$\mathcal{M}^2 \cap \mathcal{U}$, where $\mathcal{U}$ consists of all *strictly dicyclic oriented graphs*, that is, oriented graphs containing exactly one oriented cycle, which is a directed cycle. To prepare for the proofs in Section 4, we first discuss the oriented book embeddings of *oriented cycles* and *oriented trees* in Section 2.

Oriented Book Embeddings of Cycles and Trees
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In this section we discuss the oriented book embeddings of two fundamental types of oriented graphs, oriented cycles and oriented trees. We will use these oriented graphs to construct strictly uni-dicyclic graphs in Section 4.

Undirected and Oriented Cycles
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We now discuss oriented book embeddings of oriented cycles. We first discuss book embeddings of (undirected) cycles in Theorem \[cyclic\_on\_C\], and characterize the spine order of every $1$-page book embedding of a cycle. We then describe all possible $1$-page oriented book embeddings of directed cycles in Corollary \[directed cycle\].

Let $C$ be a cycle on $n$ vertices. By definition, there exists an isomorphism $\phi: [n] \rightarrow V(C)$ such that the edge set of $C$ consists of $\phi(i) \phi(i+1)$ for $i \in [n-1]$ and edge $\phi(1) \phi(n)$. For $n \in \mathbb{N}$ and $k \in \mathbb{Z}$, a [*cyclic permutation*]{} of $V(C)$ is a mapping 