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

Abstract

Given a class C of facial circuits of a graph G embedded on an orientable surface as a forbidden class, we give a necessary-sufficient condition for that an α-orientation (orientations with prescribed outdegrees) of G can be transformed into another by a sequence of flips taking on the non-forbidden circuits and further give an explicit formula for the minimum number of flips needed in such a sequence. In particular, if C is empty then two α-orientations can be transformed from each other if and only if they are homologous. Further, if C is not empty, then the set of all α-orientations of G in the same homology class carries a certain number of distributive lattices by flips taking on the non-forbidden facial circuits. This generalizes the corresponding results given by Felsner and Propp for the case that C consists of a single facial circuit.