1. Graph Theory
Unit 2

2. 2.10 Tree of Optimal Value

3. The Difference PATH You are travelling from one to the other Ex: Trips
TREE
You do not have to move from one to the other as you add up costs
Ex:
Fixing specific roads
Choosing which wires/pipes to run

4. Definition
On a network, the Tree of Minimum Value corresponds to the TREE DIAGRAM that can be created using the optimal* edges
*highest/lowest
You are removing as many edges as possible while maintaining a connected graph that does not contain any simple circuits

5. Steps - Minimum If finding MINIMUM
Select the edge with the lowest value
Of the remaining edges, select the one with the lowest value
Repeat steps 1 and 2 until your graph is connected (be careful not to create any simple circuits)

6. Example F A E B C D 12 10 8 9 5 6 11 7

7. Steps - Maximum If finding MAXMUM
Select the edge with the highest value
Of the remaining edges, select the one with the highest value
Repeat steps 1 and 2 until your graph is connected (be careful not to create any simple circuits)

8. Example F A E B C D 12 10 8 9 5 6 11 7

9. Prospecting
A mining company wants to set up 7 mines in a region, each connected by a network of roads. Because the mines are located in mountainous terrain, some roads are more difficult and expensive to build than others. The company engineers constructed a network that shows the cost of building each road in millions of dollars.

10. Questions Review of PATH of Optimal
Give the total cost of building all the roads shown in this graph
What is the least expensive path for connecting:
Wave & Swell? The Deep & Tide? Undertow & Tide?
Review of
PATH of Optimal

11. Eliminate any simple circuits
Questions
The company nonetheless wants access to all the sites from any one site. What characteristic must the desired layout have to meet this criterion?
It must be connected
To reduce costs even further, the company wants to eliminate any unnecessary roads. How can unnecessary roads be identified?
Eliminate any simple circuits
The company appears to want a connected graph without a simple circuit. What is this kind of graph called?
A TREE DIAGRAM

12. Prospecting
Construct the Tree of MINIMUM value W E S  U T D R

13. 2.10 Practice
Visions Book 2
Page 52
#2, 7, 13, 16, 20, 22

14. 2.10 Practice 2
Carousel
Page 124
#1-11