graph*]{} is a graph that has an embedding in the plane with every vertex on the unbounded face. A graph is [*hamiltonian*]{} if it contains a cycle that passes through every vertex of the graph.

$($Bernhart and Kainen, $1979$  [@BK] $)$ Let $G$ be connected, then the following hold: $1.$ $bt(G) = 0$ if and only if $G$ is a path, and\
$2.$ $bt(G) \le 1$ if and only if $G$ is outerplanar.\
$3.$ $bt(G) \le 2$ if and only if $G$ is a subgraph of a hamiltonian planar graph. \[BKthm\]

We call a graph $G$ a [*$k$-page critical graph*]{} if $bt(G) = k$, and for every edge $e \in E(G)$, we have that $bt(G \setminus e) = k-1$. By Kuratowski’s Theorem [@K] and Theorem \[BKthm\], $K_4$ and $K_{2,3}$ are $2$-page critical graphs. All $k$-page critical graphs are connected; since, if $G$ was $k$-page critical with distinct components $H_1,H_2$, then $bt(G) = max \{bt(H_1), bt(H_2) \} =k$. Assume $bt(G) = bt(H_1)$, and if we delete an edge $e' \in E(H_2)$, we have $bt(G \setminus e') =k$, and thus $G$ is not a $k$-page critical graph.

We now focus on embedding an oriented graph into a $k$-book. Since the spine $L$ 