  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
