Simplicity in Eulerian Circuits: Uniqueness and Safety

TL;DR

This paper introduces a new linear-time method to determine when a directed graph has a unique Eulerian circuit, simplifying previous complex approaches and also identifying all maximal safe walks common to all such circuits.

Contribution

It provides a simple, linear-time characterization for the uniqueness of Eulerian circuits in directed graphs, improving on prior complex polynomial-time methods.

Findings

Linear-time checkable characterization of unique Eulerian circuits

Efficient computation of all maximal safe walks in Eulerian circuits

Simplification of previous complex algorithms for Eulerian circuit analysis

Abstract

An Eulerian circuit in a directed graph is one of the most fundamental Graph Theory notions. Detecting if a graph $G$ has a unique Eulerian circuit can be done in polynomial time via the BEST theorem by de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, 1941-1951 (involving counting arborescences), or via a tailored characterization by Pevzner, 1989 (involving computing the intersection graph of simple cycles of $G$), both of which thus rely on overly complex notions for the simpler uniqueness problem. In this paper we give a new linear-time checkable characterization of directed graphs with a unique Eulerian circuit. This is based on a simple condition of when two edges must appear consecutively in all Eulerian circuits, in terms of cut nodes of the underlying undirected graph of $G$. As a by-product, we can also compute in linear-time all maximal $\textit{safe}$ walks appearing in all Eulerian circuits, for which Nagarajan and Pop proposed in 2009 a polynomial-time algorithm based on Pevzner characterization.