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Let $G$ be a graph embedded on an orientable surface. Given a class ${\calC}$ of facial circuits of $G$ as a forbidden class, we give asufficient-necessary condition for that an $\alpha$-orientation (orientationwith prescribed out-degrees) of $G$ can be transformed into another by asequence of flips on non-forbidden circuits and further give an explicitformula for the minimum number of such flips. We also consider the connectionamong all $\alpha$-orientations by defining a directed graph ${\bf D}({\calC})$, namely the ${\cal C}$-forbidden flip graph. We show that if ${\calC}\not=\emptyset$, 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 nocounterclockwise facial circuit other than that in ${\cal C}$. If ${\calC}=\emptyset$, then every component of ${\bf D}({\cal C})$ is stronglyconnected. This generalizes the corresponding results of Felsner and Propp forthe case that ${\cal C}$ consists of a single facial circuit.

Flips on homologous orientations of surface graphs with prescribed forbidden facial circuits