{"title":"Simplicity in Eulerian Circuits: Uniqueness and Safety","authors":"Nidia Obscura Acosta, Alexandru I. Tomescu","doi":"10.48550/arXiv.2208.08522","DOIUrl":null,"url":null,"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.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inf. Process. Lett.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2208.08522","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
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.
有向图中的欧拉电路是图论中最基本的概念之一。检测图$G$是否具有唯一的欧拉电路可以通过de Bruijn, van Aardenne-Ehrenfest, Smith和Tutte(1941-1951)的BEST定理在多项式时间内完成(涉及计算树形),或者通过Pevzner(1989)的定制表征(涉及计算$G$的简单循环的相交图),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。本文给出了具有唯一欧拉电路的有向图的一个新的线性时间可检性表征。这是基于一个简单的条件,即当两条边必须在所有欧拉电路中连续出现时,就底层无向图$G$的切割节点而言。作为副产品,我们还可以在线性时间内计算所有欧拉电路中出现的所有极大$\textit{safe}$行走,为此Nagarajan和Pop在2009年提出了基于Pevzner表征的多项式时间算法。