# Enumerating $k$-arc-connected orientations

## TL;DR

This paper presents efficient algorithms for enumerating all $k$-arc-connected orientations of a graph, improving computational complexity over previous methods and providing practical enumeration techniques.

## Contribution

It introduces a simple, efficient enumeration algorithm with improved time delay for $k$-arc-connected orientations, along with algorithms for related orientation and outdegree sequence enumeration.

## Key findings

- New enumeration algorithms with $O(knm^2)$ time delay.
- Improved analysis over submodular flow-based methods.
- Algorithms for $ ext{α}$-orientations and outdegree sequences.

## Abstract

We study the problem of enumerating the $k$-arc-connected orientations of a graph $G$, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with $O(knm^2)$ time delay and amortized time $O(m^2)$, which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the $\alpha$-orientations of a graph $G$ in $O(m^2)$ time delay and for the outdegree sequences attained by $k$-arc-connected orientations of $G$ in $O(knm^2)$ time delay.