#### Date of Award

Fall 2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Computational Analysis and Modeling

#### First Advisor

Jinko Kanno

#### Abstract

A book consists of a line *L* in [special characters omitted]3, called the *spine*, and a collection of half planes, called *pages*, whose common boundary is *L*. A *k-book* is book with *k* pages. A *k*-page book embedding is a continuous one-to-one mapping of a graph *G* into a book such that the vertices are mapped into *L* and the edges are each mapped to either the spine or a particular page, such that no two edges cross in any page. Each page contains a planar subgraph of *G*. The book thickness, denoted *bt*(*G*), is the minimum number of pages for a graph to have a *k*-page book embedding. We focus on oriented graphs, and propose a new way to embed oriented graphs into books, called an *oriented book embedding*, and define *oriented book thickness *.

We investigate oriented graphs having oriented book thickness *k* using *k-page critical oriented graphs*, oriented graphs with oriented book thickness equal to *k*, but, for each arc, the deletion of that arc yields an oriented graph with oriented book thickness equal to *k* –1. We discuss several classes of two-page critical oriented graphs, and use them to characterize oriented graphs with oriented book thickness equal to one that are *strictly uni-dicyclic graphs*, oriented graphs having exactly one cycle, which is a directed cycle. We give a similar result for *strictly bi-dicyclic graphs*, oriented graphs having exactly two cycles, which are directed cycles.

#### Recommended Citation

McAdams, Stacey R., "" (2016). *Dissertation*. 88.

https://digitalcommons.latech.edu/dissertations/88