#### Identifier

etd-03162015-155318

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

In this dissertation, we begin with a brief survey of the Petersen graph and its role in graph theory. We will then develop an alternative decomposition to clique sums for 3-connected graphs, called T-sums. This decomposition will be used in Chapter 2 to completely characterize those graphs which have no P_3 minor, where P_3 is a graph with 7 vertices, 12 edges, and is isomorphic to the graph created by contracting three edges of a perfect matching of the Petersen Graph. In Chapter 3, we determine the structure of any large internally 4-connected graph which has no P_2 minor, where P_2 is a graph on 8 vertices, 13 edges, and is isomorphic to the graph created by contracting two edges of a perfect matching of the Petersen Graph.

#### Date

2015

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Ferguson, Adam Beau, "Excluding Two Minors of the Petersen Graph" (2015). *LSU Doctoral Dissertations*. 63.

https://digitalcommons.lsu.edu/gradschool_dissertations/63

#### Committee Chair

Ding, Guoli