#### Date of Award

2001

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Guoli Ding

#### Abstract

Graphs are characterized by whether or not they have orientations to avoid one or more of the digraphs K&ar;3 , S&ar;3 , and P&ar;3 . K&ar;3 , S&ar;3 and P&ar;3 are created by starting with a triangle, a three point star, or a path of length three respectively, and replacing each edge with a pair of arcs in opposite directions. Conditions are described when all orientations of 3-connected and 4-connected graphs must have one or more of the above digraphs as a minor. It is shown that double wheels, and double wheels without an axle, are the only 4-connected graphs with an orientation not having a K&ar;3 -minor. For S&ar;3 , it is shown that the only 4-connected graphs which may be oriented without the minor are K5 and C26 . It is also shown that all 3-connected graphs which do not have a W5-minor have an orientation without-an S&ar;3 -minor, while every orientation of a graph with a W 6-minor has an S&ar;3 -minor. It is demonstrated that K5, C26 , and C26 plus an edge are the only 4-connected graphs with an orientation without a P&ar;3 -minor. Additionally, some restrictions on large 3-connected graphs without a P&ar;3 -minor are given, and it is shown that if a 3-connected graph has a large wheel as a minor and has an orientation without a P&ar;3 -minor, then the graph must be a wheel. Certain smaller digraphs P&ar;1 , P&ar;2 , and M = K&ar;3 \a are also considered as possible minors of orientations of graphs. It is shown that a graph has an orientation without a P&ar;1 -minor if and only if it is a forest. It is shown that every orientation of a graph has a P&ar;2 -minor if and only if the graph has T2 or K+4 as a minor. To describe graphs with an orientation without an M-minor, a similar small list of graphs is given, and it is shown that if none of the given graphs is a minor of a graph, then that graph has an orientation without an M-minor.

#### Recommended Citation

Berman, Glenn Randolph, "Orientations of Graphs Which Have Small Directed Graph Minors." (2001). *LSU Historical Dissertations and Theses*. 237.

https://digitalcommons.lsu.edu/gradschool_disstheses/237

#### ISBN

9780493213781

#### Pages

91