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Commit 1322d6e6 authored by Daniel Gonçalves's avatar Daniel Gonçalves
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few changes, and new appendix to reach the limit on the nb of lines

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comparable-box-dimension.tex
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......@@ -117,8 +117,8 @@ comparable box dimension of graphs in $\GG$ is not bounded.
Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular,
they showed that if a class $\GG$ has finite comparable box dimension, then it has polynomial strong coloring
numbers, which implies that $\GG$ has strongly sublinear separators. They also provided an example showing
that for any function $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
comparable box dimension. Dvo\v{r}\'ak et al.~\cite{wcolig}
that for many functions $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
comparable box dimension\footnote{In their construction $h(r)$ has to be at least 3, and has to tend to $+\infty$.}. Dvo\v{r}\'ak et al.~\cite{wcolig}
proved that graphs of comparable box dimension $3$ have exponential weak coloring numbers, giving the
first natural graph class with polynomial strong coloring numbers and superpolynomial weak coloring numbers
(the previous example is obtained by subdividing edges of every graph suitably many times~\cite{covcol}).
......@@ -174,19 +174,17 @@ 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 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
Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy coloring $c$ so that $c(v_i)$ is
the smallest color such that $c(v_i)\neq c(v_j)$ 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.
It remains to bound the number of colors used. Suppose we are coloring $v_i$. We shall bound the number of vertices
$v_j$ such that $j<i$ and there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B_j)/\vol(f(v_i))-1=5^d-1$.
$v_j$ such that $j<i$ and such that there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B)/\vol(f(v_i))-1=5^d-1$.
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 $v_jv_m,v_mv_i\in E(G)$
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
\[(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d\]
vertices.
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 $(5^d-1) + (3^d-1) + (3^d-1)^2$ vertices, so $2\cdot 9^d$ colors suffice.
\end{proof}
\begin{figure}
......@@ -222,7 +220,8 @@ It is clear that given a touching representation of a graph $G$, one
easily obtains a touching representation by 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.
consider other basic operations on graphs. In the following, to describe
the boxes, we are going to use the Cartesian product $\times$ defined among boxes ($A\times B$ is the box whose projection on the first dimensions gives the box $A$, while the projection on the remaing dimensions gives the box $B$) or we are going to provide its projections for every dimension ($A[i]$ is the interval obtained from projecting $A$ on its $i^\text{th}$ dimension).
\subsection{Vertex addition}\label{sec-vertad}
......@@ -312,7 +311,7 @@ different and the construction of~\cite{subconvex} uses rotated boxes,
we provide details of the argument.
\begin{lemma}\label{lemma-subg}
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+\chi^2_s(G')$.
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+\frac12 \chi^2_s(G')$.
\end{lemma}
\begin{proof}
As we can remove the boxes that represent the vertices, we can assume $V(G')=V(G)$.
......@@ -343,17 +342,17 @@ Suppose now that $uv\not\in E(G)$. If $uv\not\in E(G')$, then $f(u)$ is disjoin
$h(v)$. Hence, we can assume $uv\in E(G')\setminus E(G)$, $\varphi(u)=i$, $\varphi(v)=j$ and $i<j$. Then $u\in A_{i,j}$, $v\in A_{j,i}$,
$h(u)[d_{i,j}]=[1/3,4/3]$, $h(v)[d_{j,i}]=[-4/3,-1/3]$, and $h(u)\cap h(v)=\emptyset$.
Consequently, $h$ is a touching representation of $G$ by comparable boxes in dimension $d+\binom{c}{2}\le d+c^2$.
Consequently, $h$ is a touching representation of $G$ by comparable boxes in dimension $d+\binom{c}{2}\le d+c^2 /2$.
\end{proof}
Let us now combine Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}.
\begin{corollary}\label{cor-subg}
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+4\cdot 81^{\cbdim(G')}\le 5\cdot 81^{\cbdim(G')}$.
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+2\cdot 81^{\cbdim(G')}\le 3\cdot 81^{\cbdim(G')}$.
\end{corollary}
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 $2^{d-1}$.
but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. Indeed, for every pair $u,v$ of non-adjacent vertices there is a specific dimension $i$ such that their boxes span intervals $[a,b]$ and $[c,d]$ with $c<d$, while for every other box in the representation their $i^\text{th}$ interval contains $[b,c]$.
\subsection{Clique-sums}
......@@ -381,20 +380,19 @@ by (not necessarily comparable) boxes in $\mathbb{R}^d$ is called
\begin{itemize}
\item[] {\bf(vertices)} For each $u\in V(C^\star)$, there exists a dimension $d_u$,
such that:
\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
\subitem\emph{(v0)} $d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
\subitem\emph{(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
\subitem[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v) \subset [0,1)^d$.
\subitem\emph{(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 \[p(C)\in [0,1)^d\cap \bigcap_{v\in V(C)} h(v)\]
a point $p(C)\in [0,1)^d\cap \left( \bigcap_{v\in V(C)} h(v) \right)$
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
by setting $h^\varepsilon(C)[i] = [p(C)[i],p(C)[i]+\varepsilon]$ for every dimension
$i$, the following conditions are satisfied:
\begin{itemize}
\subitem[(c1)] For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1) \cap
% \begin{itemize}
\subitem\emph{(c1)} For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1) \cap
h^\varepsilon(C_2) = \emptyset$ (equivalently, $p(C_1) \neq p(C_2)$).
\subitem[(c2)] A box $h(v)$ intersects $h^\varepsilon(C)$ if and only if
\subitem\emph{(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 $j$,
......@@ -402,7 +400,7 @@ by (not necessarily comparable) boxes in $\mathbb{R}^d$ is called
\{p(C)[i_{C,v}] \}&\text{if $j=i_{C,v}$}\\
[p(C)[j],p(C)[j]+\varepsilon]&\text{otherwise.}
\end{cases}\]
\end{itemize}
% \end{itemize}
\end{itemize}
\end{definition}
Note that the root clique can be empty, that is the
......@@ -427,10 +425,8 @@ The clique point $p(C)$ of each clique $C$ is extended by setting $p(C)[d+1] = \
It is easy to verify that the resulting representation is $C^\star$-clique-sum extendable.
\end{proof}
The following lemma ensures that clique-sum
extendable representations behave well with respect to full
clique-sums.
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 $C^\star_1$- and a
$C^\star_2$-clique-sum extendable representations $h_1$ and $h_2$ by comparable boxes
......@@ -441,115 +437,9 @@ clique-sums.
sum extendable representation $h$ by comparable boxes in
$\mathbb{R}^{\max(d_1,d_2)}$.
\end{lemma}
\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}
The proof is in the appendix, but the idea is to translate (allowing
also exchanges of dimensions) and scale $h_2$ to fit in
$h_1^\varepsilon(C_1)$.
The following lemma enables us to pick the root clique at the expense of increasing
the dimension by $\omega(G)$.
\begin{lemma}\label{lem-apex-cs}
......@@ -557,56 +447,8 @@ the dimension by $\omega(G)$.
$C^\star$-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}
\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 suffciently 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}
The proof is also in the appendix, but it essentially the same as the one of
Lemma~\ref{lemma-apex}.
The following lemma provides an upper bound on $\ecbdim(G)$ in terms
of $\cbdim(G)$ and $\chi(G)$.
\begin{lemma}\label{lem-ecbdim-cbdim}
......@@ -640,7 +482,7 @@ of $\cbdim(G)$ and $\chi(G)$.
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_1(C) &\text{ if $i\le d$}\\
p_1(C)[i] &\text{ if $i\le d$}\\
2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\
1/2 &\text{ otherwise}
\end{cases}
......@@ -696,12 +538,12 @@ Let $\GG$ be a class of graphs of comparable box dimension at most $d$.
\begin{itemize}
\item The class $\GG'$ of graphs obtained from $\GG$ by repeatedly performing full clique-sums
has comparable box dimension at most $d + 2\cdot 3^d$.
\item The class of graphs obtained from $\GG$ by repeatedly performing clique-sums
has comparable box dimension at most $625^d$.
\item The closure of $\GG'$ by taking subgraphs
has comparable box dimension at most $1250^d$.
\end{itemize}
\end{corollary}
\begin{proof}
The former bound directly follows from Corllary~\ref{cor-csum} and the bound on the chromatic number
The former bound directly follows from Corollary~\ref{cor-csum} and the bound on the chromatic number
from Lemma~\ref{lemma-chrom}. For the latter one, we need to bound the star chromatic number of $\GG'$.
Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
For $i=1,\ldots, m$, suppose $G_i$ has an acyclic coloring $\varphi_i$ by at most $k$ colors.
......@@ -777,25 +619,10 @@ $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for suffici
\end{proof}
The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree.
Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree}
extends to graphs of treewidth at most $k$.
extends to graphs of treewidth at most $k$ (see the proof in the appendix).
\begin{corollary}\label{cor-tw}
Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
\end{corollary}
\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 $h(u)$ slighly 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}
As every planar graph $G$ has a touching representation by cubes in
$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le 3$.
......@@ -807,7 +634,7 @@ 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, \[\cbdim(G)\le 3\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.\]
\end{corollary}
Similarly, we can deal with proper minor-closed classes.
......@@ -816,7 +643,7 @@ Let $\GG$ be a proper minor-closed class. Since $\GG$ is proper, there exists $
By Theorem~\ref{thm-prod}, there exists $k$ such that every graph in $\GG$ is a subgraph of a graph obtained by repeated clique-sums
from extended $k$-tree-grids. As we have seen, $k$-tree-grids have comparable box dimension at most $k+2$,
and by Lemma~\ref{lemma-apex}, extended $k$-tree-grids have comparable box dimension at most $2k+2$.
By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le 625^{2k+2}$.
By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le 1250^{2k+2}$.
\end{proof}
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
......@@ -897,7 +724,7 @@ is fractionally treewidth-fragile, with a function $f(k) = O_{t,s,d}\bigl(k^{d}\
For a positive integer $k$, let $f(k)=(2ksd+2)^dst$.
Let $(\iota,\omega)$ be an $s$-comparable envelope representation of a graph $G$
in $\mathbb{R}^d$ of thickness at most $t$, and let $v_1$, \ldots, $v_n$ be the corresponding ordering of the vertices of $G$.
Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $|ksd\omega(v_i)[j]|$ such that $\ell_{i-1,j} / \ell_{i,j}$ is a positive integer. Formally
Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $ksd|\omega(v_i)[j]|$ such that $\ell_{i-1,j} / \ell_{i,j}$ is a positive integer. Formally
it is defined as follows.
\begin{itemize}
\item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$.
......@@ -989,4 +816,206 @@ 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 suffciently 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 $h(u)$ slighly 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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