Jinko Kanno [^2]'
date: 'February 3, 2016'
title: Oriented Book Embeddings
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Introduction
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In this work, every graph is a simple graph, i.e., a graph containing no loops or multiple edges. We follow the notation and terminology in [@West], in particular, we denote a complete graph on $n$ vertices as $K_n$, and a complete bipartite graph having two partite sets with sizes $m$ and $n$ as $K_{m,n}$. We consider orientations of simple graphs as described below.

Given a graph $D$, with finite vertex set $V(D)$ and finite edge set $E(D)$, we define an [*oriented graph*]{} as follows. For each pair of vertices $u$ and $v$ in $V(D)$ such that there exists an edge $e = uv$ in $E(D)$, we assign $u$ (resp. $v$) to be the [*tail*]{} and $v$ (resp. $u$) to be the [*head*]{}. This assignment results in an [*arc*]{}, where the tail [*is directed*]{} to the head. We denote the arc $a$ with tail $u$ and head $v$ as $(u,v)$, and call $u$ and $v$ the [*endpoints*]{} of $a$. The endpoints of an arc are [*adjacent*]{} and an arc is [*incident*]{} to both of its endpoints. We denote the resulting oriented graph $\overrightarrow{D}$ with arc set $A(\overrightarrow{D})$ 