**Graph Theory** Hamilton Paths and Hamilton Circuits

**WHAT YOU WILL LEARN** • Hamilton paths and Hamilton circuits • Traveling salesman problem • Brute force method • Nearest neighbor method

**Definitions** • A Hamilton path is a path that contains each vertex of a graph exactly once. • A Hamilton circuit is a path that begins and ends at the same vertex and passes through all other vertices of the graph exactly one time.

**Graph (a) shown has Hamilton path A, B, C, E, D. The graph** also has Hamilton path C, B, A, D, E. Can you find some others? Graph (b) shown has Hamilton path A, B, C, F, H, E, G, D. The graph also has Hamilton path G, D, E, H, F, C, B, A. Can you find some others? Example: Hamilton Path

**Graph (a) shown has Hamilton circuit ** A, B, D, G, E, H, F, C, A. Graph (b) shown has Hamilton circuit A, B, C, E, D,A. Can you find another? Example: Hamilton Circuit

**Definition** • A complete graph is a graph that has an edge between each pair of its vertices.

**B** A C D F G E Example: Finding Hamilton Circuits • Determine a Hamilton circuit for the graph shown. • One solution is A, G, F, E, C, D, B, A

**Number of Unique Hamilton Circuits in a Complete Graph** • The number of unique Hamilton circuits in a complete graph with n vertices is (n 1)! , where (n 1)! = (n 1)(n 2)(n 3)…(3)(2)(1)

**Example: Number of Hamilton Circuits** • How many unique Hamilton circuits are there in a complete graph with the following number of vertices? • a) 4 b) 9 • a) 4 = (4 1)! = 3•2•1 = 6 • b) 9 = (9 1)! = 8•7•6•5•4•3•2•1 = 40,320

**The Brute Force Method of Solving Traveling Salesman** Problems • To determine the optimal solution: 1. Represent the problem with a complete, weighted graph. 2. List all possible Hamilton circuits for this graph. 3. Determine the cost (or distance) associated with each of these Hamilton circuits. 4. The Hamilton circuit with the lowest cost (or shortest distance) is the optimal solution.

**Find a minimum Hamilton circuit for the complete, weighted** graph shown. List all the Hamilton circuits. Since there are 4 vertices there are (4 1)! = 3! = 6 possible circuits. 6 A B 2 8 2 12 D C 6 Example: Brute Force Method

**Circuit** First Leg Second Leg Third Leg Fourth Leg Total A, B, C, D, A 6 12 6 2 26 A, B, D, C, A 6 8 6 2 22 6 A B A, C, B, D, A 2 12 8 2 24 2 8 A, C, D, B, A 2 6 8 6 22 2 12 A, D, B, C, A 2 8 12 2 24 D C A, D, C, B, A 2 6 12 6 26 6 Example: Brute Force Method continued The minimum amount for the graph would be 22.

**Nearest Neighbor Method of Determining an Approximate** Solution to a Traveling Salesman Problem • To approximate a optimal solution: 1. Represent the problem with a complete, weighted graph. 2. Identify the starting vertex. 3. Of all the edges attached to the starting vertex, choose the edge that has the smallest weight. This edge is generally either the shortest distance or the lowest cost. Travel along this edge to the second vertex.

**Nearest Neighbor Method of Determining an Approximate** Solution to a Traveling Salesman Problem (continued) 4. At the second vertex, choose the edge that has the smallest weight that does not lead to a vertex already visited. Travel along this edge to the third vertex. 5. Continue this process, each time moving along the edge with the smallest weight until all vertices are visited. 6. Travel back to the original vertex.

**Example: Nearest Neighbor** • An appliance repair man must repair four appliances at the locations shown in the graph on the next slide. The estimated travel time (in minutes) is shown between the four locations. The repair man wants to visit the four locations in an order that takes the least amount of time. Use the nearest neighbor algorithm to find an approximate solution to the problem.

**Choose a starting vertex. Let’s choose vertex A.** Choose the edge that has the smallest weight. Either AD or AB, lets choose AD (8). At D choose the edge that has the smallest weight DE (4). A 8 18 8 13 3 B 7 7 9 4 D C 6 Example: Nearest Neighbor (continued) start here E 2nd vertex

**Continue moving along the edge with the smallest weight.** Circuit would be: A, D, E, B, C, A 8 + 4 + 3 + 9 + 13 = 37 This route would take about 37 minutes. A 8 18 8 13 3 B E 7 7 9 4 D C 6 Example: Nearest Neighbor (continued) start here 2nd vertex

**Starting Vertex** Circuit Total Weight B B, E, D, C, A, B 3 + 4 + 6 + 13 + 8 = 34 C C, D, E, B, A, C 6 + 4 + 3 + 8 + 13 = 34 D D, E, B, A, C, D 4 + 3 + 8 + 13 + 6 = 34 E E, B, D, C, A, E 3 + 7 + 6 + 13 + 18 = 47 Example: Nearest Neighbor (continued) • Other circuits and weights.

**In the following graph, determine a Hamilton path.** a. HGEDABCFBE b. GEDABCFH c. HGEDABCF d. FBADEGH

**In the following graph, determine a Hamilton path.** a. HGEDABCFBE b. GEDABCFH c. HGEDABCF d. FBADEGH

**Alicia lives in Washington, D.C. and wants to take a road** trip to visit her friends from college who live in the following cities: New York, Boston, Philadelphia, Dover, and Baltimore. How many ways can Alicia visit each city and return home to Washington, D.C.? a. 120 b. 60 c. 25 d. 24

**Alicia lives in Washington, D.C. and wants to take a road** trip to visit her friends from college who live in the following cities: New York, Boston, Philadelphia, Dover, and Baltimore. How many ways can Alicia visit each city and return home to Washington, D.C.? a. 120 b. 60 c. 25 d. 24

**Robert decides to take a walk to run his errands. He needs** to walk to the drug store, the post office, and the dry cleaners. The distances between these four locations are as follows: home (H) to the drug store (D) is 0.6 miles, home to the post office (P) is 0.5 miles, home to the dry cleaners (C) is 0.7 miles, the drug store to the post office is 0.2 miles, the drug store to the dry cleaner is 0.5 miles, and the dry cleaner to the post office is 0.2 miles.H to D: 0.6 H to P: 0.5 H to C: 0.7D to P: 0.2 D to C: 0.5 C to P: 0.2

**0.5** 0.5 0.7 0.2 0.2 0.6 0.2 0.5 0.5 0.6 0.2 0.7 0.2 0.6 Represent this traveling salesman problem with a complete, weighted graph.H to D: 0.6 H to P: 0.5 H to C: 0.7D to P: 0.2 D to C: 0.5 C to P: 0.2 a. c. b. d.

**0.5** 0.5 0.7 0.2 0.2 0.6 0.2 0.5 0.5 0.6 0.2 0.7 0.2 0.6 Represent this traveling salesman problem with a complete, weighted graph.H to D: 0.6 H to P: 0.5 H to C: 0.7D to P: 0.2 D to C: 0.5 C to P: 0.2 a. c. b. d.

**Use the brute force method to determine the shortest route** for Robert to visit each location once and return home. What is the distance when using this route? a. HDPCH; 1. 7 miles b. HPDCH; 1.7 miles c. HPCD; 1.2 miles d. HCDPH; 1.7 miles

**Use the brute force method to determine the shortest route** for Robert to visit each location once and return home. What is the distance when using this route? a. HDPCH; 1. 7 miles b. HPDCH; 1.7 miles c. HPCD; 1.2 miles d. HCDPH; 1.7 miles

**Use the nearest neighbor method to approximate the optimal** route for Robert to visit each location once and return home. What is the distance when using this route? a. HDPCH; 1. 7 miles b. HPDCH; 1.7 miles c. HPCD; 1.2 miles d. HCDPH; 1.7 miles

**Use the nearest neighbor method to approximate the optimal** route for Robert to visit each location once and return home. What is the distance when using this route? a. HDPCH; 1. 7 miles b. HPDCH; 1.7 miles c. HPCD; 1.2 miles d. HCDPH; 1.7 miles