exists an ordering of the edges incident to v for all v such that ...

Answer 1

The ordering of edges around each vertex allows to find the cycles that bound the faces. In particular, one can compute the number of faces. The given ordering corresponds to a planar embedding if and only if $|V| - |E| + |F| = 2$.