Let $G$ be a graph embedded on an orientable surface. Given a class ${ \cal C}$ of facial circuits of $G$ as a forbidden class, we give a sufficient-necessary condition for that an $\alpha$-orientation (orientation with prescribed out-degrees) of $G$ can be transformed into another by a sequence of flips on non-forbidden circuits and further give an explicit formula for the minimum number of such flips. We also consider the connection among all $\alpha$-orientations by defining a directed graph ${ \bf D}({ \cal C})$, namely the ${ \cal C}$-forbidden flip graph. We show that if ${ \cal C}\not=\emptyset$, then ${ \bf D}({ \cal C})$ has exactly $|O(G,{ \cal C})|$ components, each of which is the cover graph of a distributive lattice, where $|O(G,{ \cal C})|$ is the number of the $\alpha$-orientations that has no counterclockwise facial circuit other than that in ${ \cal C}$. If ${ \cal C}=\emptyset$, then every component of ${ \bf D}({ \cal C})$ is strongly connected. This generalizes the corresponding results of Felsner and Propp for the case that ${ \cal C}$ consists of a single facial circuit.
Categories: math.CO