4-4 Graph Theory Trees.

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Presentation transcript:

4-4 Graph Theory Trees

WHAT YOU WILL LEARN • Trees, spanning trees, and minimum-cost spanning trees

Definitions A tree is a connected graph in which each edge is a bridge. A spanning tree is a tree that is created from another graph by removing edges while still maintaining a path to each vertex.

Examples Graphs that are trees. Graph that are not trees.

Example: Determining Spanning Trees Determine two different spanning trees for the graph shown. A B C E F H D G

Minimum-cost spanning tree A minimum cost spanning tree is the least expensive spanning tree of all spanning trees under consideration.

Kruskal’s Algorithm To construct the minimum-cost spanning tree from a weighted graph: 1. Select the lowest-cost edge on the graph. 2. Select the next lowest-cost edge that does not form a circuit with the first edge. 3. Select the next lowest-cost edge that does not form a circuit with the previously selected edges. 4. Continue selecting the lowest-cost edges that do not form circuits with the previously selected edges. 5. When a spanning tree is complete, you have the minimum-cost spanning tree.

Example: Kruskal’s Algorithm Use Kruskal’s algorithm to determine the minimum spanning tree for the weighted graph shown. The numbers along the edges represent dollars. A B C G D E F 12 11 10 5 22 14 4 17 18

Solution Pick the lowest-cost edge of the graph, edge CD which is $4. Next we select the next lowest-cost edge that does not form a circuit; we select edge CG which is $5. A B C G D E F 12 11 10 5 22 14 4 17 18

Solution (continued) Continue selecting edges, being careful not to form a circuit. The total cost would be $12 + $10 + $5 + $14 +$18 + $4 = $63. A B C G D E F 12 11 10 5 22 14 4 17 18

Determine a spanning tree for the graph shown below. c. b. d.

Determine a spanning tree for the graph shown below. c. b. d.

Determine the minimum-cost spanning tree for the following weighted graph.

b. d. a. c.

b. d. a. c.

Kathleen is planning on installing a new computer network at her small business. Her current system has computers already in place as shown in the figure below. The numbers are shown in feet.

Determine the minimum-cost spanning tree that reaches each computer. b. d.

Determine the minimum-cost spanning tree that reaches each computer. b. d.

If the new networking system materials cost $2 If the new networking system materials cost $2.20 per foot, what is the cost of installing the system a. $79.20 b. $83.60 c. $85.80 d. $112.20

If the new networking system materials cost $2 If the new networking system materials cost $2.20 per foot, what is the cost of installing the system a. $79.20 b. $83.60 c. $85.80 d. $112.20

Practice Problems