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

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 ...

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 $|V| - |E| + |F| = 2$.