\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{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.}

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@@ -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

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 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}

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@@ -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]=

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

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

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

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

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

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@@ -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 $h(u)\sqsubseteq h(v)$.

\end{itemize}

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

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

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.