You Must Construct Additional Pylons

Notice that the exact shape of the path is irrelevant to the problem. We only need certain important points, precisely, the intersections. Let us relabel the diagram.

1. We could go straight from $A$ to $B$ : To do this, we can walk along $APQB$ , $APXQB$ , or $APYQB$ . That's three of them.
2. We could go from $A$ to $B$ , loop back to $A$ and use the remaining edge to traverse back to $B$ . Two choose the first edge, there are $3$ ways, and then to choose the back-edge, only the remaining $2$ edges, and just $1$ remaining edge to use to travel back to B. So, there are $2 \times 3 = 6$ possible such walks. $APXQPYQB$ , $APXQYPQB$ , $APQXPYQB$ , $APQYPXQB$ , $APYQXPQB$ , $APYQPXQB$

In total, there are $9$ walks from $A$ to $B$