Unit 6: Graph Theory - Assignment

Total points for Assignment: 35 points. Assignments must be submitted as a Microsoft Word document and uploaded to the Dropbox for Unit 6. All Assignments are due by Tuesday at 11:59 PM ET of the assigned Unit.
NOTE: Assignment problems should not be posted to the Discussion threads. Questions on the Assignment problems should be addressed to the instructor by sending an email or by attending office hours.
You must show your work on all problems. If a problem is worth 2 points and you only show the answer, then you will receive only 1 point credit. If you use a calculator or online website, give the source and tell me exactly what you provided as input. For example, if you used Excel to compute 16 * 16, state “I typed =16*16 into Excel and got 256. You may type your answer right into this document.

Part I. Basic Computations
1. (4 points) The plan for a four-room house is shown below. Draw a graph that models the connecting relationships between the areas in the floor plan. [Your graph does not

[Your graph does not need to be fancy. You may use any drawing software such as Visio or Creatly.com]
Answer: I used viso for graph[pic]
2.

a. Identify all the vertices in the above graph with odd degree. Identify the degree of each of these vertices. (2 points)
Answer: The odd number of edges is a odd degree vertices are D,E,F from the graph 3,1,3 it has a odd number. So D,E,F, is odd and the rest is even.
b. Describe two paths of different lengths that start at vertex A and which end at vertex F. Specify the length of each path. (2 points)
Answer: A-B-C-D length 4 A-B-C-C-D-F length 5
c. Describe a circuit of length 3. (2 points)
Answer: the circuit path is A-B-C
d. Describe two different circuits of length 4 (1 point)
Answer: Two circuits with 4 in length A-B-C-C-A and F-G-H-I-F
3. Consider this graph:
a. Find an Euler circuit in this graph that starts and ends at vertex D. (1 point)
Answer: D-B-A-C-E-G-H-I-J-G-C-D-J-F-D
b. Using Euler’s Rules, explain how you know that this graph has an Euler Circuit? (1 point)
Answer: It has all even degree vertices and a circuit

4. Paths in a zoo are located according to this map. You want to make sure that you see every exhibit along each path exactly once.
[pic]a. Where should you begin and end so that you do not need to retrace your steps? Explain how you know where to start and end. (1 point)
Answer: j-d-c-a-e-g-h-x-i-f-k
Explanation: just went all around the loop
b. Find a path such that you do not need to retrace your steps. (1 point)
Answer: F-X OR G-H Part II. Case Study The Case of the Missing Cookies

This week’s episode of “Patty Madeye Mysteries” is based on an investigation at a local Girl Sprouts Camp. Apparently, the Girl Sprout organization has been gearing up for their annual fund-raising event in which members sell cookies and candy at local shopping centers. The proceeds from the fund-raising event are then used to improve the camping facilities (tents, mess-hall, swimming area) at the camp.

In her investigation, Patty determines that the cookies and candy were delivered to the camp on Friday and stored in the camp office. Over the weekend, the camp director moved them into the refrigerator unit in the mess-hall so that they would not melt or spoil. The problem is that the camp director, then lost her keys to the refrigerator unit sometime while walking the camp paths, shown in the following diagram (triangles represent camp buildings/tents; lines represent paths):

[pic]
Task #1: (4 points) The camp director is in a hurry to find her keys and she must search along each of the paths. Can you determine a way for her to travel each trail only once, starting and ending at her office? If so, describe the path. If not, explain how you know that there is no such path, then describe a path in which the camp director MAY retrace her steps.

Answer: Is the sequence F,L,K,I,H,G,A,B,J,D,C,F

Task #2. (4 points). When camp is not in session, the camp director lives in a residence close to the camp. If she doesn’t find her keys on the camp trials, then either they have been stolen by a squirrel or they are somewhere in her house, shown below. Since she might have used her keys to open one of the many doors in her house, she will need to check each door.

Room #1

Patty has been asked to provide a sketch showing the relationships between each of the rooms and doors in the house. Can you draw a graph depicting this relationship?

Answer: [pic]

Task #3. (4 points) Can you determine a method for the camp director to search for her keys in each of the doors of the house without retracing her steps? If so, describe the path. If not, explain how you know that there is no such path.

Answer: that is no such path no door

Task #4. (8 points) The directors and producers of The Patty Madeye Mysteries need some background on graph theory, since they have not yet taken this course. Do some research on graph theory using the Kaplan Library and the internet and present a specific application for graph theory besides those presented in the course. Your answer should be in paragraph form (no more than 1 page in length) and may include properly cited or original images. Be sure to explain the specific real-world application and give a specific example of where this application has been used.

The duplicate edges added to the Eulerized graph can be thought of as actual new edges but in many contexts, the new edges of the Eulerized graph just represent a retracing of the same edge in the original graph.

Some graphs can be directed, which means the lines have an arrow and only go in one direction. Some better thanother because of coler or data. Graphs can have no edges at all, in others the edges. Eulerization involves only involves adding edges which duplicate existing edges. You cannot add "new" edges; that is, edges which do not duplicate edges already in the graph. Always Eulerize a graph using the fewest edges possible.
The duplicate edges added to a graph during the process of Eulerization are often called DEADHEAD EDGES. ("Euler Circuits Quick Summary," 2014). Graph are related to the number of vertices, edges and faces.

Reference:
Euler Circuits Quick Summary. (2014, January 1). Retrieved November 25, 2014, from http://www.ctl.ua.edu/math103/euler/quick.htm

Euler Circuits Quick Summary. (2014, January 1). Retrieved November 25, 2014, from http://www.ctl.ua.edu/math103/euler/quick.htm
Leonhard Euler. (2012, January 1). Graph Theory | World of Mathematics. Retrieved November 25, 2014, from http://world.mathigon.org/Graph_Theory

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Hall

Living
Room

Outside

Kitchen

Den

I

F

C

B

D

H

A

G

E

A

B

E

C

J

I

F

D

G

H

A

C

G

H

I

Office

E

Mess-hall

K

J

B

D

L

F

R #7

Room
#6

Room #3

Rm #8

Rm #4

Room #1

Room #2

Room #9

Room #5