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Updated with SoCG format. There is a line number restriction this year. We have to fit everything within 500 lines. Currently, we are at 602. 
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\documentclass[10pt]{article} \documentclass[a4paper,USenglish,cleveref,autoref,thm-restate]{socg-lipics-v2021}
\bibliographystyle{plainurl}
\title{On comparable box dimension}
\titlerunning{On comparable box dimension}
\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}{}{Partially 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}{orsten.ueckerdt@kit.edu}{}{}
\authorrunning{Zden\v{e}k Dvo\v{r}\'ak et al.}
\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}
\keywords{geometric graphs, minor-closed graph classes, treewidth fragility}
%\category{} %optional, e.g. invited paper
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%\supplementdetails[linktext={opt. text shown instead of the URL}, cite=DBLP:books/mk/GrayR93, subcategory={Description, Subcategory}, swhid={Software Heritage Identifier}]{General Classification (e.g. Software, Dataset, Model, ...)}{URL to related version} %linktext, cite, and subcategory are optional
%\funding{(Optional) general funding statement \dots}%optional, to capture a funding statement, which applies to all authors. Please enter author specific funding statements as fifth argument of the \author macro.
\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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\EventEditors{John Q. Open and Joan R. Access}
\EventNoEds{2}
\EventLongTitle{42nd Conference on Very Important Topics (CVIT 2016)}
\EventShortTitle{CVIT 2016}
\EventAcronym{CVIT}
\EventYear{2016}
\EventDate{December 24--27, 2016}
\EventLocation{Little Whinging, United Kingdom}
\EventLogo{}
\SeriesVolume{42}
\ArticleNo{23}
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\usepackage{amsthm} \usepackage{amsthm}
\usepackage{amsfonts} \usepackage{amsfonts}
\usepackage{amsmath} \usepackage{amsmath}
...@@ -25,25 +76,6 @@ ...@@ -25,25 +76,6 @@
\newcommand{\vol}{\brm{vol}} \newcommand{\vol}{\brm{vol}}
%%%%% %%%%%
\newcommand{\note}[1]{\textcolor{blue}{#1}}
\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}
\newtheorem{definition}[theorem]{Definition}
\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{LIRMM, Université de Montpellier, CNRS, Montpellier, France. E-mail: {\tt goncalves@lirmm.fr}. Supported by the ANR grant GATO ANR-16-CE40-0009.}\and
Abhiruk Lahiri\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{}
\begin{document} \begin{document}
\maketitle \maketitle
...@@ -142,7 +174,7 @@ For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G) ...@@ -142,7 +174,7 @@ For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G)
We focus on the star chromatic number and note that the chromatic number and the acyclic chromatic number may be bounded similarly. 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 $v_1, \ldots, v_n$ 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 $\vol(f(v_1)) \geq \ldots \geq \vol(f(v_n))$. 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 colouring $c$ so that $c(i)$ is Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy coloring $c$ so that $c(i)$ is
the smallest color such that $c(i)\neq c(j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there the smallest color such that $c(i)\neq c(j)$ 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. 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.
...@@ -153,7 +185,7 @@ contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{ ...@@ -153,7 +185,7 @@ contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{
A similar argument shows that the number of indices $m$ such that $m<i$ and $v_mv_i\in E(G)$ is at most $3^d-1$. 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 $v_jv_m,v_mv_i\in E(G)$ 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 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
$$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$ \[(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d\]
vertices. vertices.
\end{proof} \end{proof}
...@@ -240,10 +272,10 @@ of a single box; by scaling, we can without loss of generality assume this box i ...@@ -240,10 +272,10 @@ of a single box; by scaling, we can without loss of generality assume this box i
$d_G=\cbdim(G)$, and the representation $h$ of $H$. Let us define a $d_G=\cbdim(G)$, and the representation $h$ of $H$. Let us define a
representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
follows. follows.
$$f((u,v))[i]=\begin{cases} \[f((u,v))[i]=\begin{cases}
g(u)[i]&\text{ if $i\le d_G$}\\ g(u)[i]&\text{ if $i\le d_G$}\\
h(v)[i-d_G]&\text{ if $i > d_G$} h(v)[i-d_G]&\text{ if $i > d_G$}
\end{cases}$$ \end{cases}\]
Consider distinct vertices $(u,v)$ and $(u',v')$ of $G\boxtimes H$. 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 $h(v')$ 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 is a translation of $h(v)$, this implies that $f((u,v))\sqsubseteq f((u',v'))$. Hence, the boxes
...@@ -293,11 +325,11 @@ let $a_j(u)$ denote such a vertex $v$ chosen arbitrarily. ...@@ -293,11 +325,11 @@ 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, 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 for each pair $i<j$ of colors, adding a dimension $d_{i,j}$ and setting
$$h(v)[d_{i,j}]=\begin{cases} \[h(v)[d_{i,j}]=\begin{cases}
[1/3,4/3]&\text{if $v\in A_{i,j}$}\\ [1/3,4/3]&\text{if $v\in A_{i,j}$}\\
[-4/3,-1/3]&\text{if $v\in A_{j,i}$}\\ [-4/3,-1/3]&\text{if $v\in A_{j,i}$}\\
[-1/2,1/2]&\text{otherwise.} [-1/2,1/2]&\text{otherwise.}
\end{cases}$$ \end{cases}\]
Note that the boxes in this extended representation are comparable, Note that the boxes in this extended representation are comparable,
as in the added dimensions, all the boxes have size $1$. as in the added dimensions, all the boxes have size $1$.
...@@ -347,31 +379,29 @@ Consider a graph $G$ with a distinguished clique $C^\star$, called the ...@@ -347,31 +379,29 @@ Consider a graph $G$ with a distinguished clique $C^\star$, called the
by (not necessarily comparable) boxes in $\mathbb{R}^d$ is called 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 $\varepsilon>0$. \emph{$C^\star$-clique-sum extendable} if the following conditions hold for every sufficiently small $\varepsilon>0$.
\begin{itemize} \begin{itemize}
\item[{\bf(vertices)}] For each $u\in V(C^\star)$, there exists a dimension $d_u$, \item[] {\bf(vertices)} For each $u\in V(C^\star)$, there exists a dimension $d_u$,
such that: such that:
\begin{itemize} \subitem[(v0)] $d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
\item[(v0)] $d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$, \subitem[(v1)] each vertex $u\in V(C^\star)$ satisfies $h(u)[d_u] = [-1,0]$ and
\item[(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 $h(u)[i] = [0,1]$ for any dimension $i\neq d_u$, and
\item[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v) \subset [0,1)^d$. \subitem[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v) \subset [0,1)^d$.
\end{itemize} \item[] {\bf(cliques)} For every clique $C$ of $G$, there exists
\item[{\bf(cliques)}] For every clique $C$ of $G$, there exists a point \[p(C)\in [0,1)^d\cap \bigcap_{v\in V(C)} h(v)\]
a point $$p(C)\in [0,1)^d\cap \bigcap_{v\in V(C)} h(v)$$
such that, defining the \emph{clique box} $h^\varepsilon(C)$ such that, defining the \emph{clique box} $h^\varepsilon(C)$
by setting by setting
$$h^\varepsilon(C)[i] = [p(C)[i],p(C)[i]+\varepsilon]$$ for every dimension \[h^\varepsilon(C)[i] = [p(C)[i],p(C)[i]+\varepsilon]\] for every dimension
$i$, the following conditions are satisfied: $i$, the following conditions are satisfied:
\begin{itemize} \begin{itemize}
\item[(c1)] For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1) \cap \subitem[(c1)] For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1) \cap
h^\varepsilon(C_2) = \emptyset$ (equivalently, $p(C_1) \neq p(C_2)$). h^\varepsilon(C_2) = \emptyset$ (equivalently, $p(C_1) \neq p(C_2)$).
\item[(c2)] A box $h(v)$ intersects $h^\varepsilon(C)$ if and only if \subitem[(c2)] A box $h(v)$ intersects $h^\varepsilon(C)$ if and only if
$v\in V(C)$, and in that case their intersection is a facet of $v\in V(C)$, and in that case their intersection is a facet of
$h^\varepsilon(C)$ incident to $p(C)$. That is, there exists a dimension $i_{C,v}$ $h^\varepsilon(C)$ incident to $p(C)$. That is, there exists a dimension $i_{C,v}$
such that for each dimension $j$, such that for each dimension $j$,
$$h(v)[j]\cap h^\varepsilon(C)[j] = \begin{cases} \[h(v)[j]\cap h^\varepsilon(C)[j] = \begin{cases}
\{p(C)[i_{C,v}] \}&\text{if $j=i_{C,v}$}\\ \{p(C)[i_{C,v}] \}&\text{if $j=i_{C,v}$}\\
[p(C)[j],p(C)[j]+\varepsilon]&\text{otherwise.} [p(C)[j],p(C)[j]+\varepsilon]&\text{otherwise.}
\end{cases}$$ \end{cases}\]
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\end{definition} \end{definition}
...@@ -420,10 +450,10 @@ clique-sums. ...@@ -420,10 +450,10 @@ clique-sums.
h_1(v)$ for any vertex $v\in V(G_1)$. Let $V(C_1)=\{v_1,\ldots,v_k\}$; h_1(v)$ for any vertex $v\in V(G_1)$. Let $V(C_1)=\{v_1,\ldots,v_k\}$;
without loss of generality, we can assume $i_{C_1,v_i}=i$ for $i\in\{1,\ldots,k\}$, without loss of generality, we can assume $i_{C_1,v_i}=i$ for $i\in\{1,\ldots,k\}$,
and thus and thus
$$h_1(v_i)[j] \cap h_1^\varepsilon(C_1)[j] = \begin{cases} \[h_1(v_i)[j] \cap h_1^\varepsilon(C_1)[j] = \begin{cases}
\{p_1(C_1)[i]\}&\text{ if $j=i$}\\ \{p_1(C_1)[i]\}&\text{ if $j=i$}\\
[p_1(C_1)[j],p_1(C_1)[j]+\varepsilon]&\text{ otherwise.} [p_1(C_1)[j],p_1(C_1)[j]+\varepsilon]&\text{ otherwise.}
\end{cases}$$ \end{cases}\]
Now let us consider $G_2$ and its representation $h_2$. Here the Now let us consider $G_2$ and its representation $h_2$. Here the
vertices of $C^\star_2$ are also denoted $v_1,\ldots,v_k$, and vertices of $C^\star_2$ are also denoted $v_1,\ldots,v_k$, and
...@@ -601,20 +631,20 @@ of $\cbdim(G)$ and $\chi(G)$. ...@@ -601,20 +631,20 @@ of $\cbdim(G)$ and $\chi(G)$.
Now we add $\chi(G)$ dimensions to make the representation touching Now we add $\chi(G)$ dimensions to make the representation touching
again, and to ensure some space for the clique boxes again, and to ensure some space for the clique boxes
$h^\varepsilon(C)$. Formally we define $h_2$ as follows. $h^\varepsilon(C)$. Formally we define $h_2$ as follows.
$$h_2(u)[i]=\begin{cases} \[h_2(u)[i]=\begin{cases}
h_1(u)[i]&\text{ if $i\le d$}\\ h_1(u)[i]&\text{ if $i\le d$}\\
[1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\ [1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\
[0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\ [0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\
[2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)} [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)}
\end{cases}$$ \end{cases}\]
For any clique $C$ of $G$, let $c(C)$ denote the color set $\{c(u)\ |\ u\in V(C)\}$. For any clique $C$ of $G$, let $c(C)$ denote the color set $\{c(u)\ |\ u\in V(C)\}$.
We now set We now set
$$p_2(C)[i]=\begin{cases} \[p_2(C)[i]=\begin{cases}
p_1(C) &\text{ if $i\le d$}\\ p_1(C) &\text{ if $i\le d$}\\
2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\ 2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\
1/2 &\text{ otherwise} 1/2 &\text{ otherwise}
\end{cases} \end{cases}
$$ \]
As $h_2$ is an extension of $h_1$, and as in each dimension $j>d$, As $h_2$ is an extension of $h_1$, and as in each dimension $j>d$,
$h_2(v)[j]$ is an interval of length $2/5$ containing the point $2/5$ for every vertex $v$, $h_2(v)[j]$ is an interval of length $2/5$ containing the point $2/5$ for every vertex $v$,
we have that $h_2$ is an intersection representation of $G$ by comparable boxes. we have that $h_2$ is an intersection representation of $G$ by comparable boxes.
...@@ -643,10 +673,11 @@ of $\cbdim(G)$ and $\chi(G)$. ...@@ -643,10 +673,11 @@ of $\cbdim(G)$ and $\chi(G)$.
Together, the lemmas from this section show that comparable box dimension is almost preserved by Together, the lemmas from this section show that comparable box dimension is almost preserved by
full clique-sums. full clique-sums.
\begin{corollary}\label{cor-csum} \begin{corollary}
\label{cor-csum}
Let $\GG$ be a class of graphs of chromatic number at most $k$. If $\GG'$ is the class Let $\GG$ be a class of graphs of chromatic number at most $k$. If $\GG'$ is the class
of graphs obtained from $\GG$ by repeatedly performing full clique-sums, then of graphs obtained from $\GG$ by repeatedly performing full clique-sums, then
$$\cbdim(\GG')\le \cbdim(\GG) + 2k.$$ \[\cbdim(\GG')\le \cbdim(\GG) + 2k.\]
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}
Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums. Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
...@@ -776,7 +807,7 @@ Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain: ...@@ -776,7 +807,7 @@ Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain:
\begin{corollary}\label{cor-genus} \begin{corollary}\label{cor-genus}
For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$ For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$
of $G$ such that $\cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil$. of $G$ such that $\cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil$.
Consequently, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$$ Consequently, \[\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.\]
\end{corollary} \end{corollary}
Similarly, we can deal with proper minor-closed classes. Similarly, we can deal with proper minor-closed classes.
...@@ -898,7 +929,7 @@ that $a'\le a$ and $\ell_{a',j} < \ell_{a'-1,j}$ if such an index exists, ...@@ -898,7 +929,7 @@ 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 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 $\omega(v_a)[j]\le s\omega(v_{a'})[j]$. $\omega(v_a)\sqsubseteq_s\iota(v_{a'})\subseteq \omega(v_{a'})$, we have $\omega(v_a)[j]\le s\omega(v_{a'})[j]$.
Combining these inequalities, Combining these inequalities,
$$\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_{a'})[j]|}=\frac{1}{kd}.$$ \[\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_{a'})[j]|}=\frac{1}{kd}.\]
By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$. By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$.
Let us now bound the treewidth of $G-X$. Let us now bound the treewidth of $G-X$.
...@@ -925,7 +956,7 @@ is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ld ...@@ -925,7 +956,7 @@ is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ld
\item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$. \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 $b\ge 2$). \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 $b\ge 2$).
\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 integers $b\ge 2$. 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 \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 integers $b\ge 2$. 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
$$\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$ \[\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.\]
\end{itemize} \end{itemize}
Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$. 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 $\omega(v_a)\sqsubseteq_s \iota(v_i)$, 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 $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
...@@ -939,7 +970,7 @@ Since the representation has thickness at most $t$, ...@@ -939,7 +970,7 @@ Since the representation has thickness at most $t$,
&\ge \frac{\vol(\omega(v_a))(|\beta(C)|-1)}{st}. &\ge \frac{\vol(\omega(v_a))(|\beta(C)|-1)}{st}.
\end{align*} \end{align*}
Since $\vol(C')=(2ksd+2)^d\vol(\omega(v_a))$, it follows that Since $\vol(C')=(2ksd+2)^d\vol(\omega(v_a))$, it follows that
$$|\beta(C)|-1\le (2ksd+2)^dst=f(k),$$ \[|\beta(C)|-1\le (2ksd+2)^dst=f(k),\]
as required. as required.
\end{proof} \end{proof}
...@@ -957,11 +988,5 @@ with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at ...@@ -957,11 +988,5 @@ with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at
has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).$ has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).$
\end{corollary} \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.
\bibliographystyle{siam}
\bibliography{data} \bibliography{data}
\end{document} \end{document}
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