diff --git a/arxiv_cbd.tex b/arxiv_cbd.tex
index 55f0e8a5510ed0bdf6d4f285acdc06af511fd368..5f270e98135447f9beb848e238241edf681dcf6f 100644
--- a/arxiv_cbd.tex
+++ b/arxiv_cbd.tex
@@ -22,7 +22,7 @@
 
 %\category{} %optional, e.g. invited paper
 
-\relatedversion{A full version of the paper is available at \url{}} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website
+%\relatedversion{A full version of the paper is available at \url{}} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website
 %\relatedversiondetails[linktext={opt. text shown instead of the URL}, cite=DBLP:books/mk/GrayR93]{Classification (e.g. Full Version, Extended Version, Previous Version}{URL to related version} %linktext and cite are optional
 
 %\supplement{}%optional, e.g. related research data, source code, ... hosted on a repository like zenodo, figshare, GitHub, ...
@@ -100,24 +100,24 @@ This result has motivated numerous strengthenings and variations (see \cite{grap
 An attractive feature of touching representations is that it is possible to represent graph classes that are sparse
 (e.g., planar graphs, or more generally, graph classes with bounded expansion~\cite{nesbook}).
 This is in contrast to general intersection representations where the represented class always includes arbitrarily large cliques.
-Of course, whether the class of touching graphs of objects from $\OO$ is sparse or not depends on the system $\OO$.
+Of course, whether the class of touching graphs of objects from $\OO$ is sparse or not depends on the particular system $\OO$.
 For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of boxes in $\mathbb{R}^3$, where the vertices in
 one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$
-boxes (throughout the paper, by a \emph{box} we mean an axis-aligned one, i.e., the Cartesian product of closed intervals of non-zero length).
+boxes (throughout the paper, by \emph{box} we always mean \emph{axis-aligned box}, i.e., the Cartesian product of closed intervals of non-zero length).
 Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of
-long and wide boxes, a condition which can be expressed as follows. For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.
+long and wide boxes. This condition can be expressed as follows. For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.
 We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$.
 A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes
 such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. 
 Let the \emph{comparable box dimension} $\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.
-Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details.
-Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the
-comparable box dimension of graphs in $\GG$ is not bounded.
+We remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details.
+Then, for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. If the
+comparable box dimension of graphs in $\GG$ is not bounded, we write $\cbdim(\GG)=\infty$.
 
 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 many functions $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
+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
@@ -133,7 +133,7 @@ The comparable box dimension of every proper minor-closed class of graphs is fin
 
 Additionally, we show that classes of graphs with finite comparable box dimension are fractionally treewidth-fragile.
 This gives arbitrarily precise approximation algorithms for all monotone maximization problems that are
-expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx}
+expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx},
 or expressible in the first-order logic~\cite{logapx}.
 
 \section{Parameters}
@@ -145,20 +145,20 @@ For any graph $G$, we have $\omega(G)\le 2^{\cbdim(G)}$.
 \end{lemma}
 \begin{proof}
 We may assume that $G$ has bounded comparable box dimension
-witnessed by a representation $f$. To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the
+witnessed by a box representation $f$. To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the
 corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty
 intersections.  Since axis-aligned boxes have the Helly property, there
 is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap
-f(a_w)$.  As each box is full-dimensional, its interior intersects at
+f(a_w)$.  As each box is full-dimensional, their interiors each intersect at
 least one of the $2^d$ orthants at $p$. At the same time, it follows from the definition
 of a touching representation that $f(a_1),\ldots,f(a_d)$ have pairwise disjoint
 interiors, and hence $w \leq 2^d$.
 \end{proof}
 Note that a clique with $2^d$ vertices has a touching representation by comparable boxes in $\mathbb{R}^d$,
 where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$.
-Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$.
+From this together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$.
 
-In the following we consider the chromatic number $\chi(G)$, and two
+The remaining bounds pertain to the chromatic number $\chi(G)$ of a graph $G$, and two
 of its variants.  An \emph{acyclic coloring} (resp. \emph{star coloring}) of a graph $G$ is a proper
 coloring such that any two color classes induce a forest (resp. star forest, i.e., a forest in which each component is a star).  The \emph{acyclic chromatic number} $\chi_a(G)$ (resp. \emph{star chromatic
   number} $\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star)
@@ -175,11 +175,11 @@ Suppose that $G$ has comparable box dimension $d$ witnessed by a representation
 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 $c(v_i)$ 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.
+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 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 $\vol(B)/\vol(f(v_i))-1=5^d-1$.
+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 $v_jv_m,v_mv_i\in E(G)$
@@ -215,16 +215,14 @@ is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedil
 
 \section{Operations}
 
-It is clear that given a touching representation of a graph $G$, one
+It is clear that, given a touching representation of a graph $G$, one
 can 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. 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 remaining 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).
+vertices in $V(G)\setminus V(H)$.  We shall show that these representations also behave nicely under several other basic operations on graphs. To describe the boxes, we shall use the Cartesian product $\times$ defined among boxes of lower dimension (so that $A\times B$ is the box whose projection on some first number of dimensions gives the box $A$, while the projection on the remaining dimensions gives the box $B$), or specify its projections onto every dimension (and in this case write $A[i]$ for the interval obtained from projecting $A$ on its $i^\text{th}$ dimension).
 
 \subsection{Vertex addition}\label{sec-vertad}
 
-Let us start with a simple lemma saying that the addition of a vertex
+Let us start with a simple lemma which says that the addition of a vertex
 increases the comparable box dimension by at most one.  In particular,
 this implies that $\cbdim(G)\le |V(G)|$.
 \begin{lemma}\label{lemma-apex}
@@ -250,7 +248,7 @@ 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
+Indeed, in general $\cbdim(G\boxtimes H)$ 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 $\Omega(\log n)$.
@@ -264,16 +262,17 @@ of a single box; by scaling, we can without loss of generality assume this box i
   $\cbdim(G) + d_H$.
 \end{lemma}
 \begin{proof}
-  The proof simply consists in taking a product of the two
+  It suffices to take a product of the two
   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.
-  \[f((u,v))[i]=\begin{cases}
+  representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ by
+  \begin{equation*}
+  f((u,v))[i]=\begin{cases}
   g(u)[i]&\text{ if $i\le d_G$}\\
-  h(v)[i-d_G]&\text{ if $i > d_G$}
-  \end{cases}\]
+  h(v)[i-d_G]&\text{ if $i > d_G$.}
+  \end{cases}
+  \end{equation*}
   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
@@ -282,11 +281,11 @@ of a single box; by scaling, we can without loss of generality assume this box i
   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
+  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 $i$;
+  then the aforementioned intervals (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 $G\boxtimes H$.
 
@@ -301,8 +300,8 @@ of a single box; by scaling, we can without loss of generality assume this box i
 
 \subsection{Taking a subgraph}
 
-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
+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
@@ -313,7 +312,7 @@ we provide details of the argument.
 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 $V(G')=V(G)$.
+By removing boxes that represent vertices of $G$ that are not in $G'$, we may assume that $V(G')=V(G)$.
 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 $c=\chi_s(G')$.
 
@@ -350,8 +349,8 @@ Let us now combine Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}.
 If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+2\cdot 81^{\cbdim(G')}\le 3\cdot 81^{\cbdim(G')}$.
 \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 $2^{d-1}$. 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 $b<c$, while for every other box in the representation their $i^\text{th}$ interval contains $[b,c]$.
+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 $2^{d-1}$. 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 $b<c$, while the $i^\text{th}$ interval of every other box in the representation contains $[b,c]$.
 
 
 \subsection{Clique-sums}
@@ -363,8 +362,8 @@ edges of the resulting clique.  A \emph{full clique-sum} is a
 clique-sum in which we keep all the edges of the resulting clique.
 The main issue to overcome in obtaining a representation for a (full)
 clique-sum is that the representations of $G_1$ and $G_2$ can be
-``degenerate''. Consider e.g.\ the case that $G_1$ is represented by
-unit squares arranged in a grid; in this case, there is no space to
+``degenerate''. Consider, for example, the case where $G_1$ is represented by
+unit squares arranged in a grid; here there is no space to
 attach $G_2$ at the cliques formed by four squares intersecting in a
 single corner.  This can be avoided by increasing the dimension, but
 we need to be careful so that the dimension stays bounded even after
@@ -408,7 +407,7 @@ $\emptyset$.  Let $\ecbdim(G)$ be the minimum dimension such that $G$
 has an $\emptyset$-clique-sum extendable touching representation by
 comparable boxes.
 
-Let us remark that a clique-sum extendable representation in dimension $d$ implies
+Let us remark that a clique-sum extendable representation in dimension $d$ implies the existence of
 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
@@ -456,8 +455,8 @@ behave well with respect to full clique-sums.
   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 $h_1^\varepsilon(C_1)$.
+To define $h$, for $v\in V(G_1)$, set $h(v)=h_1(v)$.
+  Then 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 $h_1$,
@@ -468,19 +467,19 @@ behave well with respect to full clique-sums.
   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 $p(C)=p_1(C_1)+\varepsilon p_2(C)$,
+  If $C$ is a clique of $G_2$ then 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
+  representation by comparable boxes. Firstly, 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
+  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
+  Next, we 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
@@ -498,7 +497,7 @@ behave well with respect to full clique-sums.
   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)}
+  Finally, we verify the conditions for $C^\star_1$-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 $h(v)\subseteq h_1^\varepsilon(C_1)\subset [0,1)^d$
@@ -509,7 +508,7 @@ behave well with respect to full clique-sums.
   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 $\varepsilon'>0$,
+  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 $G$.
 
@@ -557,11 +556,11 @@ the dimension by $\omega(G)$.
 \end{lemma}
 
 \begin{proof}
-  The proof is essentially the same as the one of
+  The proof is essentially the same as that 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
+  V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We 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 $u\in V(G)\setminus V(C^\star)$,
@@ -589,8 +588,8 @@ the dimension by $\omega(G)$.
   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 $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$.
+  $h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case $v$ and $v_i$ are adjacent.
+  If instead $v_i \notin V(C)$, 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 $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. 
@@ -603,7 +602,7 @@ the dimension by $\omega(G)$.
   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. 
+  Condition (c2) is therefore fulfilled, which completes the proof of the lemma. 
 \end{proof}
 
 
@@ -621,7 +620,7 @@ of $\cbdim(G)$ and $\chi(G)$.
   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 $[a-\alpha,b+\alpha]$ for each vertex $v$ and dimension
-  $i$. We choose $\alpha$ sufficiently small so that the boxes representing
+  $i$. Here, we choose $\alpha$ to be sufficiently small so that the boxes representing
   non-adjacent vertices remain disjoint, and thus the resulting representation $h_1$ is
   an intersection representation of the same graph $G$.  Moreover, observe that
   for every clique $C$ of $G$, the intersection $I_C=\bigcap_{v\in V(C)} h_1(v)$ is
@@ -631,19 +630,19 @@ 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
-  $h^\varepsilon(C)$. Formally we define $h_2$ as follows.
+  $h^\varepsilon(C)$. Formally we define $h_2$ as
   \[h_2(u)[i]=\begin{cases}
   h_1(u)[i]&\text{ if $i\le d$}\\
   [1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\
   [0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\
-  [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)}
+  [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$).}
   \end{cases}\]
   For any clique $C$ of $G$, let $c(C)$ denote the color set $\{c(u)\ |\ u\in V(C)\}$.
   We now set
   \[p_2(C)[i]=\begin{cases}
   p_1(C)[i] &\text{ if $i\le d$}\\
   2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\
-  1/2 &\text{ otherwise}
+  1/2 &\text{ otherwise.}
   \end{cases}
   \]
   As $h_2$ is an extension of $h_1$, and as in each dimension $j>d$,
@@ -671,7 +670,7 @@ of $\cbdim(G)$ and $\chi(G)$.
 \end{proof}
 
 A touching representation of axis-aligned boxes in $\mathbb{R}^d$ is said \emph{fully touching} if any two intersecting boxes intersect on a $(d-1)$-dimensional box. Note that the construction above is fully touching.
-Indeed, two intersecting boxes corresponding to vertices $u,v$ of colors $c(u) < c(v)$, only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} lead to the following.
+Indeed, two intersecting boxes corresponding to vertices $u,v$ of colors $c(u) < c(v)$, only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} leads to the following.
 \begin{corollary}
 \label{cor-fully-touching}
 Any graph $G$ has a fully touching representation of comparable axis-aligned boxes in $\mathbb{R}^d$, where $d= \cbdim(G) + 3^{\cbdim(G)}$.
@@ -683,11 +682,11 @@ full clique-sums.
 \begin{corollary}
 \label{cor-csum}
 Let $\GG$ be a class of graphs of chromatic number at most $k$.  If $\GG'$ is the class
-of graphs obtained from $\GG$ by repeatedly performing full clique-sums, then
-\[\cbdim(\GG')\le \cbdim(\GG) + 2k.\]
+of all graphs that can be obtained from $\GG$ by repeatedly performing full clique-sums, then
+$\cbdim(\GG')\le \cbdim(\GG) + 2k.$
 \end{corollary}
 \begin{proof}
-Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
+Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by a sequence of full clique-sums.
 Without loss of generality, the labelling of the graphs is chosen so that we first
 perform the full clique-sum on $G_1$ and $G_2$, then on the resulting graph and $G_3$, and so on.
 Let $C^\star_1=\emptyset$ and for $i=2,\ldots,m$, let $C^\star_i$ be the root clique of $G_i$ on which it is
@@ -697,7 +696,7 @@ where $d=\cbdim(\GG) + 2k$.  Repeatedly applying Lemma~\ref{lem-cs}, we conclude
 $\cbdim(G)\le d$.
 \end{proof}
 
-By Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}, this gives the following bounds.
+Putting the preceding corollary together with Lemma~\ref{lemma-chrom} and Lemma~\ref{lemma-subg}, we now have the following bounds.
 \begin{corollary}\label{cor-csump}
 Let $\GG$ be a class of graphs of comparable box dimension at most $d$.
 \begin{itemize}
@@ -709,7 +708,7 @@ has comparable box dimension at most $1250^d$.
 \end{corollary}
 \begin{proof}
 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 $\GG'$.
+from Lemma~\ref{lemma-chrom}.  For the latter, 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.
 Note that the vertices of any clique get pairwise different colors, and thus by permuting the colors,
@@ -783,7 +782,7 @@ and that for $j\neq i$, $p(C)[j]$ is in the interior of $h(v_i)[j]$, implying
 $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for sufficiently small $\varepsilon>0$.
 \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}
+It is worth noting that the bound on the comparable box dimension of Theorem~\ref{thm-ktree} actually
 extends to graphs of treewidth at most $k$.
 \begin{corollary}\label{cor-tw}
 Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
@@ -841,9 +840,8 @@ Suppose $G$ is a connected planar graph and $v$ is a vertex of $G$.  For an inte
 give each vertex at distance $d$ from $v$ the color $d\bmod k$.  Then deleting the vertices of any of the $k$ colors
 results in a graph of treewidth at most $3k$. This fact (which follows from the result of Robertson and Seymour~\cite{rs3}
 on treewidth of planar graphs of bounded radius) is (in the modern terms) the basis of Baker's technique~\cite{baker1994approximation}
-for design of approximation algorithms.  However, even quite simple graph classes (e.g., strong products of three paths~\cite{gridtw})
-do not admit such a coloring (where the removal of any color class results in a graph of bounded treewidth).
-However, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx}
+for design of approximation algorithms.  However, even quite simple graph classes, such as the strong products of three paths~\cite{gridtw}, do not admit such a coloring where the removal of any color class results in a graph of bounded treewidth.
+Nonetheless, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx}
 and applies to much more general graph classes, including all graph classes with strongly sublinear separators and bounded maximum degree~\cite{twd}.
 
 We say that a class of graphs $\GG$ is \emph{fractionally treewidth-fragile} if there exists a function $f$ such that
@@ -874,23 +872,26 @@ Our main result is that all graph classes of bounded comparable box dimension ar
 We will show the result in a more general setting, motivated by concepts from~\cite{subconvex} and by applications to related
 representations. The argument is motivated by the idea used in the approximation algorithms for disk graphs
 by Erlebach et al.~\cite{erlebach2005polynomial}. Before introducing this more general setting, and as a warm-up, let us outline
-how to prove that disk graphs of thickness $t$ are fractionally treewidth-fragile. Consider first unit disk graphs. 
-By partitionning the plane with a random grid $\HH$, having squared cells of side-length $2k$, any unit disk has probability $1/2k$ 
-to intersect a vertical (resp. horizontal) line of the grid. By union bound, any disk has probability at most $1/k$ to intersect 
-the grid. Considering this probability distribution, let us now show that removing the disks intersected by the grid leads to a 
+how to prove that disk graphs of thickness $t$ are fractionally treewidth-fragile. 
+
+We first consider unit disk graphs. 
+By partitioning the plane with a random grid $\HH$ with square cells of side-length $2k$, any unit disk has probability $1/2k$ 
+of intersecting a vertical (resp. horizontal) line of the grid. Using a union bound, any disk has probability at most $1/k$ of intersecting 
+the grid. Using this probability distribution, we show that removing the disks intersected by the grid leads to a 
 unit disk graph of bounded treewidth. Indeed, in such a graph any connected component corresponds to unit disks contained in the 
-same cell of the grid. Such cell having area bounded by $4k^2$, there are at most $16tk^2/\pi$ disks contained in a cell. 
-The size of the connected components being bounded, so is the treewidth. Note that this distribution also works if we are given
-disks whose diameter lie in a certain range. If any diameter $\delta$ is such that $1/c \le \delta \le 1$, then the same process
-with a random grid of $2k\times 2k$ cells, ensures that any disk is deleted with probability at most $1/k$, while now the
-connected components have size at most $4tc^2k^2/\pi$. Dealing with arbitrary disk graphs (with any diameter $\delta$ being in the range 
-$0< \delta \le 1$) requires to delete more disks. This is why each $(2k\times 2k)$-cell is now partitionned in a quadtree-like manner. 
+same cell of the grid. Each cell has area bounded by $4k^2$, so there are at most $16tk^2/\pi$ disks contained in a cell. 
+This bounds the size of any connected component, and so the treewidth is also bounded. 
+
+Note that the above distribution also works if we are given
+disks whose diameter lie in a certain range. That is, for any diameter $\delta$ with $1/c \le \delta \le 1$, applying the same process
+with a random grid of $2k\times 2k$ cells ensures that any disk is deleted with probability at most $1/k$, and the
+connected components have size at most $4tc^2k^2/\pi$. Dealing with arbitrary disk graphs (with any diameter $\delta$ in the range 
+$0< \delta \le 1$) necessitates deleting more disks. This can be handled by partitioning each $(2k\times 2k)$-cell in a quadtree-like manner. 
 Now a disk with diameter between $\ell /2$ and $\ell$ (with $\ell =1/2^i$ for some integer $i\ge 0$) is deleted if it is not contained 
-in a $(2k\ell \times 2k\ell)$-cell of a quadtree. It is not hard to see that a disk is deleted with probability at most $1/k$.
-To prove that the remaining graph has bounded treewidth one should consider the following tree decomposition $(T,\beta)$. The 
-tree $T$ is obtained by linking the roots of the quadtrees we used (as trees) to a new common root. 
-Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all the disks of diameter at least $\ell/2$ intersecting $C$.
-To see that such bag is bounded consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any
+in a $(2k\ell \times 2k\ell)$-cell of a quadtree. It is straightforward to see that each disk is deleted with probability at most $1/k$.
+To prove that the remaining graph has bounded treewidth, one should consider the following tree decomposition $(T,\beta)$. Here, the tree $T$ is obtained by linking the roots of the quadtrees used (as trees) to a new common root. 
+Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all disks of diameter at least $\ell/2$ intersecting $C$.
+To see that such bag is bounded, consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any
 disk in $\beta(C)$ intersects $C'$ on an area at least $\pi\ell^2/16$. This implies that $|\beta(C)| \le 16t(2k+1)^2 / \pi$.
 
 Let us now give a detailed proof in a more general setting.
@@ -927,8 +928,8 @@ 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
-it is defined as follows.
+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 by the following process.
 \begin{itemize}
 \item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$.
 \item For $i=2,\ldots, n$, let $\ell_{i,j} = \ell_{i-1,j}$, if
@@ -938,17 +939,17 @@ it is defined as follows.
   \min\{\ell_{i-1,j}/b \ |\ b\in
   \mathbb{N}^+ \text{ and } \ell_{i-1,j}/b \ge ksd|\omega(v_i)[j]|\}$.
 \end{itemize}
-Let the real $x_j\in [0,\ell_{1,j}]$ be chosen uniformly at random,
+Choose $x_j\in [0,\ell_{1,j}]$ uniformly at random,
 and let $\HH^i_j$ be the set of hyperplanes in $\mathbb{R}^d$
 consisting of the points whose $j$-th coordinate is equal to
 $x_j+m\ell_{i,j}$ for some $m\in\mathbb{Z}$. As $\ell_{i,j}$ is a
 multiple of $\ell_{i',j}$ whenever $i\le i'$, we have that $\HH^i_j
 \subseteq \HH^{i'}_j$ whenever $i\le i'$.  For $i\in\{1,\ldots,n\}$,
 the \emph{$i$-grid} is $\HH^i=\bigcup_{j=1}^d \HH^i_j$, and we let the
-$0$-grid $\HH^0=\emptyset$.  Similarly as above we have that $\HH^i
+$0$-grid $\HH^0=\emptyset$. Then, as above, we have that $\HH^i
 \subseteq \HH^{i'}$ whenever $i\le i'$.
 
-We let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such
+Let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such
 that the box $\omega(v_a)$ intersects some hyperplane $H\in \HH^a$,
 that is such that $x_j+m\ell_{a,j}\in \omega(v_a)[j]$, for some
 $j\in\{1,\ldots,d\}$ and some $m\in \mathbb{Z}$.  First, let us argue
@@ -962,13 +963,13 @@ Combining these inequalities,
 \[\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_{a'})[j]|}=\frac{1}{kd}.\]
 By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$.
 
-Let us now bound the treewidth of $G-X$.  
+We now bound the treewidth of $G-X$.  
 For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus \bigl(\bigcup_{H\in \HH^a} H\bigr)$.
 A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge 0$.
 A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$.
 Note that there exists a rooted tree $T$ whose vertices are
 the non-empty cells and such that for $x,y\in V(T)$, we have $x\preceq y$ if and only if $x\subseteq y$.
-For each non-empty cell $C$, let us define $\beta(C)$ as the set of vertices $v_i\in V(G-X)$ such that
+For each non-empty cell $C$, define $\beta(C)$ to be the set of vertices $v_i\in V(G-X)$ such that
 $\iota(v)\cap C\neq\emptyset$ and $C$ is an $a$-cell for some $a\ge i$.
 
 Let us show that $(T,\beta)$ is a tree decomposition of $G-X$.  For each $v_j\in V(G-X)$, the $j$-grid is disjoint from $\omega(v_j)$,
@@ -977,10 +978,10 @@ We have $\omega(v_j)\cap \iota(v_i)\neq\emptyset$, and thus $\iota(v_i)\cap C\ne
 Finally, suppose that $v_j\in C'$ for some $C'\in V(T)$.  Then $C'$ is an $a$-cell for some $a\ge j$, and since
 $\iota(v_j)\cap C'\neq\emptyset$ and $\iota(v_j)\subset C$, we conclude that $C'\subseteq C$, and consequently $C'\preceq C$.
 Moreover, any cell $C''$ such that $C'\preceq C''\preceq C$ (and thus $C'\subseteq C''\subseteq C$) is an $a'$-cell
-for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq \iota(v_j)\cap C'\neq\emptyset$, implying $v_j\in\beta(C'')$.
+for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq \iota(v_j)\cap C'\neq\emptyset$, which implies that $v_j\in\beta(C'')$.
 It follows that $\{C':v_j\in\beta(C')\}$ induces a connected subtree of $T$.
 
-Finally, let us bound the width of the decomposition $(T,\beta)$.  Let $C$ be a non-empty cell and let $a$ be maximum such that $C$
+Finally, we bound the width of the decomposition $(T,\beta)$.  Let $C$ be a non-empty cell and let $a$ be maximum number for which $C$
 is an $a$-cell.  Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ldots, $\ell_{a,d}$.  Consider $j\in\{1,\ldots,d\}$:
 \begin{itemize}
 \item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$.
@@ -988,11 +989,11 @@ is an $a$-cell.  Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ld
 \item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integer $b\ge 2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that
 \[\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.\]
 \end{itemize}
-Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$.  Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
+Hence, in all cases we have $\ell_{a,j}<2ksd |\omega(v_a)[j]|$.  Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
 For any $v_i\in \beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
 there exists a translation $B_i$ of $\omega(v_a)$ that intersects $C\cap \iota(v_i)$ and such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$.
 Note that as $B_i$ intersects $C$, we have that $B_i\subseteq C'$.
-Since the representation has thickness at most $t$,
+Using the initial assumption that the representation has thickness at most $t$, we now have
 \begin{align*}
 \vol(C')&\ge \vol\left(C'\cap \bigcup_{v_i\in \beta(C)\setminus\{v_a\}} \iota(v_i)\right)\\
 &\ge \vol\left(\bigcup_{v_i\in \beta(C)\setminus\{v_a\}} B_i\cap\iota(v_i)\right)\\