graphs $\overrightarrow{D}$, $\overrightarrow{D}'$, via $y$ and $y'$, to be the oriented graph obtained by identifying a vertex $y \in V(\overrightarrow{D})$ with a vertex $y' \in V(\overrightarrow{D}')$; we denote the resulting oriented graph $\overrightarrow{D}(y) +_1 \overrightarrow{D'}(y')$. For convenience, we call a single vertex a *trivial oriented tree*. For $n$ distinct, possibly trivial, oriented trees $\overrightarrow{T_i}$, $1 \le i \le n$, we describe a member of $\mathcal{U}$ to be $\bigcup_{i=1}^n (\overrightarrow{D}_n(\phi(i)) +_1 \overrightarrow{T_i}(y_i))$, where $\phi(i) \in V(\overrightarrow{D}_n)$ and $y_i \in V(\overrightarrow{T_i})$. For an oriented graph $\overrightarrow{D}$ in $\mathcal{U}$, if $\overrightarrow{T_i}$ is not a trivial oriented tree, we call the vertex $\phi(i)=y_i$ in $\overrightarrow{D}$ a *heavy vertex*. In fact, each member of $\mathcal{I}$, $\mathcal{T}$, and $\mathcal{R}$ have most three heavy vertices. The next result shows that if a strictly uni-dicyclic graph has exactly one heavy vertex, then its oriented book thickness is one.

\[dicycle\_1\_tree\] Let $\overrightarrow{D} \in \mathcal{U}$. If $\overrightarrow{D}$ has exactly one heavy vertex, then there exists a $1$-page oriented book embedding of $\overrightarrow{D}$ such that the page is upwards and the endpoints of the loose arc of $\overrightarrow{D_n}$ are half-loose in the embedding.

Let $\overrightarrow{D}_n$ be an $n$-dicycle with arcs $(\phi(i), \phi(i+1))$, for $1 \le i \le n-1$, and arc $(\phi(n),\phi(1))$. 