Oriented Book Embeddings

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