book embedding of a graph $D$, we call an edge in $D$ that is embedded into the spine a [*tight edge*]{}, and we call an edge in $D$ that is not a tight edge a [*loose edge*]{}. In a given book embedding of a graph $D$, we call a vertex $v$ of $D$ a [*tight vertex*]{} if it is the common endpoint of exactly two tight edges; we call $v$ a [*half-loose vertex*]{} if it is the endpoint of exactly one tight edge, and we call $v$ a [*loose vertex*]{} if it is the endpoint of no tight edge. For a particular book embedding of a graph $D$, we say that an edge $uv$ in $D$ [*covers*]{} a vertex $x \in V(D)$ if $x$ is between $u$ and $v$ in the spine order of the embedding. For two arcs $a_1,a_2$ that are in the same page, if $a_1$ and $a_2$ share an endpoint, and $a_1$ covers the other endpoint of $a_2$, or if $a_1$ covers both endpoints of $a_2$, we say that $a_1$ and $a_2$ are *nested* or that *$a_2$ is nested inside $a_1$*.

A [*planar graph*]{} is a graph that has an embedding in the plane, and an [*outerplanar 