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%!TEX root = main.tex
\section{A-Graphs}
\seclabel{quad}
\seclabel{quadrangulations}
In this section, we study a special class of graphs that are closely
related to quadrangulations in which every edge crosses $Y$. (See \figref{a-graph} for an example.)
\begin{defn}\deflabel{a-graph}
An \emph{A-graph}, $G$, is a plane straight-line graph with $n\ge 3$ vertices that has the following properties:
\begin{compactenum}
\item Every edge of $G$ intersects $Y$ in exactly one
point, possibly an endpoint.
\label{p1}
\item Every face of $G$, including the outer face, is
a quadrilateral or a triangle (not containing any
disconnected components inside).
\item Every quadrilateral face of $G$ is non-convex.
\item Every triangular face contains one vertex
on $Y$, one in $L$, and one in $R$.
\item Every vertex $v$ on $Y$ is incident to precisely
two triangular faces, one ``above $v$'', which
contains the line segment between $v$ and
$v+(0,\epsilon)$ for some $\epsilon>0$, and one ``below
$v$'', containing the line segment between $v$ and $v-(0,\epsilon)$ for some $\epsilon >0$.
\label{p-last}
\end{compactenum}
\end{defn}
%\begin{wrapfigure}[11]{r}{.3\textwidth}
% \Vspace{-1mm}
\begin{figure}
\centering{\includegraphics[scale = 0.95]{figs/a-graph-new}}
\caption{An A-graph with 2 vertices on $Y$.}
\figlabel{a-graph}
\end{figure}
% \end{wrapfigure}
In the special case where $G$ has no vertices in $Y$, the graph $G$ is a quadrangulation in which every edge crosses $Y$. Further, Property~5 applies even if $v$ is on the outer face of $G$ (in which case it implies that the outer face of $G$ must be a triangle).
Some additional properties of $G$ follow from % \defref{a-graph}:
Properties~\ref{p1}--\ref{p-last}.
\begin{compactenum}\setcounter{enumi}{5}
\item $G$ is connected.
\item Every vertex of $G$ has degree at least~2.
\item If $n\ge 4$, then every vertex in $Y$ has degree at least~3.
\end{compactenum}
Property~6 follows directly from Property~2.
Property~7 follows from the fact that every vertex is incident to at
least one face and every face is a simple cycle.
Property~8 follows from the fact that every vertex on $Y$ is incident
to at least two triangular faces, which involve at least 4 vertices,
unless $n=3$.
(Property~3 and 4 are actually redundant---Property 3 follows from Properties~1 and 5; Property~4 follows from Property~1.)
%We will show that every A-graph $G$ has a \Fary\ drawing with prescribed
%intersections with $Y$ and a prescribed outer face. Since the outer
%face of an A-graph can be a triangle or a quadrilateral, in the following
%theorem, $\Delta$ is a triangle or quadrilateral defined as follows:
%\begin{enumerate}
% \item If $(0,y_1)$ is a vertex of $G$, then $\Delta$ is a triangle
% with one vertex at $(0,y_1)$ and the opposite edge edge crossing $Y$ at $y_m$.
%
% \item If $(0,y_m)$ is a vertex of $G$, then $\Delta$ is a triangle
% with one vertex at $(0,y_m)$ and the opposite edge crossing $Y$ at $y_1$.
%
% \item Otherwise,
% $\Delta$ is a quadrilateral whose edges cross $Y$ at $y_1$, $y_a$,
% $y_b$, and $y_m$, where $e_1$, $e_a$, $e_b$, and $e_m$ are the four edges on the outer face of $G$
%\end{enumerate}
In the following theorem,
we will show that every A-graph $G$ has a \Fary\ drawing with prescribed intersections with $Y$ and a prescribed outer face.
\begin{figure*}
\begin{center}\begin{tabular}{ccc}
\includegraphics{figs/outerface-cases-2} &
\includegraphics{figs/outerface-cases-3} &
\includegraphics{figs/outerface-cases-4} \\
(a) & (b) & (c)
\end{tabular}\end{center}
\caption{The three possibilities for the outer face in
\thmref{a-graph}.
%(In (a) and (b), we have assumed that the edges $e_1$ and $e_m$ are on the
%outer face, although the order of edges that are
%incident to the same vertex on $Y$ is not specified in the theorem.)
}
\figlabel{outerface-cases}
\end{figure*}
\begin{thm}\thmlabel{a-graph}\ %\newline
\begin{compactitem}
\item Let $G$ be an A-graph.
\item Let $e_1,\ldots,e_m$ be the sequence of edges in $G$,
in the order they are intersected by $Y$. Ties between edges having
a common endpoint on $Y$ are broken arbitrarily,
except that $e_1$ and $e_m$ are
always edges on the outer face.
\item Let $y_1\le\cdots\le y_m$ be any sequence of numbers where, for
each $i\in\{1,\ldots,m-1\}$, $y_i=y_{i+1}$ if and only if $e_i$
and $e_{i+1}$ have a common endpoint in $Y$.
% \item Let $\Delta$ be a triangle or quadrilateral, where:
% \begin{compactenum}
% \item If $(0,y_1)$ is a vertex of $G$, then $\Delta$ is a triangle
% with a vertex at $(0,y_1)$ and the opposite edge crossing $Y$ at $y_m$.
% \item If $(0,y_m)$ is a vertex of $G$, then $\Delta$ is a triangle
% with a vertex at $(0,y_m)$ and the opposite edge crossing $Y$ at $y_1$.
% \item Otherwise, $\Delta$ is a quadrilateral whose edges cross $Y$ at $y_1$, $y_a$,
% $y_b$, and $y_m$, where $e_1$, $e_a$, $e_b$, and $e_m$ are the four edges on the outer face of $G$.
% \end{compactenum}
\end{compactitem}
Then $G$ has a \Fary\ drawing in which the intersection
between $e_i$ and $Y$ is the single point $(0,y_i)$, for each
$i\in\{1,\ldots,m\}$.
Moreover, the shape $\Delta$ of the outer face can be prescribed,
subject only to the constraint that $\Delta$
has to be consistent with the graph and the data $y_1,\ldots,y_m$.
Specifically, we have three possibilities, which
are illustrated in \figref{outerface-cases}.
\begin{compactenum}[a)]
\item If the outer face of $G$ is a triangle containing the lowest
vertex on $Y$, then $\Delta$ must be a triangle with a vertex at
$(0,y_1)$, and the opposite edge crosses $Y$ at $(0,y_m)$.
\item Symmetrically, if the outer face of $G$ is a triangle containing
the highest vertex on $Y$, then $\Delta$ must be a triangle with a vertex
at $(0,y_m)$, and the opposite edge crosses $Y$ at $(0,y_1)$.
\item Otherwise, the outer face of $G$ is a quadrilateral. Let $e_1$,
$e_a$, $e_b$, and $e_m$ be the %four
edges of the outer face. Then $\Delta$ has to be a quadrilateral
whose edges cross $Y$ at $(0,y_1)$, $(0,y_a)$, $(0,y_b)$, and $(0,y_m)$.
\end{compactenum}
%If the outer face is a quadrilateral, the drawing with specified outer
%face is unique. If the outer face is a triangle, there is one degree
%of freedom.
\end{thm}
It would have been more natural to represent the intersection of $Y$ with $G$
as a mixed sequence of vertices and edges. However, to simplify the
statement of the theorem and its proof, we have chosen to specify the
desired drawing by a number $y_i$ for every edge, subject to equality
constraints. %The reason for this choice will shortly become apparent.
The convention in Condition~2 about $e_1$ and $e_m$ being boundary
edges is introduced only for
notational convenience.
The rest of this section is devoted to proving \thmref{a-graph}. We
are going to prove \thmref{a-graph} in its strongest form, in which the
outer face $\Delta$ is prescribed. We begin by making some simplifying
assumptions, all without loss of generality.
%
% The truth of the theorem is independent of the order of edges that are
% incident to the same vertex on $Y$. For notational convenience, we
% can therefore assume that $e_1$ and $e_m$ are edges on the outer face.
% With this assumption, the three possibilities, 4(a--c), for the
% outer face are illustrated in \figref{outerface-cases}.
%
% The theorem does not specify the order of edges that are
% incident to the same vertex on $Y$. For notational convenience, we
% refine the edge order such that that $e_1$ and $e_m$ are
% always edges on the outer face. (This convention has been used in
% \figref{outerface-cases}.)
%
First, we assume w.l.o.g.\ that $\Delta$ and all vertices of $G$
are contained in the strip $[-1,1]\times(-\infty,+\infty)$. This can
be achieved by a uniform scaling. Second, if the outer face of $G$
is a quadrilateral, we assume w.l.o.g.\ that the common vertex
of $e_1$ and $e_m$ in the given drawing of $G$ is in $L$, as in
Figures~\ref{fig:a-graph} and \ref{fig:outerface-cases}c, and the
vertex of desired output shape $\Delta$ incident to $e_1$ and $e_m$
is also in $L$; this can be achieved by a reflection of $G$ or $\Delta$
with respect to $Y$.
% this is
% also not a loss of generality, since if the vertex of $\Delta$
% incident to both $e_1$ and $e_m$ is in $R$, then $\Delta$ can be
% reflected with respect to $Y$, obtaining a quadrilateral $\Delta'$
% whose vertex incident to both $e_1$ and $e_m$ is in $L$, then a \Fary\
% drawing of $G$ can be constructed in which the outer face is
% $\Delta'$, and finally the \Fary\ drawing can be reflected with
% respect to $Y$, thus obtaining a \Fary\ drawing of $G$ in which the
% outer face is $\Delta$.
If $m=3$ or $m=4$, then $G$ is a 3- or a 4-cycle, respectively, hence it suffices to draw it as $\Delta$. Therefore we assume, from now on, that $m\ge 5$.
%%%%% copia fino a qui %%%%%%
% Before continuing, we pause to fully specify the ordering
% $e_1,\ldots,e_m$. This ordering is unambiguous except where
% some vertex $v\in Y$ is incident to several edges
% $e_{i},\ldots,e_{i+d}$, where $d\ge 2$ by Property~8 of A-graphs. Refer to \figref{ab}. In this case we partition $v$'s neighbors into two
% sets $\alpha_1,\ldots,\alpha_k\in L$ and $\beta_1,\ldots,\beta_\ell\in
% R$, where $\alpha_1,\ldots,\alpha_k$ are ordered clockwise around $v$
% and $\beta_1,\ldots,\beta_\ell$ are ordered counterclockwise. We then use
% the convention that $e_i,\ldots,e_{i+k-1}=v\alpha_1,\ldots,v\alpha_k$
% and $e_{i+k},\ldots,e_{i+d}=v\beta_1,\ldots,v\beta_\ell$.
We will describe the desired \Fary\ drawing by assigning a slope $s_i$
to each edge $e_i\in E(G)$. Since there can be no vertical edges,
each slope $s_i$ is well-defined. We have $m=|E(G)|$ slope variables,
$s_1,\ldots,s_m$.
We can see that these variables determine the drawing:
Since every edge $e_i$ contains the point $(0,y_i)$,
the slope $s_i$ fixes the line through $e_i$. Since every vertex $v$
not on $Y$ is incident to at least two edges that contain distinct points
on $Y$, the location of $v$ is fixed by any two of $v$'s incident edges.
(The location of each vertex on
$Y$ is fixed by definition.) Our strategy is to construct a
system of $m$ linear equations in the $m$ variables $s_1,\ldots,s_m$,
and to show that
this system is feasible and that its solution gives the desired \Fary\
drawing of $G$.
A necessary condition for the slopes to determine a F\'ary drawing of
$G$ is that the %supporting lines of
edges
with a common vertex should be concurrent. Let $v$ be a vertex
not on $Y$, and let $e_i, e_j, e_k$ be three edges incident to $v$.
The fact that the supporting lines of $e_i$, $e_j$, and $e_k$
meet at a common point (the location of $v$) is expressed by the following
\emph{concurrency constraint} in terms of the slopes $s_i,s_j,s_k$:
\ifSODA
\begin{align}\eqlabel{slope0}
\left|
\begin{matrix}
1&1&1\\
s_i&s_j&s_k\\
y_i&y_j&y_k
\end{matrix}
\right|&=
({y_j{-}y_k}) s_i + ({y_k{-}y_i}) s_j
+ ({y_i{-}y_j})s_k %\\&
= 0
%\nonumber
\end{align}
\else
\begin{equation}\eqlabel{slope0}
\left|
\begin{matrix}
1&1&1\\
s_i&s_j&s_k\\
y_i&y_j&y_k
\end{matrix}
\right|=
({y_j-y_k}) s_i + ({y_k-y_i}) s_j
+ ({y_i-y_j})s_k = 0
\end{equation}
\fi
Since $y_1,\ldots,y_m$ are given, this is a linear equation
in $s_1,\ldots,s_m$.
Writing this equation for all triplets of edges incident to a common
vertex $v$ will include many redundant equations. Indeed,
if $v$ has degree $d_v$,
it suffices to take $d_v-2$ equations: For each vertex $v\in V(G)$, we choose two fixed
incident edges $e_i$ and $e_j$ and run $e_k$ through the remaining
$d_v-2$ edges, specifying that $e_k$ should go through the common vertex
of $e_i$ and $e_j$.
%We call the resulting collection of $\sum_{v\in V(G)\setminus Y} d_v-2$ equations the \emph{concurrency constraints}.
Whenever convenient, we will use edges of $G$
as indices so that, if $e=e_i$ is an edge of $G$, then $s_e=s_i$
and $y_e=y_i$. Further, if $e$ is a line segment that
intersects $Y$ in a point, we will use $y_e$ to denote the $y$-coordinate
of the intersection of $e$ and $Y$ and $s_e$ to denote the slope of~$e$. %'s supporting line.
%It will be important to have as many equations as variables;
%thus,
We now introduce additional equations for the edges that emanate from a
vertex on $Y$; refer to \figref{proportional}.
Suppose that a vertex $v\in Y$ is incident to edges $a_1,\ldots,a_k\in L\cup Y$
and $b_1,\ldots,b_\ell\in Y\cup R$, ordered from bottom to top as in \figref{ab}.
\begin{figure*}
\begin{center}
\includegraphics{figs/proportional}
\end{center}
\caption{The proportionality constraints on slopes of edges incident to a vertex $v\in Y$.}
\figlabel{proportional}
\end{figure*}
From Property~4 of A-graphs we have $k,\ell\ge1$ and in addition
$k+\ell\ge 3$ by Property~8.
Let us first look at the slopes on the right side.
We want these slopes to be increasing:
$s_{b_1} < s_{b_2} < \dots <s_{b_\ell}$. We stipulate a stronger
condition:
We require that the slopes
$s_{b_2}, \dots, s_{b_{\ell-1}}$ partition the interval
$[s_{b_1},s_{b_\ell}]$ in fixed proportions. In other words:
\begin{equation}
\label{eq:proportion}
s_{b_i} = s_{b_1} + \lambda_i(s_{b_{\ell}}-s_{b_1}),
\end{equation}
for some fixed sequence $0<\lambda_2<\cdots<\lambda_{\ell-1}<1$.
For example, we might set $\lambda_i := (i-1)/(l-1)$.
This gives $\ell-2$ equations, for $\ell\ge 2$. Similarly, we get
$k-2$ equations for the slopes
$s_{a_1}, \dots, s_{a_{k}}$ of the edges on the left side, for $k\ge 2$.
In addition, for $k\ge 2$ and $\ell\ge 2$, we require that the \emph{range} of
slopes
on the two sides are in a fixed proportion:
\begin{equation}
\label{eq:proportion2}
s_{a_1}-s_{a_{k}} = \mu (s_{b_{\ell}}-s_{b_1}),
\end{equation}
for some fixed value $\mu>0$.
We call the equations
\thetag{\ref{eq:proportion}--\ref{eq:proportion2}} the
\emph{proportionality constraints}.
There are $(k+\ell)-3$ such equations for the $k+\ell$ slopes, hence
we have three degrees of freedom for the slopes incident to a vertex.
\figref{proportional} illustrates these degrees of freedom:
Namely, we can shear the edges on the right side vertically, adding the same constant to all
slopes. We can independently shear all edges on the left side.
In addition, we can vertically scale {all} lines jointly (both to
the left and to the right), multiplying all slopes by the same constant factor.
If this factor is negative, we would reverse the order of the
slopes, simultaneously on the left and on the right. We will later see
that this undesirable possibility is prevented in conjunction with
other constraints that we are going to impose. We can already observe
that any two slopes on one side determine all remaining slopes on that side. Moreover, the range of slopes on the other side ($s_{a_1}-s_{a_{k}}$ or $s_{b_{\ell}}-s_{b_1}$) is also determined.
%
The notations $\lambda_i$ and $\mu$ are here used in a local sense;
for a different vertex $v$, we may choose different constants.
%\begin{figure}
% \centering{\includegraphics{figs/proportional}}
% \caption{The degrees of freedom provided by the proportionality constraints}
% \figlabel{proportional}
%\end{figure}
%We have the following.
\begin{lem} \label{le:number-of-equations}
The total number of equations \thetag{\ref{eq:slope0}},~\thetag{\ref{eq:proportion}}, and~\thetag{\ref{eq:proportion2}} is $m-4$.
\end{lem}
\begin{proof}
Let $n=|V|$ and let $n_0$ be the number
of vertices on $Y$. Assume that $G$ has $f_3$ triangular and $f_4$
quadrangular faces.
We have two triangles for every vertex on $Y$ (Properties 4 and 5 of A-graphs):
\begin{equation}
\label{eq:f3}
f_3 = 2n_0
\end{equation}
Euler's formula gives
\begin{equation}
\label{eq:Euler}
n + f_3+f_4 = m+2.
\end{equation}
Double-counting of edge-face incidences leads to the relation
\begin{equation}
\label{eq:edge-face}
3f_3+4f_4=2m.
\end{equation}
Denoting the degree of a vertex $v$ by $d_v$,
we have $d_v-3$ equations for each of the $n_0$ vertices $v$ on $Y$. For each of the
$n-n_0$ vertices $v$ not on $Y$,
we have $d_v-2$ equations.
The total number of equations is therefore
\ifSODA
\begin{align*}
% \label{eq:number-equations2}
P &=
\sum_{v\in V\cap Y}(d_v-3)+
\sum_{v\in V\cap(L\cup R)}(d_v-2)
\\&
=
\sum_{v\in V}(d_v-2)-n_0
=
2m-2n-n_0.
\end{align*}
\else
\begin{align*}
% \label{eq:number-equations2}
P &=
\sum_{v\in V\cap Y}(d_v-3)+
\sum_{v\in V\cap(L\cup R)}(d_v-2)
=
\sum_{v\in V}(d_v-2)-n_0
=
2m-2n-n_0.
\end{align*}
\fi
Using \thetag{\ref{eq:f3}--\ref{eq:edge-face}}, this can be
simplified to
\ifSODA
\begin{align*}
P&
=
2m-2n-n_0\\
&
= 2m -2n -(2f_3+2f_4) +(2f_3+2f_4)-n_0\\
&= 2m -2(n +f_3+f_4) +\tfrac12(4f_3+4f_4-f_3)\\
&= 2m -2(m+2) +\tfrac12(2m) = m-4.
\qedhere
\end{align*}
\vspace{-2,2\baselineskip}
\hrule height 0pt
\else
\begin{align*}
P%&
=
2m-2n-n_0%\\
&
= 2m -2n -(2f_3+2f_4) +(2f_3+2f_4)-n_0\\
&= 2m -2(n +f_3+f_4) +\tfrac12(4f_3+4f_4-f_3)\\
&= 2m -2(m+2) +\tfrac12(2m) = m-4.
\qedhere
\end{align*}
\fi
%This concludes the proof of the lemma.
\end{proof}
To achieve the desired number $m$ of equations, we add
four \emph{boundary equations}. If the outer face is a quadrilateral,
the desired slopes of the boundary edges already give us four equations:
We set
the slopes $s_1$, $s_a$, $s_b$, and $s_m$ of the boundary edges $e_1$, $e_a$,
$e_b$, and $e_m$ to the fixed values of the slopes of the edges of
$\Delta$.
If the outer face is a triangle, the shape $\Delta$ gives us
only three constraints for the slopes of the three edges $e_1$, $e_a$,
$e_m$. If the triangle is $\alpha\beta\gamma$ with $\gamma\in Y$,
we arbitrarily pick another (non-boundary) edge $e_b$ incident to $\gamma$
and set its slope $s_b$ to an appropriate fixed value; this value has
to be either larger or smaller than each of $s_1$, $s_a$, and $s_m$
depending on whether $\gamma$ is the topmost or the bottommost point
on $Y$ and whether $e_b$ lies in $L$ or $R$. Together with the
proportionality constraints, this effectively pins \emph{all} slopes
incident to $\gamma$ to fixed values.
In both cases, we get 4 equations of the form
\begin{equation}
\label{eq:boundary}
s_i = h_i, %\text{ for $i=1,a,b,m.$}
\end{equation}
where $i\in\{1,a,b,m\}$.
Altogether, we now have a system of $m$ linear equations
in the $m$ unknowns $s=(s_1,\ldots,s_m)$, which we can write
compactly as
$A\cdot s = b$, with a square matrix $A$ whose entries come from
\thetag{\ref{eq:slope0}--\ref{eq:proportion2}}
and \eqref{eq:boundary}.
%are the variables we wish to solve for, and $b$ is a column $m$-vector
%whose entries also come from \eqref{slope}.
Only four entries of
the right-hand side vector
$b$
are non-zero, due to the four boundary equations.
We will show that $A\cdot s=b$ has a unique
solution and that this solution gives a \Fary\ drawing of $G$.
\subsection{Setting the Proportionality Constraints}
\label{sec:setting}
% Our plan is to morph the given drawing of $G$
%into the desired drawing.
Our plan is to construct the desired drawing by a continuous morph,
starting from the given drawing of $G$. Since the proportionality
constraints are not part of the output specification but were
artificially added to achieve the right number of equations, we can make
our life easy by just setting their coefficients so that they are
satisfied by the initial drawing.
Specifically, the statement of \thmref{a-graph} assumes that $G$ is a
\Fary\ drawing. In this drawing, every edge $e$
%intersects $Y$
%in a single point $(0,y_e')$ and
has a slope $s_e'$.
We use these slopes to set the
coefficients in the proportionality constraints.
Consider a
vertex $v\in Y$, incident to edges $a_1,\ldots,a_k$ and $b_1,\ldots,b_\ell$
as described above.
In the notation used
in \eqref{eq:proportion}, we set
\[
\lambda_i = (s_{b_i}'-s_{b_1}')/(s_{b_\ell}'-s_{b_1}') .
\]
The coefficients for the edges
$a_1,\ldots,a_k$ on the left side are set similarly.
If $k\ge2$ and $l\ge 2$,
we set
\[
\mu = (s_{a_1}'-s_{a_k}')/(s_{b_\ell}'-s_{b_1}')
\]
in \eqref{eq:proportion2}.
This ensures that the initial slopes $s_{1}',\ldots,s_{m}'$ satisfy the
proportionality constraints.
\subsection{Ordering constraints}
We define a relation $\prec$ on the edges of $G$, where $e_1 \prec e_2$ if and only if
\begin{itemize}
\item $y_{e_1} < y_{e_2}$ and $e_1$ and $e_2$ have a common endpoint $v\in L$; or
\item $y_{e_1} > y_{e_2}$ and $e_1$ and $e_2$ have a common endpoint $v\in R$.
\end{itemize}
We say that a vector $s=(s_1,\ldots,s_m)$ \emph{satisfies the ordering
constraints} if $s_{e_1} < s_{e_2}$ for every pair $e_1,e_2\in E(G)$
such that $e_1\prec e_2$. This definition captures the condition that vertices of $G$ in $L$ (respectively, $R$) should be drawn so that they remain in $L$ (respectively, $R$), as in the following.
\begin{obs}\obslabel{left-right}
If a solution $s$ to $A\cdot s=b$ satisfies the ordering constraints, then every vertex that is in $L$ (in $R$) in $G$ is also in $L$ (respectively in $R$) in the drawing corresponding to $s$.
\end{obs}
\begin{proof}
Consider any vertex $v$ that is in $L$ in $G$ and that is incident to
(at least) two edges $e_1$ and $e_2$ with $y_{e_1} < y_{e_2}$, and hence $e_1 \prec e_2$. Since $s$ satisfies the ordering constraints we have $s_{e_1} < s_{e_2}$, hence the lines with slopes $s_{e_1}$ and $s_{e_2}$ through $(0,y_{e_1})$ and $(0,y_{e_2})$, respectively, meet in $L$. The argument for the vertices in $R$ is analogous.
\end{proof}
By construction, the slopes $s_{e_1}',\ldots,s_{e_m}'$ of edges
in $G$ satisfy the ordering constraints, so the relation $\prec$ is acyclic.
% Indeed, $i_1\prec
% \cdots \prec i_r$ implies that, for each $j\in\{3,\ldots,r\}$, $y_{i_j}\in
% (\min\{y_{i_{j-1}},y_{i_{j-2}}\}, \max\{y_{i_{j-1}},y_{i_{j-2}}\})$. Thus,
% a chain in $\prec$ corresponds to a sequence of strictly nested intervals.
\begin{lem}\lemlabel{order-gives-drawing}
Any solution $s$ to $A\cdot s=b$ satisfying
the ordering constraints % $\prec$
yields a
\Fary\ drawing of $G$ with $\Delta$ as the outer face.
\end{lem}
\begin{proof}
If $G$ is a plane drawing of a 2-connected graph, then another
straight-line drawing $G'$ of the same graph $G$ is a \Fary\ drawing provided
that two conditions are met:
(i) For every vertex~$v$, the clockwise order of the
edges around $v$ in $G'$ is the same as in $G$; and
(ii) in the drawing $G'$, every face cycle of $G$ is drawn without crossings
(Devillers, Liotta, Preparata, and Tamassia \cite[Lemma~16]{devillers.liotta.ea:checking}; see Felsner and Rote~\cite[Lemma~6 in conjunction with Section~4.1.2]{fr-pdcr-19} for a complete proof).
In our case, $G'$ is a straight-line drawing of $G$ given by a solution
to $A\cdot s = b$ that satisfies the ordering constraints.
First we show that $G'$ satisfies condition (i).
More specifically, we establish the following stronger
property for every vertex $v$.
\begin{quote}
\thetag{$*$}
The edges going to the right from $v$ are the same in $G$ and $G'$,
and their slopes have the same order in $G$ and $G'$.
The same properties hold for the edges to the left.
\end{quote}
We distinguish the following cases:
\begin{enumerate}
\item $v\not\in Y$. Since $s$ satisfies the ordering
constraints, by \obsref{left-right} we know that
$v$ is on the same side ($L$ or $R$)
in $G$ and
in $G'$.
All incident edges go to one side.
This, together with the fact that the
orders in which the edges incident to $v$ intersect
$Y$ in $G$ and $G'$ agree implies that the
slope orders of the edges around $v$ in $G$ and $G'$ agree.
\item
$v\in Y$, with incident edges $a_1,\ldots,a_k\in
L\cup Y$ and $b_1,\ldots,b_\ell\in Y\cup R$ as in
\figref{ab}.
Again, \obsref{left-right} ensures that
these edges remain on the same side in $G'$.
\begin{enumerate}
\item If $v$ is a boundary vertex, % say that $v$ is at $(0,y_1)$ in
% $G'$,
then the boundary equations fix the slopes of the two incident
boundary edges, $a_1, b_1$ or $a_k, b_l$, plus a third edge. As we
already observed when the proportionality constraints were defined,
these constraints then fix the slopes of all edges incident to $v$,
so that their ordering agrees with that of $G$.
\item If $v$ is an interior vertex then,
%, by \obsref{left-right}, the edges $a_1,\ldots,a_k$ are in $L\cup
%Y$ and the edges $b_1,\ldots,b_\ell$ are in $R\cup Y$ in
%$G'$. Further,
as discussed above, the proportionality constraints ensure that the
slope order of $v$'s incident edges in $G'$ either matches that of
$G$ on each side, or it is completely reversed on both sides.
% , so that the counterclockwise
%order of vertices around $v$ in $G'$
%is
%$a_1,\ldots,a_k,b_\ell,\ldots,b_1$.
Let us assume for contradiction that the latter case happens:
\begin{equation}
\label{eq:not-ordered}
s_{b_1}\ge s_{b_\ell}
\text{ and }
s_{a_k}\ge s_{a_1}
\end{equation}
Let $e$ be the third edge of the triangle with edges $a_1$ and
$b_1$, and let $f$ be the third edge of the triangle with edges
$a_k$ and $b_\ell$, see \figref{ab}.
Then the ordering constraints for the endpoints of $e$ imply
\begin{math}
s_{b_1}<s_e<s_{a_1}
\end{math},
and the ordering constraints for the endpoints of $f$ imply
\begin{math}
s_{a_k}<s_f<s_{b_\ell}
\end{math}.
Together with \eqref{eq:not-ordered}, this leads to a contradiction.
\end{enumerate}
\end{enumerate}
\begin{figure}
\centering
{\includegraphics[scale = 1]{figs/abx}}
\caption{The ordering of the edges incident to a vertex $v$ on $Y$.}
\figlabel{ab}
\end{figure}
From the statement \thetag{$*$}, it is now easy to derive that
$G'$ satisfies condition (ii).
The graph $G$ has triangle and quadrilateral faces.
%
For a triangular face, \thetag{$*$} ensures that the triangle does not
degenerate, and is therefore non-crossing, in $G'$.
%The graph $G$ has two kinds of
%faces:
A quadrilateral face $q$ must be non-convex in $G$ by Property 3 of
A-graphs, and for each vertex, the two incident edges of~$q$ go in
the same direction (left or right).
Thus, Property \thetag{$*$} ensures that $q$ is non-crossing in~$G'$.
% \begin{enumerate}
% \item Let $q = abcd$ be a quadrilateral face of
% $G$. By Property 3 of A-graphs, $q$ is non-convex in
% $G$;
% let $c$ be the reflex vertex of $q$ in $G$. Note that $a\notin Y$ and $c\notin Y$ by Properties 1 and 5 of A-graphs, respectively. Conversely, each of $b$ and $d$ might or might not be on $Y$. Then the ordering constraints and the proportionality constraints at $b$ and $d$ ensure that $q$ is non-crossing in $G'$.
% % If $q=abcd$ is a quadrilateral face in $G$ with no vertex on $Y$,
% % then the ordering constraints imply that $q$ is non-crossing in $G'$.
% % Otherwise, by Property~3 of A-graphs, $c$ is reflex vertex of $q$ and,
% % by Property~5 of A-graphs, $c\not\in Y$. The vertex $a$ opposite $c$
% % is also not in $T$, so so $b\in Y$ and/or $d\in Y$. In this case,
% % the ordering constraints and the proportionality constraints at $b$
% % and/or $d$ ensure that $q$ is non-crossing in $G'$.
% \item
% For a triangular face $\alpha\beta\gamma$, with $\gamma\in Y$,
% the ordering constraints on the vertices $\alpha$ and $\beta$
% ensure that the triangle does not degenerate, and is therefore non-crossing, in $G'$.
% \end{enumerate}
Therefore, by the result of Devillers \etal\ cited above, $G'$
is a \Fary\ drawing. That $G'$ has $\Delta$ as the outer
face follows from the inclusion of the boundary equations in
$A\cdot s = b$.
\end{proof}
Any solution $s$ to $A\cdot s=b$ has the outer face drawn as $\Delta$,
by the boundary equations, and the intersection between $e_i$ and $Y$ is $(0,y_i)$ by construction. Hence, by~\lemref{order-gives-drawing}, ensuring the existence of a solution $s$ to $A\cdot s=b$ satisfying the ordering constraints is enough to prove~\thmref{a-graph}.
\subsection{Strong Ordering Constraints}
\label{strong}
For some $\epsilon > 0$, we say that $s=(s_1,\ldots,s_m)$ satisfies
the \emph{$\epsilon$-strong ordering constraints} if, for each
$i,j\in\{1,\ldots,m\}$ such that $e_i\prec e_j$, the inequality
$s_j-s_i \ge \epsilon$ holds.
% A solution $s$ that satisfies
Clearly, any $s$ satisfying the $\epsilon$-strong ordering constraints
also satisfies the ordering constraints.
The converse holds, for a suitably small $\epsilon$ (the inequalities
being strict in the definition of ordering constraints). The following
lemma tells us that this $\epsilon$ can be determined by $\Delta$ and by the
sequence $y_1,\ldots,y_m$.
%
%The converse holds when
%equations
%$A\cdot s=b$
%% ~\thetag{\ref{eq:slope0}}--\thetag{\ref{eq:proportion2}}
%are satisfied, for a suitably small $\epsilon$:
\begin{lem}\lemlabel{weak-to-strong}
If
$\Delta\subset[-1,1]\times(-\infty,+\infty)$, then
any solution $s$ to $A\cdot s=b$ that satisfies
the ordering constraints
also satisfies
the $\epsilon$-strong ordering constraints
for all $\epsilon\le\min\{\,|y_i-y_j| : e_i\prec e_j\,\}$.
\end{lem}
\begin{proof}
By \lemref{order-gives-drawing} every vertex is contained in
$\Delta$. %\subset[-1,1]\times(-\infty,+\infty)$.
Hence, every $x$-coordinate is in the interval $[-1,1]$.
If $e_i\prec e_j$, then the common vertex of $e_i$ and $e_j$ has $x$-coordinate
$(y_j-y_i)/(s_i-s_j)$. From $|(y_j-y_i)/(s_i-s_j)|\le 1$ we
derive $|s_i-s_j|\ge|y_j-y_i| \ge \epsilon$.
\end{proof}
\subsection{Uniqueness of Solutions Satisfying Ordering Constraints}
\lemref{weak-to-strong} and the $\epsilon$-strong ordering
constraints play a crucial role in our proof because they
allow us to appeal to continuity: If the slopes change continuously,
it is impossible
to violate
the ordering constraints without first violating the
$\epsilon$-strong ordering constraints.
But since the ordering constraints imply the
$\epsilon$-strong ordering constraints,
it is impossible
to violate
the ordering constraints at all.
% by showing that, if $A\cdot s=b$
%were to have some undesireable property, then some function which we
%know to be continuous would have a discontinuity.
An example of this argument will be seen in the following proof.
\begin{lem}\lemlabel{unique}
If $s$ is a solution to $A\cdot s=b$ that satisfies the ordering
constraints, % $\prec$,
then $s$ is
the unique solution to $A\cdot s=b$.
\end{lem}
\begin{proof}
Assume that $\epsilon$ is fixed so that $0<\epsilon\le\min\{\,|y_i-y_j| : e_i\prec e_j\,\}$.
Suppose, for contradiction, that there is a solution $s$ to
$A\cdot s=b$ that satisfies the ordering
constraints, % $\prec$,
but is not unique. Since $A\cdot s=b$ is a linear system, it
must then have a 1-parameter family of solutions $s+\lambda r$,
$\lambda\in\mathbb R$, for some non-zero $m$-vector $r$.
%such that, for every
%$\lambda\in\mathbb R$, $A(s+\lambda r)=b$.
Define the continuous (in fact, piecewise linear) function
\begin{equation*}
f(\lambda) := \min \{\, (s_j+\lambda r_j)-(s_i+\lambda r_i) : e_i \prec
e_j\,\}
.
\end{equation*}
Let $\lambda^*$ be the value with the smallest absolute value
$|\lambda^*|$ such that
$f(\lambda^*)\le\epsilon/2$. In order to prove that
$\lambda^*$ exists, it suffices to prove that $f(\lambda)\le
0$ can be achieved. The vector $r=(r_1,\ldots,r_m)$ has at least four zero entries
$r_1=r_a=r_b=r_m=0$ since the slopes $s_1$, $s_a$, $s_b$, and $s_m$
are fixed.
Since $G$ is connected and $m\geq 5$, there is at least one vertex $v$ with two incident edges $e_k$
and $e_\ell$ such that $r_k=0$ and $r_\ell\neq 0$.
We can thus pick $\lambda$ so that $(s_\ell+\lambda r_\ell)-(s_k+\lambda r_k)=s_\ell-s_k+\lambda r_\ell=0$,
and then $f(\lambda)\le 0$. It follows that $\lambda^*$ exists.
Now we know that, for any $\lambda$ between $0$ and $\lambda^*$ and for any $i$ and $j$ such that $e_i\prec e_j$, the difference $(s_j+\lambda r_j)-(s_i+\lambda r_i)$ has the same sign as $s_j-s_i$. It follows that the slopes satisfy the ordering constraints throughout
this interval, but then
\lemref{weak-to-strong} implies that $f(\lambda^*)\ge\epsilon$, a contradiction.
\end{proof}
%The proof of \lemref{unique} was quite explicit (perhaps overly so)
%in showing the discontinuity caused by the $\epsilon$-strong ordering
%constraints. In subsequent arguments we will not be quite so explicit.
\subsection{A Parametric Family of Linear Systems}
We now define a parametric family of linear systems
$A^t\cdot s = b^t$, parameterized by $0\le t\le 1$
by varying the intersection points $y=(y_1,\ldots,y_m)$
and the boundary slopes $h=(h_1,h_a,h_b,h_m)$.
Let us first see how the coefficients $A$ and right-hand sides $b$ of the system change when these data are changed.
% The coefficients
% in $A^t$
% depend on $y=(y_1,\ldots,y_m)$:
% More precisely,
The coefficients of the
concurrency constraints
\eqref{eq:slope0} depend linearly on $y$,
whereas
the
proportionality constraints
\thetag{\ref{eq:proportion}--\ref{eq:proportion2}} remain unchanged,
and the boundary constraints~\eqref{eq:boundary} have just the constant coefficient~1.
In the right-hand sides $b^t$,
the four nonzero entries
%depend linearly on
are the four slopes
$h=(h_1,h_a,h_b,h_m)$.
We derive the intermediate systems
$A^t\cdot s = b^t$ by linear interpolation between the initial data
and the target data:
%We denote by $A^0\cdots s = b^0$
For the ``starting system'', we use
the intercepts
$y^0=(y_1^0,\ldots,y_m^0)$
and the slopes
$h^0=(h_1^0,h_a^0,h_b^0,h_m^0)$
% $s'=(s_1',\ldots,s_m')$
of the edges in the initial drawing~$G$.
% We denote by $A^1\cdots s = b^1$
In the ``target system'', we use
the specified target intercepts
$y^1=(y_1,\ldots,y_m)$
and the slopes
$h^1=(h_1,h_a,h_b,h_m)$ from the target shape~$\Delta$.
(If $\Delta$ is a triangle, this vector includes $h_b$ as an arbitrarily chosen
additional slope, as described earlier.)
We define the intermediate data $y^t$ and $h^t$
by linear interpolation:
%We define the intermediate systems
%$A^t\cdot s = b^t$ by linear interpolation:
\begin{equation*}
y^t = (1-t)y^0 + ty^1,
% A^t = (1-t)A^1 + tA^1,
\qquad
h^t = (1-t)h^0 + th^1.
% b^t = (1-t)b^1 + tb^1.
\end{equation*}
This defines the corresponding intermediate systems
$A^t\cdot s = b^t$, whose coefficients and right-hand sides depend
linearly on the parameter~$t$.
It is important to note that
the starting system
$A^0\cdot s = b^0$
has at least one solution, % $s^0$,
namely the slopes
$s^0=(s_1^0,\ldots,s_m^0)$
of the edges in the initial drawing $G$.
The proportionality constraints
\thetag{\ref{eq:proportion}--\ref{eq:proportion2}} were designed
in this way, as described in \secref{setting}.
The
concurrency constraints
\eqref{eq:slope0} are fulfilled because the initial drawing $G$ is a
straight-line drawing.
The boundary constraints~\eqref{eq:boundary} are fulfilled by
construction.
We will show that \lemref{weak-to-strong} can be applied to the
system $A^t\cdot s=b^t$, for every $0\le t\le 1$. We define
an appropriate threshold value $ \epsilon^*$ by
\begin{align*}
\epsilon^*&
=
\min_{e_i\prec e_j}
\min_{0\le t\le 1}
|y_j^t-y_i^t| %: e_i\prec e_j\,\}
\\ &
=%\min_{0\le t\le 1} %\bigl\{\,
\min_{e_i\prec e_j}
\min\{|y_j^0-y_i^0|,|y_j^1-y_i^1|\}%: e_i\prec e_j\,
% \bigr\}
> 0
\end{align*}
% Note that $A$
% and $b$ are functions of $y=(y_1,\ldots,y_m)$ and of the four slopes
% $h=(h_1,h_a,h_b,h_m)$. We make this explicit, by writing $A_1=A(y,h)$
% and $b_1=b(y,h)$.
% Let $y'=(y_1',\ldots,y_m')$ and $s'=(h_1',\ldots,h_m')$ denote the
% $y$-intercepts and the slopes of the edges in the initial drawing of $G$
% and let $h'=(h_1',h_a',h_b',h_m')$.
% Consider the system $A(y',h')\cdot s = b(y',h')$. This system has
% at least one solution $s=s'$. We now define
% a continuous family of linear systems that interpolates between $A(y',h')\cdot s=b(y',h')$ and $A(y,h)\cdot s=b(y,h)$. Suppose first that the outer face of $G$ is delimited by a quadrilateral.
% For $0\le t\le 1$ and $i\in\{1,a,b,m\}$, define $h_i(t)=(1-t)h_i' + th_i$ and $h(t)=(h_1(t),h_a(t),h_b(t),h_m(t))$.
% Note that
% \[
% h_1(t)-h_a(t) = (1-t)(h_1'-h_a') + t(h_1-h_a) > 0.
% \]
% Inequalities $h_1'-h_a'>0$ and $h_1-h_a>0$ come from the assumption that the vertex incident to $e_1$ and $e_m$ is in $L$ both in $G$ and in $\Delta$. Similarly, we have $h_m(t)-h_1(t)>0$ and $h_b(t)-h_m(t)>0$. Then we can define \[
% \epsilon_1 = \min_{0\le t\le 1}\min\{h_1(t)-h_a(t), h_m(t)-h_1(t), h_b(t)-h_m(t)\}
% \]
% and observe that $\epsilon_1>0$.
% Analogously, for every $0\le t\le 1$ and for each $i\in\{1,\ldots,m\}$, define $y_i(t) = (1-t)y_i' + ty_i$ and define $y(t)=(y_1(t),\ldots,y_m(t))$.
% Observe that, for any
% $1\le i< j\le m$ and any $0\le t\le 1$,
% \[
% y_j(t) - y_i(t) = (1-t)(y'_j-y'_i) + t(y_j-y_i) > 0.
% \]
% Let
% \[ \epsilon_2=\min_{0\le t\le 1}\min\{|y_j(t)-y_i(t)|: e_i\prec e_j\}
% \]
% and observe that $\epsilon_2 >0$.
% The entries in $A_t$ and $b^t$ are obtained in the same way as the entries of $A$ and $b$ were derived earlier, however each entry is now a linear function of~$t$, as $y$ and $h$ are replaced by $y(t)$ and $h(t)$, respectively, in the determination of the equations represented by $A_t$ and $b^t$.
% Consider the unique quadrilateral $\Delta(t)$ whose edges cross $Y$ at
% $y_1(t)$, $y_a(t)$, $y_b(t)$, $y_m(t)$ and have slopes $h_1(t)$,
% $h_a(t)$, $h_b(t)$, and $h_m(t)$, respectively.
% If the outer face of $G$ is delimited by a triangle, the arguments are analogous, however the inequalities $h_1(t)-h_a(t)>0$ and $h_m(t)-h_1(t)>0$ become either $h_m(t)-h_1(t)>0$ and $h_a(t)-h_m(t)>0$ or $h_a(t)-h_1(t)>0$ and $h_1(t)-h_m(t)>0$, depending on whether $(0,y_1)$ or $(0,y_m)$ is a vertex of $G$, respectively. Further, the inequality involving $h_b(t)$ now states either $h_b(t)-h_a(t)>0$ or that $h_b(t)$ is smaller than the smallest between $h_1(t)$ and $h_m(t)$, depending on which of the two holds in $G$ (as when determining the boundary equations).
% Next we show that, for every $0\le t\le 1$, \lemref{weak-to-strong} holds for the system $A^t\cdot s=b^t$.
\begin{lem}
For every $0\le t\le 1$, a solution $s$ to $A^t\cdot s = b^t$
that satisfies the ordering constraints also satisfies the
$\epsilon^*$-strong ordering constraints.
\end{lem}
\begin{proof}
We denote by $\Delta^t$ the shape of the outer face as specified by
$y^t$ and $h^t$.
It suffices to prove that this shape
is contained in
$[-1,1]\times(-\infty,+\infty)$,
at which point \lemref{weak-to-strong} applies.
We show that each vertex of $\Delta^t$ is in $[-1,1]\times[-\infty,+\infty]$.
If such a vertex $v$ does not lie on $Y$ and is incident to
the two outer edges
$e_i$ and $e_j$, with $i,j\in \{1,a,b,m\}$ and $e_i \prec e_j$, it
has $x$-coordinate $( y_i^t - y_j^t ) / ( s_j^t -
s_i^t )$.
%$e_i$ and $e_j$ are the two outer edges
%incident to $v$.
Consider the case that $v\in R$. So we want to show that
\begin{equation}
( y_i^t - y_j^t ) / ( s_j^t - s_i^t ) \le 1 \enspace . \eqlabel{pff}
\end{equation}
By the ordering constraints, $s_j^t - s_i^t > 0$, so \eqref{eq:pff} is
equivalent to
\begin{equation*}
y_i^t - y_j^t \le s_j^t - s_i^t.
% \\
%y_i^t - y_j^t + s_i^t - s_j^t \le 0\\
%y'_i + t(y_i-y'_i) - y'_j - t(y_j-y'_j) + s'_i + t(s_i-s'_i) - s'_j -
%t(s_j-s'_j) \le 0\\
\end{equation*}
This inequality holds for $t=0$ and for $t=1$. The left side is linear
in $t$.
Since $e_i$ and $e_j$ are boundary edges, the right side is also
linear in $t$.
So the inequality holds for
every $t\in [0,1]$.
In the case $v\in L$, the proof that $v$'s $x$-coordinate is at
least $-1$ is similar.
\end{proof}
\subsection{Existence (and uniqueness) of solutions to $A^t\cdot s=b^t$}
We now prove the following lemma which, together with \lemref{order-gives-drawing}, completes the proof of \thmref{a-graph}.
\begin{lem}\lemlabel{uniqueness}
For every $0\le t\le 1$, the system $A^t\cdot s=b^t$ has a
unique solution $s^t$, and this solution satisfies the ordering
constraints.
\end{lem}
\begin{proof}
Since $A^t$ is an $m\times m$ matrix, the system $A^t\cdot
s=b^t$ has a unique solution~$s^t$ if and only if $\det A^t \neq 0$.
When $\det A^t =0$, the system may have no solutions or
multiple solutions.
When $\det A^t\neq 0$,
Cramer's Rule states that
the solution
is $s^t=(s_1^t,\ldots,s_m^t)$ where, for each
$i\in\{1,\ldots,m\}$,
\[
s_i^t = \frac{\det A^t_i}{\det A^t }
\]
and $A^t_i$ denotes the matrix $A^t$ with its $i$-th column replaced
by $b^t$.
The numerators $\det A^t_i$ and the common
denominator $\det A^t $ are polynomials in $t$, and therefore
continuous
functions of $t$.
The solution $s^t=(s_1^t,\ldots,s_m^t)$ depends continuously on $t$
as long as $\det A^t\ne0 $.
We have already established that $A^0\cdot s=b^0$ has a
solution $s^0$ that satisfies the ordering constraints. By
\lemref{unique}, this solution is unique, so $\det A^0\neq 0$.