Commit fe97fd5d authored by Daniel Gonçalves's avatar Daniel Gonçalves
Browse files

New section "Parameters"

Replacing the tree-decomposition by clique-sum operations. 
Introduction of clique-sum extendable representation.
Replacing treewidth by k-trees
parent d42d6198
......@@ -20,6 +20,7 @@
\newcommand{\bb}[1]{\mathbb{#1}}
\newcommand{\brm}[1]{\operatorname{#1}}
\newcommand{\cbdim}{\brm{dim}_{cb}}
\newcommand{\ecbdim}{\brm{dim}^{ext}_{cb}}
\newcommand{\tw}{\brm{tw}}
\newcommand{\vol}{\brm{vol}}
%%%%%
......@@ -33,11 +34,12 @@
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{question}[theorem]{Question}
\newtheorem{definition}[theorem]{Definition}
\title{On comparable box dimension}
\author{Zden\v{e}k Dvo\v{r}\'ak\thanks{Computer Science Institute, Charles University, Prague, Czech Republic. E-mail: {\tt rakdver@iuuk.mff.cuni.cz}.
Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}\and
Daniel Gon\c{c}alves\thanks{...}\and
Daniel Gon\c{c}alves\thanks{LIRMM, Université de Montpellier, CNRS, Montpellier, France. E-mail: {\tt goncalves@lirmm.fr}. Supported by the ANR grant GATO ANR-16-CE40-0009.}\and
Abhiruk Lahiri\thanks{...}\and
Jane Tan\thanks{Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK. E-mail: {\tt jane.tan@maths.ox.ac.uk}}\and
Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten.ueckerdt@kit.edu}}}
......@@ -47,8 +49,12 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten
\maketitle
\begin{abstract}
Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded
comparable box dimension and explore further properties of this notion.
Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset
of a translation of the other. The comparable box dimension of a graph
$G$ is the minimum integer $d$ such that $G$ can be represented as a
touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We
show that proper minor-closed classes have bounded comparable box
dimension and explore further properties of this notion.
\end{abstract}
\section{Introduction}
......@@ -98,237 +104,40 @@ This gives arbitrarily precise approximation algorithms for all monotone maximiz
expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx}
or expressible in the first-order logic~\cite{logapx}.
\section{Operations}
Let us start with a simple lemma saying that the addition of a vertex increases the comparable box dimension by at most one.
In particular, this implies that $\cbdim(G)\le |V(G)|$.
\begin{lemma}\label{lemma-apex}
For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$.
\end{lemma}
\begin{proof}
Let $f$ be a touching representation of $G-v$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G-v)$.
We define a representation $h$ of $G$ as follows.
For each $u\in V(G)\setminus\{v\}$, let $h(u)=[0,1]\times f(u)$ if $uv\in E(G)$ and
$h(u)=[1/2,3/2]\times f(u)$ if $uv\not\in E(G)$. Let $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$,
where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \times [-M,M]$ for every $u\in V(G)\setminus\{v\}$.
Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$.
\end{proof}
\section{Parameters}
We need a bound on the clique number in terms of the comparable box dimension.
For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_i$.
Let us first bound the clique number $\omega(G)$ in terms of
$\cbdim(G)$.
\begin{lemma}\label{lemma-cliq}
If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$.
\end{lemma}
\begin{proof}
To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty intersection.
Since axis-aligned boxes have the Helly property, there is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap f(a_w)$.
As each box is full-dimensional, its interior intersects at least one of the $2^d$ orthants at $p$.
Since $f$ is a touching representation, $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors and hence $w \leq 2^d$.
\end{proof}
A \emph{tree decomposition} of a graph $G$
is a pair $(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$ assigns a \emph{bag} to each of its nodes,
such that
\begin{itemize}
\item for each $uv\in E(G)$, there exists $x\in V(T)$ such that $u,v\in\beta(x)$, and
\item for each $v\in V(G)$, the set $\{x\in V(T):v\in\beta(x)\}$ is non-empty and induces a connected subtree of $T$.
\end{itemize}
For nodes $x,y\in V(T)$, we write $x\preceq y$ if $x=y$ or $x$ is a descendant of $y$ in $T$.
For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ and $x$ is nearest to the root of $T$.
The \emph{adhesion} of the tree decomposition is the maximum of $|\beta(x)\cap\beta(y)|$ over distinct $x,y\in V(T)$,
and its \emph{width} is the maximum of the sizes of the bags minus $1$. The \emph{treewidth} of a graph is the minimum
of the widths of its tree decompositions. We will need to know that graphs of bounded treewidth have bounded comparable box dimension.
In fact, we will prove the following stronger fact. \note{(TODO: Was this published somewhere before? I am only aware of the upper bound of $t+2$ on the boxicity of $G$~\cite{box-treewidth}.)}
\begin{lemma}\label{lemma-tw}
Let $(T,\beta)$ be a tree decomposition of a graph $G$ of width $t$.
Then $G$ has a touching representation $h$ by axis-aligned hypercubes in $\mathbb{R}^{t+1}$ such that
for $u,v\in V(G)$, if $p(u)\prec p(v)$, then $h(u)\sqsubset h(v)$.
Moreover, the representation can be chosen so that no two hypercubes have the same size.
\end{lemma}
\begin{proof}
Without loss of generality, we can assume that the root has a bag of size one
and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta(y)|=1$
(if $\beta(x)\subseteq \beta(y)$, we can contract the edge $xy$; if $|\beta(x)\setminus\beta(y)|>1$,
we can subdivide the edge $xy$, introducing $|\beta(x)\setminus\beta(y)|-1$ new vertices,
and set their bags appropriately). It is now natural to relabel the vertices of $G$
so that $V(G)=V(T)$, by giving the unique vertex in $\beta(x)\setminus\beta(y)$
the label $x$. In particular, $p(x)=x$. Furthermore, switching to this new labeling, we can assume that $y\in\beta(x)$.
Otherwise, if $y$ is not the root of $T$, we can replace the edge $xy$ by the edge from $x$
to the parent of $y$. If $y$ is the root of $T$, then the subtree rooted in $x$ induces a
union of connected components in $G$, and we can process this subtree separately from the
rest of the graph (being careful to only use hypercubes smaller than the one representing $y$
and of different sizes from those used on the rest of the graph).
Let us now greedily color $G$ by giving $x$ a color different from the colors of
all other vertices in $\beta(x)$; such a coloring $\varphi$ uses only colors
$\{1,\ldots,t+1\}$.
Let $D=4\Delta(T)+1$. Let $V(G)=V(T)=\{x_1,x_2,\ldots, x_n\}$, where
for every $i<j$, $x_i$ and $x_j$ are either incomparable in $\prec$ or
$x_j\prec x_i)$; in particular, $x_1$ is the root of $T$. Let $\varepsilon=D^{-n-1}$.
Let $s_i=D^{-i}$; we will represent $x_i$ by a hypercube $h(x_i)$ with edges of length $s_i$.
Additionally, we will need to consider larger hypercubes around $h(x_i)$; let $h'(x_i)$
be the hypercube with sides of length $2s_i$ and with $\min(h'(x_i)[j])=\min(h(x_i)[j])$
for $j\in\{1,\ldots, t+1\}$, and $h''(x_i)$ the hypercube with sides of length $2s_i+\epsilon$
and with $\min(h''(x_i)[j])=\min(h(x_i)[j])-\varepsilon$. We will construct the representation $h$ so that the following
invariant is satisfied:
\begin{itemize}
\item[(a)] For each $x,z\in V(T)$ such that $x\prec z$, we have $h'(x)\subset h''(z)$.
\item[(b)] For each $y\in V(T)$ and distinct children $x$ and $z$ of $y$, we have $h''(x)\cap h''(z)=\emptyset$.
\end{itemize}
Note that this ensures that if $x$ and $z$ are vertices of $T$ and $h(x)\cap h(z)\neq\emptyset$, then $x\prec z$ or $z\prec x$.
We now construct the representation $h$. For the root $x_1$ of $T$, $h(r)$ is an arbitrary hypercube with sides
of length $s_1$. Assuming now we have already selected $h(y)$ for a vertex $y\in V(T)$, the hypercube $h(x_i)$ with sides of length $s_i$
for a child $x_i$ of $y$ is chosen as follows. For $j\in\{1,\ldots, t+1\}$,
\begin{itemize}
\item[(i)] if $j=\varphi(w)$ for $w\in\beta(x_i)\setminus\{x_i\}$, we choose $h(x_i)[j]$ so that
$\min(h(x_i)[j])=\max(h(w)[j])$ if $xw\in E(G)$ and so that $\min(h(x_i)[j])=\max(h(w)[j]) + \varepsilon$ otherwise.
\item[(ii)] if $j$ is different from the colors of all vertices in $\beta(x_i)\setminus\{x_i\}$,
then we choose $h(x_i)[j]$ so that $h''(x_i)[j]$ is a subset of the interior of $h(y)[j]$. The interval $h''(x_i)[[j]$
is furthermore chosen to be disjoint from $h''(x_m)[j]$ for any other child $x_m$ of $y$;
this is always possible by the choice of $D$, $s_i$, and $s_m$.
\end{itemize}
Note that (ii) always applies for $j=\varphi(x_i)$ and this ensures that the invariant (b) holds.
For the invariant (a), note that in the case (ii), we ensure $h''(x_i)[[j]\subseteq h(y)[j]$ and
we have $h(y)[j]\subseteq h''(z)[j]$ by the invariant (a) for $y$ and $z$. In the case (i),
if $z\prec w$, then we have $w\in\beta(z)\setminus\{z\}$ and
$\min(h(x_i)[j]),\min(h(z)[j])\in\{\max(h(w)[j]),\max(h(w)[j])+\varepsilon\}$.
If $w\preceq z$, then note we choose $h'(x_i)[j]\subseteq h'(w)[j]$ by (i) and that
we have $h'(w)[j]\subset h''(z)[j]$ by (a). This verifies that the invariant (a)
also holds at $x_i$.
Consider now two adjacent vertices of $G$, say $x_i$ and $w$. Note that any two
adjacent vertices are comparable in $\prec$, and thus we can assume $x_i\prec w$
and $w\in\beta(x_i)$.
By (i), for $j=\varphi(w)$, the intervals $h(x_i)[j]$ and $h(w)[j]$
intersect in a single point. If $j\neq \varphi(w)$, then let $w_1$ be the child of $w$ on the path in $T$ from
$w$ to $x_i$. If no vertex in $\beta(w_1)\setminus\{w_1\}$ has color $j$, then by (ii), we have $h''(w_1)[j]\subset h(w)[j]$,
Otherwise, $z\in \beta(w_1)\setminus\{w_1\}$ such that $\varphi(z)=j$; clearly, $w\prec z$ and $z\in\beta(w)\setminus\{w\}$.
We have $\min(h(w_1)[j]),\min(h(w)[j])\in\{\max(h(z)[j]),\max(h(z)[j])+\varepsilon\}$ by (i).
Hence, $\min(h(w)[j])-2\epsilon\le \min(h''(w_1)[j])\le \max(h''(w_1)[j])\le \max(h(w)[j])$.
Since $h(x_i)[j]\subseteq h''(w_1)[j]$ by (a) and the length of the interval $h(x_i)[j]$ is greater than $2\varepsilon$,
we conclude that $h(x_i)[j]\cap h(w)[j]\neq\emptyset$. Therefore, the boxes $h(w)$ and $h(x_i)$ touch.
Finally, consider two non-adjacent vertices of $G$, say $x_i$ and $w$. As we noted before, if $x_i$ and $w$ are
incomparable in $\prec$, then (a) and (b) implies that the boxes $h(x_i)$ and $h(w)$ are disjoint.
Suppose now that say $x_i\prec w$, and let $j=\varphi(w)$. If $w\in\beta(x_i)$, then $h(x_i)[j]$ and $h(w)[j]$ are disjoint by (i).
Otherwise, let $y$ be the last vertex on the path from $w$ to $x_i$ in $T$ such that $w\in\beta(y)$ and let $z$ be the child of $y$
on this path. As we argued in the first paragraph, $y\neq w$, and by (i), the interior of $h(y)[j]$ is disjoint from $h(w)[j]$.
By (ii), $h''(z)[j]$ is contained in the interior of $h(y)[j]$. By (a), we conclude that $h(x_i)[j]\subseteq h''(z)[j]$,
implying that the boxes $h(x_i)$ and $h(w)$ are disjoint. Therefore, $h$ is a touching representation of $G$.
\end{proof}
Next, let us deal with clique-sums. A \emph{clique-sum} of two graphs $G_1$ and $G_2$ is obtained from their disjoint union
by identifying vertices of a clique in $G_1$ and a clique of the same size in $G_2$ and possibly
deleting some of the edges of the resulting clique. The main issue to overcome in obtaining a representation for a clique-sum
is that the representations of $G_1$ and $G_2$ can be ``degenerate''. Consider e.g.\ the case that $G_1$ is represented
by unit squares arranged in a grid; in this case, there is no space to attach $G_2$ at the cliques formed by four squares intersecting
in a single corner. This can be avoided by increasing the dimension, but we need to be careful so that the dimension stays bounded
even after an arbitrary number of clique-sums.
It will be convenient to work in the setting of tree decompositions.
Consider a tree decomposition $(T,\beta)$ of a graph $G$.
For each $x\in V(T)$, the \emph{torso} $G_x$ of $x$ is the graph obtained from $G[\beta(x)]$ by adding a clique on $\beta(x)\cap\beta(y)$
for each $y\in V(T)$. For a class of graphs $\GG$, we say that the tree decomposition is \emph{over $\GG$} if all the torsos belong to $\GG$.
We use the following well-known fact.
\begin{observation}
A graph $G$ is obtained from graphs in a class $\GG$ by repeated clique-sums if and only if $G$ has a tree decomposition over $\GG$.
\end{observation}
For each note $x\in V(T)$, let $\pi(x)=\{x\}\cup \{p(v):v\in\beta(x)\}$. Let $T_\beta$ be the graph with vertex set $V(T)$ such that
$xy\in E(T_\beta)$ if and only if $x\in\pi(y)$ or $y\in\pi(x)$.
\begin{lemma}\label{lemma-legraf}
If $(T,\beta)$ is a tree decomposition of $G$ of adhesion $a$, then $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$.
Moreover, $\pi(x)$ is a clique in $T_\beta$ and $p(x)=x$ for each $x\in V(T)$.
For any graph $G$, then $\omega(G)\le 2^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
For each edge $xy\in E(T_\beta)$, we have $x,y\in \pi(x)$ or $x,y\in \pi(y)$ by definition.
Moreover, for each $x\in V(T_\beta)$, we have
$$\{y:x\in\pi(y)\}=\{x\}\cup \bigcup_{v\in V(G):p(v)=x} \{y:v\in\beta(y)\},$$
and all the sets on the right-hand size induce connected subtrees containing $x$,
implying that $\{y:x\in\pi(y)\}$ also induces a connected subtree containing $x$.
Hence, $(T,\pi)$ is a tree decomposition of $T_\beta$.
Consider a node $x\in V(T)$. Note that for each $v\in \beta(x)$, the vertex $p(v)$ is an ancestor of $x$ in $T$.
In particular, if $x$ is the root of $T$, then $\pi(x)=\{x\}$. Otherwise, if $y$ is the parent of $x$ in $T$, then
$p(v)=x$ for every $v\in \beta(x)\setminus\beta(y)$, and thus $|\pi(x)|\le |\beta(x)\cap \beta(y)|+1\le a+1$.
Hence, the width of $(T,\pi)$ is at most $a$.
Suppose now that $y$ and $z$ are distinct vertices in $\pi(x)$. Then both $y$ and $z$ are ancestors of $x$ in $T$,
and thus without loss of generality, we can assume that $y\preceq z$. If $y=x$, then $yz\in E(T_\beta)$ by definition.
Otherwise, there exist vertices $u,v\in \beta(x)$ such that $p(u)=y$ and $p(v)=z$. Since $v\in \beta(x)\cap\beta(z)$
and $y$ is on the path in $T$ from $x$ to $z$, we also have $v\in\beta(y)$. This implies $z\in\pi(y)$ and $yz\in E(T_\beta)$.
Hence, $\pi(x)$ is a clique in $T_\beta$. Moreover, note that $x\not\in \pi(w)$ for any ancestor $w\neq x$ of $x$ in $T$,
and thus $p(x)=x$.
To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the
corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty
intersection. Since axis-aligned boxes have the Helly property, there
is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap
f(a_w)$. As each box is full-dimensional, its interior intersects at
least one of the $2^d$ orthants at $p$. Since $f$ is a touching
representation, $f(a_1),\ldots,f(a_d)$ have pairwise disjoint
interiors and hence $w \leq 2^d$.
\end{proof}
We are now ready to deal with the clique-sums.
\begin{theorem}\label{thm-cs}
If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then there exists a graph $G'$ such that
$G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$.
\end{theorem}
\begin{proof}
Let $(T,\beta)$ be a tree decomposition of $G$ over $\GG$; the adhesion $a$ of $(T,\beta)$ is at most $\omega(\GG)$.
By Lemma~\ref{lemma-legraf}, $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$. By Lemma~\ref{lemma-tw},
$T_\beta$ has a touching representation $h$ by axis-aligned hypercubes in $\mathbb{R}^{a+1}$ such that
$h(x)\sqsubseteq h(y)$ whenever $x\preceq y$.
Since $T_\beta$ has treewidth at most $a$, it has a proper coloring $\varphi$ by colors $\{0,\ldots,a\}$.
For every $x\in V(T)$, let $f_x$ be a touching representation of the torso $G_x$ of $x$ by comparable boxes in $\mathbb{R}^d$,
where $d=\cbdim(\GG)$. We scale and translate the representations so that for every $x\in V(T)$ and $i\in\{0,\ldots,a\}$,
there exists a box $E_i(x)$ such that
\begin{itemize}
\item[(a)] for distinct $x,y\in V(T)$, if $h(x)\sqsubset h(y)$, then $E_i(x) \sqsubset E_i(y)$ and $E_i(x) \sqsubset f_y(v)$ for every $v\in \beta(y)$,
\item[(b)] if $x\prec y$, then $E_i(x)\subseteq E_i(y)$,
\item[(c)] if $i=\varphi(x)$, then $f_x(v)\subseteq E_i(x)$ for every $v\in\beta(x)$, and
\item[(d)] if $i\neq \varphi(x)$ and $K=\{v\in\beta(x):\varphi(p(v))=i\}$ is non-empty, then letting $y=p(v)$ for $v\in K$,
the box $E_i(x)$ contains a point belonging to $\bigcap_{v\in K} f_y(v)$.
\end{itemize}
Some explanation is in order for the last point: Firstly, since $\pi(x)$ is a clique in $T_\beta$, there
exists only one vertex $y\in \pi(x)$ of color $i$, and thus $y=p(v)$ for all $v\in K$.
Moreover, $K$ is a clique in $G_y$, and thus $\bigcap_{v\in K} f_y(v)$ is non-empty. Lastly,
note that if $x\prec z\prec y$, then $K=\beta(x)\cap\beta(y)\subseteq \beta(z)\cap \beta(y)$,
and thus $E_i(z)$ was also chosen to contain a point of $\bigcap_{v\in K} f_y(v)$;
hence, a choice of $E_i(x)$ satisfying $E_i(x)\subseteq E_i(z)$ as required by (b) is possible.
Let us now define $f(v)=h(p(v))\times E_0(v)\times \cdots\times E_a(v)$ for each $v\in V(G)$,
where $E_i(v)=f_{p(v)}(v)$ if $i=\varphi(p(v))$ and $E_i(v)=E_i(p(v))$ otherwise. We claim this gives a touching representation
of a supergraph of $G$ by comparable boxes in $\mathbb{R}^{(d+1)(a+1)}$. First, note that the boxes are indeed comparable;
if $p(u)=p(v)$, then this is the case since $f_{p(v)}$ is a representation by comparable boxes, and if say
$p(u)\prec p(v)$, then this is due to (a) and (c). Next, let us argue $f(u)$ and $f(v)$ have disjoint
interiors. If $p(u)=p(v)$, this is the case since $f_{p(v)}$ is a touching representation, and if $p(u)\neq p(v)$,
then this is the case because $h$ is a touching representation. Finally, suppose that $uv\in E(G)$. Let $x$
be the node of $T$ nearest to the root such that $u,v\in \beta(x)$. Without loss of generality, $p(u)=x$.
Let $y=p(v)$. If $x=y$, then $f(u)\cap f(v)\neq\emptyset$, since $f_x$ is a touching representation of $G_x$ and $uv\in E(G_x)$.
If $x\neq y$, then $y\in\pi(x)$ and $xy\in E(T_\beta)$, implying that $h(x)\cap h(y)\neq\emptyset$.
Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\ldots,a$ by (b) if $\varphi(x)\neq i\neq \varphi(y)$,
(b) and (c) if $\varphi(x)=i\neq\varphi(y)$, and (d) if $\varphi(x)\neq i=\varphi(y)$ (we cannot have $\varphi(x)=i=\varphi(y)$, since
$xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$.
\end{proof}
We can now combine Theorem~\ref{thm-cs} with Lemma~\ref{lemma-cliq}.
\begin{corollary}\label{cor-cs}
If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then there exists a graph $G'$ such that
$G\subseteq G'$ and \[\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}.\]
\end{corollary}
Note that only bound the comparable box dimension of a supergraph
of $G$. To deal with this issue, we show that the comparable box dimension of a subgraph
is at most exponential in the comparable box dimension of the whole graph.
This is essentially Corollary~25 in~\cite{subconvex}, but since the setting is somewhat
different and the construction of~\cite{subconvex} uses rotated boxes,
we provide details of the argument.
A \emph{star coloring} of a graph $G$ is a proper coloring such that any two color classes induce
a star forest (i.e., a graph not containing any 4-vertex path). The \emph{star chromatic number} $\chi_s(G)$
of $G$ is the minimum number of colors in a star coloring of $G$.
We will need the fact that the star chromatic number is at most exponential in the comparable box dimension;
this follows from~\cite{subconvex} and we include the argument to make the dependence clear.
In the following we consider the chromatic number $\chi(G)$, and one
of its variant. A \emph{star coloring} of a graph $G$ is a proper
coloring such that any two color classes induce a star forest (i.e., a
graph not containing any 4-vertex path). The \emph{star chromatic
number} $\chi_s(G)$ of $G$ is the minimum number of colors in a star
coloring of $G$. We will need the fact that the star chromatic number
is at most exponential in the comparable box dimension; this follows
from~\cite{subconvex} and we include the argument to make the
dependence clear.
\begin{lemma}\label{lemma-chrom}
If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ has star chromatic number at most $2\cdot 9^d$.
For any graph $G$, then $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot
9^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
Let us focus on the star chromatic number.
Let $v_1$, \ldots, $v_n$ be the vertices of $G$ ordered non-increasingly by the size of the boxes that represent them;
i.e., so that $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. We greedily color the vertices in order, giving $v_i$ the smallest
color different from the colors of all vertices $v_j$ such that $j<i$ and either $v_jv_i\in E(G)$, or there exists $m>j$
......@@ -346,9 +155,100 @@ is at most $(3^d-1)^2$.
Consequently, when choosing the color of $v_i$ greedily, we only need to avoid colors of at most
$$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$
vertices.
vertices. We proceed similarly to bound the chromatic number.
\end{proof}
\section{Operations}
It is clear that given a touching representation of a graph $G$, one
easily obtains a touching representation with boxes of an induced
subgraph $H$ of $G$ by simply deleting the boxes corresponding to the
vertices in $V(G)\setminus V(H)$. In this section we are going to
consider other basic operations on graphs.
\subsection{Vertex addition}
Let us start with a simple lemma saying that the addition of a vertex
increases the comparable box dimension by at most one. In particular,
this implies that $\cbdim(G)\le |V(G)|$.
\begin{lemma}\label{lemma-apex}
For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$.
\end{lemma}
\begin{proof}
Let $f$ be a touching representation of $G-v$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G-v)$.
We define a representation $h$ of $G$ as follows.
For each $u\in V(G)\setminus\{v\}$, let $h(u)=[0,1]\times f(u)$ if $uv\in E(G)$ and
$h(u)=[1/2,3/2]\times f(u)$ if $uv\not\in E(G)$. Let $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$,
where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \times [-M,M]$ for every $u\in V(G)\setminus\{v\}$.
Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$.
\end{proof}
\subsection{Strong product}
Let $G\boxtimes H$ denote the \emph{strong product} of the graphs $G$
and $H$, i.e., the graph with vertex set $V(G)\times V(H)$ and with
distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if
either $u_1=u_2$ or $u_1u_2\in E(G)$ and either $v_1=v_2$ or
$v_1v_2\in E(G)$. To obtain a touching representation of $G\boxtimes
H$ it suffice to take a product of representations of $G$ and $H$, but
the obtained representation may contain uncomparable boxes. Thus,
bounding $\cbdim(G\boxtimes H)$ in terms of $\cbdim(G)$ and
$\cbdim(H)$ seems to be a complicated task. In the following lemma we
overcome this issue, by constraining one of the representations.
\begin{lemma}\label{lemma-sp}
Consider a graph $H$ having a touching representation $h$ in
$\mathbb{R}^{d_H}$ with hypercubes of unit size. Then for any graph
$G$, the strong product of these graphs is such that
$\cbdim(G\boxtimes H) \le \cbdim(G) + d_H$.
\end{lemma}
\begin{proof}
The proof simply consists in taking a product of the two
representations. Indeed, consider a touching respresentation with
comparable boxes $g$ of $G$ in $\mathbb{R}^{d_G}$, with
$d_G=\cbdim(G)$, and the depresentation $h$ of $H$. Let us define a
representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
follows.
$$f((u,v))[i]=\begin{cases}
g(u)[i]&\text{ if $i\le d_G$}\\
h(u)[i-d_G]&\text{ if $i > d_G$}
\end{cases}$$
Notice first that the boxes of $f$ are comparable as $f((u,v))
\sqsubset f((u',v'))$ if and only if $g(u) \sqsubset g(u')$.
Now let us observe that for any two vertices $u, u'$ of $G$, there
is an hyperplane separating the interiors of $g(u)$ and $g(u')$, and
similarly for $h$ and $H$. This implies that the boxes in $f$ are
interior disjoint. Indeed, the same hyperplane that separates $g(u)$
and $g(u')$, when extended to $\mathbb{R}^{d_G+d_H}$, now separates
any two boxes $f((u,v))$ and $f((u',v'))$. This implies that $f$ is
a touching representation of a subgraph of $G\boxtimes H$.
Similarly, one can also observe that there is a point $p$ in the
intersection of $f((u,v))$ and $f((u',v'))$, if and only if there is
a point $p_G$ in the intersection of $g(u)$ and $g(u')$, and a point
$p_H$ in the intersection of $h(v)$ and $h(v')$. Indeed, one can
obtain $p_G$ and $p_H$ by projecting $p$ in $\mathbb{R}^{d_G}$ or in
$\mathbb{R}^{d_G}$ respectively, and conversely $p$ can be obtained
by taking the product of $p_G$ and $p_H$. Thus $f$ is indeed a
touching representation of $G\boxtimes H$.
\end{proof}
\subsection{Subgraph}
Examples show that the comparable box dimension of a graph $G$ may be
larger than the one a subgraph $H$ of $G$. However we show that the
comparable box dimension of a subgraph is at most exponential in the
comparable box dimension of the whole graph. This is essentially
Corollary~25 in~\cite{subconvex}, but since the setting is somewhat
different and the construction of~\cite{subconvex} uses rotated boxes,
we provide details of the argument.
Next, let us show a bound on the comparable box dimension of subgraphs.
\begin{lemma}\label{lemma-subg}
......@@ -387,65 +287,344 @@ If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+4\cdot 81^{\
Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim{K_{2^d}}=d$,
but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$.
Corollaries~\ref{cor-cs} and~\ref{cor-subg} now give the main result of this section.
\begin{corollary}\label{cor-comb}
If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then
$\cbdim(G)\le 5\cdot 81^{6^{\cbdim(\GG)}}$.
\end{corollary}
\section{The product structure and minor-closed classes}
\subsection{Clique-sums}
A \emph{clique-sum} of two graphs $G_1$ and $G_2$ is obtained from
their disjoint union by identifying vertices of a clique in $G_1$ and
a clique of the same size in $G_2$ and possibly deleting some of the
edges of the resulting clique. A \emph{full clique-sum} is a
clique-sum in which we keep all the edges of the resulting clique.
The main issue to overcome in obtaining a representation for a (full)
clique-sum is that the representations of $G_1$ and $G_2$ can be
``degenerate''. Consider e.g.\ the case that $G_1$ is represented by
unit squares arranged in a grid; in this case, there is no space to
attach $G_2$ at the cliques formed by four squares intersecting in a
single corner. This can be avoided by increasing the dimension, but
we need to be careful so that the dimension stays bounded even after
an arbitrary number of clique-sums. We thus introduce the notion of
\emph{clique-sum extendable} representations.
\begin{definition}
Consider a graph $G$ with a distinguished clique $C^*$, called the
\emph{root clique} of $G$. A touching representation (with comparable
boxes or not) $h$ of $G$ in $\mathbb{R}^d$ is called
\emph{$C^*$-clique-sum extendable} if the following conditions hold.
\begin{itemize}
\item[{\bf(vertices)}] There are $|V(C^*)|$ dimensions, denoted $d_u$ for each
vertex $u\in V(C^*)$, such that:
\begin{itemize}
\item[(v1)] for each vertex $u\in V(C^*)$, $h(u)[d_u] = [-1 ,0]$ and
$h(u)[i] = [0,1]$, for any dimension $i\neq d_u$, and
\item[(v2)] for any vertex $v\notin V(C^*)$, $h(v) \subset [0,1)^d$.
\end{itemize}
\item[{\bf(cliques)}] For every clique $C$ of $G$ we define a point
$p(C)\in I_C \cap [0,1)^d$, where $I_C =\cap_{v\in V(C)} h(v)$, and
we define the box $h^\epsilon(C)$, for any $\epsilon > 0$, by
$h^\epsilon(C)[i] = [p(C)[i],p(C)[i]+\epsilon]$, for every dimension
$i$. Furthermore, for a sufficiently small $\epsilon > 0$ these
\emph{clique boxes} verify the following conditions.
\begin{itemize}
\item[(c1)] For any two cliques $C_1\neq C_2$, $h^\epsilon(C_1) \cap
h^\epsilon(C_2) = \emptyset$ (i.e. $p(C_1) \neq p(C_2)$).
\item[(c2)] A box $h(v)$ intersects $h^\epsilon(C)$ if and only if
$v\in V(C)$, and in that case their intersection is a facet of
$h^\epsilon(C)$ incident to $p(C)$ (i.e. if we denote this
intersection $I$, then $I[i] = \{p(C)[i] \}$ for some dimension
$i$, and $I[j] = [p(C)[j],p(C)[j]+\epsilon]$ for the other
dimensions $j\neq i$).
\end{itemize}
\end{itemize}
\end{definition}
Note that we may consider that the root clique is empty, that is the
empty subgraph with no vertices. In that case the clique is denoted
$\emptyset$. Let $\ecbdim(G)$ be the minimum dimension such that $G$
has a $\emptyset$-clique-sum extendable touching representation with
comparable boxes. The following lemma ensures that clique-sum
extendable representations behave well with respect to full
clique-sums.
\begin{lemma}\label{lem-cs}
Consider two graphs $G_1$ and $G_2$, given with a $C^*_1$- and a
$C^*_2$-clique-sum extendable representations with comparable boxes
$h_1$ and $h_2$, in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$
respectively. Let $G$ be the graph obtained after performing a full
clique-sum of these two graphs on any clique $C_1$ of $G_1$, and on
the root clique $C^*_2$ of $G_2$. Then $G$ admits a $C^*_1$-clique
sum extendable representation with comparable boxes $h$ in
$\mathbb{R}^{\max(d_1,d_2)}$. Furthermore, we have that the aspect
ratio of $h(v)$ is the same as the one of $h_1(v)$, if $v\in
V(G_1)$, or the same as $h_2(v)$ if $v\in V(G_2)\setminus V(C^*_2)$.
\note{ shall we remove the aspect ratio thing ? only needed for the
hypercubes of the k-trees, but those are not really needed...}
\end{lemma}
\begin{proof}
The idea is to translate (allowing also exchanges of dimensions) and
scale $h_2$ to fit in $h_1^\epsilon(C_1)$. Consider an $\epsilon >0$
sufficiently small so that, $h_1^\epsilon(C_1)$ verifies all the
(cliques) conditions, and such that $h_1^\epsilon(C_1) \sqsubseteq
h_1(v)$ for any vertex $v\in V(G_1)$. Without loss of generality,
let us assume that $V(C_1)=\{v_1,\ldots,v_k\}$, and we also assume
that $h_1(v_i)$ and $h_1^\epsilon(C_1)$ touch in dimension $i$ (i.e.
$h_1(v_i)[i] \cap h_1^\epsilon(C_1)[i] = \{p(C_1)[i]\}$, and
$h_1(v_i)[j] \cap h_1^\epsilon(C_1)[j] =
[p(C_1)[j],p(C_1)[j]+\epsilon] $ for $j\neq i$.
Now let us consider $G_2$ and its representation $h_2$. Here the
vertices of $C^*_2$ are also denoted $v_1,\ldots,v_k$, and let us
denote $d_{v_i}$ the dimension in $h_2$ that fulfills condition (v1)
with respect to $v_i$.
Let $d=\max(d_1,d_2)$. We are now ready for defining $h$. For the
vertices of $G_1$ it is almost the same representation as $h_1$, as
we set $h(v)[i] = h_1(v)[i]$ for any dimension $i\le d_1$. If $d=d_2
> d_1$, then for any dimension $i>d_1$ we set $h(v)[i] = [0,1]$ if
$v\in V(C^*_1)$, and $h(v)[i] = [0,\frac12]$ if $v\in
V(G_1)\setminus V(C^*_1)$. Similarly the clique points $p_1(C)$
become $p(C)$ by setting $p(C)[i] = p_1(C)[i]$ for $i\le d_1$, and
$p(C)[i] = \frac14$ for $i> d_1$.
For $h_2$ and the vertices in $V(G_2) \setminus \{v_1,\ldots,v_k\}$
we have to consider a mapping $\sigma$ from $\{1,\ldots,d_2\}$ to
$\{1,\ldots,d\}$ such that $\sigma(d_{v_i}) = i$. This mapping
describes the changes of dimension we have to perform. We also have
to perform a scaling in order to make $h_2$ fit inside
$h_1^\epsilon(C_1)$. This is ensured by multiplying the coordinates
by $\epsilon$. More formally, for any vertex $v\in V(G_2) \setminus
\{v_1,\ldots,v_k\}$, we set $h(v)[\sigma(i)] = p(C_1)[\sigma(i)] +
\epsilon h_2(v)[i]$ for $i\in \{1 ,\ldots,d_2\}$, and $h(v)[j] =
p(C_1)[j] + [0, \epsilon/2]$, otherwise (for any $j$ not in the
image of $\sigma$). Note that if we apply the same mapping from
$h_2$ to $h$, to the vertices of $\{v_1,\ldots,v_k\}$, then the
image of $h_2(v_i)$ fit inside the (previously defined) box
$h(v_i)$. Similarly the clique points $p_2(C)$ become $p(C)$ by
setting $p(C)[\sigma(i)] = p(C_1)[\sigma(i)] + \epsilon p_2(C)[i]$
for $i\in \{1 ,\ldots,d_2\}$, and $p(C)[j] = p(C_1)[j] + [0,
\frac14]$, otherwise.
Note that we have defined (differently) both $h^\epsilon(C_1)$
(resp. $p(C_1)$) and $h^\epsilon(C^*_2)$ (resp. $p(C^*_2)$), despite
the fact that those cliques were merged. In the following we use
$h^\epsilon(C_1)$ and $p(C_1)$ only for the purpose of the
proof. The point and the box corresponding to these clique in $h$ is
$p(C^*_2)$ and $h^\epsilon(C^*_2)$.
Let us now check that $h$ is a $C^*_1$-clique sum extendable
representation with comparable boxes. The fact that the boxes are
comparable follows from the fact that those of $V(G_1)$
(resp. $V(G_1)$) are comparable in $h_1$ (resp. $h_2$) with the
boxes of $V(C^*_1)$ (resp. $V(C^*_2)$) being hypercubes of side one,
and the other boxes being smaller. Clearly, by construction both
$h_1(u) \sqsubseteq h_1(v)$ or $h_2(u) \sqsubseteq h_2(v)$, imply
$h(u) \sqsubseteq h(v)$, and for any vertex $u\in V(G_1)$ and any
vertex $v\in V(G_2) \setminus \{