Parallelism of stable traces

A $d$-stable trace is a closed walk which traverses every edge of a graph exactly twice and for every vertex $v$, there is no subset $N \subseteq N(v)$ ($1 \leq |N| \leq d$), such that every time the walk enters $v$ from $N$, it also exits to a vertex in $N$. In addition, in a parallel $d$-stable trace, every edge is traversed twice in the same direction. $d$-stable traces were investigated in [Strong traces model of self-assembly polypeptide structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 199-212] as a mathematical model for an innovative biotechnological procedure. It was proven there, that graphs that admit parallel $d$-stable traces are precisely Eulerian graphs with minimum degree strictly higher than $d$. In present paper we give an alternative proof of this result by thoroughly examining the special case for $d = 2$. We also explain its importance for synthetic biology and present two algorithms which can be in special cases used for their construction.