\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{...}
Abhiruk Lahiri\thanks{...}\and
Jane Tan\thanks{...}\and
Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten.ueckerdt@kit.edu}}}
\date{}
\begin{document}
\maketitle
\begin{abstract}
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 boxes in $\mathbb{R}^d$ (two boxes are comparable if one of them is
a subset of a translation of the other one). We show that proper minor-closed classes have bounded
comparable box dimension and explore further properties of this notion.
\end{abstract}
\section{Introduction}
For a system $\OO$ of subsets of $\mathbb{R}^d$, we say that a graph $G$ is a \emph{touching graph of objects from $\OO$}
if there exists a function $f:V(G)\to\OO$ (called the \emph{touching representation by objects from $\OO$})
such that for distinct $u,v\in V(G)$, the interiors of $f(u)$ and $f(v)$ are disjoint
and $f(u)\cap f(v)\neq\emptyset$ if and only if $uv\in E(G)$.
Famously, Koebe~\cite{koebe} proved that a graph is planar if and only if it is a touching graph of balls in $\mathbb{R}^2$.
This result motivated a number of strenthenings and variations~\cite{...}; most relevantly for us, every planar graph is
a touching graph of cubes in $\mathbb{R}^3$~\cite{felsner2011contact}.
An attractive feature of touching representation is that it makes it possible to represent graph classes that are sparse
(e.g., planar graphs, or more generally, graph classes with bounded expansion theory~\cite{nesbook}),
whereas in a general intersection representation, the represented class always includes arbitrarily large cliques.
Of course, whether the class of touching graph of objects from $\OO$ is sparse or not depends on the system $\OO$.
For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis-aligned boxes in $\mathbb{R}^d$, where the vertices in
one part are represented by $m\times1\times1$ boxes and the vertices of the other part are represented by $1\times n\times1$
boxes (a \emph{box} is the cartesian product of intervals of non-zero length).
Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of
long and wide boxes: Two boxes are \emph{comparable} if a translation of one of them is a subset of the other one.
A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes
such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. For a graph $G$, let the \emph{comparable box dimension}$\cbdim(G)$
of $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.
For a class $\GG$ of graphs, let $\cbdim(\GG)=\sup\{\cbdim(G):G\in\GG\}$; note that $\cbdim(\GG)=\infty$ if the
comparable box dimension of graphs in $\GG$ is not bounded.
Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular,
they proved that if a class $\GG$ has finite comparable box dimension, then it has polynomial strong coloring
numbers, which implies that $\GG$ has strongly sublinear separators. They also provided an example showing
that for any function $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
comparable box dimension. Dvo\v{r}\'ak et al.~\cite{wcolig}
proved that graphs of comparable box dimension $3$ have exponential weak coloring number, giving the
first natural graph class with olynomial strong coloring numbers and superpolynomial weak coloring numbers
(the previous example is obtained by subdividing edges of every graph suitably many times~\cite{covcol}).
We show that the comparable box dimension behaves well under the operations of addition of apex vertices,
clique-sums, and taking subgraphs. Together with known results on product structure~\cite{DJM+}, this implies
the main result of this paper.
\begin{theorem}\label{thm-minor}
The comparable box dimension of every proper minor-closed class of graphs is finite.
\end{theorem}
Additionally, we show that classes of graphs with finite comparable box dimension are fractionally treewidth-fragile.
This gives arbitrarily precise approximation algorithms for alll monotone maximization problems that are
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 addition of a vertex increases the comparable box dimension by at most one.
\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 comparable box representation of $G-v$ in $\mathbb{R}^d$, where $d=\cbdim(G-v)$.
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 comparable box representation of $G$ in $\mathbb{R}^{d+1}$.
\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 arrangef 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, motivating the following definition.
For a box $B=I_1\times\cdot\times I_d$ and $i\in\{1,\ldots,d\}$ let $B[i]$ denote the interval $I_i$.
Let $B_1$, \ldots, $B_k$ be pairwise touching boxes in $\mathbb{R}^d$.
A box $B$\emph{touches $B_1$, \ldots, $B_k$ generically} if there exist distinct $i_1,\ldots, i_k\in\{1,\ldots, d\}$ such that for
$j=1,\ldots, k$,
\begin{itemize}
\item$B[i_j]\cap B_j[i_j]$ consists of a single point, and
\item for every $i\in\{1,\ldots,d\}\setminus\{i_j\}$, $B[i]\cap B_j[i]$ is a non-empty interval of non-zero length.
\end{itemize}
A clique $K$ in a touching box representation $f$ of a graph $G$ in $\mathbb{R}^d$ is \emph{exposed} if there exists a box $B$
such that
\begin{itemize}
\item the interior of $B$ is disjoint from $f(v)$ for every $v\in V(G)$,
\item$B$ touches the boxes $\{f(u):u\in K\}$ generically, and
\item there exists $i\in\{1,\ldots,d\}$ such that $B[i]\subseteq f(u)[i]$ for every $u\in K$.
\end{itemize}
The representation is \emph{exposed} if all cliques are exposed.
\begin{lemma}\label{lemma-expose}
Every graph $G$ has an exposed comparable box representation in $\mathbb{R}^{\chi(G)}$.
\end{lemma}
\begin{proof}
...
\end{proof}
Note that the chromatic number of $G$ 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 $G$ is $3^d$-colorable.
\end{lemma}
\begin{proof}
We actually show that $G$ is $(3^d-1)$-degenerate. Since every induced subgraph of $G$ also
has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$
is less than $3^d$. Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume. For every neighbor $u$ of $v$,
there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$.
Note that $f(v)\cup\bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in
a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+|N(v)|\le3^d$.
