1. Write a summary of what you think are the important points of thischapter.

2. Is the following graph planar or nonplanar? If it is planar, redraw itwithout edge crossings.

3. Show that the graph described by the following adjacency matrix isplanar.

4. What is the chromatic number for each of the following?

a. any tree with five vertices

b. any tree with an odd number of vertices

c. any tree with an even number of vertices

d. any tree with two or more vertices

Chapter 5 Review

A B C D E

A

B

C

D

E

� � � � � � � �

0 1 1 1 0

1 0 0 1 1

1 0 0 1 1

1 1 1 0 1

0 1 1 1 0

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290 Chapter 5 • More Graphs, Subgraphs, and Trees

5.

a. Explain why this graph is a bipartite graph.

b. Is this graph a complete bipartite graph? Explain why or whynot.

c. Is this graph planar? If so, find a planar drawing for the graph.

d. What is the chromatic number for this graph?

6. For the following graph, draw a subgraph that has 4 vertices and 4 edges.

7. Mr. Gonzalez, the principal at Central High School, leaves his officeonce an hour to visit the math, science, and civics classrooms, andthen returns to his office. The distances between rooms are shownon the following graph.

T

SR

W V

U

Science

Civics

Math

96 ft

50 ft

104 ft

Office

60 ft

100 ft

110 ft

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291Chapter 5 • Review

a. Find the shortest route possible for Mr. Gonzalez.

b. What is the total distance of the shortest route in part a?

c. What kind of circuit does Mr. Gonzalez make?

8. For the following complete graph, find the total weight of thenearest-neighbor route starting at A.

9. Find a spanning tree for the following graph if one exists.

10. How many different spanning trees are there for a cycle with 3vertices? With 4 vertices? With 5 vertices? With n vertices?

11. For the following graph, explain why the darkened edges are not aspanning tree.

A

B

C

DE

9

19

35

38

2332

28

42

31

37

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292 Chapter 5 • More Graphs, Subgraphs, and Trees

12. When given the position where it is currently located and itsdestination, a certain robot car is programmed to find the shortestpath for the trip. The routes that the car can travel are shown onthe following graph.

a. Use inspection to find the shortest path from Home to BigBurger.

b. What is the minimum distance from Home to Big Burger?

c. Use the shortest path algorithm (page 248) to find the shortestpaths from Home to each of the other locations on the graph.

13. Assume that all locations represented by the graph in Exercise 12need to be connected by cable. Find the minimum amount of cableneeded to link the nine locations.

14. Are the following graphs trees? Explain why or why not.

a. b.

TheaterPool

Grocery

Hospital

BigBurger

Cleaners

Home

SchoolLibrary

6

6

2

4

3 7

2 3 miles

2

342

5

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293Chapter 5 • Review

c. d.

15. What happens to a tree if the number of edges is increased by one?

16. Use the breadth-first search algorithm from page 264 to find aspanning tree for the following graph. Begin the algorithm at thevertex labeled S.

17. The vertices of the following graph represent buildings on a smallcollege campus. Administrators at the campus want to connect thebuildings with fiber-optic cable and are interested in finding theleast expensive way of doing so. The costs of connecting buildings(in thousands of dollars) are shown as weighted edges of the graph.

a. Use one of the spanning tree algorithms to find a minimumspanning tree for the graph.

b. What is the total cost of connecting the buildings?

S

4

6

5

8

9

3

5

5

42

23

3

4

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294 Chapter 5 • More Graphs, Subgraphs, and Trees

18. Draw a tree with eight vertices that has exactly four vertices ofdegree 1.

19. Create a problem that can be solved by using one of the minimumspanning tree algorithms and find the solution to your problem.Then give your problem to a classmate and ask him or her to solveit. If the answer differs from yours, determine the correct solution.

20. You roll a die with faces numbered 1–6. Then you flip a coin. Drawa tree diagram to show the possible outcomes.

21. Represent (4 – 3) * 8 + 5 as a binary expression tree.

22. Find a postorder listing for the following binary tree.

23. Evaluate the following reverse Polish notation.

7 1 + 3 * 2 4 + –

24. Create a mathematical expression of your own, represent it with anexpression tree, and find the postorder listing for the tree. Give yourlisting to another student in your class and have him or herevaluate the reverse Polish notation. Check the value of thenotation with the value of your original mathematical expression.

2

36

4 5

+

+

*

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295Chapter 5 • Review

BibliographyChartrand, Gary. 1985. Introductory Graph Theory. Mineola, NY: Dover

Publications, Inc.

Chavey, Darrah. 1992. Drawing Pictures with One Line. (HistoMAP Module21). Lexington, MA: COMAP, Inc.

COMAP. 2013. For All Practical Purposes: Mathematical Literacy in Today’sWorld. 9th ed. New York: W. H. Freeman.

Cook, William J. 2012. In Pursuit of the Traveling Salesman: Mathematics atthe Limits of Computation. Princeton, NJ: Princeton University Press.

Copes, W., C. Sloyer, R. Stark, and W. Sacco. 1987. Graph Theory: Euler’sRich Legacy. Providence, RI: Janson.

Cozzens, Margaret B., and R. Porter. 1987. Mathematics and ItsApplications. Lexington, MA: D. C. Heath and Company.

Cozzens, Margaret B., and R. Porter. 1987. Problem Solving Using Graphs.Lexington, MA: COMAP, Inc.

Crisler, Nancy, and Walter Meyer. 1993. Shortest Paths. Lexington, MA:COMAP, Inc.

Dossey, John, A. Otto, L. Spence, and C. Vanden Eynden. 2006. DiscreteMathematics. 5th ed. Upper Saddle River, NJ: Pearson.

Francis, Richard L. 1989. The Mathematician’s Coloring Book. (HiMAPModule 10). Lexington, MA: COMAP, Inc.

Ore, O. 1990. Graphs and Their Uses. Washington, DC: MathematicalAssociation of America.

Queen Mary, University of London. "How bumblebees tackle the travelingsalesman problem." ScienceDaily, 29 Jun. 2011. Web. 8 Feb. 2013.

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