1. Chapter 2: Business Efficiency Lesson Plan
For All Practical Purposes
Business Efficiency
Visiting Vertices-Graph Theory Problem
Hamiltonian Circuits
Vacation Planning Problem
Minimum Cost-Hamiltonian Circuit
Method of Trees
Fundamental Principle of Counting
Traveling Salesman Problem
Helping Traveling Salesmen
Nearest Neighbor and Sorted Edges Algorithms
Minimum-Cost Spanning Trees
Kruskal’s Algorithm
Critical-Path Analysis
Mathematical Literacy in Today’s World, 8th ed.

2. Chapter 2: Business Efficiency Business Efficiency
Visiting Vertices
In some graph theory problems, it is only necessary to visit specific locations (using the travel routes, or streets available).
Problem: Find an efficient route along distinct edges of a graph that visits each vertex only once in a simple circuit.
Applications:
Salesman visiting particular cities
Delivering mail to drop-off boxes
Pharmaceutical representative visiting doctors

3. Chapter 2: Business Efficiency Hamiltonian Circuit
A tour that starts and ends at the same vertex (circuit definition).
Visits each vertex once. (Vertices cannot be reused or revisited.)
Circuits can start at any location.
Use wiggly edges to show the circuit.

4. Chapter 2: Business Efficiency Business Efficiency
Starting at vertex A, the tour can be written as ABDGIHFECA, or starting at E, it would be EFHIGDBACE.

5. Chapter 2: Business Efficiency Business Efficiency
A different circuit visiting each vertex once and only once would be CDBIGFEHAC (starting at vertex C).

6. Chapter 2: Business Efficiency Hamiltonian Circuit vs. Euler Circuits
A circuit that traverses each edge of a graph exactly once and starts and stops at the same point.

7. Chapter 2: Business Efficiency Hamiltonian Circuit vs. Euler Circuits
A tour (showed by wiggly edges) that starts at a vertex of a graph and visits each vertex once and only once, returning to where it started.

8. Chapter 2: Business Efficiency Hamiltonian Circuit vs. Euler Circuits
Hamiltonian vs. Euler Circuits
Similarities
Both forbid re-use.
Hamiltonian do not reuse vertices.
Euler do not reuse edges.

9. Hamiltonian is a circuit of vertices. Euler is a circuit of edges.
Chapter 2: Business Efficiency Hamiltonian Circuit vs. Euler Circuits
Differences
Hamiltonian is a circuit of vertices.
Euler is a circuit of edges.
Euler graphs are easy to spot (connectedness and even valence).
Hamiltonian circuits are NOT as easy to determine upon inspection.
Some certain family of graphs can be known to have or not have Hamiltonian circuits.

10. Vacation Planning Problem
Chapter 2: Business Efficiency Hamiltonian Circuits
Example 1
Vacation Planning Problem

11. Chapter 2: Business Efficiency Hamiltonian Circuits
Let’s imagine that you are a college student in Chicago. During spring break you and a group of friends have decided to take a car trip to visit other friends in Minneapolis, Cleveland, and St. Louis. There are many choices as to the order of visiting cities and returning to Chicago, but you want to design a route that minimizes the distance you have to travel. This will also cut costs because of the cost of gasoline for the trip.

12. Chapter 2: Business Efficiency Hamiltonian Circuits
Similar problems with complications would arise for bus, railroad, or airplane trips.
Now the local automobile club has provided you with the inter-city driving distances between Chicago, Minneapolis, Cleveland, and St. Louis.
We can construct a graph model with this information, representing each city by a vertex and the legs of the journey between cities by edges joining the vertices.
To complete the model, we add a number called a weight to each graph edge.

13. Chapter 2: Business Efficiency Hamiltonian Circuits
Road mileage between four cities

14. Chapter 2: Business Efficiency Hamiltonian Circuits
Hamiltonian circuit concept is used to find the best route that minimizes the total distance traveled to visit friends in different cities (assume less mileage  less gas  minimizes costs)

15. Hamiltonian circuit with weighted edges
Chapter 2: Business Efficiency Hamiltonian Circuits
Hamiltonian circuit with weighted edges
Edges of the graph are given weights, or in this case mileage or distance between cities.
As you travel from vertex to vertex, add the numbers (mileage in this case).
Each Hamiltonian circuit will produce a particular sum.

16. Minimum-Cost Hamiltonian Circuit
Chapter 2: Business Efficiency Hamiltonian Circuit
Minimum-Cost Hamiltonian Circuit
A Hamiltonian circuit with the lowest possible sum of the weights of its edges.
Algorithm is a step-by-step description of how to solve a problem.

17. Algorithm (step-by-step process) for Solving This Problem
Chapter 2: Business Efficiency Hamiltonian Circuit
Algorithm (step-by-step process) for Solving This Problem
Generate all possible Hamiltonian tours (starting with Chicago).
Add up the distances on the edges of each tour.
Choose the tour of minimum distance.

18. Chapter 2: Business Efficiency Hamiltonian Circuit
Steps 2 and 3 of the algorithm are straightforward. Thus, we need worry only about Step 1, generating all the possible Hamiltonian circuits in a systematic way.
To find the Hamiltonian tours, we will use the method of trees. In this step we disregard the distances.

19. Chapter 2: Business Efficiency Hamiltonian Circuit
1st Stage
This stage is where you select a starting vertex, say Chicago, and making a tree-diagram showing the next possible locations.

20. Chapter 2: Business Efficiency Hamiltonian Circuit

21. Chapter 2: Business Efficiency Hamiltonian Circuit
2nd Stage
At each subsequent stage down, there will be one less choice .

22. Chapter 2: Business Efficiency Hamiltonian Circuit22

23. Chapter 2: Business Efficiency Hamiltonian Circuit
3rd Stage
At each subsequent stage down, there will be one less choice .

24. Chapter 2: Business Efficiency Hamiltonian Circuits

25. Name the 6 tours from the tree.
Chapter 2: Business Efficiency Hamiltonian Circuits
Name the 6 tours from the tree.
C-M-S-CL-C
C-M-CL-S-C
C-S-M-CL-C
C-S-CL-M-C
C-CL-M-S-C
C-CL-S-M-C
Notice that 1 and 6 are the same tour, 2 and 4 are the same tour, and 3 and 5 are the same tour.
So there are only three are unique circuits.
Find the total distance for each unique circuit.

26. Chapter 2: Business Efficiency Hamiltonian Circuits
The three Hamiltonian circuits’ sums of the tours
The optimal tour is Chicago, Minneapolis, St. Louis, Cleveland, Chicago.
These graphs are called complete graphs.

27. Assignment Pg , 8, 10, 12, 19