More tracings

You have to start from the straight line from either vertex as they both have odd connections(YOU C.ANT START ON THE CIRCLE , you cant trace it then)if we start from the left one we will still end up on the right one so , the ways from right vertex = ways from left vertex.

So now we are on right vertex & we have 2 choices , either to go up or go down on the circle , lets say we go up & then on next intersection we have same situation. So we have two choices on each intersection & we have 4 of these so total ways = 2x2x2x2 = 16 ( When coming back we take the path we left so there is no choice when coming back) . So 16 from right vertex & 16 from left one so the answer is 32.

There are two points which have an odd number of lines connecting to them: either end of the straight line. These will have to be the starting and end points. Where to start then is a binary choice. The first three circles have to be made half, and can only be completed after the last one. But for each circle we have a binary choice: clockwise or counterclockwise. All 5 choices can be made indepently. $2^5=\boxed{32}$