### Added details to the proof of Lemma 10.

parent 0d75f91f
 ... ... @@ -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 graphG$, the strong product$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 $i0$. \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