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\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
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 a \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 strengthenings 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 $\mathbb{R}^3$, where the vertices in
one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$
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: For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if a translation of $B_1$ is a subset of $B_2$.
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. 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 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{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 $d=\cbdim(G-v)$.
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}
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 $B[i]=I_i$.
\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}
For 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 (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 $R^{t+1}$ 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, 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 $ij$. We greedily color the vertices in order, giving $v_i$ the smallest
color different from the colors of all vertices $v_j$ such that $j*j$
such that $v_jv_m,v_mv_i\in E(G)$. Note this gives a star coloring: A path on four vertices always contains a 3-vertex subpath
$v_{i_1}v_{i_2}v_{i_3}$ such that $i_1i$ such that $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box that is five times larger than $f(v)$
and has the same center as $f(v)$. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
contained in $f(v_j)\cap B$. The boxes $B_j$ for different $j$ have disjoint interiors and their interiors are also disjoint from
$f(v_i)\subset B$, and thus the number of such indices $j$ is at most $\vol(B_j)/\vol(f(v_i))-1=5^d-1$.
A similar argument shows that the number of indices $m$ such that $m**t+2$.}
\end{cases}$$
Clearly, this is a touching representation by comparable boxes.
\end{proof}
Combining Theorem~\ref{thm-prod}, Lemma~\ref{lemma-ps}, and the results of the previous section,
we obtain the following corollary, which in particular implies Theorem~\ref{thm-minor}.
\begin{corollary}\label{cor-minor}
For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$ of $G$ such that
$\cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil$. Consequently,
$$\cbdim(G)\le 5\cdot 81^7 \cdot (2g+3)^{\log_2 81}.$$
Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, then
$\cbdim(G)\le d$.
\end{corollary}
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
and comparable box dimension $n$, showing that the dependence of the comparable box dimension cannot be
subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-minor}
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 this part of Corollary~\ref{cor-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\ge 2$,
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\ge 2$, 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]\le 1/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 $\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)\to 2^{\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 $i1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $b_{a,j}=1$, implying $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$.
\item If $a>1$ and $\ell_{a,j}>\ell_{a-1,j}$, then $\ell_{a-1,j}\ge 2ksd|\omega(v_a)[j]|$ and
$$\ell_{a,j}=\frac{\ell_{a-1,j}}{\lfloor \frac{\ell_{a-1,j}}{ksd|\omega(v_a)[j]|}\rfloor}<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$
\end{itemize}
Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
For any $v_i\in \beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
there exists a translation $B_i\subseteq C'$ of $\omega(v_a)$ such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$.
Since the representation has thickness at most $t$,
\begin{align*}
\vol(C')&\ge \vol\left(C'\cap \bigcup_{v_i\in \beta(C)\setminus\{v_a\}} \iota(v_i)\right)\\
&\ge \vol\left(\bigcup_{v_i\in \beta(C)\setminus\{v_a\}} B_i\cap\iota(v_i)\right)\\
&\ge \frac{1}{t}\sum_{v_i\in \beta(C)\setminus\{v_a\}} \vol(B_i\cap\iota(v_i))\\
&\ge \frac{\vol(\omega(v_a))(|\beta(C)|-1)}{st}.
\end{align*}
Since $\vol(C')=(2ksd+2)^d\vol(\omega(v_a))$, it follows that
$$|\beta(C)|-1\le (2ksd+2)^dst=f(k),$$
as required.
\end{proof}
The proof that (generalizations of) graphs with bounded comparable box dimensions have sublinear separators in~\cite{subconvex}
is indirect; it is established that these graphs have polynomial coloring numbers, which in turn implies they have polynomial
expansion, which then gives sublinear separators using the algorithm of Plotkin, Rao, and Smith~\cite{plotkin}.
The existence of sublinear separators is known to follow more directly from fractional treewidth-fragility. Indeed, since $\text{Pr}[v\in X]\le 1/k$,
there exists $X\subseteq V(G)$ such that $\tw(G-X)\le f(k)$ and $|X|\le |V(G)|/k$. The graph $G-X$ has a balanced separator of size
at most $\tw(G-X)+1$, which combines with $X$ to a balanced separator of size at most $V(G)|/k+f(k)+1$ in $G$.
Optimizing the value of $k$ (choosing it so that $V(G)|/k=f(k)$), we obtain the following corollary of Theorem~\ref{thm-twfrag}.
\begin{corollary}
For positive integers $t$, $s$, and $d$, every graph $G$
with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at most $t$
has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).$
\end{corollary}
\subsection*{Acknowledgments}
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.
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