From 5ada19374292f6934ec8f0b48ae83e7a6e88e5b0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Zden=C4=9Bk=20Dvo=C5=99=C3=A1k?= <rakdver@iuuk.mff.cuni.cz>
Date: Fri, 19 Nov 2021 15:52:44 +0100
Subject: [PATCH] Added details to the proof of Lemma 10.
---
comparable-box-dimension.tex | 454 ++++++++++++++++++++---------------
1 file changed, 256 insertions(+), 198 deletions(-)
diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index 8ecc63d..e51273a 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -210,58 +210,65 @@ Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^
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.
+distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if
+$u_1$ is equal to or adjacent to $u_2$ in $G$
+and $v_1$ is equal to or adjacent to $v_2$ in $H$.
+To obtain a touching representation of $G\boxtimes
+H$ it suffices to take a product of representations of $G$ and $H$, but
+the resulting representation may contain incomparable boxes.
+Indeed, $\cbdim(G\boxtimes H)$ in general is not bounded by a function
+of $\cbdim(G)$ and $\cbdim(H)$; for example, every star has comparable box dimension
+at most two, but the strong product of the star $K_{1,n}$ with itself contains
+$K_{n,n}$ as an induced subgraph, and thus its comparable box dimension is at least $\Omega(\log n)$.
+However, as shown in the following lemma, this issue does not arise if the representation of $H$ consists of translates
+of a single box; by scaling, we can without loss of generality assume this box is a unit hypercube.
\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
- $\cbdim(G\boxtimes H) \le \cbdim(G) + d_H$.
+ $\mathbb{R}^{d_H}$ by axis-aligned hypercubes of unit size. Then for any graph
+ $G$, the strong product $G\boxtimes H$ of these graphs has comparable box dimension at most
+ $\cbdim(G) + d_H$.
\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
+ representations. Indeed, consider a touching respresentation $g$ of $G$ by
+ comparable boxes in $\mathbb{R}^{d_G}$, with
$d_G=\cbdim(G)$, and the representation $h$ of $H$. Let us define a
representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
follows.
$$f((u,v))[i]=\begin{cases}
g(u)[i]&\text{ if $i\le d_G$}\\
- h(u)[i-d_G]&\text{ if $i > d_G$}
+ h(v)[i-d_G]&\text{ if $i > d_G$}
\end{cases}$$
- 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
- interiorly 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
- $\mathbb{R}^{d_H}$ 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 $G\boxtimes H$.
+ Consider distinct vertices $(u,v)$ and $(u',v')$ of $G\boxtimes H$.
+ The boxes $g(u)$ and $g(u')$ are comparable, say $g(u)\sqsubseteq g(u')$. Since $h(v')$
+ is a translation of $h(v)$, this implies that $f((u,v))\sqsubseteq f((u',v'))$. Hence, the boxes
+ of the representation $f$ are pairwise comparable.
+
+ The boxes of the representations $g$ and $h$ have pairwise disjoint interiors.
+ Hence, if $u\neq u'$, then there exists $i\le d_G$ such that the interiors
+ of the intervals $f((u,v))[i]=g(u)[i]$ and $f((u',v'))[i]=g(u')[i]$ are disjoint;
+ and if $v\neq v'$, then there exists $i\le d_H$ such that the interiors
+ of the intervals $f((u,v))[i+d_G]=h(v)[i]$ and $f((u',v'))[i+d_G]=h(v')[i]$ are disjoint.
+ Consequently, the interiors of boxes $f((u,v))$ and $f((u',v'))$ are pairwise disjoint.
+ Moreover, if $u\neq u'$ and $uu'\not\in E(G)$, or if $v\neq v'$ and $vv'\not\in E(G)$,
+ then the intervals discussed above (not just their interiors) are disjoint for some $i$;
+ hence, if $(u,v)$ and $(u',v')$ are not adjacent in $G\boxtimes H$, then $f((u,v))\cap f((u',v'))=\emptyset$.
+ Therefore, $f$ is a touching representation of a subgraph of $G\boxtimes H$.
+
+ Finally, suppose that $(u,v)$ and $(u',v')$ are adjacent in $G\boxtimes H$.
+ Then there exists a point $p_G$ in the intersection of $g(u)$ and $g(u')$,
+ since $u=u'$ or $uu'\in E(G)$ and $g$ is a touching representation of $G$;
+ and similarly, there exists a point $p_H$ in the intersection of $h(v)$ and $h(v')$.
+ Then $p_G\times p_H$ is a point in the intersection of $f((u,v))$ and $f((u',v'))$.
+ Hence, $f$ is indeed a touching representation of $G\boxtimes H$.
\end{proof}
-\subsection{Subgraph}
+\subsection{Taking a 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
+The comparable box dimension of a subgraph of a graph $G$ may be larger than $\cbdim(G)$, see the end of this
+section for an example. 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
@@ -273,24 +280,31 @@ If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+\chi^2_s(G')
\end{lemma}
\begin{proof}
As we can remove the boxes that represent the vertices, we can assume $V(G')=V(G)$.
-Let $f$ be a touching representation by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G')$. Let $\varphi$
+Let $f$ be a touching representation of $G'$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G')$. Let $\varphi$
be a star coloring of $G'$ using colors $\{1,\ldots,c\}$, where $c=\chi_s(G')$.
-For any distinct colors $i,j\in\{1,\ldots,c\}$, let $A_{i,j}\subseteq V(G)$ consist of vertices $u$ of color $i$
-such that there exists a vertex $v$ of color $j$ such that $uv\in E(G')$ and $uv\not\in E(G)$.
+For any distinct colors $i,j\in\{1,\ldots,c\}$, let $A_{i,j}\subseteq V(G)$ be the set of vertices $u$ of color $i$
+such that there exists a vertex $v$ of color $j$ such that $uv\in E(G')\setminus E(G)$. For each $u\in A_{i,j}$,
+let $a_j(u)$ denote such a vertex $v$ chosen arbitrarily.
+
Let us define a representation $h$ by boxes in $\mathbb{R}^{d+\binom{c}{2}}$ by starting from the representation $f$ and,
for each pair $i<j$ of colors, adding a dimension $d_{i,j}$ and setting
-$h(v)[d_{i,j}]=[1/3,4/3]$ for $v\in A_{i,j}$, $h(v)[d_{i,j}]=[-4/3,-1/3]$ for $v\in A_{j,i}$,
-and $h(v)[d_{i,j}](v)=[-1/2,1/2]$ otherwise. Note that the boxes in this extended representation are comparable,
+$$h(v)[d_{i,j}]=\begin{cases}
+[1/3,4/3]&\text{if $v\in A_{i,j}$}\\
+[-4/3,-1/3]&\text{if $v\in A_{j,i}$}\\
+[-1/2,1/2]&\text{otherwise.}
+\end{cases}$$
+Note that the boxes in this extended representation are comparable,
as in the added dimensions, all the boxes have size $1$.
-Suppose $uv\in E(G)$, where $\varphi(u)=i$ and $\varphi(v)=j$ and say $i<j$. The boxes $f(u)$ and $f(v)$ touch.
-We cannot have $u\in A_{i,j}$ and $v\in A_{j,u}$, as then $G'$ would contain a 4-vertex path in colors $i$ and $j$.
-Hence, in any added dimension $d'$, at least one of $h(u)$ and $h(v)$ is represented by the interval $[-1/2,1/2]$,
-and thus $h(u)[d']\cap h(v)[d']\neq\emptyset$. Therefore, the boxes $h(u)$ and $h(v)$ touch.
+Suppose $uv\in E(G)$, where $\varphi(u)=i$ and $\varphi(v)=j$ and say $i<j$.
+We cannot have $u\in A_{i,j}$ and $v\in A_{j,i}$, as then $a_j(u)uva_i(v)$ would be a 4-vertex path in $G'$ in colors $i$ and $j$.
+Hence, in any added dimension $d'$, we have $h(u)[d']=[-1/2,1/2]$ or $h(v)[d']=[-1/2,1/2]$,
+and thus $h(u)[d']\cap h(v)[d']\neq\emptyset$.
+Since the boxes $f(u)$ and $f(v)$ touch, it follows that the boxes $h(u)$ and $h(v)$ touch as well.
Suppose now that $uv\not\in E(G)$. If $uv\not\in E(G')$, then $f(u)$ is disjoint from $f(v)$, and thus $h(u)$ is disjoint from
-$h(v)$. Hence, we can assume $uv\in E(G')$, $\varphi(u)=i$, $\varphi(v)=j$ and $i<j$. Then $u\in A_{i,j}$, $v\in A_{j,i}$,
+$h(v)$. 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 $d+\binom{c}{2}\le d+c^2$.
@@ -302,7 +316,7 @@ Let us now combine Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}.
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+4\cdot 81^{\cbdim(G')}\le 5\cdot 81^{\cbdim(G')}$.
\end{corollary}
-Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim{K_{2^d}}=d$,
+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 $2^{d-1}$.
@@ -324,162 +338,206 @@ 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.
+Consider a graph $G$ with a distinguished clique $C^\star$, called the
+\emph{root clique} of $G$. A touching representation $h$ of $G$
+by (not necessarily comparable) boxes in $\mathbb{R}^d$ is called
+\emph{$C^\star$-clique-sum extendable} if the following conditions hold for every sufficiently small $\varepsilon>0$.
\begin{itemize}
-\item[{\bf(vertices)}] There are $|V(C^*)|$ dimensions, denoted $d_u$ for each
- vertex $u\in V(C^*)$, such that:
+\item[{\bf(vertices)}] For each $u\in V(C^\star)$, there exists a dimension $d_u$,
+ 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 $v\notin V(C^*)$, $h(v) \subset [0,1)^d$.
+ \item[(v0)] $d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
+ \item[(v1)] each vertex $u\in V(C^\star)$ satisfies $h(u)[d_u] = [-1,0]$ and
+ $h(u)[i] = [0,1]$ for any dimension $i\neq d_u$, and
+ \item[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v) \subset [0,1)^d$.
\end{itemize}
-\item[{\bf(cliques)}] For every clique $C$ of $G$ 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.
+\item[{\bf(cliques)}] For every clique $C$ of $G$, there exists
+ a point $$p(C)\in [0,1)^d\cap \bigcap_{v\in V(C)} h(v)$$
+ such that, defining the \emph{clique box} $h^\varepsilon(C)$
+ by setting
+ $$h^\varepsilon(C)[i] = [p(C)[i],p(C)[i]+\varepsilon]$$ for every dimension
+ $i$, the following conditions are satisfied:
\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
+ \item[(c1)] For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1) \cap
+ h^\varepsilon(C_2) = \emptyset$ (equivalently, $p(C_1) \neq p(C_2)$).
+ \item[(c2)] A box $h(v)$ intersects $h^\varepsilon(C)$ if and only if
$v\in V(C)$, 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 $j\neq i$).
+ $h^\varepsilon(C)$ incident to $p(C)$. That is, there exists a dimension $i_{C,v}$
+ such that for each dimension $j$,
+ $$h(v)[j]\cap h^\varepsilon(C)[j] = \begin{cases}
+ \{p(C)[i_{C,v}] \}&\text{if $j=i_{C,v}$}\\
+ [p(C)[j],p(C)[j]+\varepsilon]&\text{otherwise.}
+ \end{cases}$$
\end{itemize}
\end{itemize}
\end{definition}
-Note that we may consider that the root clique is empty, that is the
+Note that the root clique can be 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 by
-comparable boxes. The following lemma ensures that clique-sum
+has an $\emptyset$-clique-sum extendable touching representation by
+comparable boxes.
+
+Let us remark that a clique-sum extendable representation in dimension $d$ implies
+such a representation in higher dimensions as well.
+\begin{lemma}\label{lemma-add}
+Let $G$ be a graph with a root clique $C^\star$ and let $h$ be
+a $C^\star$-clique-sum extendable touching representation of $G$ by comparable boxes in $\mathbb{R}^d$.
+Then $G$ has such a representation in $\mathbb{R}^{d'}$ for every $d'\ge d$.
+\end{lemma}
+\begin{proof}
+It clearly suffices to consider the case that $d'=d+1$.
+Note that the \textbf{(vertices)} conditions imply that $h(v')\sqsubseteq h(v)$ for every $v'\in V(G)\setminus V(C^\star)$
+and $v\in V(C^\star)$. We extend the representation $h$
+by setting $h(v)[d+1] = [0,1]$ for $v\in V(C^\star)$ and $h(v)[d+1] = [0,\frac12]$ for $v\in V(G)\setminus V(C^\star)$.
+The clique point $p(C)$ of each clique $C$ is extended by setting $p(C)[d+1] = \frac14$.
+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.
\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
+ 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 $C^*_2$ of $G_2$. Then $G$ admits a $C^*_1$-clique
- sum extendable representation by comparable boxes $h$ in
+ 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}
\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 $h_1^\epsilon(C_1)$. 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 $j\neq i$.
+ 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 $C^*_2$ 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 $h(v)[j] =
- [p(C_1)[j], p(C_1)[j] +\epsilon/2]$, 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 boxes $h_2(v_i)$ for $i\in \{1,\ldots,k\}$, then the
- image of $h_2(v_i)$ fits 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 $p(C)[j] = p(C_1)[j] + \epsilon/4$, 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 this 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
+ 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 $V(G_1)$
- (resp. $V(G_2)$) 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 $h(v)
- \sqsubseteq h^\epsilon(C_1) \sqsubseteq h(u)$.
+ 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. $u,v \in V(G_2) \setminus \{v_1,\ldots,v_k\}$) it
+ \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 $v \in V(G_2) \setminus \{v_1,\ldots,v_k\}$. As
- $h(v)$ fits inside $h^\epsilon(C_1)$, we have that $h(u)$ and $h(v)$
- have disjoint interiors. 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 (vertices) conditions hold. For the (cliques)
- conditions, as the mapping from $p_2(C)$ to $p(C)$ (extended to a
- mapping from $\mathbb{R}^{d_2}$ to $\mathbb{R}^{d}$) is injective,
- we have that (c1) clearly holds. For (c2) one has to notice that if
- $d=d_2$, then the mapping from $h_2$ to $h$ extended to the clique boxes
- would lead to the same clique boxes $h^{\epsilon'}(C)$, with the same point $p(C)$ in their lower corner.
- If there are extra dimensions, that is if $d> d_2$, then for any such
- dimension $j$ that is not in the image of $\sigma$, we have that
- $h^{\epsilon'}(C)[j] = [p(C_1)[j] + \epsilon/4, p(C_1)[j] + \epsilon/4 + \epsilon']
- \subset [p(C_1)[j], p(C_1)[j] + \epsilon/2] = h(v)[j]$.
+ 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 following lemma shows that any graphs has a $C^*$-clique-sum
+The following lemma shows that any graphs has a $C^\star$-clique-sum
extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
-\ecbdim(G)$ and for any clique $C^*$.
+\ecbdim(G)$ and for any clique $C^\star$.
\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 by comparabe
- boxes in $\mathbb{R}^d$, for $d = |V(C^*)| + \ecbdim(G\setminus
- V(C^*))$.
+ For any graph $G$ and any clique $C^\star$, we have that $G$ admits a
+ $C^\star$-clique-sum extendable touching representation by comparabe
+ 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 $G\setminus V(C^*)$ by
+ extendable touching representation $h'$ of $G\setminus V(C^\star)$ 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
+ 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
- $v_i\in V(C^*)$ 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
+ $v_i\in V(C^\star)$ let $h(v_i)$ be the box fulfilling (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 $h(u)[i] =
\alpha_i h'(u)[i-k]$, for some $\alpha_i>0$. The values $\alpha_i>0$
- are chosen suffciently small so that $h(u)[i] \subset [0,1)$, whenever $u\notin V(C^*)$.
+ are 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 have
@@ -487,30 +545,30 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
\{v_1,\ldots,v_k\}$, as we set $p(C)[i] = \alpha_i p'(C')[i-k]$.
As $h'(u) \sqsubset h'(v)$ implies that $h(u) \sqsubset h(v)$, and as
- $h(u) \sqsubset h(v_i)$, for every $u\in V(G)\setminus V(C^*)$ and every
- $v_i \in V(C^*)$, we have that $h$ is a touching representation by comparable boxes.
+ $h(u) \sqsubset h(v_i)$, for every $u\in V(G)\setminus V(C^\star)$ and every
+ $v_i \in V(C^\star)$, we have that $h$ is a touching representation by comparable boxes.
By the construction, it is clear that $h$ is a representation of $G$.
- For the $C^*$-clique-sum extendability, it is clear that the (vertices) conditions hold.
+ For the $C^\star$-clique-sum extendability, it is clear that the (vertices) conditions hold.
For the (cliques) condition (c1), let us first consider two distinct cliques $C_1$ and $C_2$
- of $G$ such that $|V(C_1)| \ge |V(C_2)|$, and let $C'_i=C_i\setminus V(C^*)$. If $C'_1 = C'_2$,
+ 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 $C'_1 \neq C'_2$, we have that $p'(C'_1) \neq p'(C'_2)$, which leads to
$p(C_1) \neq p(C_2)$ by construction.
- For the (cliques) condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^*)$ and a clique $C$ of $G$ containing $v$.
- In the first dimensions $i \le k$, we always have $h^\epsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$ we have
- $h^\epsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$ (as in that case $v$ and $v_i$ are adjacent), and if $v_i \notin V(C)$
- we have $h^\epsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. Then for the last $d'$ dimensions, by definition of $h'$,
- we have that $h^\epsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
- for which $h^\epsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. This completes the first case
- and we now consider a vertex $v\in V(G)\setminus V(C^*)$ and a clique $C$ of $G$ not containing $v$.
+ For the (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 first dimensions $i \le k$, we always have $h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$ we have
+ $h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$ (as in that case $v$ and $v_i$ are adjacent), and if $v_i \notin V(C)$
+ we have $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. Then for the last $d'$ dimensions, by definition of $h'$,
+ we have that $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
+ for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. This completes the first case
+ and we now consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
As $v\notin V(C')$, there is an hyperplane
${\mathcal H}' = \{ p\in \mathbb{R}^{d'}\ |\ p[i] = c\}$ that separates $p'(C')$ and $h'(v)$.
This implies that the following hyperplane
${\mathcal H} = \{ p\in \mathbb{R}^{d}\ |\ p[k+i] = \alpha_{k+i} c\}$ separates $p(C)$ and $h(v)$.
- Now we consider a vertex $v_i \in V(C^*)$, and we note that for any clique $C$ containing $v_i$
- we have that $h^\epsilon(C)[i] \cap h(v_i)[i] = [0,\epsilon]\cap [-1,0] = \{0\}$, and $h^\epsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$
+ Now we consider a vertex $v_i \in V(C^\star)$, and we 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
- $h^\epsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$.
+ $h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$.
Condition (c2) is thus fulfilled and this completes the proof of the lemma.
\end{proof}
@@ -526,7 +584,7 @@ of $\cbdim(G)$ and $\chi(G)$.
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
+ by $[a-\varepsilon,b+\varepsilon]$ 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,
@@ -536,7 +594,7 @@ of $\cbdim(G)$ and $\chi(G)$.
Now we add $\chi(G)$ dimensions to make the representation touching
again, and to ensure some space for the clique boxes
- $h^\epsilon(C)$. Formally we define $h_2$ as follows.
+ $h^\varepsilon(C)$. Formally we define $h_2$ as follows.
$$h_2(u)[i]=\begin{cases}
h_1(u)[i]&\text{ if $i\le d$}\\
[1/5,3/5]&\text{ if $c(u) < i-d$}\\
@@ -566,13 +624,13 @@ of $\cbdim(G)$ and $\chi(G)$.
which clique points $p_2(C_1)$ and $p_2(C_2)$ are based on distinct maximum cliques, necessarily lead to distinct points.
In the case that $C_1$ and $C_2$ belong to some maximal clique $C$, we have that $c(C_1) \neq c(C_2)$
and this implies by construction that $p_2(C_1)$ and $p_2(C_2)$ are distinct. Thus (c1) holds.
- By construction of $h_1$, we have that if $h_2^{\epsilon}(C')[i] \cap h_2(v)[i]$ is non-empty for every $i\le d$,
- then we have that $h_2^{\epsilon}(C')[i] \subset h_2(v)[i]$ for every $i\le d$,
+ By construction of $h_1$, we have that if $h_2^{\varepsilon}(C')[i] \cap h_2(v)[i]$ is non-empty for every $i\le d$,
+ then we have that $h_2^{\varepsilon}(C')[i] \subset h_2(v)[i]$ for every $i\le d$,
and we have that $v$ belongs to some maximal clique $C$ containing $C'$. If $v\notin V(C')$ note that
$p_2(C')[d+c(v)] = 1/2 \notin[0,2/5]=h_2(v)[d+c(v)]$, while if $v\in V(C')$ we have that
- $h_2^{\epsilon}(C')[i] \subset [2/5,1/2+\epsilon] \subset h_2(v)[i]$ for every dimension $i>d$,
- except if $c(v)=i-d$, and in that case $h_2(v)[i] \cap h_2^{\epsilon}(C')[i] =
- [0,2/5]\cap[2/5,2/5+\epsilon] = \{2/5\}$. We thus have that (c2) holds, and this concludes the proof of the lemma.
+ $h_2^{\varepsilon}(C')[i] \subset [2/5,1/2+\varepsilon] \subset h_2(v)[i]$ for every dimension $i>d$,
+ except if $c(v)=i-d$, and in that case $h_2(v)[i] \cap h_2^{\varepsilon}(C')[i] =
+ [0,2/5]\cap[2/5,2/5+\varepsilon] = \{2/5\}$. We thus have that (c2) holds, and this concludes the proof of the lemma.
\end{proof}
@@ -605,14 +663,14 @@ in $\mathbb{R}^{\lceil \log_2 m \rceil}$ respectively, by
For any $k$-tree $G$, $\cbdim(G) \le \ecbdim(G) \le k+1$.
\end{theorem}
\begin{proof}
- 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
+ Note that there exists a $k$-tree $G'$ having a $k$-clique $C^\star$
+ such that $G'\setminus V(C^\star)$ corresponds to $G$. Let us construct
+ a $C^\star$-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
+containing $C^\star$, 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
@@ -641,15 +699,15 @@ 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
+$h^\varepsilon(C)$ 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 $p(C)\in h^\varepsilon(C)$, 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.
+(resp. $h(v_i)[i] = [0, \frac12]$ if $i= k+1$) and $h^\varepsilon(C)[i] =
+[\frac14,\frac14+\varepsilon]$ (resp. $h^\varepsilon(C)[i] =
+[\frac34,\frac34+\varepsilon]$). Finally, if $v_i\in V(C)$ we have that
+$h(v_i)[i] \cap h^\varepsilon(C)[i] = \{p(C)[i]\}$ and that $h(v_i)[j]
+\cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for any $j\neq i$
+and any $\varepsilon <\frac14$. This concludes the proof of the theorem.
\end{proof}
Note that actually the bound on the comparable boxes dimension of Theorem~\ref{thm-ktree}
extends to graphs of treewidth $k$. For this, note that the construction in this proof can
@@ -659,7 +717,7 @@ with $h(u) \sqsubset h(v)$ say, the intersection $I = h(u) \cap h(v)$ is a facet
Actually $I[i] = h(u)[i]$ for every dimension, except one that we denote $j$. For this
dimension we have that $I[j]=\{c\}$ for some $c$, and that $h(u)[j]=[c,c+s]$,
where $s$ is the length of the sides of $h(u)$. In that context to delete an edge $uv$
-one can simply replace $h(u)[j]=[c,c+s]$ with $[c+\epsilon,c+s]$, for a sufficiently small $\epsilon$.
+one can simply replace $h(u)[j]=[c,c+s]$ with $[c+\varepsilon,c+s]$, for a sufficiently small $\varepsilon$.
One can proceed similarly for any subset of edges, and note that as the hypercubes in $h$ have
distinct sizes these small perturbations give rise to boxes that are still comparable.
Thus for any treewidth $k$ graph $H$ (that is a subgraph of a $k$-tree $G$) we have $\cbdim(H)\le k+1$.
@@ -689,7 +747,7 @@ 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
+a root clique $C^\star$, they have a $C^\star$-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
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