Then we begin at one arbitrary point and go along the graph until we get a a point where the graph crosses itself. We add that point to the graph (twice) and then we change the branch, that is we ar now going (again in the given direction)
I thought about that yesterday, but I couldn't figure out how to decide which way to go at an intersection. I think I've got it figured out now though. Using your "loop" as an example, at the start of the path, when facing in the direction of travel, the green line is on the "right", and the red line is on the "left" (not exactly a mathematical way of putting it, but I'm after the concept at this point). If we start on the green line then when we get to an intersection we always make a "right turn". This forces the path to be on the outer perimeter, and we will eventually get back to the start. If we start on the red line, we always make a "left turn' at an intersection. So to summarize, if we start on the left we always make make left turns, and if we start on the right we always make right turns.