diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index a4e7bf6c160b11f4f3c5971d1d1e7812940be9eb..a2b9669d4f40be6763e1be70cde0c4314fe86cb1 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -20,6 +20,7 @@
\newcommand{\bb}[1]{\mathbb{#1}}
\newcommand{\brm}[1]{\operatorname{#1}}
\newcommand{\cbdim}{\brm{dim}_{cb}}
+\newcommand{\ecbdim}{\brm{dim}^{ext}_{cb}}
\newcommand{\tw}{\brm{tw}}
\newcommand{\vol}{\brm{vol}}
%%%%%
@@ -33,11 +34,12 @@
\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{...}\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}}}
@@ -47,8 +49,12 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten
\maketitle
\begin{abstract}
-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.
+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}
\section{Introduction}
@@ -98,237 +104,40 @@ This gives arbitrarily precise approximation algorithms for all monotone maximiz
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 the addition of a vertex increases the comparable box dimension by at most one.
-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 $u\in V(G)\setminus\{v\}$.
-Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$.
-\end{proof}
+\section{Parameters}
-We need a bound on the clique number in terms of the comparable box dimension.
-For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_i$.
+Let us first bound the clique number $\omega(G)$ in terms of
+$\cbdim(G)$.
\begin{lemma}\label{lemma-cliq}
-If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$.
-\end{lemma}
-\begin{proof}
-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 $w \leq 2^d$.
-\end{proof}
-
-A \emph{tree decomposition} of a graph $G$
-is a pair $(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$ assigns a \emph{bag} to each of its nodes,
-such that
-\begin{itemize}
-\item for each $uv\in E(G)$, there exists $x\in V(T)$ such that $u,v\in\beta(x)$, and
-\item for each $v\in V(G)$, the set $\{x\in V(T):v\in\beta(x)\}$ is non-empty and induces a connected subtree of $T$.
-\end{itemize}
-For nodes $x,y\in V(T)$, we write $x\preceq y$ if $x=y$ or $x$ is a descendant of $y$ in $T$.
-For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ and $x$ is nearest to the root of $T$.
-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. \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 $\mathbb{R}^{t+1}$ 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}
-\begin{proof}
-Without loss of generality, we can assume that the root has a bag of size one
-and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta(y)|=1$
-(if $\beta(x)\subseteq \beta(y)$, we can contract the edge $xy$; if $|\beta(x)\setminus\beta(y)|>1$,
-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, 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
-rest of the graph (being careful to only use hypercubes smaller than the one representing $y$
-and of different sizes from those used on the rest of the graph).
-
-Let us now greedily color $G$ by giving $x$ a color different from the colors of
-all other vertices in $\beta(x)$; such a coloring $\varphi$ uses only colors
-$\{1,\ldots,t+1\}$.
-
-Let $D=4\Delta(T)+1$. Let $V(G)=V(T)=\{x_1,x_2,\ldots, x_n\}$, where
-for every $i<j$, $x_i$ and $x_j$ are either incomparable in $\prec$ or
-$x_j\prec x_i)$; in particular, $x_1$ is the root of $T$. Let $\varepsilon=D^{-n-1}$.
-Let $s_i=D^{-i}$; we will represent $x_i$ by a hypercube $h(x_i)$ with edges of length $s_i$.
-Additionally, we will need to consider larger hypercubes around $h(x_i)$; let $h'(x_i)$
-be the hypercube with sides of length $2s_i$ and with $\min(h'(x_i)[j])=\min(h(x_i)[j])$
-for $j\in\{1,\ldots, t+1\}$, and $h''(x_i)$ the hypercube with sides of length $2s_i+\epsilon$
-and with $\min(h''(x_i)[j])=\min(h(x_i)[j])-\varepsilon$. We will construct the representation $h$ so that the following
-invariant is satisfied:
-\begin{itemize}
-\item[(a)] For each $x,z\in V(T)$ such that $x\prec z$, we have $h'(x)\subset h''(z)$.
-\item[(b)] For each $y\in V(T)$ and distinct children $x$ and $z$ of $y$, we have $h''(x)\cap h''(z)=\emptyset$.
-\end{itemize}
-Note that this ensures that if $x$ and $z$ are vertices of $T$ and $h(x)\cap h(z)\neq\emptyset$, then $x\prec z$ or $z\prec x$.
-
-We now construct the representation $h$. For the root $x_1$ of $T$, $h(r)$ is an arbitrary hypercube with sides
-of length $s_1$. Assuming now we have already selected $h(y)$ for a vertex $y\in V(T)$, the hypercube $h(x_i)$ with sides of length $s_i$
-for a child $x_i$ of $y$ is chosen as follows. For $j\in\{1,\ldots, t+1\}$,
-\begin{itemize}
-\item[(i)] if $j=\varphi(w)$ for $w\in\beta(x_i)\setminus\{x_i\}$, we choose $h(x_i)[j]$ so that
-$\min(h(x_i)[j])=\max(h(w)[j])$ if $xw\in E(G)$ and so that $\min(h(x_i)[j])=\max(h(w)[j]) + \varepsilon$ otherwise.
-\item[(ii)] if $j$ is different from the colors of all vertices in $\beta(x_i)\setminus\{x_i\}$,
-then we choose $h(x_i)[j]$ so that $h''(x_i)[j]$ is a subset of the interior of $h(y)[j]$. The interval $h''(x_i)[[j]$
-is furthermore chosen to be disjoint from $h''(x_m)[j]$ for any other child $x_m$ of $y$;
-this is always possible by the choice of $D$, $s_i$, and $s_m$.
-\end{itemize}
-Note that (ii) always applies for $j=\varphi(x_i)$ and this ensures that the invariant (b) holds.
-For the invariant (a), note that in the case (ii), we ensure $h''(x_i)[[j]\subseteq h(y)[j]$ and
-we have $h(y)[j]\subseteq h''(z)[j]$ by the invariant (a) for $y$ and $z$. In the case (i),
-if $z\prec w$, then we have $w\in\beta(z)\setminus\{z\}$ and
-$\min(h(x_i)[j]),\min(h(z)[j])\in\{\max(h(w)[j]),\max(h(w)[j])+\varepsilon\}$.
-If $w\preceq z$, then note we choose $h'(x_i)[j]\subseteq h'(w)[j]$ by (i) and that
-we have $h'(w)[j]\subset h''(z)[j]$ by (a). This verifies that the invariant (a)
-also holds at $x_i$.
-
-Consider now two adjacent vertices of $G$, say $x_i$ and $w$. Note that any two
-adjacent vertices are comparable in $\prec$, and thus we can assume $x_i\prec w$
-and $w\in\beta(x_i)$.
-By (i), for $j=\varphi(w)$, the intervals $h(x_i)[j]$ and $h(w)[j]$
-intersect in a single point. If $j\neq \varphi(w)$, then let $w_1$ be the child of $w$ on the path in $T$ from
-$w$ to $x_i$. If no vertex in $\beta(w_1)\setminus\{w_1\}$ has color $j$, then by (ii), we have $h''(w_1)[j]\subset h(w)[j]$,
-Otherwise, $z\in \beta(w_1)\setminus\{w_1\}$ such that $\varphi(z)=j$; clearly, $w\prec z$ and $z\in\beta(w)\setminus\{w\}$.
-We have $\min(h(w_1)[j]),\min(h(w)[j])\in\{\max(h(z)[j]),\max(h(z)[j])+\varepsilon\}$ by (i).
-Hence, $\min(h(w)[j])-2\epsilon\le \min(h''(w_1)[j])\le \max(h''(w_1)[j])\le \max(h(w)[j])$.
-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.
-
-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 $y$
-on this path. As we argued in the first paragraph, $y\neq w$, and by (i), the interior of $h(y)[j]$ is disjoint from $h(w)[j]$.
-By (ii), $h''(z)[j]$ is contained in the interior of $h(y)[j]$. By (a), we conclude that $h(x_i)[j]\subseteq h''(z)[j]$,
-implying that the boxes $h(x_i)$ and $h(w)$ are disjoint. Therefore, $h$ is a touching representation of $G$.
-\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 arranged 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 an arbitrary number of clique-sums.
-
-It will be convenient to work in the setting of tree decompositions.
-Consider a tree decomposition $(T,\beta)$ of a graph $G$.
-For each $x\in V(T)$, the \emph{torso} $G_x$ of $x$ is the graph obtained from $G[\beta(x)]$ by adding a clique on $\beta(x)\cap\beta(y)$
-for each $y\in V(T)$. For a class of graphs $\GG$, we say that the tree decomposition is \emph{over $\GG$} if all the torsos belong to $\GG$.
-We use the following well-known fact.
-\begin{observation}
-A graph $G$ is obtained from graphs in a class $\GG$ by repeated clique-sums if and only if $G$ has a tree decomposition over $\GG$.
-\end{observation}
-For each note $x\in V(T)$, let $\pi(x)=\{x\}\cup \{p(v):v\in\beta(x)\}$. Let $T_\beta$ be the graph with vertex set $V(T)$ such that
-$xy\in E(T_\beta)$ if and only if $x\in\pi(y)$ or $y\in\pi(x)$.
-\begin{lemma}\label{lemma-legraf}
-If $(T,\beta)$ is a tree decomposition of $G$ of adhesion $a$, then $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$.
-Moreover, $\pi(x)$ is a clique in $T_\beta$ and $p(x)=x$ for each $x\in V(T)$.
+For any graph $G$, then $\omega(G)\le 2^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
-For each edge $xy\in E(T_\beta)$, we have $x,y\in \pi(x)$ or $x,y\in \pi(y)$ by definition.
-Moreover, for each $x\in V(T_\beta)$, we have
-$$\{y:x\in\pi(y)\}=\{x\}\cup \bigcup_{v\in V(G):p(v)=x} \{y:v\in\beta(y)\},$$
-and all the sets on the right-hand size induce connected subtrees containing $x$,
-implying that $\{y:x\in\pi(y)\}$ also induces a connected subtree containing $x$.
-Hence, $(T,\pi)$ is a tree decomposition of $T_\beta$.
-
-Consider a node $x\in V(T)$. Note that for each $v\in \beta(x)$, the vertex $p(v)$ is an ancestor of $x$ in $T$.
-In particular, if $x$ is the root of $T$, then $\pi(x)=\{x\}$. Otherwise, if $y$ is the parent of $x$ in $T$, then
-$p(v)=x$ for every $v\in \beta(x)\setminus\beta(y)$, and thus $|\pi(x)|\le |\beta(x)\cap \beta(y)|+1\le a+1$.
-Hence, the width of $(T,\pi)$ is at most $a$.
-
-Suppose now that $y$ and $z$ are distinct vertices in $\pi(x)$. Then both $y$ and $z$ are ancestors of $x$ in $T$,
-and thus without loss of generality, we can assume that $y\preceq z$. If $y=x$, then $yz\in E(T_\beta)$ by definition.
-Otherwise, there exist vertices $u,v\in \beta(x)$ such that $p(u)=y$ and $p(v)=z$. Since $v\in \beta(x)\cap\beta(z)$
-and $y$ is on the path in $T$ from $x$ to $z$, we also have $v\in\beta(y)$. This implies $z\in\pi(y)$ and $yz\in E(T_\beta)$.
-Hence, $\pi(x)$ is a clique in $T_\beta$. Moreover, note that $x\not\in \pi(w)$ for any ancestor $w\neq x$ of $x$ in $T$,
-and thus $p(x)=x$.
+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 $w \leq 2^d$.
\end{proof}
-We are now ready to deal with the clique-sums.
-
-\begin{theorem}\label{thm-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 $\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$.
-\end{theorem}
-\begin{proof}
-Let $(T,\beta)$ be a tree decomposition of $G$ over $\GG$; the adhesion $a$ of $(T,\beta)$ is at most $\omega(\GG)$.
-By Lemma~\ref{lemma-legraf}, $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$. By Lemma~\ref{lemma-tw},
-$T_\beta$ has a touching representation $h$ by axis-aligned hypercubes in $\mathbb{R}^{a+1}$ such that
-$h(x)\sqsubseteq h(y)$ whenever $x\preceq y$.
-Since $T_\beta$ has treewidth at most $a$, it has a proper coloring $\varphi$ by colors $\{0,\ldots,a\}$.
-For every $x\in V(T)$, let $f_x$ be a touching representation of the torso $G_x$ of $x$ by comparable boxes in $\mathbb{R}^d$,
-where $d=\cbdim(\GG)$. We scale and translate the representations so that for every $x\in V(T)$ and $i\in\{0,\ldots,a\}$,
-there exists a box $E_i(x)$ such that
-\begin{itemize}
-\item[(a)] for distinct $x,y\in V(T)$, if $h(x)\sqsubset h(y)$, then $E_i(x) \sqsubset E_i(y)$ and $E_i(x) \sqsubset f_y(v)$ for every $v\in \beta(y)$,
-\item[(b)] if $x\prec y$, then $E_i(x)\subseteq E_i(y)$,
-\item[(c)] if $i=\varphi(x)$, then $f_x(v)\subseteq E_i(x)$ for every $v\in\beta(x)$, and
-\item[(d)] if $i\neq \varphi(x)$ and $K=\{v\in\beta(x):\varphi(p(v))=i\}$ is non-empty, then letting $y=p(v)$ for $v\in K$,
-the box $E_i(x)$ contains a point belonging to $\bigcap_{v\in K} f_y(v)$.
-\end{itemize}
-Some explanation is in order for the last point: Firstly, since $\pi(x)$ is a clique in $T_\beta$, there
-exists only one vertex $y\in \pi(x)$ of color $i$, and thus $y=p(v)$ for all $v\in K$.
-Moreover, $K$ is a clique in $G_y$, and thus $\bigcap_{v\in K} f_y(v)$ is non-empty. Lastly,
-note that if $x\prec z\prec y$, then $K=\beta(x)\cap\beta(y)\subseteq \beta(z)\cap \beta(y)$,
-and thus $E_i(z)$ was also chosen to contain a point of $\bigcap_{v\in K} f_y(v)$;
-hence, a choice of $E_i(x)$ satisfying $E_i(x)\subseteq E_i(z)$ as required by (b) is possible.
-
-Let us now define $f(v)=h(p(v))\times E_0(v)\times \cdots\times E_a(v)$ for each $v\in V(G)$,
-where $E_i(v)=f_{p(v)}(v)$ if $i=\varphi(p(v))$ and $E_i(v)=E_i(p(v))$ otherwise. We claim this gives a touching representation
-of a supergraph of $G$ by comparable boxes in $\mathbb{R}^{(d+1)(a+1)}$. First, note that the boxes are indeed comparable;
-if $p(u)=p(v)$, then this is the case since $f_{p(v)}$ is a representation by comparable boxes, and if say
-$p(u)\prec p(v)$, then this is due to (a) and (c). Next, let us argue $f(u)$ and $f(v)$ have disjoint
-interiors. If $p(u)=p(v)$, this is the case since $f_{p(v)}$ is a touching representation, and if $p(u)\neq p(v)$,
-then this is the case because $h$ is a touching representation. Finally, suppose that $uv\in E(G)$. Let $x$
-be the node of $T$ nearest to the root such that $u,v\in \beta(x)$. Without loss of generality, $p(u)=x$.
-Let $y=p(v)$. If $x=y$, then $f(u)\cap f(v)\neq\emptyset$, since $f_x$ is a touching representation of $G_x$ and $uv\in E(G_x)$.
-If $x\neq y$, then $y\in\pi(x)$ and $xy\in E(T_\beta)$, implying that $h(x)\cap h(y)\neq\emptyset$.
-Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\ldots,a$ by (b) if $\varphi(x)\neq i\neq \varphi(y)$,
-(b) and (c) if $\varphi(x)=i\neq\varphi(y)$, and (d) if $\varphi(x)\neq i=\varphi(y)$ (we cannot have $\varphi(x)=i=\varphi(y)$, since
-$xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$.
-\end{proof}
-
-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 \[\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}.\]
-\end{corollary}
-
-Note that only bound the comparable box dimension of a supergraph
-of $G$. To deal with this issue, we show that the comparable box dimension of a subgraph
-is at most exponential in the comparable box dimension of the whole graph.
-This is essentially Corollary~25 in~\cite{subconvex}, but since the setting is somewhat
-different and the construction of~\cite{subconvex} uses rotated boxes,
-we provide details of the argument.
-
-A \emph{star coloring} of a graph $G$ is a proper coloring such that any two color classes induce
-a star forest (i.e., a graph not containing any 4-vertex path). The \emph{star chromatic number} $\chi_s(G)$
-of $G$ is the minimum number of colors in a star coloring of $G$.
-We will need the fact that the star chromatic number is at most exponential in the comparable box dimension;
-this follows from~\cite{subconvex} and we include the argument to make the dependence clear.
+In the following we consider the chromatic number $\chi(G)$, and one
+of its variant. A \emph{star coloring} of a graph $G$ is a proper
+coloring such that any two color classes induce a star forest (i.e., a
+graph not containing any 4-vertex path). The \emph{star chromatic
+ number} $\chi_s(G)$ of $G$ is the minimum number of colors in a star
+coloring of $G$. We will need the fact that the star chromatic number
+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 $G$ has star chromatic number at most $2\cdot 9^d$.
+For any graph $G$, then $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot
+9^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
+ Let us focus on the star chromatic number.
Let $v_1$, \ldots, $v_n$ be the vertices of $G$ ordered non-increasingly by the size of the boxes that represent them;
i.e., so that $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. We greedily color the vertices in order, giving $v_i$ the smallest
color different from the colors of all vertices $v_j$ such that $j<i$ and either $v_jv_i\in E(G)$, or there exists $m>j$
@@ -346,9 +155,100 @@ is at most $(3^d-1)^2$.
Consequently, when choosing the color of $v_i$ greedily, we only need to avoid colors of at most
$$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$
-vertices.
+vertices. We proceed similarly to bound the chromatic number.
\end{proof}
+
+\section{Operations}
+
+It is clear that given a touching representation of a graph $G$, one
+easily obtains a touching representation with boxes of an induced
+subgraph $H$ of $G$ by simply deleting the boxes corresponding to the
+vertices in $V(G)\setminus V(H)$. In this section we are going to
+consider other basic operations on graphs.
+
+\subsection{Vertex addition}
+
+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 $\cbdim(G)\le |V(G)|$.
+\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 $u\in V(G)\setminus\{v\}$.
+Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$.
+\end{proof}
+
+
+\subsection{Strong product}
+
+Let $G\boxtimes H$ denote the \emph{strong product} of the graphs $G$
+and $H$, i.e., the graph with vertex set $V(G)\times V(H)$ and with
+distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if
+either $u_1=u_2$ or $u_1u_2\in E(G)$ and either $v_1=v_2$ or
+$v_1v_2\in E(G)$. To obtain a touching representation of $G\boxtimes
+H$ it suffice to take a product of representations of $G$ and $H$, but
+the obtained representation may contain uncomparable boxes. Thus,
+bounding $\cbdim(G\boxtimes H)$ in terms of $\cbdim(G)$ and
+$\cbdim(H)$ seems to be a complicated task. In the following lemma we
+overcome this issue, by constraining one of the representations.
+
+\begin{lemma}\label{lemma-sp}
+ Consider a graph $H$ having a touching representation $h$ in
+ $\mathbb{R}^{d_H}$ with hypercubes of unit size. Then for any graph
+ $G$, the strong product of these graphs is such that
+ $\cbdim(G\boxtimes H) \le \cbdim(G) + d_H$.
+\end{lemma}
+\begin{proof}
+ The proof simply consists in taking a product of the two
+ representations. Indeed, consider a touching respresentation with
+ comparable boxes $g$ of $G$ in $\mathbb{R}^{d_G}$, with
+ $d_G=\cbdim(G)$, and the depresentation $h$ of $H$. Let us define a
+ representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
+ follows.
+ $$f((u,v))[i]=\begin{cases}
+ g(u)[i]&\text{ if $i\le d_G$}\\
+ h(u)[i-d_G]&\text{ if $i > d_G$}
+ \end{cases}$$
+ Notice first that the boxes of $f$ are comparable as $f((u,v))
+ \sqsubset f((u',v'))$ if and only if $g(u) \sqsubset g(u')$.
+
+ Now let us observe that for any two vertices $u, u'$ of $G$, there
+ is an hyperplane separating the interiors of $g(u)$ and $g(u')$, and
+ similarly for $h$ and $H$. This implies that the boxes in $f$ are
+ interior disjoint. Indeed, the same hyperplane that separates $g(u)$
+ and $g(u')$, when extended to $\mathbb{R}^{d_G+d_H}$, now separates
+ any two boxes $f((u,v))$ and $f((u',v'))$. This implies that $f$ is
+ a touching representation of a subgraph of $G\boxtimes H$.
+
+ Similarly, one can also observe that there is a point $p$ in the
+ intersection of $f((u,v))$ and $f((u',v'))$, if and only if there is
+ a point $p_G$ in the intersection of $g(u)$ and $g(u')$, and a point
+ $p_H$ in the intersection of $h(v)$ and $h(v')$. Indeed, one can
+ obtain $p_G$ and $p_H$ by projecting $p$ in $\mathbb{R}^{d_G}$ or in
+ $\mathbb{R}^{d_G}$ respectively, and conversely $p$ can be obtained
+ by taking the product of $p_G$ and $p_H$. Thus $f$ is indeed a
+ touching representation of $G\boxtimes H$.
+\end{proof}
+
+
+\subsection{Subgraph}
+
+Examples show that the comparable box dimension of a graph $G$ may be
+larger than the one a subgraph $H$ of $G$. However we show that the
+comparable box dimension of a subgraph is at most exponential in the
+comparable box dimension of the whole graph. This is essentially
+Corollary~25 in~\cite{subconvex}, but since the setting is somewhat
+different and the construction of~\cite{subconvex} uses rotated boxes,
+we provide details of the argument.
+
+
+
Next, let us show a bound on the comparable box dimension of subgraphs.
\begin{lemma}\label{lemma-subg}
@@ -387,65 +287,344 @@ If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+4\cdot 81^{\
Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim{K_{2^d}}=d$,
but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$.
-Corollaries~\ref{cor-cs} and~\ref{cor-subg} now give the main result of this section.
-\begin{corollary}\label{cor-comb}
-If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then
-$\cbdim(G)\le 5\cdot 81^{6^{\cbdim(\GG)}}$.
-\end{corollary}
-\section{The product structure and minor-closed classes}
+\subsection{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. A \emph{full clique-sum} is a
+clique-sum in which we keep all the edges of the resulting clique.
+The main issue to overcome in obtaining a representation for a (full)
+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 arranged 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
+an arbitrary number of clique-sums. We thus introduce the notion of
+\emph{clique-sum extendable} representations.
+
+\begin{definition}
+Consider a graph $G$ with a distinguished clique $C^*$, called the
+\emph{root clique} of $G$. A touching representation (with comparable
+boxes or not) $h$ of $G$ in $\mathbb{R}^d$ is called
+\emph{$C^*$-clique-sum extendable} if the following conditions hold.
+\begin{itemize}
+\item[{\bf(vertices)}] There are $|V(C^*)|$ dimensions, denoted $d_u$ for each
+ vertex $u\in V(C^*)$, such that:
+ \begin{itemize}
+ \item[(v1)] for each vertex $u\in V(C^*)$, $h(u)[d_u] = [-1 ,0]$ and
+ $h(u)[i] = [0,1]$, for any dimension $i\neq d_u$, and
+ \item[(v2)] for any vertex $v\notin V(C^*)$, $h(v) \subset [0,1)^d$.
+ \end{itemize}
+\item[{\bf(cliques)}] For every clique $C$ of $G$ we define a point
+ $p(C)\in I_C \cap [0,1)^d$, where $I_C =\cap_{v\in V(C)} h(v)$, and
+ we define the box $h^\epsilon(C)$, for any $\epsilon > 0$, by
+ $h^\epsilon(C)[i] = [p(C)[i],p(C)[i]+\epsilon]$, for every dimension
+ $i$. Furthermore, for a sufficiently small $\epsilon > 0$ these
+ \emph{clique boxes} verify the following conditions.
+ \begin{itemize}
+ \item[(c1)] For any two cliques $C_1\neq C_2$, $h^\epsilon(C_1) \cap
+ h^\epsilon(C_2) = \emptyset$ (i.e. $p(C_1) \neq p(C_2)$).
+ \item[(c2)] A box $h(v)$ intersects $h^\epsilon(C)$ if and only if
+ $v\in V(C)$, and in that case their intersection is a facet of
+ $h^\epsilon(C)$ incident to $p(C)$ (i.e. if we denote this
+ intersection $I$, then $I[i] = \{p(C)[i] \}$ for some dimension
+ $i$, and $I[j] = [p(C)[j],p(C)[j]+\epsilon]$ for the other
+ dimensions $j\neq i$).
+ \end{itemize}
+\end{itemize}
+\end{definition}
+Note that we may consider that the root clique is empty, that is the
+empty subgraph with no vertices. In that case the clique is denoted
+$\emptyset$. Let $\ecbdim(G)$ be the minimum dimension such that $G$
+has a $\emptyset$-clique-sum extendable touching representation with
+comparable boxes. The following lemma ensures that clique-sum
+extendable representations behave well with respect to full
+clique-sums.
+
+\begin{lemma}\label{lem-cs}
+ Consider two graphs $G_1$ and $G_2$, given with a $C^*_1$- and a
+ $C^*_2$-clique-sum extendable representations with comparable boxes
+ $h_1$ and $h_2$, in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$
+ respectively. Let $G$ be the graph obtained after performing a full
+ clique-sum of these two graphs on any clique $C_1$ of $G_1$, and on
+ the root clique $C^*_2$ of $G_2$. Then $G$ admits a $C^*_1$-clique
+ sum extendable representation with comparable boxes $h$ in
+ $\mathbb{R}^{\max(d_1,d_2)}$. Furthermore, we have that the aspect
+ ratio of $h(v)$ is the same as the one of $h_1(v)$, if $v\in
+ V(G_1)$, or the same as $h_2(v)$ if $v\in V(G_2)\setminus V(C^*_2)$.
+ \note{ shall we remove the aspect ratio thing ? only needed for the
+ hypercubes of the k-trees, but those are not really needed...}
+\end{lemma}
+\begin{proof}
+ The idea is to translate (allowing also exchanges of dimensions) and
+ scale $h_2$ to fit in $h_1^\epsilon(C_1)$. Consider an $\epsilon >0$
+ sufficiently small so that, $h_1^\epsilon(C_1)$ verifies all the
+ (cliques) conditions, and such that $h_1^\epsilon(C_1) \sqsubseteq
+ h_1(v)$ for any vertex $v\in V(G_1)$. Without loss of generality,
+ let us assume that $V(C_1)=\{v_1,\ldots,v_k\}$, and we also assume
+ that $h_1(v_i)$ and $h_1^\epsilon(C_1)$ touch in dimension $i$ (i.e.
+ $h_1(v_i)[i] \cap h_1^\epsilon(C_1)[i] = \{p(C_1)[i]\}$, and
+ $h_1(v_i)[j] \cap h_1^\epsilon(C_1)[j] =
+ [p(C_1)[j],p(C_1)[j]+\epsilon] $ for $j\neq i$.
+
+ Now let us consider $G_2$ and its representation $h_2$. Here the
+ vertices of $C^*_2$ are also denoted $v_1,\ldots,v_k$, and let us
+ denote $d_{v_i}$ the dimension in $h_2$ that fulfills condition (v1)
+ with respect to $v_i$.
+
+ Let $d=\max(d_1,d_2)$. We are now ready for defining $h$. For the
+ vertices of $G_1$ it is almost the same representation as $h_1$, as
+ we set $h(v)[i] = h_1(v)[i]$ for any dimension $i\le d_1$. If $d=d_2
+ > d_1$, then for any dimension $i>d_1$ we set $h(v)[i] = [0,1]$ if
+ $v\in V(C^*_1)$, and $h(v)[i] = [0,\frac12]$ if $v\in
+ V(G_1)\setminus V(C^*_1)$. Similarly the clique points $p_1(C)$
+ become $p(C)$ by setting $p(C)[i] = p_1(C)[i]$ for $i\le d_1$, and
+ $p(C)[i] = \frac14$ for $i> d_1$.
+
+ For $h_2$ and the vertices in $V(G_2) \setminus \{v_1,\ldots,v_k\}$
+ we have to consider a mapping $\sigma$ from $\{1,\ldots,d_2\}$ to
+ $\{1,\ldots,d\}$ such that $\sigma(d_{v_i}) = i$. This mapping
+ describes the changes of dimension we have to perform. We also have
+ to perform a scaling in order to make $h_2$ fit inside
+ $h_1^\epsilon(C_1)$. This is ensured by multiplying the coordinates
+ by $\epsilon$. More formally, for any vertex $v\in V(G_2) \setminus
+ \{v_1,\ldots,v_k\}$, we set $h(v)[\sigma(i)] = p(C_1)[\sigma(i)] +
+ \epsilon h_2(v)[i]$ for $i\in \{1 ,\ldots,d_2\}$, and $h(v)[j] =
+ p(C_1)[j] + [0, \epsilon/2]$, otherwise (for any $j$ not in the
+ image of $\sigma$). Note that if we apply the same mapping from
+ $h_2$ to $h$, to the vertices of $\{v_1,\ldots,v_k\}$, then the
+ image of $h_2(v_i)$ fit inside the (previously defined) box
+ $h(v_i)$. Similarly the clique points $p_2(C)$ become $p(C)$ by
+ setting $p(C)[\sigma(i)] = p(C_1)[\sigma(i)] + \epsilon p_2(C)[i]$
+ for $i\in \{1 ,\ldots,d_2\}$, and $p(C)[j] = p(C_1)[j] + [0,
+ \frac14]$, otherwise.
+
+ Note that we have defined (differently) both $h^\epsilon(C_1)$
+ (resp. $p(C_1)$) and $h^\epsilon(C^*_2)$ (resp. $p(C^*_2)$), despite
+ the fact that those cliques were merged. In the following we use
+ $h^\epsilon(C_1)$ and $p(C_1)$ only for the purpose of the
+ proof. The point and the box corresponding to these clique in $h$ is
+ $p(C^*_2)$ and $h^\epsilon(C^*_2)$.
+
+ Let us now check that $h$ is a $C^*_1$-clique sum extendable
+ representation with comparable boxes. The fact that the boxes are
+ comparable follows from the fact that those of $V(G_1)$
+ (resp. $V(G_1)$) are comparable in $h_1$ (resp. $h_2$) with the
+ boxes of $V(C^*_1)$ (resp. $V(C^*_2)$) being hypercubes of side one,
+ and the other boxes being smaller. Clearly, by construction both
+ $h_1(u) \sqsubseteq h_1(v)$ or $h_2(u) \sqsubseteq h_2(v)$, imply
+ $h(u) \sqsubseteq h(v)$, and for any vertex $u\in V(G_1)$ and any
+ vertex $v\in V(G_2) \setminus \{v_1,\ldots,v_k\}$, we have $h(v)
+ \sqsubseteq h^\epsilon(C_1) \sqsubseteq h(u)$.
+
+ We now check that $h$ is a contact representation of $G$. For $u,v
+ \in V(G_1)$ (resp. $u,v \in V(G_2) \setminus \{v_1,\ldots,v_k\}$) it
+ is clear that $h(u)$ and $h(v)$ are interior disjoint, 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 $v \in V(G_2) \setminus \{v_1,\ldots,v_k\}$. As
+ $h(v)$ fits inside $h^\epsilon(C_1)$, we have that $h(u)$ and $h(v)$
+ are interior disjoint. Furthermore, if they intersect then $u\in
+ V(C_1)$, say $u=v_1$, and $h(v)[1] = [p(C_1)[1], p(C_1)[1]+\alpha]$
+ for some $\alpha>0$. By construction, this implies that $h_2(v_1)$
+ and $h_2(v)$ intersect.
+
+ Finally for the $C^*_1$-clique-sum extendability, one can easily
+ check that the (vertex) conditions hold, such as the (clique)
+ conditions.
+\end{proof}
+
+The following lemma shows that any graphs has a $C^*$-clique-sum
+extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
+\ecbdim(G)$ and for any clique $C^*$.
+\begin{lemma}\label{lem-apex-cs}
+ For any graph $G$ and any clique $C^*$, we have that $G$ admits a
+ $C^*$-clique-sum extendable touching representation with comparabe
+ boxes in $\mathbb{R}^d$, for $d = |V(C^*)| + \ecbdim(G\setminus
+ V(C^*))$.
+\end{lemma}
+\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 $G\setminus V(C^*)$ by
+ comparable boxes in $\mathbb{R}^{d'}$, with $d' = \cbdim(G\setminus
+ V(C^*))$, and let $V(C^*) = \{v_1,\ldots,v_k\}$. We now construct
+ the desired representation $h$ of $G$ as follows. For each vertex
+ $v_i\in V(C^*)$ let $h(v_i)$ be the box fulfilling (v1) with
+ $d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^*)$, 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 $h(u)[i] =
+ h'(u)[i-k]$. 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 have
+ to refer to the clique point $p'(C')$ of $C'=C\setminus
+ \{v_1,\ldots,v_k\}$, as we set $p(C)[i] = p'(C')[i-k]$. One can
+ easily check that $h$ is as desired.
+\end{proof}
+
+The following lemma provides an upper bound on $\ecbdim(G)$ in terms
+of $\cbdim(G)$ and $\chi(G)$.
+\begin{lemma}\label{lem-ecbdim-cbdim}
+ For any graph $G$, $\ecbdim(G) \le \cbdim(G) + \chi(G)$.
+\end{lemma}
+\begin{proof}
+ Let $h$ be a touching representation with comparable boxes of $G$ in
+ $\mathbb{R}^d$, with $d=\cbdim(G)$, and let $c$ be a
+ $\chi(G)$-coloring of $G$. We start with a slightly modified version
+ of $h$. We first scale $h$ to fit in $(0,1)^d$, and for a
+ sufficiently small real $\alpha>0$ we increase each box in $h$, by
+ $2\alpha$ in every dimension, that is we replace $h(v)[i] = [a,b]$
+ by $[a-\epsilon,b+\epsilon]$ for each vertex $v$ and dimension
+ $i$. Furthermore $\alpha$ is chosen sufficiently small, so that no
+ new intersection was created. The obtained representation $h_1$ is
+ thus an intersection representation of the same graph $G$ such that,
+ for every clique $C$ of $G$, the intersection $I_C= \cap_{v\in V(C)
+ h_1(v)}$ is $d$-dimensional. For any maximal clique $C$ of $G$, let
+ $p_1(C)$ be a point in the interior of $I_C$.
+
+ Now we add $\chi(G)$ dimensions to make the representation touching
+ again, and to ensure some space for the clique boxes
+ $h^\epsilon(C)$. Formally we define $h_2$ as follows.
+ $$h_2(u)[i]=\begin{cases}
+ h_1(u)[i]&\text{ if $i\le d$}\\
+ [1/5,3/5]&\text{ if $c(u) < i-d$}\\
+ [0,2/5]&\text{ if $c(u) = i-d$}\\
+ [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)}
+ \end{cases}$$
+ For any clique $C'$ of $G$, let us denote $c(C)$, the set
+ $\{c(u)\ |\ u\in V(C')\}$, and let $C$ be one of the maximal cliques
+ containing $C'$.We now set
+ $$p_2(C')[i]=\begin{cases}
+ p_1(C) &\text{ if $i\le d$}\\
+ 2/5 &\text{ if $i-d \in c(C')$}\\%\text{if $\exists u\in V(C)$ with $c(u) = i-d$}\\
+ 1/2 &\text{ otherwise}
+ \end{cases}
+ $$ One can now check that this is a $\emptyset$-clique-sum
+ extendable touching representation with comparable boxes. In particular,
+ one should notice that for a vertex $u$ of a clique $C'$ we have that
+ $h_2^{\epsilon}(C')[i] \subset h_2(u)[i]$ for every dimension $i$,
+ except if $c(u)=i-d$. If $c(u)=i-d$, then $h_2(u)[i] \cap
+ h_2^{\epsilon}(C')[i] = \{2/5\}$.
+\end{proof}
+
+
+
+
+
+\section{The strong product structure and minor-closed classes}
-Let $G\boxtimes H$ denote the \emph{strong product} of the graphs $G$ and $H$, i.e., the graph with vertex
-set $V(G)\times V(H)$ and with distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if
-either $u_1=u_2$ or $u_1u_2\in E(G)$ and either $v_1=v_2$ or $v_1v_2\in E(G)$.
Dujmovi{\'c} et al.~\cite{DJM+} proved the following result.
\begin{theorem}\label{thm-prod}
-Any graph $G$ is a subgraph of the strong product of a path, a graph of threewidth at most $t$, and $K_m$, where
+Any graph $G$ is a subgraph of the strong product of a $k$-tree, a path, and $K_m$, where
\begin{itemize}
-\item $t=3$ and $m=3$ if $G$ is planar, and
-\item $t=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.
+\item $k=3$ and $m=3$ if $G$ is planar, and
+\item $k=4$ 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
-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.
+Moreover, for every $t$, there exists a $k$ such that any
+$K_t$-minor-free graph $G$ is a subgraph of a graph obtained from
+successive clique-sums of graphs, that are obtained from the strong
+product of a path and a $k$-tree, by adding at most $k$ apex vertices.
\end{theorem}
-The connection to the comparable box dimension comes from the following observation.
-\begin{lemma}\label{lemma-ps}
-The strong product of a path $P$, a graph $T$ of treewidth at most $t$, and $K_m$ has
-comparable box dimension at most $t+2+\lceil \log_2 m\rceil$.
-\end{lemma}
+Let us first bound the comparable box dimension of a graph in terms of
+its Euler genus. As paths and $m$-clique admit touching
+representations with hypercubes of unit size in $\mathbb{R}^{1}$ and
+in $\mathbb{R}^{\lceil \log_2 m \rceil}$ respectively, by
+ Lemma~\ref{lemma-sp} it suffice to bound the comparable box
+ dimension of $k$-trees.
+
+\begin{theorem}\label{thm-ktree}
+ For any $k$-tree $G$, $\cbdim(G) \le \ecbdim(G) \le k+1$.
+\end{theorem}
\begin{proof}
-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}. 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
-$$f(p,v,x_1, \ldots, x_k)[j]=\begin{cases}
-h(v)[j]&\text{ for $j=1,\ldots, t+1$}\\
-[p,p+1]&\text{ if $j=t+2$}\\
-[x_{j-t-2},x_{j-t-2}+1]&\text{ if $j>t+2$.}
-\end{cases}$$
-Clearly, this is a touching representation by comparable boxes.
+ Note that there exists a $k$-tree $G'$ having a $k$-clique $C^*$
+ such that $G'\setminus V(C^*)$ corresponds to $G$. Let us construct
+ a $C^*$-clique-sum extendable representation of $G'$ and note that
+ it induces a $\emptyset$-clique-sum extendable representation of
+ $G$.
+
+Note that $G'$ can be obtained by starting with a $(k+1)$-clique
+containing $C^*$, and by performing successive full clique-sums of
+$K_{k+1}$ on a $K_k$ subclique. By Lemma~\ref{lem-cs}, it suffice to
+show that $K_{k+1}$, the $(k+1)$-clique with vertex set $\{v_1,
+\ldots, v_{k+1}\}$, has a $(K_{k+1} -\{v_{k+1}\})$-clique-sum
+extendable touching representation with hypercubes. Let us define such
+touching representation $h$ as follows:
+\begin{itemize}
+ \item $h(v_i)[i] = [-1,0] $ if $i\le k$
+ \item $h(v_i)[j] = [0,1] $ if $i\le k$ and $i\neq j$
+ \item $h(v_{k+1})[j] = [0,\frac12]$ for any $j$
+\end{itemize}
+One can easily check that the (vertices)
+conditions are fulfilled. For the (cliques) conditions let us set
+the point $p(C)$ for every clique $C$ as follows:
+\begin{itemize}
+ \item $p(C)[i] = 0 $ for every $i\le k$ and if $v_i\in C$
+ \item $p(C)[i] = \frac14 $ for every $i\le k$ and if $v_i\notin C$
+ \item $p(C)[k+1] = \frac12 $ if $v_{k+1}\in C$
+ \item $p(C)[k+1] = \frac34 $ if $v_{k+1}\notin C$
+\end{itemize}
+By construction, it is clear that $p(C) \in h(v_i)$ if and only if
+$v_i\in V(C)$. Let us check the other (cliques) conditions.
+
+For any two distinct cliques $C_1$ and $C_2$ the points $p(C_1)$ and
+$p(C_2)$ are distinct. Indeed, if $|V(C_1)|\ge |V(C_2)|$ there is a
+vertex $v_i\in V(C_1)\setminus V(C_2)$, and this implies that
+$p(C_1)[i] < p(C_2)[i]$.
+
+For a vertex $v_i$ and a clique $C$, the boxes $h(v_i)$ and
+$h^\epsilon(C)$ intersect if and only if $v_i\in V(C)$. Indeed, if
+$v_i\in V(C)$ then $p(C)\in h(v_i)$ and $p(C)\in h^\epsilon(C)$, and
+if $v_i\notin V(C)$ then $h(v_i)[i] = [-1,0]$ if $i\le k$
+(resp. $h(v_i)[i] = [0 \frac12]$ if $i= k+1$) and $h^\epsilon(C)[i] =
+[\frac14,\frac14+\epsilon]$ (resp. $h^\epsilon(C)[i] =
+[\frac34,\frac34+\epsilon]$). Finally, if $v_i\in V(C)$ we have that
+$h(v_i)[i] \cap h^\epsilon(C)[i] = \{p(C)[i]\}$ and that $h(v_i)[j]
+\cap h^\epsilon(C)[j] = [p(C)[j],p(C)[j]+\epsilon]$ for any $j\neq i$
+and any $\epsilon <\frac14$. This concludes the proof of the theorem.
\end{proof}
-Combining Theorem~\ref{thm-prod}, Lemma~\ref{lemma-ps}, and the results of the previous section,
-we obtain the following corollary, which in particular implies Theorem~\ref{thm-minor}.
-
-\begin{corollary}\label{cor-minor}
-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,
-$$\cbdim(G)\le 5\cdot 81^7 \cdot (2g+3)^{\log_2 81}.$$
-Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, then
-$\cbdim(G)\le d$.
+As every planar graph $G$ has a touching representation with cubes in
+$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le
+3$. For the graphs with higher Euler genus we can also derive upper
+bounds. Indeed, combining the previous observation on the
+representations of paths and $K_m$, with Theorem~\ref{thm-ktree},
+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 $\cbdim(G')\le 5+1+\lceil \log_2 \max(2g,3)\rceil$.
+Consequently, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2
+ 81}.$$
\end{corollary}
+
+
+Let us now finally prove Theorem~\ref{thm-minor}, using the structure
+provided by Theorem~\ref{thm-prod}. We have seen that the strong
+product of a path and a $k$-tree has bounded comparable boxes
+dimension, and by Lemma~\ref{lemma-apex} adding at most $k$ apex
+vertices keeps the dimension bounded. Then by Lemma~\ref{lemma-chrom}
+and Lemma~\ref{lem-ecbdim-cbdim}, these graphs admit a
+$\emptyset$-clique-sum extendable representations in bounded
+dimensions. As the obtained graphs have bounded dimension, by
+Lemma~\ref{lemma-cliq} and Lemma~\ref{lem-apex-cs}, for any choice of
+a root clique $C^*$, they have a $C^*$-clique-sum extendable
+representation in bounded dimension. Thus by Lemma~\ref{lem-cs} any
+sequence of clique sum from these graphs leads to a graph with bounded
+dimension. Finally, we have seen that taking a subgraph does not lead
+to undounded dimension, and we obtain Theorem~\ref{thm-minor}.
+
+
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
and comparable box dimension $n$. It follows that the dependence of the comparable box dimension cannot be
-subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-minor}
+subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-genus}
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}.
-It would be interesting to prove this part of Corollary~\ref{cor-minor} without using the structure theorem.
+It would be interesting to prove Theorem~\ref{thm-minor} without using the structure theorem.
\section{Fractional treewidth-fragility}