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To solve each part of this problem using the "nearest-neighbor" method, we need to select the path with the lowest cost at each step, beginning from the specified starting vertex and visiting each vertex only once until we return to the starting point.

Let’s analyze each case:

### (a) Nearest-neighbor tour starting with vertex $ B $
1. **From $ B $**, the nearest vertex is $ C $ with a weight of $119.
2. **From $ C $**, the nearest vertex is $ A $ with a weight of $152.
3. **From $ A $**, the nearest vertex is $ E $ with a weight of $133.
4. **From $ E $**, the nearest vertex is $ D $ with a weight of $199.
5. **From $ D $**, we return to $ B $ with a weight of $150.

   **Total weight** = $ 119 + 152 + 133 + 199 + 150 = 753 $

### (b) Nearest-neighbor tour starting with vertex $ C $
1. **From $ C $**, the nearest vertex is $ B $ with a weight of $119.
2. **From $ B $**, the nearest vertex is $ A $ with a weight of $185.
3. **From $ A $**, the nearest vertex is $ E $ with a weight of $133.
4. **From $ E $**, the nearest vertex is $ D $ with a weight of $199.
5. **From $ D $**, we return to $ C $ with a weight of $174.

   **Total weight** = $ 119 + 185 + 133 + 199 + 174 = 810 $

### (c) Nearest-neighbor tour starting with vertex $ D $
1. **From $ D $**, the nearest vertex is $ B $ with a weight of $150.
2. **From $ B $**, the nearest vertex is $ C $ with a weight of $119.
3. **From $ C $**, the nearest vertex is $ A $ with a weight of $152.
4. **From $ A $**, the nearest vertex is $ E $ with a weight of $133.
5. **From $ E $**, we return to $ D $ with a weight of $199.

   **Total weight** = $ 150 + 119 + 152 + 133 + 199 = 753 $

### (d) Nearest-neighbor tour starting with vertex $ E $
1. **From $ E $**, the nearest vertex is $ A $ with a weight of $133.
2. **From $ A $**, the nearest vertex is $ C $ with a weight of $152.
3. **From $ C $**, the nearest vertex is $ B $ with a weight of $119.
4. **From $ B $**, the nearest vertex is $ D $ with a weight of $150.
5. **From $ D $**, we return to $ E $ with a weight of $199.

   **Total weight** = $ 133 + 152 + 119 + 150 + 199 = 753 $

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Would you like further details on any of these tours or have additional questions?

### Follow-up Questions:
1. How would the nearest-neighbor method change if the weights were different?
2. What would be the impact of choosing a different starting vertex on the tour weight?
3. Can we confirm that the tours found are optimal?
4. How does the nearest-neighbor method compare to other methods like the greedy or brute-force approach?
5. Could the graph have multiple tours with the same weight if the vertices are visited in different orders?

### Tip:
In optimization problems involving weighted graphs, consider multiple approaches for comparison, as methods like the nearest-neighbor may not always yield the optimal solution.

For the weighted graph shown, find the indicated tour and give its weight:
(a) The nearest-neighbor tour with starting vertex B.
(b) The nearest-neighbor tour with starting vertex C.
(c) The nearest-neighbor tour with starting vertex D.
(d) The nearest-neighbor tour with starting vertex E.

Explore how to find the nearest-neighbor tour on a weighted graph starting from various vertices. This solution demonstrates step-by-step calculations for determining tour weights from vertices B, C, D, and E using the nearest-neighbor heuristic.

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