### few changes, and new appendix to reach the limit on the nb of lines

parent d9f8f497
 ... ... @@ -117,8 +117,8 @@ comparable box dimension of graphs in $\GG$ is not bounded. 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 any function $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite comparable box dimension. Dvo\v{r}\'ak et al.~\cite{wcolig} 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 (the previous example is obtained by subdividing edges of every graph suitably many times~\cite{covcol}). ... ... @@ -174,19 +174,17 @@ For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G) We focus on the star chromatic number and note that the chromatic number and the acyclic chromatic number may be bounded similarly. Suppose that$G$has comparable box dimension$d$witnessed by a representation$f$, and let$v_1, \ldots, v_n$be the vertices of$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(i)$is the smallest color such that$c(i)\neq c(j)$for any$jj$. Now define a greedy coloring$c$so that$c(v_i)$is the smallest color such that$c(v_i)\neq c(v_j)$for any$jj$such that$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_1i$for which$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_j)/\vol(f(v_i))-1=5^d-1$.$v_j$such that$ji$for which$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$. A similar argument shows that the number of indices$m$such that$m0$sufficiently small so that$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$C^\star_2$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$h_1$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.$u,v \in V(G_2) \setminus V(C^\star_2)$) 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$v \in V(G_2) \setminus V(C^\star_2)$. 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$h_2(u)$and$h_2(v)$intersect, and by (v1) and (v2) for$h_2$, we conclude that$h_2(v)=[0,\alpha]$for some positive$\alpha<1$. Therefore,$p_1(C_1)\in h(v)$. 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$p_1(C_1)\in h(u)\cap h(v)$. 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$h_2^\varepsilon(C)\subseteq (0,1)$, and thus$h^{\varepsilon'}(C)\subseteq h_1^\varepsilon(C_1)\setminus\{p(C_1)\}$. By (c2) for$h_1$, we have$h(v)\cap h_1^\varepsilon(C_1)=h_1(v)\cap h_1^\varepsilon(C_1)=p(C_1)$, 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$h_2(v)=[-1,0]$and$h_2^\varepsilon(C)\subseteq [0,1)$is the single point$p_2(C)=0$, and thus$p(C)=p_1(C_1)$and$h^{\varepsilon'}(C)=[p_1(C_1),p_1(C_1)+\varepsilon']$. Recall that the property (c2) of$h_1$implies$h(v)\cap h_1^\varepsilon(C_1)=\{p(C_1)\}$, and thus$h(v)\cap h^{\varepsilon'}(C)=\{p(C)\}$. 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 proof is in the appendix, but the idea is to translate (allowing also exchanges of dimensions) and scale$h_2$to fit in$h_1^\varepsilon(C_1)$. The following lemma enables us to pick the root clique at the expense of increasing the dimension by$\omega(G)$. \begin{lemma}\label{lem-apex-cs} ... ... @@ -557,56 +447,8 @@ the dimension by$\omega(G)$.$C^\star$-clique-sum extendable touching representation by comparable 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$G\setminus V(C^\star)$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 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$u\in V(G)\setminus V(C^\star)$, 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$u\notin V(C^\star)$. 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$C'=C\setminus \{v_1,\ldots,v_k\}$, and we set$p(C)[i] = \alpha p'(C')[i-k]$. By the construction, it is clear that$h$is a touching representation of$G$. As$h'(u) \sqsubset h'(v)$implies that$h(u) \sqsubset h(v)$, and as$h(u) \sqsubset h(v_i)$for every$u\in V(G)\setminus V(C^\star)$and every$v_i \in V(C^\star)$, we have that$h$is a representation by comparable boxes. For the$C^\star$-clique-sum extendability, the \textbf{(vertices)} conditions hold by the construction. For the \textbf{(cliques)} condition (c1), let us consider distinct cliques$C_1$and$C_2$of$G$such that$|V(C_1)| \ge |V(C_2)|$, and let$C'_i=C_i\setminus V(C^\star)$. 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$C'_1 \neq C'_2$, then$p'(C'_1) \neq p'(C'_2)$, which implies$p(C_1) \neq p(C_2)$by construction. 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$h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. By the property (c2) of$h'$, we have$h^\varepsilon(C)[i] \subseteq h(v)[i]$for every$i>k$, except one, for which$h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. Next, let us consider a vertex$v\in V(G)\setminus V(C^\star)$and a clique$C$of$G$not containing$v$. As$v\notin V(C')$, the condition (c2) for$h'$implies that$p'(C')$is disjoint from$h'(v)$, and thus$p(C)$is disjoint from$h(v)$. Finally, we consider a vertex$v_i \in V(C^\star)$. 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$h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$. Condition (c2) is thus fulfilled and this completes the proof of the lemma. \end{proof} The proof is also in the appendix, but it essentially the same as the one of Lemma~\ref{lemma-apex}. The following lemma provides an upper bound on$\ecbdim(G)$in terms of$\cbdim(G)$and$\chi(G)$. \begin{lemma}\label{lem-ecbdim-cbdim} ... ... @@ -640,7 +482,7 @@ of$\cbdim(G)$and$\chi(G)$. 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) &\text{ if i\le d}\\ 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} \end{cases} ... ... @@ -696,12 +538,12 @@ Let \GG be a class of graphs of comparable box dimension at most d. \begin{itemize} \item The class \GG' of graphs obtained from \GG by repeatedly performing full clique-sums has comparable box dimension at most d + 2\cdot 3^d. \item The class of graphs obtained from \GG by repeatedly performing clique-sums has comparable box dimension at most 625^d. \item The closure of \GG' by taking subgraphs has comparable box dimension at most 1250^d. \end{itemize} \end{corollary} \begin{proof} The former bound directly follows from Corllary~\ref{cor-csum} and the bound on the chromatic number 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'. 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. ... ... @@ -777,25 +619,10 @@ h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon] for suffici \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} extends to graphs of treewidth at most k. extends to graphs of treewidth at most k (see the proof in the appendix). \begin{corollary}\label{cor-tw} Every graph G satisfies \cbdim(G)\le\tw(G)+1. \end{corollary} \begin{proof} Let k=\tw(G). Observe that there exists a k-tree T with the root clique C^\star such that G\subseteq T-V(C^\star). Inspection of the proof of Theorem~\ref{thm-ktree} (and Lemma~\ref{lem-cs}) shows that we obtain a representation h of T-V(C^\star) in \mathbb{R}^{k+1} such that \begin{itemize} \item the vertices are represented by hypercubes of pairwise different sizes, \item if uv\in E(T-V(C^\star)) and h(u)\sqsubseteq h(v), then h(u)\cap h(v) is a facet of h(u) incident with its point with minimum coordinates, and \item for each vertex u and each facet of h(u) incident with its point with minimum coordinates, there exists 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). Since the hypercubes of h have pairwise different sizes, the resulting touching representation of G is by comparable boxes. \end{proof} As every planar graph G has a touching representation by cubes in \mathbb{R}^3~\cite{felsner2011contact}, we have that \cbdim(G)\le 3. ... ... @@ -807,7 +634,7 @@ Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain: \begin{corollary}\label{cor-genus} For every graph G of Euler genus g, there exists a supergraph G' of G such that \cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil. Consequently, \[\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$ Consequently, $\cbdim(G)\le 3\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$ \end{corollary} Similarly, we can deal with proper minor-closed classes. ... ... @@ -816,7 +643,7 @@ Let$\GG$be a proper minor-closed class. Since$\GG$is proper, there exists$ By Theorem~\ref{thm-prod}, there exists $k$ such that every graph in $\GG$ is a subgraph of a graph obtained by repeated clique-sums from extended $k$-tree-grids. As we have seen, $k$-tree-grids have comparable box dimension at most $k+2$, and by Lemma~\ref{lemma-apex}, extended $k$-tree-grids have comparable box dimension at most $2k+2$. By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le 625^{2k+2}$. By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le 1250^{2k+2}$. \end{proof} Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$ ... ... @@ -897,7 +724,7 @@ 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}$is a positive integer. Formally Let us define$\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}$is a positive integer. Formally it is defined as follows. \begin{itemize} \item Let$\ell_{1,j}=ksd|\omega(v_1)[j]|$. ... ... @@ -989,4 +816,206 @@ has a sublinear separator of size$O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr). \end{corollary} \bibliography{data} \appendix \section{Omitted proofs} \newtheorem*{lemma-A}{Lemma~\ref{lem-cs}} \begin{lemma-A} 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 $C^\star_2$ 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-A} \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 $h_1^\varepsilon(C_1)$. 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 $C^\star_2$ 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 $h_1$ 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. $u,v \in V(G_2) \setminus V(C^\star_2)$) 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 $v \in V(G_2) \setminus V(C^\star_2)$. 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 $h_2(u)$ and $h_2(v)$ intersect, and by (v1) and (v2) for $h_2$, we conclude that $h_2(v)=[0,\alpha]$ for some positive $\alpha<1$. Therefore, $p_1(C_1)\in h(v)$. 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 $p_1(C_1)\in h(u)\cap h(v)$. 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 $h_2^\varepsilon(C)\subseteq (0,1)$, and thus $h^{\varepsilon'}(C)\subseteq h_1^\varepsilon(C_1)\setminus\{p(C_1)\}$. By (c2) for $h_1$, we have $h(v)\cap h_1^\varepsilon(C_1)=h_1(v)\cap h_1^\varepsilon(C_1)=p(C_1)$, 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 $h_2(v)=[-1,0]$ and $h_2^\varepsilon(C)\subseteq [0,1)$ is the single point $p_2(C)=0$, and thus $p(C)=p_1(C_1)$ and $h^{\varepsilon'}(C)=[p_1(C_1),p_1(C_1)+\varepsilon']$. Recall that the property (c2) of $h_1$ implies $h(v)\cap h_1^\varepsilon(C_1)=\{p(C_1)\}$, and thus $h(v)\cap h^{\varepsilon'}(C)=\{p(C)\}$. 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} \newtheorem*{lemma-B}{Lemma~\ref{lem-apex-cs}} \begin{lemma-B} For any graph $G$ and any clique $C^\star$, the graph $G$ admits a $C^\star$-clique-sum extendable touching representation by comparable boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$. \end{lemma-B} \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 $G\setminus V(C^\star)$ by