\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{...}
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}}}
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@@ -53,20 +53,20 @@ comparable box dimension and explore further properties of this notion.
\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 $\OO$})
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 strenthenings and variations~\cite{...}; most relevantly for us, every planar graph is
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 $\mathbb{R}^d$, where the vertices in
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 $m\times1\times1$ 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).
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 $B_2\sqsubseteq B_1$.
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@@ -82,7 +82,7 @@ numbers, which implies that $\GG$ has strongly sublinear separators. They also
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
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,
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@@ -94,13 +94,13 @@ The comparable box dimension of every proper minor-closed class of graphs is fin
\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
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 addition of a vertex increases the comparable box dimension by at most one.
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 $\cbdim(G)\le\cbdim(G-v)+1$.
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@@ -114,12 +114,15 @@ Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^
\end{proof}
We need a bound on the clique number in terms of the comparable box dimension.
For a box $B=I_1\times\cdots I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_i$.
For a box $B=I_1\times\cdots\timesI_d$ 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 $\omega(G)\le2^d$.
\end{lemma}
\begin{proof}
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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 $w \leq2^d$.
\end{proof}
A \emph{tree decomposition} of a graph $G$
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@@ -130,11 +133,11 @@ such that
\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 $v\in\beta(x)$ nearest to the root of $T$.
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 dimension.
In fact, we will prove the following stronger fact (TODO: Was this published somehere before?)
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 $t$.
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@@ -219,13 +222,13 @@ implying that the boxes $h(x_i)$ and $h(w)$ are disjoint. Therefore, $h$ is a t
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
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 arbitrary number of clique-sums.
even after an arbitrary number of clique-sums.
It will be convenient to work in the setting of tree decompositions.
Consider a tree decompostion $(T,\beta)$ of a graphs$G$.
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.
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@@ -235,7 +238,7 @@ A graph $G$ is obtained from graphs in a class $\GG$ by repeated clique-sums if
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 decompositon of $G$ of adhesion $a$, then $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$.
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 $x\in V(T)$.
