Simplicity in Eulerian circuits: Uniqueness and safety

An Eulerian circuit in a directed graph is one of the most fundamental Graph Theory notions. Detecting if G has unique can be done polynomial time via BEST theorem by de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte (1941–1951) [15], [16] (involving counting arborescences), or tailored characterization Pevzner, 1989 computing intersection simple cycles G), both which thus rely on overly complex notions for simpler uniqueness problem. In this paper we give new linear-time checkable graphs with circuit. This based condition when two edges must appear consecutively all circuits, terms cut nodes underlying undirected G. As by-product, also compute maximal safe walks appearing Nagarajan Pop proposed 2009 [12] polynomial-time algorithm Pevzner characterization.