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\documentclass[a4paper,USenglish,cleveref,autoref,thm-restate]{socg-lipics-v2021}
\documentclass[a4paper,english,cleveref,autoref,thm-restate]{socg-lipics-v2021}
\bibliographystyle{plainurl}
\title{On comparable box dimension}
\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{Abhiruk Lahiri}{Charles University, Prague, Czech Republic}{abhiruk@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{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}{torsten.ueckerdt@kit.edu}{}{}
......@@ -22,7 +22,7 @@
%\category{} %optional, e.g. invited paper
%\relatedversion{} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website
\relatedversion{A full version of the paper is available at \url{}} %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, ...
......@@ -32,22 +32,22 @@
\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
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\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
%\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}
\EventEditors{Xavier Goaoc and Michael Kerber}
\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}
\EventLongTitle{38th International Symposium on Computational Geometry (SoCG 2022)}
\EventShortTitle{SoCG 2022}
\EventAcronym{SoCG}
\EventYear{2022}
\EventDate{June 7--10, 2022}
\EventLocation{Berlin, Germany}
\EventLogo{socg-logo}
\SeriesVolume{224}
\ArticleNo{XX} % <-- This will be filled in by the typesetters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsthm}
......@@ -815,206 +815,4 @@ has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).
\end{corollary}
\bibliography{data}
\appendix
\section{Omitted proofs}
\newtheorem*{lemma-A}{Lemma~\ref{lem-cs}}
\begin{lemma-A}
Consider two graphs $G_1$ and $G_2$, given with a $C^\star_1$- and a
$C^\star_2$-clique-sum extendable representations $h_1$ and $h_2$ by comparable boxes
in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$,
respectively. Let $G$ be the graph obtained by performing a full
clique-sum of these two graphs on any clique $C_1$ of $G_1$, and on
the root clique $C^\star_2$ of $G_2$. Then $G$ admits a $C^\star_1$-clique
sum extendable representation $h$ by comparable boxes in
$\mathbb{R}^{\max(d_1,d_2)}$.
\end{lemma-A}
\begin{proof}
By Lemma~\ref{lemma-add}, we can assume that $d_1=d_2$; let $d=d_1$.
The idea is to translate (allowing also exchanges of dimensions) and
scale $h_2$ to fit in $h_1^\varepsilon(C_1)$. Consider an $\varepsilon >0$
sufficiently small so that $h_1^\varepsilon(C_1)$ satisfies all the
\textbf{(cliques)} conditions, and such that $h_1^\varepsilon(C_1) \sqsubseteq
h_1(v)$ 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
\[h_1(v_i)[j] \cap h_1^\varepsilon(C_1)[j] = \begin{cases}
\{p_1(C_1)[i]\}&\text{ if $j=i$}\\
[p_1(C_1)[j],p_1(C_1)[j]+\varepsilon]&\text{ otherwise.}
\end{cases}\]
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
without loss of generality, the \textbf{(vertices)} conditions are
satisfied by setting $d_{v_i}=i$ for $i\in\{1,\ldots,k\}$
We are now ready to define $h$. For $v\in V(G_1)$, we set $h(v)=h_1(v)$.
We now scale and translate $h_2$ to fit inside $h_1^\varepsilon(C_1)$.
That is, we fix $\varepsilon>0$ small enough so that
\begin{itemize}
\item the conditions \textbf{(cliques)} hold for $h_1$,
\item $h_1^\varepsilon(C_1)\subset [0,1)^d$, and
\item $h_1^\varepsilon(C_1)\sqsubseteq h_1(u)$ for every $u\in V(G_1)$,
\end{itemize}
and for each $v\in V(G_2) \setminus V(C^\star_2)$,
we set $h(v)[i]=p_1(C_1)[i] + \varepsilon h_2(v)[i]$ for $i\in\{1,\ldots,d\}$.
Note that the condition (v2) for $h_2$ implies $h(v)\subset h_1^\varepsilon(C_1)$.
Each clique $C$ of $H$ is a clique of $G_1$ or $G_2$.
If $C$ is a clique of $G_2$, we set $p(C)=p_1(C_1)+\varepsilon p_2(C)$,
otherwise we set $p(C)=p_1(C)$. In particular, for subcliques of $C_1=C^\star_2$,
we use the former choice.
Let us now check that $h$ is a $C^\star_1$-clique sum extendable
representation by comparable boxes. The fact that the boxes are
comparable follows from the fact that those of $h_1$ and $h_2$
are comparable and from the scaling of $h_2$: By construction both
$h_1(v) \sqsubseteq h_1(u)$ and $h_2(v) \sqsubseteq h_2(u)$ imply
$h(v) \sqsubseteq h(u)$, and for any vertex $u\in V(G_1)$ and any
vertex $v\in V(G_2) \setminus V(C^\star_2)$, we have $h(v) \subset h_1^\varepsilon(C_1) \sqsubseteq h(u)$.
We now check that $h$ is a contact representation of $G$. For $u,v
\in V(G_1)$ (resp. $u,v \in V(G_2) \setminus V(C^\star_2)$) it
is clear that $h(u)$ and $h(v)$ have disjoint interiors, and that they
intersect if and only if $h_1(u)$ and $h_1(v)$ intersect (resp. if
$h_2(u)$ and $h_2(v)$ intersect). Consider now a vertex $u \in
V(G_1)$ and a vertex $v \in V(G_2) \setminus V(C^\star_2)$. As
$h(v)\subset h^\varepsilon(C_1)$, the condition (v2) for $h_1$ implies
that $h(u)$ and $h(v)$ have disjoint interiors.
Furthermore, if $uv\in E(G)$, then $u\in V(C_1)=V(C^\star_2)$, say $u=v_1$.
Since $uv\in E(G_2)$, the intervals $h_2(u)[1]$ and $h_2(v)[1]$ intersect,
and by (v1) and (v2) for $h_2$, we conclude that $h_2(v)[1]=[0,\alpha]$ for some positive $\alpha<1$.
Therefore, $p_1(C_1)[1]\in h(v)[1]$. Since $p_1(C_1)\in \bigcap_{x\in V(C_1)} h_1(x)$,
we have $p_1(C_1)\in h(u)$, and thus $p_1(C_1)[1]\in h(u)[1]\cap h(v)[1]$.
For $i\in \{2,\ldots,d\}$, note that $i\neq 1=i_{C_1,u}$, and thus
by (c2) for $h_1$, we have $h_1^\varepsilon(C_1)[i]\subseteq h_1(u)[i]=h(u)[i]$.
Since $h(v)[i]\subseteq h_1^\varepsilon(C_1)[i]$, it follows that $h(u)$ intersects $h(v)$.
Finally, let us consider the $C^\star_1$-clique-sum extendability. The \textbf{(vertices)}
conditions hold, since (v0) and (v1) are inherited from $h_1$, and
(v2) is inherited from $h_1$ for $v\in V(G_1)\setminus V(C^\star_1)$
and follows from the fact that $h(v)\subseteq h_1^\varepsilon(C_1)\subset [0,1)^d$
for $v\in V(G_2)\setminus V(C^\star_2)$. For the \textbf{(cliques)} condition (c1),
the mapping $p$ inherits injectivity when restricted to cliques of $G_2$,
or to cliques of $G_1$ not contained in $C_1$. For any clique $C$ of $G_2$,
the point $p(C)$ is contained in $h_1^\varepsilon(C_1)$, since $p_2(C)\in [0,1)^d$.
On the other hand, if $C'$ is a clique of $G_1$ not contained in $C_1$, then there
exists $v\in V(C')\setminus V(C_1)$, we have $p(C')=p_1(C')\in h_1(v)$, and
$h_1(v)\cap h_1^\varepsilon(C_1)=\emptyset$ by (c2) for $h_1$.
Therefore, the mapping $p$ is injective, and thus for sufficiently small $\varepsilon'>0$,
we have $h^{\varepsilon'}(C)\cap h^{\varepsilon'}(C')=\emptyset$ for any distinct
cliques $C$ and $C'$ of $G$.
The condition (c2) of $h$ is (for sufficiently small $\varepsilon'>0$)
inherited from the property (c2) of $h_1$ and $h_2$
when $C$ is a clique of $G_2$ and $v\in V(G_2)\setminus V(C^\star_2)$, or
when $C$ is a clique of $G_1$ not contained in $C_1$ and $v\in V(G_1)$.
If $C$ is a clique of $G_1$ not contained in $C_1$ and $v\in V(G_2)\setminus V(C^\star_2)$,
then by (c1) for $h_1$ we have $h_1^\varepsilon(C)\cap h_1^\varepsilon(C_1)=\emptyset$,
and since $h^{\varepsilon'}(C)\subseteq h_1^\varepsilon(C)$ and $h(v)\subseteq h_1^\varepsilon(C_1)$,
we conclude that $h(v)\cap h^{\varepsilon'}(C)=\emptyset$.
It remains to consider the case that $C$ is a clique of $G_2$ and $v\in V(G_1)$.
Note that $h^{\varepsilon'}(C)\subseteq h_1^\varepsilon(C_1)$.
\begin{itemize}
\item If $v\not\in V(C_1)$, then by the property (c2) of $h_1$, the box $h(v)=h_1(v)$ is disjoint from $h_1^\varepsilon(C_1)$,
and thus $h(v)\cap h^{\varepsilon'}(C)=\emptyset$.
\item Otherwise $v\in V(C_1)=V(C^\star_2)$, say $v=v_1$.
Note that by (v1), we have $h_2(v)=[-1,0]\times [0,1]^{d-1}$.
\begin{itemize}
\item If $v\not\in V(C)$, then by the property (c2) of $h_2$, the box $h_2(v)$ is disjoint from $h_2^\varepsilon(C)$.
Since $h_2^\varepsilon(C)[i]\subseteq[0,1]=h_2(v)[i]$ for $i\in\{2,\ldots,d\}$,
it follows that $h_2^\varepsilon(C)[1]\subseteq (0,1)$, and thus $h^{\varepsilon'}(C)[1]\subseteq h_1^\varepsilon(C_1)[1]\setminus\{p(C_1)[1]\}$.
By (c2) for $h_1$, we have $h(v)[1]\cap h_1^\varepsilon(C_1)[1]=h_1(v)[1]\cap h_1^\varepsilon(C_1)[1]=p(C_1)[1]$,
and thus $h(v)\cap h^{\varepsilon'}(C)=\emptyset$.
\item If $v\in V(C)$, then by the property (c2) of $h_2$, the intersection of
$h_2(v)[1]=[-1,0]$ and $h_2^\varepsilon(C)[1]\subseteq [0,1)$ is the single point $p_2(C)[1]=0$,
and thus $p(C)[1]=p_1(C_1)[1]$ and $h^{\varepsilon'}(C)[1]=[p_1(C_1)[1],p_1(C_1)[1]+\varepsilon']$.
Recall that the property (c2) of $h_1$ implies $h(v)[1]\cap h_1^\varepsilon(C_1)[1]=\{p(C_1)[1]\}$,
and thus $h(v)[1]\cap h^{\varepsilon'}(C)[1]=\{p(C)[1]\}$. For $i\in\{2,\ldots, d\}$,
the property (c2) of $h_1$ implies $h_1^\varepsilon(C_1)[i]\subseteq h_1(v)[i]=h(v)[i]$, and
since $h^{\varepsilon'}(C)[i]\subseteq h_1^\varepsilon(C_1)[i]$, it follows that
$h^{\varepsilon'}(C)[i]\subseteq h(v)[i]$.
\end{itemize}
\end{itemize}
\end{proof}
\newtheorem*{lemma-B}{Lemma~\ref{lem-apex-cs}}
\begin{lemma-B}
For any graph $G$ and any clique $C^\star$, the graph $G$ admits a
$C^\star$-clique-sum extendable touching representation by
comparable boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| +
\ecbdim(G\setminus V(C^\star))$.
\end{lemma-B}
\begin{proof}
The proof is essentially the same as the one of
Lemma~\ref{lemma-apex}. Consider a $\emptyset$-clique-sum
extendable touching representation $h'$ of $G\setminus V(C^\star)$ by
comparable boxes in $\mathbb{R}^{d'}$, with $d' = \cbdim(G\setminus
V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We now construct
the desired representation $h$ of $G$ as follows. For each vertex
$v_i\in V(C^\star)$, let $h(v_i)$ be the box in $\mathbb{R}^d$ uniquely determined
by the condition (v1) with $d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$,
if $i\le k$ then let $h(u)[i] = [0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i] =
[1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i] =
\alpha h'(u)[i-k]$, for some $\alpha>0$. The value $\alpha>0$
is chosen sufficiently small so that $h(u)[i] \subset [0,1)$ whenever $u\notin V(C^\star)$.
We proceed similarly for the clique points. For any
clique $C$ of $G$, if $i\le k$ then let $p(C)[i] = 0$ if $v_i \in V(C)$,
and $p(C)[i] = 1/4$ if $v_i \notin V(C)$. For $i>k$ we refer to the clique point $p'(C')$ of $C'=C\setminus
\{v_1,\ldots,v_k\}$, and we set $p(C)[i] = \alpha p'(C')[i-k]$.
By the construction, it is clear that $h$ is a touching representation of $G$.
As $h'(u) \sqsubset h'(v)$ implies that $h(u) \sqsubset h(v)$, and as
$h(u) \sqsubset h(v_i)$ for every $u\in V(G)\setminus V(C^\star)$ and every
$v_i \in V(C^\star)$, we have that $h$ is a representation by comparable boxes.
For the $C^\star$-clique-sum extendability, the \textbf{(vertices)} conditions hold by the construction.
For the \textbf{(cliques)} condition (c1), let us consider distinct cliques $C_1$ and $C_2$
of $G$ such that $|V(C_1)| \ge |V(C_2)|$, and let $C'_i=C_i\setminus V(C^\star)$. If $C'_1 = C'_2$,
there is a vertex $v_i \in V(C_1) \setminus V(C_2)$, and $p(C_1)[i] = 0 \neq 1/4 = p(C_2)[i]$.
Otherwise, if $C'_1 \neq C'_2$, then $p'(C'_1) \neq p'(C'_2)$, which implies
$p(C_1) \neq p(C_2)$ by construction.
For the \textbf{(cliques)} condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and
a clique $C$ of $G$ containing $v$. In the dimensions $i\in\{1,\ldots,k\}$, we always have
$h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$, then
$h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case $v$ and $v_i$ are adjacent;
and if $v_i \notin V(C)$, then $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$.
By the property (c2) of $h'$,
we have $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$.
Next, let us consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
As $v\notin V(C')$, the condition (c2) for $h'$ implies that $p'(C')$ is disjoint from $h'(v)$,
and thus $p(C)$ is disjoint from $h(v)$.
Finally, we consider a vertex $v_i \in V(C^\star)$. Note that for any clique $C$ containing $v_i$,
we have that $h^\varepsilon(C)[i] \cap h(v_i)[i] = [0,\varepsilon]\cap [-1,0] = \{0\}$, and $h^\varepsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$
for any $j\neq i$. For a clique $C$ that does not contain $v_i$ we have that
$h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$.
Condition (c2) is thus fulfilled and this completes the proof of the lemma.
\end{proof}
\newtheorem*{corollary-C}{Corollary~\ref{cor-tw}}
\begin{corollary-C}
Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
\end{corollary-C}
\begin{proof}
Let $k=\tw(G)$. Observe that there exists a $k$-tree $T$ with the root clique $C^\star$ such that $G\subseteq T-V(C^\star)$.
Inspection of the proof of Theorem~\ref{thm-ktree} (and Lemma~\ref{lem-cs}) shows that we obtain
a representation $h$ of $T-V(C^\star)$ in $\mathbb{R}^{k+1}$ such that
\begin{itemize}
\item the vertices are represented by hypercubes of pairwise different sizes,
\item if $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$, then $h(u)\cap h(v)$ is a facet of $h(u)$ incident
with its point with minimum coordinates, and
\item for each vertex $u$ and each facet of $h(u)$ incident with its point with minimum coordinates, there exists
at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.
\end{itemize}
If for some $u,v\in V(G)$, we have $uv\in E(T)\setminus E(G)$, where without loss of generality $h(u)\sqsubseteq h(v)$,
we now alter the representation by shrinking $h(u)$ slightly away from $h(v)$ (so that all other touchings are preserved).
Since the hypercubes of $h$ have pairwise different sizes, the resulting touching representation of $G$ is by comparable boxes.
\end{proof}
\end{document}
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