diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index 3d6c9151de550a4188577afc4d8ab06c39b63bf0..f03f158a2e49cf7fa65314e5c662baebce70cf3d 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -1,4 +1,55 @@
-\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
+
+%\relatedversion{} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website
+%\relatedversiondetails[linktext={opt. text shown instead of the URL}, cite=DBLP:books/mk/GrayR93]{Classification (e.g. Full Version, Extended Version, Previous Version}{URL to related version} %linktext and cite are optional
+
+%\supplement{}%optional, e.g. related research data, source code, ... hosted on a repository like zenodo, figshare, GitHub, ...
+%\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
+
+%\nolinenumbers %uncomment to disable line numbering
+
+%\hideLIPIcs %uncomment to remove references to LIPIcs series (logo, DOI, ...), e.g. when preparing a pre-final version to be uploaded to arXiv or another public repository
+
+%Editor-only macros:: begin (do not touch as author)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amsmath}
@@ -25,25 +76,6 @@
\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}
\maketitle
@@ -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.
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 $\vol(f(v_1)) \geq \ldots \geq \vol(f(v_n))$.
-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
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_{
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 $v_jv_m,v_mv_i\in E(G)$
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.
\end{proof}
@@ -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
representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
follows.
- $$f((u,v))[i]=\begin{cases}
+ \[f((u,v))[i]=\begin{cases}
g(u)[i]&\text{ if $i\le d_G$}\\
h(v)[i-d_G]&\text{ if $i > d_G$}
- \end{cases}$$
+ \end{cases}\]
Consider distinct vertices $(u,v)$ and $(u',v')$ of $G\boxtimes H$.
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
@@ -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,
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 $v\in A_{i,j}$}\\
[-4/3,-1/3]&\text{if $v\in A_{j,i}$}\\
[-1/2,1/2]&\text{otherwise.}
-\end{cases}$$
+\end{cases}\]
Note that the boxes in this extended representation are comparable,
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
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 $\varepsilon>0$.
\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:
- \begin{itemize}
- \item[(v0)] $d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
- \item[(v1)] each vertex $u\in V(C^\star)$ satisfies $h(u)[d_u] = [-1,0]$ and
+ \subitem[(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
$h(u)[i] = [0,1]$ for any dimension $i\neq d_u$, and
- \item[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v) \subset [0,1)^d$.
- \end{itemize}
-\item[{\bf(cliques)}] For every clique $C$ of $G$, there exists
- a point $$p(C)\in [0,1)^d\cap \bigcap_{v\in V(C)} h(v)$$
+ \subitem[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v) \subset [0,1)^d$.
+\item[] {\bf(cliques)} For every clique $C$ of $G$, there exists
+ a point \[p(C)\in [0,1)^d\cap \bigcap_{v\in V(C)} h(v)\]
such that, defining the \emph{clique box} $h^\varepsilon(C)$
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:
\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, $p(C_1) \neq p(C_2)$).
- \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
$h^\varepsilon(C)$ incident to $p(C)$. That is, there exists a dimension $i_{C,v}$
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)[j],p(C)[j]+\varepsilon]&\text{otherwise.}
- \end{cases}$$
+ \end{cases}\]
\end{itemize}
\end{itemize}
\end{definition}
@@ -420,10 +450,10 @@ clique-sums.
h_1(v)$ for any vertex $v\in V(G_1)$. Let $V(C_1)=\{v_1,\ldots,v_k\}$;
without loss of generality, we can assume $i_{C_1,v_i}=i$ for $i\in\{1,\ldots,k\}$,
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)[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
vertices of $C^\star_2$ are also denoted $v_1,\ldots,v_k$, and
@@ -601,20 +631,20 @@ of $\cbdim(G)$ and $\chi(G)$.
Now we add $\chi(G)$ dimensions to make the representation touching
again, and to ensure some space for the clique boxes
$h^\varepsilon(C)$. 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 $i\le d$}\\
[1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\
[0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\
[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 $\{c(u)\ |\ u\in V(C)\}$.
We now set
- $$p_2(C)[i]=\begin{cases}
+ \[p_2(C)[i]=\begin{cases}
p_1(C) &\text{ if $i\le d$}\\
2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\
1/2 &\text{ otherwise}
\end{cases}
- $$
+ \]
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 $v$,
we have that $h_2$ is an intersection representation of $G$ by comparable boxes.
@@ -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
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
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}
\begin{proof}
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:
\begin{corollary}\label{cor-genus}
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, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$$
+Consequently, \[\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.\]
\end{corollary}
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,
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 $\omega(v_a)[j]\le s\omega(v_{a'})[j]$.
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 $\text{Pr}[v_a\in X]\le 1/k$.
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
\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 $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ 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}
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 $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
@@ -939,7 +970,7 @@ Since the representation has thickness at most $t$,
&\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
-$$|\beta(C)|-1\le (2ksd+2)^dst=f(k),$$
+\[|\beta(C)|-1\le (2ksd+2)^dst=f(k),\]
as required.
\end{proof}
@@ -957,11 +988,5 @@ with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at
has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).$
\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}
-
\end{document}