Oriented Book Embeddings

TL;DR

This paper explores the properties of oriented graphs with directed cycles using oriented book embeddings, defining critical classes and providing complete characterizations for small cases.

Contribution

It introduces the concept of oriented book thickness and characterizes critical oriented graphs for small thickness values, especially for graphs with a single directed cycle.

Findings

Complete list of 1-page critical oriented graphs.

Characterization of 2-page critical oriented graphs with a single directed cycle.

Abstract

A graph $G$ has a $k$-page book embedding if $G$ can be embedded into a $k$-page book. The minimum $k$ such that $G$ has a $k$-page book embedding is the book thickness of $G$, denoted $bt(G)$. Most of the work on this subject has been done for unoriented graphs and oriented acyclic graphs (no directed cycles). In this work we discuss oriented graphs $\overrightarrow{D}$ containing directed cycles by using oriented book embeddings and oriented book thickness, $obt(\overrightarrow{D})$. To characterize $\overrightarrow{D}$ such that $obt(\overrightarrow{D}) = k$, we define the class $\mathcal{M}^k$ of $k$-page critical oriented graphs to be all oriented graphs $\overrightarrow{D}$ with $obt(\overrightarrow{D}) =k$, but for every proper oriented subgraph of $\overrightarrow{D}$, denoted $\overrightarrow{D}'$, we have that $obt(\overrightarrow{D}') < k$. Determining $\mathcal{M}^k$ for general $k$ is challenging; we narrow down the list of oriented graphs in $\mathcal{M}^k$ for small $k$. In this work we show complete lists for $\mathcal{M}^1$ and for $\mathcal{M}^2 \cap \mathcal{U}$, where $\mathcal{U}$ consists of all strictly dicyclic oriented graphs, that is, oriented graphs containing exactly one oriented cycle, which is a directed cycle. Keywords: book embedding, book thickness, oriented book embedding, oriented book thickness, directed cycle, critical graph