V(\overrightarrow{T})$, and two empty lists $L$ and $S$ A list $S$ that yields the spine order of a $1$-page oriented book embedding of $\overrightarrow{T}$ such that each vertex is loose in the embedding, the direction of all arcs agree, and $x$ is uncovered

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Let $N^+(x) = \{u_1, u_2,...,u_n \}$ Add $u_1$ to the end of $L$ Add $u_1$ to the beginning of $S$

add $u_i$ to $L$ between $u_{i-1}$ and $x$ add $u_i$ to $S$ between $u_{i-1}$ and $x$

Let $N^-(x) = \{v_1, v_2,...,v_n \}$ Add $u_v$ to the end of $L$ Add $v_1$ to the end of $S$

add $v_i$ to $L$ between $v_{i-1}$ and $x$ add $v_i$ to $S$ between $v_{i-1}$ and $x$

\[tree\_sink\] If a vertex $x$ is a sink in $\overrightarrow{T}$ then there exists a spine order of $V(\overrightarrow{T})$ which yields a $1$-page oriented book embedding of $\overrightarrow{T}$ such that $x$ is the top vertex in the spine and each vertex is loose in the oriented book embedding.

Since $x$ is a sink, let $N(x)=N^-(x)= \{v_1,v_2,...,v_k \}$. Delete $x$ to obtain $k$ disconnected oriented trees, denoted $\overrightarrow{T_{v_i}}$, with root $v_i$, $1 \le i \le k$. Let $\beta_i$ be the spine order of $\overrightarrow{T_{v_i}}$ in the 