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

TL;DR

This paper characterizes when one surface graph orientation can be transformed into another via flips avoiding certain facial circuits, providing explicit formulas and analyzing the structure of the flip graph.

Contribution

It introduces a necessary and sufficient condition for flips between orientations with forbidden facial circuits and describes the structure of the associated flip graph.

Findings

The flip graph has exactly |O(G,C)| components when C is non-empty.

Each component of the flip graph forms a distributive lattice.

If C is empty, the flip graph is strongly connected.

Abstract

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.