#### Date of Award

8-1-2000

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Computer Science

#### First Advisor

Dr. Hesham H. Ali

#### Abstract

Graphs are diagrams made up of nodes and edges. The nodes are the points on the graph. The edges are the lines connecting the nodes. These graphs are useful in that they allow for the modeling of real world problems into a format that can be readily solved by computers. Graph theory can be used in fields as diverse as chemistry, transportation, and music. However, graph theory is not being fully utilized because of the level of knowledge required to use it. The first of three goals of this thesis is to make graph theory accessible to a larger audience by developing a graphical application. This application allows a user to create a graph, apply a graph algorithm, and display the results through a graphical user interface. The second goal of this thesis is to implement the ost useful graph algorithms. This includes basic algorithms that have been well researched and can be solved in polynomial time. Advanced algorithms for the class of graphs known as perfect graphs will also be implemented. The third goal is to add to graph theory to make it more practical. A relatively new class of graphs known as tolerance graphs allows for variations that occur in real world problems. The nodes on a tolerance graph correspond to intervals on a real line. Each interval has a tolerance value. Edges are drawn between two nodes if the intersection of the two intervals is greater than the tolerance of either interval. This thesis examines known algorithms for tolerance graphs. There are still some open problems dealing with tolerance graphs. Among them are the problem of recognizing tolerance graphs and converting a known tolerance graph into a tolerance representation. These two problems will be explored within this thesis.

#### Recommended Citation

Hazelwood, Michael D., "Graph theory and tolerance graphs." (2000). *Student Work*. 3540.

https://digitalcommons.unomaha.edu/studentwork/3540

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## Comments

A Thesis Presented to the Department of Computer Science and the Faculty of the Graduate College University of Nebraska In Partial Fulfillment of the Requirements for the Degree Master of Science University of Nebraska at Omaha. Copyright 2000 Michael D. Hazelwood