# Flip-distance between \alpha-orientations of graphs embedded on plane and sphere

## TL;DR

This paper provides an explicit formula for the minimum number of flip operations needed to transform one lpha-orientation into another in graphs embedded on the plane or sphere, extending previous lattice structure results.

## Contribution

It introduces a formula for the flip-distance between lpha-orientations on planar and spherical graphs, advancing understanding of their combinatorial structure.

## Key findings

- Derived an explicit flip-distance formula for lpha-orientations.
- Extended lattice structure analysis to include flip distances.
- Applicable to graphs embedded on both plane and sphere.

## Abstract

Felsner introduced a cycle reversal, namely the `flip' reversal, for \alpha-orientations (i.e., each vertex admits a prescribed out-degree) of a graph G embedded on the plane and further proved that the set of all the \alpha-orientations of G carries a distributive lattice with respect to the flip reversals. In this paper, we give an explicit formula for the minimum number of flips needed to transform one \alpha-orientation into another for graphs embedded on the plane or sphere, respectively.