Graph Theory Unit 2

2.10 Tree of Optimal Value

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

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

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)

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

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

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.

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

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

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

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

2.10 Practice 2 Carousel Page 124 #1-11