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

In her investigation, Patty evaluates 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 could not spoil or melt. The problem is that the camp director, then lost her keys to the refrigerator unit sometime while walking the camp paths, shown in the given diagram (triangles indicate camp buildings/tents; lines indicate paths):

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

Task #2. When camp is not in session, the camp director lives in a residence near to the camp. If she doesn't get her keys on the camp trials, then either they have been stolen by a squirrel or they are somewhere in her house, given 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.

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

Task #3. Can you evaluate 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, explain the path. If not, describe how you know that there is no such path.

Task #4. The directors and producers of The Patty Madeye Mysteries require 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 show a definite application for graph theory besides those presented in the course. Your answer could be in paragraph form and can include properly cited or original images. Be sure to describe the specific real-world application and provide a specific example of where this application has been used.