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.

翻译：有向图中的欧拉回路是图论中最基本的概念之一。判定图$G$是否具有唯一欧拉回路可在多项式时间内完成，依据de Bruijn、van Aardenne-Ehrenfest、Smith与Tutte（1941-1951年）提出的BEST定理（涉及对树状图的计数），或采用Pevzner（1989年）的定制化特征刻画（涉及计算$G$中简单环的交图）。然而，对于更简单的唯一性判定问题，上述两种方法均依赖过于复杂的理论。本文给出一种新的线性时间可判定的有向图唯一欧拉回路特征刻画。该特征基于一个简单条件：在$G$的底层无向图的割节点作用下，任意两条边在所有欧拉回路中必须相邻出现的条件。作为副产品，我们还可在线性时间内计算出所有欧拉回路中的最大$\textit{安全}$游走——针对此问题，Nagarajan与Pop于2009年基于Pevzner特征提出了多项式时间算法。