Beginning of the writeup.

comparable-box-dimension.tex

+\documentclass[10pt]{article}
+\usepackage{amsthm}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{colonequals}
+\usepackage{epsfig}
+\usepackage{url}
+\newcommand{\GG}{{\cal G}}
+\newcommand{\CC}{{\cal C}}
+\newcommand{\OO}{{\cal O}}
+\newcommand{\PP}{{\cal P}}
+\newcommand{\RR}{{\cal R}}
+\newcommand{\col}{\text{col}}
+\newcommand{\vol}{\text{vol}}
+
+\newcommand{\eps}{\varepsilon}
+\newcommand{\mc}[1]{\mathcal{#1}}
+\newcommand{\mf}[1]{\mathfrak{#1}}
+\newcommand{\bb}[1]{\mathbb{#1}}
+\newcommand{\brm}[1]{\operatorname{#1}}
+\newcommand{\cbdim}{\brm{dim}_{cb}}
+%%%%%
+
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{example}[theorem]{Example}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{observation}[theorem]{Observation}
+\newtheorem{question}[theorem]{Question}
+
+\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{...}
+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 $m\times 1\times 1$ 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: 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 $1+|N(v)|\le 3^d$.
+\end{proof}
+
+\section{Exploiting the product structure}
+
+\bibliographystyle{siam}
+
+\bibliography{data}
+
+\end{document}