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

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}}}

\date{}

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@@ -45,9 +47,7 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten

\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

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}

...

...

@@ -58,18 +58,17 @@ if there exists a function $f:V(G)\to \OO$ (called a \emph{touching representati

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}.

This result motivated a number of strengthenings and variations (see \cite{graphsandgeom, sachs94} for some classical examples); 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.

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 theory~\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 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 $m\times1\times1$ 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, in particular axis-aligned).

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 $B_2$.

long and wide boxes: 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. For a graph $G$, let the \emph{comparable box dimension}$\cbdim(G)$

...

...

@@ -102,12 +101,13 @@ 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 $\cbdim(G)\le |V(G)$.

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 $u\in V(G)\setminus\{v\}$.

...

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@@ -120,7 +120,7 @@ For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_

If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le2^d$.

\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.

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 $w \leq2^d$.

...

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@@ -138,11 +138,11 @@ For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in

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}.)

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 $R^{t+1}$ such that

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}

...

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@@ -153,7 +153,7 @@ and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta(

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 $y\in\beta(x)$.

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

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.

Consider now two non-adjacent vertices of $G$, say $x_i$ and $w$. As we noted before, if $x_i$ and $w$ are

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 $y$

...

...

@@ -310,7 +310,7 @@ $xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$.

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 $\cbdim(G')\le(\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le6^{\cbdim(\GG)}$.

$G\subseteq G'$ and \[\cbdim(G')\le(\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le6^{\cbdim(\GG)}.\]

\end{corollary}

Note that only bound the comparable box dimension of a supergraph

...

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@@ -407,7 +407,7 @@ Any graph $G$ is a subgraph of the strong product of a path, a graph of threewid

\item$t=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.

\end{itemize}

Moreover, for every $k$, there exists $t$ such that if $K_k\not\preceq_m G$, then $G$ is a subgraph of a clique-sum

for graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding

of graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding

at most $t$ apex vertices.

\end{theorem}

...

...

@@ -420,13 +420,10 @@ comparable box dimension at most $t+2+\lceil \log_2 m\rceil$.

Without loss of generality, we can assume that $k=\log_2 m$ is an integer and the

vertices of $K_m$ are elements of $\{0,1\}^k$. Moreover, we can assume that $V(P)=\{1,\ldots,n\}$

with $ij\in E(P)$ iff $|i-j|=1$. Let $h$ be the touching representation of $T$ by

hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}.

The representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$

is obtained as follows: For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,

hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}. Then an explicit representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$ can be obtained as follows. For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,

we set

$$f(p,v,x_1, \ldots, x_k)[j]=\begin{cases}

f(v)[j]&\text{ for $j=1,\ldots, t+1$}\\

h(v)[j]&\text{ for $j=1,\ldots, t+1$}\\

[p,p+1]&\text{ if $j=t+2$}\\

[x_{j-t-2},x_{j-t-2}+1]&\text{ if $j>t+2$.}

\end{cases}$$

...

...

@@ -444,7 +441,7 @@ Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, the

$\cbdim(G)\le d$.

\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

and comparable box dimension $n$. It follows that the dependence of the comparable box dimension cannot be

subpolynomial (though the degree $\log_281$ 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}.