Enumerating Cyclic Orientations of a Graph

TL;DR

This paper introduces an efficient method to enumerate all cyclic orientations of a connected undirected graph, providing algorithms with near-linear delay and manageable space complexity, filling a gap in graph orientation enumeration.

Contribution

It presents the first algorithm for enumerating cyclic orientations of graphs, with optimized delay and space complexity, distinct from existing acyclic orientation enumeration methods.

Findings

Enumeration of cyclic orientations is feasible with $ ilde{O}(m)$ delay.

Space complexity is $O(m)$ with a setup cost of $O(n^2)$.

Alternative setup with $O(mn)$ space and $ ilde{O}(m)$ time.

Abstract

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the problem of efficiently enumerating all the \emph{cyclic} orientations of an undirected connected graph with $n$ vertices and $m$ edges, observing that it cannot be solved using algorithmic techniques previously employed for enumerating acyclic orientations.We show that the problem is of independent interest from both combinatorial and algorithmic points of view, and that each cyclic orientation can be listed with $\tilde{O}(m)$ delay time. Space usage is $O(m)$ with an additional setup cost of $O(n^2)$ time before the enumeration begins, or $O(mn)$ with a setup cost of $\tilde{O}(m)$ time.