11.  Does each of these graphs have an Euler circuit? If so, find it.
12.  Does each of these graphs have an Euler circuit? If so, find it. 
13.  Eulerize this graph using as few edge duplications as possible. Then, find an Euler circuit. 
17.  Does each of these graphs have at least one Hamiltonian circuit? If so, find one. 
19.  A company needs to deliver product to each of their 5 stores around the Dallas, TX area. Driving distances between the stores are shown below. Find a route for the driver to follow, returning to the distribution center in Fort Worth: 
a. Using Nearest Neighbor starting in Fort Worth 
b. Using Repeated Nearest Neighbor 
c. Using Sorted Edges 
21. When installing fiber optics, some companies will install a sonet ring; a full loop of cable connecting multiple locations. This is used so that if any part of the cable is damaged it does not interrupt service, since there is a second connection to the hub. A company has 5 buildings. Costs (in thousands of dollars) to lay cables between pairs of buildings are shown below. Find the circuit that will minimize cost: 
a. Using Nearest Neighbor starting at building A 
b. Using Repeated Nearest Neighbor 
c. Using Sorted Edges 
23.  Find a minimum cost spanning tree for the graph you created in problem #3Graph Theory 117

© David Lippman Creative Commons BY-SA

Graph Theory and Network Flows
In the modern world, planning efficient routes is essential for business and industry, with

applications as varied as product distribution, laying new fiber optic lines for broadband

internet, and suggesting new friends within social network websites like Facebook.

This field of mathematics started nearly 300 years ago as a look into a mathematical puzzle

(we’ll look at it in a bit). The field has exploded in importance in the last century, both

because of the growing complexity of business in a global economy and because of the

computational power that computers have provided us.

Graphs

Drawing Graphs

Example 1

Here is a portion of a housing development from Missoula, Montana1. As part of her job, the

development’s lawn inspector has to walk down every street in the development making sure

homeowners’ landscaping conforms to the community requirements.

1 Sam Beebe. http://www.flickr.com/photos/sbeebe/2850476641/, CC-BY

118

Naturally, she wants to minimize the amount of walking she has to do. Is it possible for her

to walk down every street in this development without having to do any backtracking?

While you might be able to answer that question just by looking at the picture for a while, it

would be ideal to be able to answer the question for any picture regardless of its complexity.

To do that, we first need to simplify the picture into a form that is easier to work with. We

can do that by drawing a simple line for each street. Where streets intersect, we will place a

dot.

This type of simplified picture is called a graph.

Graphs, Vertices, and Edges

A graph consists of a set of dots, called vertices, and a set of edges connecting pairs of

vertices.

While we drew our original graph to correspond with the picture we had, there is nothing

particularly important about the layout when we analyze a graph. Both of the graphs below

are equivalent to the one drawn above since they show the same edge connections between

the same vertices as the original graph.

You probably already noticed that we are using the term graph differently than you may have

used the term in the past to describe the graph of a mathematical function.

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Graph Theory 119

Example 2

Back in the 18th century in the Prussian city of

Königsberg, a river ran through the city and seven

bridges crossed the forks of the river. The river

and the bridges are highlighted in the picture to

the right2.

As a weekend amusement, townsfolk would see if

they could find a route that would take them

across every bridge once and return them to

where they started.

Leonard




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