B

Step 1: Identify the structure of the circuit with vertices A, B, C, D, E, and F.
Step 2: Analyze the paths between points A and B.
- There are three main paths: A to E to D to B, A to F to D to B, and A to E to C to B.
Step 3: Calculate the resistance for each path based on the equal resistances (r) of the wires.
Path AE to DB: 3r (E to D is one resistance, and two connections contribute to this total).
Path AF to DB: 3r.
Path AE to CB to B: This includes a combination respect to the triangle CED.
Step 4: Use the formula for equivalent resistance in parallel circuits: \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots
Step 5: After calculations, we find the equivalent resistance R_{eq} = \frac{r}{2}. Therefore, since the paths are in parallel, R_{eq} = \frac{r}{3}
Therefore, the answer is B.