  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.

Combinatorics