Chapter 4 Review. 1. Write a summary of what you think are the important points of this chapter. 2. Draw a graph for the following task table.

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1 hapter Review 1. Write a summary of what you think are the important points of this chapter.. raw a graph for the following task table. Task Time Prerequisites Start 0 None 1 5,, I, inish. a. List the vertices of the following graph and give their earlieststart time. b. etermine the minimum project time. 5 Start 0 J 5 inish 1 6 I

2 hapter raphs and Their pplications. omplete the task table for the following graph. Start inish Task Time Prerequisites Start 0 inish 5. Use your graph from xercise. a. Recopy it and label each vertex with its ST. b. etermine the critical path and the minimum project time.

3 hapter Review 6. Task Time Prerequisites Start 0 6 None 9 None 8 None 5, 10 5, 7 I 8 inish 7. a. raw a graph using the information in the table. b. Label each vertex with its earliest-start time. c. etermine the minimum project time. d. etermine the critical path(s). a. Is this graph connected? xplain why or why not. b. Is this graph complete? xplain why or why not. c. Name two vertices that are adjacent to vertex. d. Name a path from to of length. e. What is the degree of vertex? f. etermine an adjacency matrix for the graph.

4 hapter raphs and Their pplications 8. a. raw a K 5 graph. Label the vertices,,,, and I. b. eg() =?. c. oes the graph have an uler circuit? xplain. 9. Try to construct a graph with five vertices that is not connected so that two of the vertices have degree and two of the vertices have degree. No loops or multiple edges may be used. 10. Tell whether the following graphs have an uler circuit, an uler path, or neither. xplain your answers. a. b. 11. onstruct a graph for each of the following. a. V = {,,,, } = {,,,, } b

5 hapter Review 5 1. ollowing is a multigraph that represents the downtown area of a small city. The local post office has decided that the mail drop boxes, which are located at the intersection of each street, must be monitored twice daily. a. Is it possible to find a circuit that begins and ends at the same intersection and visits each drop box exactly once? b. If not, is there a path that begins at one drop box, visits each drop box exactly once, and ends at a different drop box? c. If either route exists, copy the figure onto your paper and darken the edges of your proposed route. 1. Use the graph in xercise 1. a. Is it possible for the local street inspector to begin at an intersection and inspect each street exactly once? b. Is it possible for the inspector to finish her route at the same intersection from which she began? xplain why or why not. 1. ind an adjacency matrix for the following digraph.

6 6 hapter raphs and Their pplications 15. onsider the following set of preference schedules a. Represent this election with a cumulative preference tournament. b. Is there a ondorcet winner? xplain why or why not. c. ind several amiltonian paths for your graph. d. Show how to use a amiltonian path to construct a pairwise voting scheme (see xercise, Lesson 1., page 0) that results in winning the election. 16. raw a connected graph with four vertices that has a amiltonian circuit but no uler circuits or paths. 17. In scheduling the final exam for summer school at entral igh, six different tests must be scheduled. The following table shows the exams that are needed for seven different students. Students xams (M) Math () rt (S) Science () istory () rench (R) Reading a. raw a graph that models which exams have students in common with other exams. b. What is the minimum number of time slots needed to schedule the six exams? 0

7 hapter Review The ederal ommunications ommission () is in charge of assigning frequencies to radio stations so that broadcasts from one station do not interfere with broadcasts from other stations. Suppose the needs to assign frequencies to eight stations. The following table shows which stations cannot share frequencies. Station annot Share with,,,,,,,,,,,,,, a. Model this situation with a graph. b. ind the minimum number of frequencies needed by the. 19. onsider the following digraph. a. oes it have a directed uler circuit? xplain why or why not. If it does, list one. b. oes it have a directed uler path? xplain why or why not. If it does, list one.

8 8 hapter raphs and Their pplications 0. a. Represent the following map with a graph. b. olor your graph. c. What is the minimum number of colors needed to color the map? KY V TN N S MS L L ibliography iggs, N. L.,. K. Lloyd, and R. J. Wilson raph Theory Oxford: larendon Press. usch, The New ritical Path Method. hicago: Probus. havey, arrah rawing Pictures with One Line. (istomp Module 1). Lexington, M: OMP, Inc. OMP, Inc. 01. or ll Practical Purposes: Mathematical Literacy in Today s World. 9th ed. New York: W.. reeman. opes, W.,. Sloyer, R. Stark, and W. Sacco raph Theory: uler s Rich Legacy. Providence, RI: Janson. ozzens, Margaret., and R. Porter Mathematics and Its pplications. Lexington, M:.. eath and ompany. ozzens, Margaret., and R. Porter Problem Solving Using raphs. (imp Module 6). Lexington, M: OMP, Inc.

9 hapter Review 9 ossey, John,. Otto, L. Spence, and. Vanden ynden iscrete Mathematics. 5th ed. Upper Saddle River, NJ: Pearson. rancis, Richard L The Mathematicians oloring ook. (imp Module #10). Lexington, M: OMP, Inc. Kenney, Margaret J., ed iscrete Mathematics across the urriculum, K Yearbook of the National ouncil of Teachers of Mathematics. Reston, V: NTM. Malkevitch, J., and W. Meyer raphs, Models, and inite Mathematics. nglewood liffs, NJ: Prentice all. Tannenbaum, P., and R. rnold. 01. xcursions in Modern Mathematics. 8th ed. Upper Saddle River, NJ: Pearson. Wilson, Robin. 00. our olors Suffice. Princeton, NJ: Princeton University Press.