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.

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