the same page; an oriented book embedding may have an upwards page and a downwards page. The [*oriented book thickness*]{}, denoted $obt(\overrightarrow{D})$ is the minimum $k$ such that $\overrightarrow{D}$ has a $k$-page oriented book embedding. From the definition of oriented book thickness and its requirements, for an oriented graph $\overrightarrow{D}$ with underlying graph $D$, we have that $bt(D) \le obt(\overrightarrow{D})$. The definitions for spine order, tight or loose arcs, and whether an arc covers a vertex in a particular oriented book embedding are all similar to their definitions for book embeddings in the unoriented case.

We call an oriented graph $\overrightarrow{D}$ a [*$k$-page critical oriented graph*]{} if $obt(\overrightarrow{D}) = k$, and for every arc $a \in A(\overrightarrow{D})$, we have that $obt(\overrightarrow{D} \setminus a) = k-1$. The class of all $k$-page critical oriented graphs is denoted $\mathcal{M}^k$.

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 Section 3 we show complete list for $\mathcal{M}^1$, and in Section 4 we discuss $\mathcal{M}^2$. For undirected graphs, there are exactly two $2$-page critical graphs, $K_4$ and $K_{2,3}$; however, in the oriented case is becomes more complicated. We show a complete list for 