graph $P_n$ with the added edge having endpoints $1$ and $n$. A [*cycle on $n$ vertices*]{} (resp. an [*n-cycle*]{}) is a graph isomorphic to $C_n$ (for some $n$).

The [*standard directed path*]{}, $\overrightarrow{P_n}$, is the oriented graph whose underlying graph is isomorphic to $P_n$, and for each edge $ \{i, i+1 \}$, we have the arc $(i, i+1)$. An oriented graph isomorphic to $\overrightarrow{P_n}$ is called an [*n-dipath*]{}. The [*standard directed cycle*]{}, denoted $\overrightarrow{C_n}$, is the oriented graph consisting of $\overrightarrow{P_n}$ and the arc $(n,1)$. An oriented graph isomorphic to $\overrightarrow{C_n}$ is referred to as an [*n-dicycle*]{}. One may notice that we specifically stated [*directed*]{} path or [*directed*]{} cycle. This is because the directions of the arcs “follow” each other along the path or the cycle. If this is not the case, and the underlying graph is isomorphic to a path or a cycle, but arbitrarily directed, we refer to each as an [*oriented n-path*]{} or [*oriented n-cycle*]{}, respectively.

For an integer $k \geq 0$, a [*$k$-book*]{}, or [*a book with $k$ pages*]{}, consists of a line $L$ in $\mathbb{R}^3$, called the [*spine*]{}, where $L$ is identified with the $z$-axis, i.e. $L = \{(0,0,z) : z \in \mathbb{R} \}$ and 