Euler-cycle / path that doesn't cross itself

Hello
I don’t follow where you are now, but for

If you have a graph that is self intersecting, you can use it in order to suppress self intersection.
The idea is quite simple at each self intersection flip the direction of the curve . I have done this tool for curve in my Nautilus Plugin (not online at the moment, but could be send).
If you have that


You could get that