Added details to the proof of Lemma 10.

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