\author{Daniel Gon\c{c}alves}{LIRMM, Université de Montpellier, CNRS, Montpellier, France}{goncalves@lirmm.fr}{}{Supported by the ANR grant GATO ANR-16-CE40-0009.}

\author{Daniel Gon\c{c}alves}{LIRMM, Université de Montpellier, CNRS, Montpellier, France}{goncalves@lirmm.fr}{}{Supported by the ANR grant GATO ANR-16-CE40-0009.}

\author{Abhiruk Lahiri}{Charles University, Prague, Czech Republic}{abhiruk@iuuk.mff.cuni.cz}{}{Partially supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}

\author{Abhiruk Lahiri}{Charles University, Prague, Czech Republic}{abhiruk@iuuk.mff.cuni.cz}{}{Partially supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}

\author{Jane Tan}{Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom}{jane.tan@maths.ox.ac.uk}{}{}

\author{Jane Tan}{Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom}{jane.tan@maths.ox.ac.uk}{}{}

\author{Torsten Ueckerdt}{Karlsruhe Institute of Technology, Karlsruhe, Germany}{orsten.ueckerdt@kit.edu}{}{}

\author{Torsten Ueckerdt}{Karlsruhe Institute of Technology, Karlsruhe, Germany}{torsten.ueckerdt@kit.edu}{}{}

\authorrunning{Zden\v{e}k Dvo\v{r}\'ak et al.}

\authorrunning{Zden\v{e}k Dvo\v{r}\'ak et al.}

...

@@ -221,7 +221,7 @@ easily obtains a touching representation by boxes of an induced

...

@@ -221,7 +221,7 @@ easily obtains a touching representation by boxes of an induced

subgraph $H$ of $G$ by simply deleting the boxes corresponding to the

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

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

consider other basic operations on graphs. In the following, to describe

the boxes, we are going to use the Cartesian product $\times$ defined among boxes ($A\times B$ is the box whose projection on the first dimensions gives the box $A$, while the projection on the remaing dimensions gives the box $B$) or we are going to provide its projections for every dimension ($A[i]$ is the interval obtained from projecting $A$ on its $i^\text{th}$ dimension).

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).

\subsection{Vertex addition}\label{sec-vertad}

\subsection{Vertex addition}\label{sec-vertad}

...

@@ -960,7 +960,7 @@ has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).

...

@@ -960,7 +960,7 @@ has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).

if $i\le k$ then let $h(u)[i]=[0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i]=

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]=

[1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i]=

\alpha h'(u)[i-k]$, for some $\alpha>0$. The value $\alpha>0$

\alpha h'(u)[i-k]$, for some $\alpha>0$. The value $\alpha>0$

is chosen suffciently small so that $h(u)[i]\subset[0,1)$ whenever $u\notin V(C^\star)$.

is chosen sufficiently small so that $h(u)[i]\subset[0,1)$ whenever $u\notin V(C^\star)$.

We proceed similarly for the clique points. For any

We proceed similarly for the clique points. For any

clique $C$ of $G$, if $i\le k$ then let $p(C)[i]=0$ if $v_i \in V(C)$,

clique $C$ of $G$, if $i\le k$ then let $p(C)[i]=0$ if $v_i \in V(C)$,

and $p(C)[i]=1/4$ if $v_i \notin V(C)$. For $i>k$ we refer to the clique point $p'(C')$ of $C'=C\setminus

and $p(C)[i]=1/4$ if $v_i \notin V(C)$. For $i>k$ we refer to the clique point $p'(C')$ of $C'=C\setminus

...

@@ -1014,7 +1014,7 @@ with its point with minimum coordinates, and

...

@@ -1014,7 +1014,7 @@ with its point with minimum coordinates, and

at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.

at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.

\end{itemize}

\end{itemize}

If for some $u,v\in V(G)$, we have $uv\in E(T)\setminus E(G)$, where without loss of generality $h(u)\sqsubseteq h(v)$,

If for some $u,v\in V(G)$, we have $uv\in E(T)\setminus E(G)$, where without loss of generality $h(u)\sqsubseteq h(v)$,

we now alter the representation by shrinking $h(u)$ slighly away from $h(v)$ (so that all other touchings are preserved).

we now alter the representation by shrinking $h(u)$ slightly away from $h(v)$ (so that all other touchings are preserved).

Since the hypercubes of $h$ have pairwise different sizes, the resulting touching representation of $G$ is by comparable boxes.

Since the hypercubes of $h$ have pairwise different sizes, the resulting touching representation of $G$ is by comparable boxes.