### Problem on Graph Theory and the Travelling Salesman Problem#### Problem Statement:This exercise revolves around the roads between towns and the length of each road in kilometers. A delivery driver resides in town D and needs to visit every other town before returning to D.#### Graph Description:- The graph has five vertices (A, B, C, D, and E) representing towns.- The edges between the vertices represent roads, and the numbers on the edges indicate the distance between towns in kilometers.#### Questions:a) Complete the graph with edges showing the least distance between each pair of towns.b) Use the nearest neighbor algorithm starting at D to find an upper bound for the Travelling Salesman Problem (TSP).c) The delivery van can travel 600 km on one tank of fuel. By applying the deleted vertex algorithm with vertex A deleted, show that the driver must refuel during the trip.#### Graph Analysis:- **Vertices (Towns):** A, B, C, D, E- **Edges (Roads with distances):** - A-B: 150 km - B-C: 250 km - C-E: 100 km - E-A: 200 km - B-D: 110 km - D-C: 80 km#### Detailed Explanation of the Graph:The graph illustrates a network where the least distances between towns have already been provided. Here are the connections with distances:1. A is connected to B with a road of 150 km2. B is connected to D with a road of 110 km3. B is connected to C with a road of 250 km4. D is connected to C with a road of 80 km5. C is connected to E with a road of 100 km6. E is connected to A with a road of 200 km#### Task Breakdown:1. **Part A: Completing the Graph:** Ensure every pair of towns (vertices) has an edge denoting the least distance between them. According to the provided graph, this step is already done.2. **Part B: Upper Bound for TSP Using Nearest Neighbour Algorithm:** Apply the nearest neighbor algorithm starting at town D: - Start at D, move to the nearest neighbor (which is C, 80 km) - From C, move to the nearest unvisited town

we are given a graph that shows the roads between towns, and the length of each road in kilometres…

This graph shows the roads between towns, and the length of each road in kilometres. В 150 A delivery driver lives in town D, and must visit every other town before returning to D. 110 a Complete the graph with edges showing the least distance between each pair of towns. 25 200 b Use the nearest neighbour algorithm starting at D to find an upper bound for the TSP. C The delivery van can travel 600 km on one tank of fuel. By applying the deleted vertex algorithm with vertex A 80 E 100 deleted show that the driver must refuel during the trip

This graph shows the roads between towns, and the length of each road in kilometres. В 150 A delivery driver lives in town D, and must visit every other town before returning to D. 110 a Complete the graph with edges showing the least distance between each pair of towns. 25 200 b Use the nearest neighbour algorithm starting at D to find an upper bound for the TSP. C The delivery van can travel 600 km on one tank of fuel. By applying the deleted vertex algorithm with vertex A 80 E 100 deleted show that the driver must refuel during the trip

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Problem on Graph Theory and the Travelling Salesman Problem

#### Problem Statement:
This exercise revolves around the roads between towns and the length of each road in kilometers. A delivery driver resides in town D and needs to visit every other town before returning to D.

#### Graph Description:
- The graph has five vertices (A, B, C, D, and E) representing towns.
- The edges between the vertices represent roads, and the numbers on the edges indicate the distance between towns in kilometers.

#### Questions:
a) Complete the graph with edges showing the least distance between each pair of towns.

b) Use the nearest neighbor algorithm starting at D to find an upper bound for the Travelling Salesman Problem (TSP).

c) The delivery van can travel 600 km on one tank of fuel. By applying the deleted vertex algorithm with vertex A deleted, show that the driver must refuel during the trip.

#### Graph Analysis:
- **Vertices (Towns):** A, B, C, D, E
- **Edges (Roads with distances):**
- A-B: 150 km
- B-C: 250 km
- C-E: 100 km
- E-A: 200 km
- B-D: 110 km
- D-C: 80 km

#### Detailed Explanation of the Graph:
The graph illustrates a network where the least distances between towns have already been provided. Here are the connections with distances:

1. A is connected to B with a road of 150 km
2. B is connected to D with a road of 110 km
3. B is connected to C with a road of 250 km
4. D is connected to C with a road of 80 km
5. C is connected to E with a road of 100 km
6. E is connected to A with a road of 200 km

#### Task Breakdown:
1. **Part A: Completing the Graph:**
Ensure every pair of towns (vertices) has an edge denoting the least distance between them. According to the provided graph, this step is already done.

2. **Part B: Upper Bound for TSP Using Nearest Neighbour Algorithm:**
Apply the nearest neighbor algorithm starting at town D:
- Start at D, move to the nearest neighbor (which is C, 80 km)
- From C, move to the nearest unvisited town

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