subgraph isomorphic to either $S^+$ or $S^-$; therefore, the only other oriented graphs to consider are oriented cycles. If an oriented cycle is not a directed cycle, it must contain an oriented subgraph that is isomorphic to either $S^+$ or $S^-$. While an $n$-dicycle, denoted $\overrightarrow{D}_n$, contains no $S^+$ or $S^-$, its oriented book thickness is one, which will be discussed more in Section 2. However, the deletion of any arc in $\overrightarrow{D}_n$ results in a dipath; therefore, $\overrightarrow{D}_n$, $n \geq 3$, is in $\mathcal{M}^1$, giving the following result.

\[M\_1\] $\mathcal{M}^1 = \{ S^+, S^-, \overrightarrow{D}_n \vert n \geq 3\}$.

Strictly Uni-dicyclic Critical Graphs
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In this section, we give the complete description for the class of *strictly uni-dicyclic graphs*, denoted $\mathcal{U}$, and discuss three subclasses, $\mathcal{I},\mathcal{T},\mathcal{R}$, of $\mathcal{U}$ that are $2$-page critical, which we use to characterize $\mathcal{M}^2 \cap \mathcal{U}$ in Theorem \[unicyclic\_list\].

Let $\overrightarrow{D}_n$, $n \geq 3$, be an $n$-dicycle, with arcs $(\phi(i),\phi(i+1))$, for $1 \le i \le n-1$, and arc $(\phi(n),\phi(1))$; then $\overrightarrow{D}_n$ is a member of $\mathcal{U}$ and we can construct every oriented graph in $\mathcal{U}$ as follows. Following the definition of $1$-sum for undirected graphs, found in [@sum], we define the [*$1$-sum*]{} of two oriented 