$E$-restricted double traces

For a graph $G$ and $E \subseteq E(G)$, $E$-restricted strong trace is a closed walk which traverses every edge from $E$ once in each direction and every other edge twice in the same direction. In addition, every time a strong trace come to a vertex $v$ from $N \subseteq N(v)$ it continues to $u \notin N$, for $1 \leq |N| < d(v)$. We characterize graphs admitting $E$-restricted strong traces and explain how this result can be used as an upgrade of mathematical model for self-assembling nanostructure design first presented by Gradi\v{s}ar et al. in [Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments, Nature Chemical Biology 9 (2013) 362--366].