We may assume the heavy vertex in $\overrightarrow{D}$ is $\phi(1)$. Then there exists a non-trivial oriented tree $\overrightarrow{T}$, containing a vertex $y$, such that $\overrightarrow{D}= \overrightarrow{D_n}(\phi(1)) +_1 \overrightarrow{T}(y)$. Applying Theorem \[oriented tree\], embed $\overrightarrow{T}$ into a $1$-page book such that each vertex is loose and the direction of the page is upwards. Since $y= \phi(1)$ is loose in the spine, we can insert the directed path $\phi(1), \phi(2),...,\phi(n)$ into the spine below $y$, so that each arc of the dipath is downwards in the spine, and place the arc $(\phi(n),\phi(1))$ into the interior of the page. Thus we have a $1$-page oriented book embedding of $\overrightarrow{D}$ with $\phi(1)$ and $\phi(n)$ half-loose in the embedding.

We now define the first subclass of strictly uni-dicyclic graphs, $\mathcal{I}$, as follows. An oriented graph $\overrightarrow{D}$ is a member of $\mathcal{I}$ if $\overrightarrow{D}$ contains an $n$-dicycle, $\overrightarrow{D_n}$, with $n \geq 4$, and for two vertices $\phi(i), \phi(j) \in V(\overrightarrow{D_n})$ with $\vert i - j \vert > 1$, there exist two arcs $a_i,a_j \in A(\overrightarrow{D})$, having one endpoint of degree one and other endpoint $\phi(i), \phi(j)$, respectively, in $\overrightarrow{D_n}$, such that $\overrightarrow{D} = \overrightarrow{D_n} \cup \{a_i,a_j \}$. Three members of $\mathcal{I}$, with $n =4$, are shown 