In 2013 a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in Gradišar et al. (Nat Chem Bio 9:362–366, 2013 ). It was since observed that a polyhedron P can be realized by interlocking pairs of polypeptide chains if its corresponding graph G ( P ) admits a strong trace. It was since also demonstrated that a similar strategy can also be expanded to self-assembly of designed DNA (Kočar, Nat commun 7:1–8, 2016 ). In this direction, in the present paper we characterize graphs which admit closed walk which traverses every edge exactly once in each direction and for every vertex v , there is no subset N of its neighbors, with 1 ≤ | N | ≤ d , such that every time the walk enters v from N , it also exits to a vertex in N . This extends Thomassen’s characterization (Thomassen, J Combin Theory Ser B 50:198–207, 1990 ) for the case d = 1 .