\author{Zden\v{e}k Dvo\v{r}\'ak}{Charles University, Prague, Czech Republic}{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.}
\author{Daniel Gon\c{c}alves}{LIRMM, Université de Montpellier, CNRS, Montpellier, France}{goncalves@lirmm.fr}{}{Supported by the ANR grant GATO ANR-16-CE40-0009.}
\author{Abhiruk Lahiri}{Charles University, Prague, Czech Republic}{abhiruk@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.}
\author{Jane Tan}{Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom}{jane.tan@maths.ox.ac.uk}{}{}
\author{Torsten Ueckerdt}{Karlsruhe Institute of Technology, Karlsruhe, Germany}{torsten.ueckerdt@kit.edu}{}{}
\authorrunning{Z. Dvo\v{r}\'ak, D. Gon\c{c}alves, A. Lahiri, J. Tan, and T. Ueckerdt}
\Copyright{Zden\v{e}k Dvo\v{r}\'ak Daniel Gon\c{c}alves, Abhiruk Lahiri, Jane Tan and Torsten Ueckerdt}
\ccsdesc[500]{Theory of computation~Computational geometry}
\ccsdesc[500]{Mathematics of computing~Graphs and surfaces}
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\acknowledgements{This research was carried out at the workshop on Geometric Graphs and Hypergraphs organized by Yelena Yuditsky and Torsten Ueckerdt in September 2021. We would like to thank the organizers and all participants for creating a friendly and productive environment.}%optional
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%Editor-only macros:: begin (do not touch as author)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\EventEditors{Xavier Goaoc and Michael Kerber}
\EventNoEds{2}
\EventLongTitle{38th International Symposium on Computational Geometry (SoCG 2022)}
\EventShortTitle{SoCG 2022}
\EventAcronym{SoCG}
\EventYear{2022}
\EventDate{June 7--10, 2022}
\EventLocation{Berlin, Germany}
\EventLogo{socg-logo}
\SeriesVolume{224}
\ArticleNo{XX}% <-- This will be filled in by the typesetters
Two boxes in $\mathbb{R}^d$ are \emph{comparable} if one of them is a subset
of a translation of the other one. The \emph{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}
Given 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 a \emph{touching representation by objects from $\OO$})
such that the interiors of $f(u)$ and $f(v)$ are disjoint for all distinct $u,v\in V(G)$, 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 has motivated numerous strengthenings and variations (see \cite{graphsandgeom, sachs94} for some classical examples); most relevantly for us, Felsner and Francis~\cite{felsner2011contact} showed that every planar graph is a touching graph of cubes in $\mathbb{R}^3$.
An attractive feature of touching representations is that it is possible to represent graph classes that are sparse
(e.g., planar graphs, or more generally, graph classes with bounded expansion~\cite{nesbook}).
This is in contrast to general intersection representations where the represented class always includes arbitrarily large cliques.
Of course, whether the class of touching graphs 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 boxes in $\mathbb{R}^3$, 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 (throughout the paper, by a \emph{box} we mean an axis-aligned one, i.e., the Cartesian product of closed 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, a condition which can be expressed as follows. For two boxes $B_1$ and $B_2$, we write $B_1\sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.
We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$.
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.
Let the \emph{comparable box dimension}$\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.
Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details.
Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. 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 showed 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 many functions $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
comparable box dimension\footnote{In their construction $h(r)$ has to be at least 3, and has to tend to $+\infty$.}. Dvo\v{r}\'ak et al.~\cite{wcolig}
proved that graphs of comparable box dimension $3$ have exponential weak coloring numbers, giving the
first natural graph class with polynomial 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 all 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{Parameters}
In this section we bound some basic graph parameters in terms of comparable box dimension.
The first result bounds the clique number $\omega(G)$ in terms of $\cbdim(G)$.
\begin{lemma}\label{lemma-cliq}
For any graph $G$, we have $\omega(G)\le2^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
We may assume that $G$ has bounded comparable box dimension
witnessed by a representation $f$. 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
intersections. 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 $p$. At the same time, it follows from the definition
of a touching representation that $f(a_1),\ldots,f(a_d)$ have pairwise disjoint
interiors, and hence $w \leq2^d$.
\end{proof}
Note that a clique with $2^d$ vertices has a touching representation by comparable boxes in $\mathbb{R}^d$,
where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$.
Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$.
In the following we consider the chromatic number $\chi(G)$, and two
of its variants. An \emph{acyclic coloring} (resp. \emph{star coloring}) of a graph $G$ is a proper
coloring such that any two color classes induce a forest (resp. star forest, i.e., a forest in which each component is a star). The \emph{acyclic chromatic number}$\chi_a(G)$ (resp. \emph{star chromatic
number}$\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star)
coloring of $G$. We will need the fact that all the variants of the chromatic number
are at most exponential in the comparable box dimension; this follows
from~\cite{subconvex}, although we include an argument to make the
dependence clear.
\begin{lemma}\label{lemma-chrom}
For any graph $G$ we have $\chi(G)\le3^{\cbdim(G)}$, $\chi_a(G)\le5^{\cbdim(G)}$ and $\chi_s(G)\le2\cdot9^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
We focus on the star chromatic number and note that the chromatic number and the acyclic chromatic number may be bounded similarly.
Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$
be the vertices of $G$ written so that $\vol(f(v_1))\geq\ldots\geq\vol(f(v_n))$.
Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy coloring $c$ so that $c(v_i)$ is
the smallest color such that $c(v_i)\neq c(v_j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there
exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path.
It remains to bound the number of colors used. Suppose we are coloring $v_i$. We shall bound the number of vertices
$v_j$ such that $j<i$ and such that there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B)/\vol(f(v_i))-1=5^d-1$.
A similar argument shows that the number of indices $m$ such that $m<i$ and $v_mv_i\in E(G)$ is at most $3^d-1$.
Consequently, the number of indices $j<i$ for which there exists $m$ such that $j<m<i$ and $v_jv_m,v_mv_i\in E(G)$
is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedily, we only need to avoid colors of at most $(5^d-1)+(3^d-1)+(3^d-1)^2$ vertices, so $2\cdot9^d$ colors suffice.
\end{proof}
\begin{figure}
\centering
\begin{tikzpicture}[xscale=1, yscale=0.6]
\footnotesize
\draw[dashed] (0,0) rectangle (5,5);
\filldraw[red!20!white] (2,2) rectangle (3,3);
\draw (1,0) rectangle (3,2);
\node [fill=none, color=red] at (2.5, 2.5) {$f(v_i)$};
\node [fill=none, color=blue] at (0.7, 3.3) {$B_2$};
\node [fill=white] at (3.1, 1) {$f(v_3)$};
\node [fill=none, color=blue] at (2, 1) {$B_3$};
\end{tikzpicture}
\caption{Nearby boxes obstructing colors at $v_i$.}
\label{fig:lowercolcount}
\end{figure}
\section{Operations}
It is clear that given a touching representation of a graph $G$, one
can easily obtains a touching representation by 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. In the following, to describe
the boxes, we are going to use the Cartesian product $\times$ defined among boxes ($A\times B$ is the box whose projection on the first dimensions gives the box $A$, while the projection on the remaining dimensions gives the box $B$) or we are going to provide its projections for every dimension ($A[i]$ is the interval obtained from projecting $A$ on its $i^\text{th}$ dimension).
\subsection{Vertex addition}\label{sec-vertad}
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
$u_1$ is equal to or adjacent to $u_2$ in $G$
and $v_1$ is equal to or adjacent to $v_2$ in $H$.
To obtain a touching representation of $G\boxtimes
H$ it suffices to take a product of representations of $G$ and $H$, but
the resulting representation may contain incomparable boxes.
Indeed, $\cbdim(G\boxtimes H)$ in general is not bounded by a function
of $\cbdim(G)$ and $\cbdim(H)$; for example, every star has comparable box dimension
at most two, but the strong product of the star $K_{1,n}$ with itself contains
$K_{n,n}$ as an induced subgraph, and thus its comparable box dimension is at least $\Omega(\log n)$.
However, as shown in the following lemma, this issue does not arise if the representation of $H$ consists of translates
of a single box; by scaling, we can without loss of generality assume this box is a unit hypercube.
\begin{lemma}\label{lemma-sp}
Consider a graph $H$ having a touching representation $h$ in
$\mathbb{R}^{d_H}$ by axis-aligned hypercubes of unit size. Then for any graph
$G$, the strong product $G\boxtimes H$ of these graphs has comparable box dimension at most
$\cbdim(G)+ d_H$.
\end{lemma}
\begin{proof}
The proof simply consists in taking a product of the two
representations. Indeed, consider a touching respresentation $g$ of $G$ by
comparable boxes in $\mathbb{R}^{d_G}$, with
$d_G=\cbdim(G)$, and the representation $h$ of $H$. Let us define a
representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
follows.
\[f((u,v))[i]=\begin{cases}
g(u)[i]&\text{ if $i\le d_G$}\\
h(v)[i-d_G]&\text{ if $i > d_G$}
\end{cases}\]
Consider distinct vertices $(u,v)$ and $(u',v')$ of $G\boxtimes H$.
The boxes $g(u)$ and $g(u')$ are comparable, say $g(u)\sqsubseteq g(u')$. Since $h(v')$
is a translation of $h(v)$, this implies that $f((u,v))\sqsubseteq f((u',v'))$. Hence, the boxes
of the representation $f$ are pairwise comparable.
The boxes of the representations $g$ and $h$ have pairwise disjoint interiors.
Hence, if $u\neq u'$, then there exists $i\le d_G$ such that the interiors
of the intervals $f((u,v))[i]=g(u)[i]$ and $f((u',v'))[i]=g(u')[i]$ are disjoint;
and if $v\neq v'$, then there exists $i\le d_H$ such that the interiors
of the intervals $f((u,v))[i+d_G]=h(v)[i]$ and $f((u',v'))[i+d_G]=h(v')[i]$ are disjoint.
Consequently, the interiors of boxes $f((u,v))$ and $f((u',v'))$ are pairwise disjoint.
Moreover, if $u\neq u'$ and $uu'\not\in E(G)$, or if $v\neq v'$ and $vv'\not\in E(G)$,
then the intervals discussed above (not just their interiors) are disjoint for some $i$;
hence, if $(u,v)$ and $(u',v')$ are not adjacent in $G\boxtimes H$, then $f((u,v))\cap f((u',v'))=\emptyset$.
Therefore, $f$ is a touching representation of a subgraph of $G\boxtimes H$.
Finally, suppose that $(u,v)$ and $(u',v')$ are adjacent in $G\boxtimes H$.
Then there exists a point $p_G$ in the intersection of $g(u)$ and $g(u')$,
since $u=u'$ or $uu'\in E(G)$ and $g$ is a touching representation of $G$;
and similarly, there exists a point $p_H$ in the intersection of $h(v)$ and $h(v')$.
Then $p_G\times p_H$ is a point in the intersection of $f((u,v))$ and $f((u',v'))$.
Hence, $f$ is indeed a touching representation of $G\boxtimes H$.
\end{proof}
\subsection{Taking a subgraph}
The comparable box dimension of a subgraph of a graph $G$ may be larger than $\cbdim(G)$, see the end of this
section for an example. 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.
\begin{lemma}\label{lemma-subg}
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le\cbdim(G')+\frac12\chi^2_s(G')$.
\end{lemma}
\begin{proof}
As we can remove the boxes that represent the vertices, we can assume $V(G')=V(G)$.
Let $f$ be a touching representation of $G'$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G')$. Let $\varphi$
be a star coloring of $G'$ using colors $\{1,\ldots,c\}$, where $c=\chi_s(G')$.
For any distinct colors $i,j\in\{1,\ldots,c\}$, let $A_{i,j}\subseteq V(G)$ be the set of vertices $u$ of color $i$
such that there exists a vertex $v$ of color $j$ such that $uv\in E(G')\setminus E(G)$. For each $u\in A_{i,j}$,
let $a_j(u)$ denote such a vertex $v$ chosen arbitrarily.
Let us define a representation $h$ by boxes in $\mathbb{R}^{d+\binom{c}{2}}$ by starting from the representation $f$ and,
for each pair $i<j$ of colors, adding a dimension $d_{i,j}$ and setting
\[h(v)[d_{i,j}]=\begin{cases}
[1/3,4/3]&\text{if $v\in A_{i,j}$}\\
[-4/3,-1/3]&\text{if $v\in A_{j,i}$}\\
[-1/2,1/2]&\text{otherwise.}
\end{cases}\]
Note that the boxes in this extended representation are comparable,
as in the added dimensions, all the boxes have size $1$.
Suppose $uv\in E(G)$, where $\varphi(u)=i$ and $\varphi(v)=j$ and say $i<j$.
We cannot have $u\in A_{i,j}$ and $v\in A_{j,i}$, as then $a_j(u)uva_i(v)$ would be a 4-vertex path in $G'$ in colors $i$ and $j$.
Hence, in any added dimension $d'$, we have $h(u)[d']=[-1/2,1/2]$ or $h(v)[d']=[-1/2,1/2]$,
and thus $h(u)[d']\cap h(v)[d']\neq\emptyset$.
Since the boxes $f(u)$ and $f(v)$ touch, it follows that the boxes $h(u)$ and $h(v)$ touch as well.
Suppose now that $uv\not\in E(G)$. If $uv\not\in E(G')$, then $f(u)$ is disjoint from $f(v)$, and thus $h(u)$ is disjoint from
$h(v)$. Hence, we can assume $uv\in E(G')\setminus E(G)$, $\varphi(u)=i$, $\varphi(v)=j$ and $i<j$. Then $u\in A_{i,j}$, $v\in A_{j,i}$,
$h(u)[d_{i,j}]=[1/3,4/3]$, $h(v)[d_{j,i}]=[-4/3,-1/3]$, and $h(u)\cap h(v)=\emptyset$.
Consequently, $h$ is a touching representation of $G$ by comparable boxes in dimension $d+\binom{c}{2}\le d+c^2/2$.
\end{proof}
Let us now combine Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}.
\begin{corollary}\label{cor-subg}
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le\cbdim(G')+2\cdot81^{\cbdim(G')}\le3\cdot81^{\cbdim(G')}$.
\end{corollary}
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 $2^{d-1}$. Indeed, for every pair $u,v$ of non-adjacent vertices there is a specific dimension $i$ such that their boxes span intervals $[a,b]$ and $[c,d]$ with $b<c$, while for every other box in the representation their $i^\text{th}$ interval contains $[b,c]$.
\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 $C^\star$, called the
\emph{root clique} of $G$. A touching representation $h$ of $G$
by (not necessarily comparable) boxes in $\mathbb{R}^d$ is called
\emph{$C^\star$-clique-sum extendable} if the following conditions hold for every sufficiently small $\varepsilon>0$.
\begin{itemize}
\item[]{\bf(vertices)} For each $u\in V(C^\star)$, there exists a dimension $d_u$,
such that:
\subitem\emph{(v0)}$d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
\subitem\emph{(v1)} each vertex $u\in V(C^\star)$ satisfies $h(u)[d_u]=[-1,0]$ and
$h(u)[i]=[0,1]$ for any dimension $i\neq d_u$, and
\subitem\emph{(v2)} each vertex $v\notin V(C^\star)$ satisfies $h(v)\subset[0,1)^d$.
\item[]{\bf(cliques)} For every clique $C$ of $G$, there exists
a point $p(C)\in[0,1)^d\cap\left(\bigcap_{v\in V(C)} h(v)\right)$
such that, defining the \emph{clique box}$h^\varepsilon(C)$
by setting $h^\varepsilon(C)[i]=[p(C)[i],p(C)[i]+\varepsilon]$ for every dimension
$i$, the following conditions are satisfied:
% \begin{itemize}
\subitem\emph{(c1)} For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1)\cap
Similarly, we can deal with proper minor-closed classes.
\begin{proof}[Proof of Theorem~\ref{thm-minor}]
Let $\GG$ be a proper minor-closed class. Since $\GG$ is proper, there exists $t$ such that $K_t\not\in\GG$.
By Theorem~\ref{thm-prod}, there exists $k$ such that every graph in $\GG$ is a subgraph of a graph obtained by repeated clique-sums
from extended $k$-tree-grids. As we have seen, $k$-tree-grids have comparable box dimension at most $k+2$,
and by Lemma~\ref{lemma-apex}, extended $k$-tree-grids have comparable box dimension at most $2k+2$.
By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le1250^{2k+2}$.
\end{proof}
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
and comparable box dimension $n$. It follows that the dependence of the comparable box dimension on the Euler genus cannot be
subpolynomial (though the degree $\log_281$ of the polynomial established in Corollary~\ref{cor-genus}
certainly can be improved). The dependence of the comparable box dimension on the size of the forbidden minor that we
established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}.
It would be interesting to prove Theorem~\ref{thm-minor} without using the structure theorem.
\section{Fractional treewidth-fragility}
Suppose $G$ is a connected planar graph and $v$ is a vertex of $G$. For an integer $k\ge2$,
give each vertex at distance $d$ from $v$ the color $d\bmod k$. Then deleting the vertices of any of the $k$ colors
results in a graph of treewidth at most $3k$. This fact (which follows from the result of Robertson and Seymour~\cite{rs3}
on treewidth of planar graphs of bounded radius) is (in the modern terms) the basis of Baker's technique~\cite{baker1994approximation}
for design of approximation algorithms. However, even quite simple graph classes (e.g., strong products of three paths~\cite{gridtw})
do not admit such a coloring (where the removal of any color class results in a graph of bounded treewidth).
However, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx}
and applies to much more general graph classes, including all graph classes with strongly sublinear separators and bounded maximum degree~\cite{twd}.
We say that a class of graphs $\GG$ is \emph{fractionally treewidth-fragile} if there exists a function $f$ such that
for every graph $G\in\GG$ and integer $k\ge2$, there exist sets $X_1, \ldots, X_m\subseteq V(G)$ such that
each vertex belongs to at most $m/k$ of them and $\tw(G-X_i)\le f(k)$ for every $i$
(equivalently, there exists a probability distribution on the set $\{X\subseteq V(G):\tw(G-X)\le f(k)\}$
such that $\text{Pr}[v\in X]\le1/k$ for each $v\in V(G)$).
For example, the class of planar graphs is (fractionally) treewidth-fragile, since we can let $X_i$ consist of the
vertices of color $i-1$ in the coloring described at the beginning of the section.
Our main result is that all graph classes of bounded comparable box dimension are fractionally treewidth-fragile.
We will show the result in a more general setting, motivated by concepts from~\cite{subconvex} and by applications to related
representations. The argument is motivated by the idea used in the approximation algorithms for disk graphs
by Erlebach et al.~\cite{erlebach2005polynomial}.
For a measurable set $A\subseteq\mathbb{R}^d$, let $\vol(A)$ denote the Lebesgue measure of $A$.
For two measurable subsets $A$ and $B$ of $\mathbb{R}^d$ and a positive integer $s$, we write $A\sqsubseteq_s B$
if for every $x\in B$, there exists a translation $A'$ of $A$ such that $x\in A'$ and $\vol(A'\cap B)\ge\tfrac{1}{s}\vol(A)$.
Note that for two boxes $A$ and $B$, we have $A\sqsubseteq_1 B$ if and only if $A\sqsubseteq B$.
An \emph{$s$-comparable envelope representation}$(\iota,\omega)$ of a graph $G$ in $\mathbb{R}^d$ consists of
two functions $\iota,\omega:V(G)\to2^{\mathbb{R}^d}$ such that for some ordering $v_1$, \ldots, $v_n$ of vertices of $G$,
\begin{itemize}
\item for each $i$, $\omega(v_i)$ is a box, $\iota(v_i)$ is a measurable set, and $\iota(v_i)\subseteq\omega(v_i)$,
\item if $i<j$, then $\omega(v_j)\sqsubseteq_s \iota(v_i)$, and
\item if $i<j$ and $v_iv_j\in E(G)$, then $\omega(v_j)\cap\iota(v_i)\neq\emptyset$.
\end{itemize}
We say that the representation has \emph{thickness at most $t$} if for every point $x\in\mathbb{R}^d$, there
exist at most $t$ vertices $v\in V(G)$ such that $x\in\iota(v)$.
For example:
\begin{itemize}
\item If $f$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^d$, then
$(f,f)$ is a $1$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2^d$.
\item If $f$ is a touching representation of $G$ by balls in $\mathbb{R}^d$ and letting $\omega(v)$ be
the smallest axis-aligned hypercube containing $f(v)$, then there exists a positive integer $s_d$ depending only on $d$ such that
$(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2$.
\end{itemize}
Let us recall some notions about treewidth.
A \emph{tree decomposition} of a graph $G$ is a pair
$(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to2^{V(G)}$
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 $T$.
The \emph{width} of the tree decomposition 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. Let us remark that the value of treewidth obtained via this definition coincides
with the one via $k$-trees which we used in the previous section.
\begin{theorem}\label{thm-twfrag}
For positive integers $t$, $s$, and $d$, the class of graphs
with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at most $t$
is fractionally treewidth-fragile, with a function $f(k)= O_{t,s,d}\bigl(k^{d}\bigr)$.
\end{theorem}
\begin{proof}
For a positive integer $k$, let $f(k)=(2ksd+2)^dst$.
Let $(\iota,\omega)$ be an $s$-comparable envelope representation of a graph $G$
in $\mathbb{R}^d$ of thickness at most $t$, and let $v_1$, \ldots, $v_n$ be the corresponding ordering of the vertices of $G$.
Let us define $\ell_{i,j}\in\mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $ksd|\omega(v_i)[j]|$ such that $\ell_{i-1,j}/\ell_{i,j}$ is a positive integer. Formally
it is defined as follows.
\begin{itemize}
\item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$.
\item For $i=2,\ldots, n$, let $\ell_{i,j}=\ell_{i-1,j}$, if
$\ell_{i-1,j} < ksd|\omega(v_i)[j]|$, and otherwise let
$\ell_{i,j}$ be lowest fraction of $\ell_{i-1,j}$ that is
greater than $ksd|\omega(v_i)[j]|$, formally $\ell_{i,j}=
\min\{\ell_{i-1,j}/b \ |\ b\in
\mathbb{N}^+\text{ and }\ell_{i-1,j}/b \ge ksd|\omega(v_i)[j]|\}$.
\end{itemize}
Let the real $x_j\in[0,\ell_{1,j}]$ be chosen uniformly at random,
and let $\HH^i_j$ be the set of hyperplanes in $\mathbb{R}^d$
consisting of the points whose $j$-th coordinate is equal to
$x_j+m\ell_{i,j}$ for some $m\in\mathbb{Z}$. As $\ell_{i,j}$ is a
multiple of $\ell_{i',j}$ whenever $i\le i'$, we have that $\HH^i_j
\subseteq\HH^{i'}_j$ whenever $i\le i'$. For $i\in\{1,\ldots,n\}$,
the \emph{$i$-grid} is $\HH^i=\bigcup_{j=1}^d \HH^i_j$, and we let the
$0$-grid $\HH^0=\emptyset$. Similarly as above we have that $\HH^i
\subseteq\HH^{i'}$ whenever $i\le i'$.
We let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such
that the box $\omega(v_a)$ intersects some hyperplane $H\in\HH^a$,
that is such that $x_j+m\ell_{a,j}\in\omega(v_a)[j]$, for some
$j\in\{1,\ldots,d\}$ and some $m\in\mathbb{Z}$. First, let us argue
that $\text{Pr}[v_a\in X]\le1/k$. Indeed, the set $[0,\ell_{1,j}]\cap\bigcup_{m\in\mathbb{Z}}(\omega(v_a)[j]-m\ell_{a,j})$
has measure $\tfrac{\ell_{1,j}}{\ell_{a,j}}\cdot |\omega(v_a)[j]|$, implying that for fixed $j$, this happens with probability
$|\omega(v_a)[j]|/\ell_{a,j}$. Let $a'$ be the largest integer such
that $a'\le a$ and $\ell_{a',j} < \ell_{a'-1,j}$ if such an index exists,
and $a'=1$ otherwise; note that $\ell_{a,j}=\ell_{a',j}\ge ksd|\omega(v_{a'})[j]|$. Moreover, since
$\omega(v_a)\sqsubseteq_s\iota(v_{a'})\subseteq\omega(v_{a'})$, we have $\omega(v_a)[j]\le s\omega(v_{a'})[j]$.
By the union bound, we conclude that $\text{Pr}[v_a\in X]\le1/k$.
Let us now bound the treewidth of $G-X$.
For $a\ge0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus\bigl(\bigcup_{H\in\HH^a} H\bigr)$.
A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge0$.
A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$.
Note that there exists a rooted tree $T$ whose vertices are
the non-empty cells and such that for $x,y\in V(T)$, we have $x\preceq y$ if and only if $x\subseteq y$.
For each non-empty cell $C$, let us define $\beta(C)$ as the set of vertices $v_i\in V(G-X)$ such that
$\iota(v)\cap C\neq\emptyset$ and $C$ is an $a$-cell for some $a\ge i$.
Let us show that $(T,\beta)$ is a tree decomposition of $G-X$. For each $v_j\in V(G-X)$, the $j$-grid is disjoint from $\omega(v_j)$,
and thus $\iota(v_j)\subseteq\omega(v_j)\subset C$ for some $j$-cell $C\in V(T)$ and $v_j\in\beta(C)$. Consider now an edge $v_iv_j\in E(G-X)$, where $i<j$.
We have $\omega(v_j)\cap\iota(v_i)\neq\emptyset$, and thus $\iota(v_i)\cap C\neq\emptyset$ and $v_i\in\beta(C)$.
Finally, suppose that $v_j\in C'$ for some $C'\in V(T)$. Then $C'$ is an $a$-cell for some $a\ge j$, and since
$\iota(v_j)\cap C'\neq\emptyset$ and $\iota(v_j)\subset C$, we conclude that $C'\subseteq C$, and consequently $C'\preceq C$.
Moreover, any cell $C''$ such that $C'\preceq C''\preceq C$ (and thus $C'\subseteq C''\subseteq C$) is an $a'$-cell
for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq\iota(v_j)\cap C'\neq\emptyset$, implying $v_j\in\beta(C'')$.
It follows that $\{C':v_j\in\beta(C')\}$ induces a connected subtree of $T$.
Finally, let us bound the width of the decomposition $(T,\beta)$. Let $C$ be a non-empty cell and let $a$ be maximum such that $C$
is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ldots, $\ell_{a,d}$. Consider $j\in\{1,\ldots,d\}$:
\begin{itemize}
\item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$.
\item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$ (otherwise $\ell_{a,j}=\ell_{a-1,j}/b$ for some integer $b\ge2$).
\item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integer $b\ge2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that
Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
For any $v_i\in\beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
there exists a translation $B_i$ of $\omega(v_a)$ that intersects $C\cap\iota(v_i)$ and such that $\vol(B_i\cap\iota(v_i))\ge\tfrac{1}{s}\vol(\omega(v_a))$.
Note that as $B_i$ intersects $C$, we have that $B_i\subseteq C'$.
Since the representation has thickness at most $t$,
