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) [15], [16] (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 safe walks appearing in all Eulerian circuits, for which Nagarajan and Pop proposed in 2009 [12] a polynomial-time algorithm based on Pevzner characterization.

有向图中的欧拉回路是最基本的图论概念之一。检测图G是否具有唯一的欧拉回路可以通过 de Bruijn、van Aardenne-Ehrenfest、Smith 和 Tutte (1941–1951) [15]、[16]（涉及树状计数）的 BEST 定理在多项式时间内完成，或者通过 Pevzner, 1989 的定制表征（涉及计算G的简单循环的交集图），因此这两者都依赖于过于复杂的概念来解决更简单的唯一性问题。

在本文中，我们给出了具有独特欧拉电路的有向图的新的线性时间可检查表征。这是基于一个简单的条件，即两条边必须连续出现在所有欧拉电路中，就G的底层无向图的割节点而言。作为副产品，我们还可以在线性时间内计算出现在所有欧拉电路中的所有最大安全游走，为此 Nagarajan 和 Pop 在 2009 年[12]提出了一种基于 Pevzner 表征的多项式时间算法。