Antiparallel d-stable Traces and a Stronger Version of Ore Problem

In $2013$ a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in [Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments, Nature Chem. Bio. 9 (2013) 362--366]. 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 [Design principles for rapid folding of knotted DNA nanostructures, Nature communications 7 (2016) 1--8.]. 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 $1 \leq |N| \leq d$, such that every time the walk enters $v$ from $N$, it also exits to a vertex in $N$. This extends C. Thomassen's characterization [Bidirectional retracting-free double tracings and upper embeddability of graphs, J. Combin. Theory Ser. B 50 (1990) 198--207] for the case $d = 1$.