Cyclically Orientable Graphs

TL;DR

This paper provides a simple recursive characterization of cyclically orientable graphs and introduces an efficient O(n) algorithm to determine if a graph is cyclically orientable, advancing understanding in graph theory and cluster algebras.

Contribution

It offers a new recursive description of cyclically orientable graphs and an O(n) algorithm for testing this property, filling a gap in prior research.

Findings

Recursive description of cyclically orientable graphs

O(n) algorithm for testing cyclically orientability

Most results independently obtained by Gurvich

Abstract

Barot, Geiss and Zelevinsky define a notion of a ``cyclically orientable graph'' and use it to devise a test for whether a cluster algebra is of finite type. Barot, Geiss and Zelivinsky's work leaves open the question of giving an efficient characterization of cyclically orientable graphs. In this paper, we give a simple recursive description of cyclically orientable graphs, and use this to give an O(n) algorithm to test whether a graph on $n$ vertices is cyclically orientable. Shortly after writing this paper, I learned that most of its results had been obtained independently by Gurvich; I am placing this paper on the arXiv to spread knowledge of these results.