Let $G$ be a finite, simple, undirected graph.
What conditions on $G$ ensure that every rotation system of $G$ corresponds to an cellular embedding of $G$ on an oriened surface of small genus?
(e.g. genus at most 2)
In particular, what conditions on $G$ ensure that every rotation system of $G$ corresponds to a plane embedding of $G$?
Context: I am interested in rotation systems with a specific property (biparite dual), let us call it property $P$. I would like to find at least a small class of planar graphs $G$ such that it suffices to check all plane embeddings of $G$ in order to test existence of rotation systems of $G$ with property $P$.