### added justifications in the proofs wrt clique-sum extendability

parent fe97fd5d
 ... ... @@ -162,7 +162,7 @@ vertices. We proceed similarly to bound the chromatic number. \section{Operations} It is clear that given a touching representation of a graph $G$, one easily obtains a touching representation with boxes of an induced 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. ... ... @@ -189,8 +189,8 @@ 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$or distinct vertices$(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$, but the obtained representation may contain uncomparable boxes. Thus, ... ... @@ -208,7 +208,7 @@ overcome this issue, by constraining one of the representations. The proof simply consists in taking a product of the two representations. Indeed, consider a touching respresentation with comparable boxes$g$of$G$in$\mathbb{R}^{d_G}$, with$d_G=\cbdim(G)$, and the depresentation$h$of$H$. Let us define a$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} ... ... @@ -221,7 +221,7 @@ overcome this issue, by constraining one of the representations. 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 interior disjoint. Indeed, the same hyperplane that separates g(u) 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. ... ... @@ -231,7 +231,7 @@ overcome this issue, by constraining one of the representations. 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_G} respectively, and conversely p can be obtained \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. \end{proof} ... ... @@ -247,10 +247,6 @@ Corollary~25 in~\cite{subconvex}, but since the setting is somewhat different and the construction of~\cite{subconvex} uses rotated boxes, we provide details of the argument. Next, let us show a bound on the comparable box dimension of subgraphs. \begin{lemma}\label{lemma-subg} If G is a subgraph of a graph G', then \cbdim(G)\le \cbdim(G')+\chi^2_s(G'). \end{lemma} ... ... @@ -340,7 +336,7 @@ boxes or not) h of G in \mathbb{R}^d is called Note that we may consider that the root clique is 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 with has a \emptyset-clique-sum extendable touching representation by comparable boxes. The following lemma ensures that clique-sum extendable representations behave well with respect to full clique-sums. ... ... @@ -352,12 +348,8 @@ clique-sums. respectively. Let G be the graph obtained after 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 with comparable boxes h in \mathbb{R}^{\max(d_1,d_2)}. Furthermore, we have that the aspect ratio of h(v) is the same as the one of h_1(v), if v\in V(G_1), or the same as h_2(v) if v\in V(G_2)\setminus V(C^*_2). \note{ shall we remove the aspect ratio thing ? only needed for the hypercubes of the k-trees, but those are not really needed...} sum extendable representation by comparable boxes h in \mathbb{R}^{\max(d_1,d_2)}. \end{lemma} \begin{proof} The idea is to translate (allowing also exchanges of dimensions) and ... ... @@ -394,26 +386,25 @@ clique-sums. 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] + [0, \epsilon/2], otherwise (for any j not in the [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 vertices of \{v_1,\ldots,v_k\}, then the image of h_2(v_i) fit inside the (previously defined) box 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] + [0, \frac14], otherwise. 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 these clique in h is 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 representation with comparable boxes. The fact that the boxes are representation by comparable boxes. The fact that the boxes are comparable follows from the fact that those of V(G_1) (resp. V(G_1)) are comparable in h_1 (resp. h_2) with the (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 ... ... @@ -423,19 +414,27 @@ clique-sums. 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 is clear that h(u) and h(v) are interior disjoint, and that they 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) are interior disjoint. Furthermore, if they intersect then u\in have disjoint interiors. Furthermore, if they intersect then u\in V(C_1), say u=v_1, and h(v) = [p(C_1), p(C_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 (vertex) conditions hold, such as the (clique) conditions. 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]. \end{proof} The following lemma shows that any graphs has a C^*-clique-sum ... ... @@ -443,7 +442,7 @@ extendable representation in \mathbb{R}^d, for d= \omega(G) + \ecbdim(G) and for any clique C^*. \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 with comparabe C^*-clique-sum extendable touching representation by comparabe boxes in \mathbb{R}^d, for d = |V(C^*)| + \ecbdim(G\setminus V(C^*)). \end{lemma} ... ... @@ -458,12 +457,40 @@ extendable representation in \mathbb{R}^d, for d= \omega(G) + d_{v_i} = i. For each vertex u\in V(G)\setminus V(C^*), 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] = h'(u)[i-k]. We proceed similarly for the clique points. For any \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^*). 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 to refer to the clique point p'(C') of C'=C\setminus \{v_1,\ldots,v_k\}, as we set p(C)[i] = p'(C')[i-k]. One can easily check that h is as desired. \{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. 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 (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, 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. 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_i \in V(C^*), and we note that for any clique C containing v_i we have that h^\epsilon(C)[i] \cap h(v_i)[i] = [0,\epsilon]\cap [-1,0] = \{0\}, and h^\epsilon(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^\epsilon(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 following lemma provides an upper bound on \ecbdim(G) in terms ... ... @@ -472,7 +499,7 @@ of \cbdim(G) and \chi(G). For any graph G, \ecbdim(G) \le \cbdim(G) + \chi(G). \end{lemma} \begin{proof} Let h be a touching representation with comparable boxes of G in Let h be a touching representation by comparable boxes of G in \mathbb{R}^d, with d=\cbdim(G), and let c be a \chi(G)-coloring of G. We start with a slightly modified version of h. We first scale h to fit in (0,1)^d, and for a ... ... @@ -495,20 +522,36 @@ of \cbdim(G) and \chi(G). [0,2/5]&\text{ if c(u) = i-d}\\ [2/5,4/5]&\text{ otherwise (if c(u) > i-d > 0)} \end{cases}$$ For any clique$C'$of$G$, let us denote$c(C)$, the set For any clique$C'$of$G$, let us denote$c(C')$, the color set$\{c(u)\ |\ u\in V(C')\}$, and let$C$be one of the maximal cliques containing$C'$.We now set containing$C'$. We now set $$p_2(C')[i]=\begin{cases} p_1(C) &\text{ if i\le d}\\ 2/5 &\text{ if i-d \in c(C')}\\%\text{if \exists u\in V(C) with c(u) = i-d}\\ 1/2 &\text{ otherwise} \end{cases}$$ One can now check that this is a$\emptyset$-clique-sum extendable touching representation with comparable boxes. In particular, one should notice that for a vertex$u$of a clique$C'$we have that$h_2^{\epsilon}(C')[i] \subset h_2(u)[i]$for every dimension$i$, except if$c(u)=i-d$. If$c(u)=i-d$, then$h_2(u)[i] \cap h_2^{\epsilon}(C')[i] = \{2/5\}$.$$As$h_2$is an extension of$h_1$, and as for all the extra dimensions$j$($j>d$) we have that$2/5 \in h_2(v)[j]$for every vertex$v$, we have that$h_2$is an intersection representation of$G$. To prove that it is touching consider two adjacent vertices$u$and$v$such that$c(u)d$) the length of$h_2(v)[j]$is$2/5$for every vertex$v$, we have that the boxes in$h_2$are comparables boxes. For the$\emptyset$-clique-sum extendability, the (vertices) conditions clearly hold. For the (cliques) conditions, let us first note that the points$p_1(C)$, defined for the maximum cliques, are necessarily distinct. This impies that two cliques$C_1$and$C_2$, which clique points$p_2(C_1)$and$p_2(C_2)$are based on distinct maximum cliques, necessarily lead to distinct points. In the case that$C_1$and$C_2$belong to some maximal clique$C$, we have that$c(C_1) \neq c(C_2)$and this implies by construction that$p_2(C_1)$and$p_2(C_2)$are distinct. Thus (c1) holds. By construction of$h_1$, we have that if$h_2^{\epsilon}(C')[i] \cap h_2(v)[i]$is non-empty for every$i\le d$, then we have that$h_2^{\epsilon}(C')[i] \subset h_2(v)[i]$for every$i\le d$, and we have that$v$belongs to some maximal clique$C$containing$C'$. If$v\notin V(C')$note that$p_2(C')[d+c(v)] = 1/2 \notin[0,2/5]=h_2(v)[d+c(v)]$, while if$v\in V(C')$we have that$h_2^{\epsilon}(C')[i] \subset [2/5,1/2+\epsilon] \subset h_2(v)[i]$for every dimension$i>d$, except if$c(v)=i-d$, and in that case$h_2(v)[i] \cap h_2^{\epsilon}(C')[i] = [0,2/5]\cap[2/5,2/5+\epsilon] = \{2/5\}$. We thus have that (c2) holds, and this concludes the proof of the lemma. \end{proof} ... ... @@ -531,7 +574,7 @@ product of a path and a$k$-tree, by adding at most$k$apex vertices. \end{theorem} Let us first bound the comparable box dimension of a graph in terms of its Euler genus. As paths and$m$-clique admit touching its Euler genus. As paths and$m$-cliques admit touching representations with hypercubes of unit size in$\mathbb{R}^{1}$and in$\mathbb{R}^{\lceil \log_2 m \rceil}$respectively, by Lemma~\ref{lemma-sp} it suffice to bound the comparable box ... ... @@ -552,7 +595,7 @@ containing$C^*$, and by performing successive full clique-sums of$K_{k+1}$on a$K_k$subclique. By Lemma~\ref{lem-cs}, it suffice to show that$K_{k+1}$, the$(k+1)$-clique with vertex set$\{v_1, \ldots, v_{k+1}\}$, has a$(K_{k+1} -\{v_{k+1}\})$-clique-sum extendable touching representation with hypercubes. Let us define such extendable touching representation by hypercubes. Let us define such touching representation$h$as follows: \begin{itemize} \item$h(v_i)[i] = [-1,0] $if$i\le k$... ... @@ -580,15 +623,28 @@ For a vertex$v_i$and a clique$C$, the boxes$h(v_i)$and$h^\epsilon(C)$intersect if and only if$v_i\in V(C)$. Indeed, if$v_i\in V(C)$then$p(C)\in h(v_i)$and$p(C)\in h^\epsilon(C)$, and if$v_i\notin V(C)$then$h(v_i)[i] = [-1,0]$if$i\le k$(resp.$h(v_i)[i] = [0 \frac12]$if$i= k+1$) and$h^\epsilon(C)[i] = (resp. $h(v_i)[i] = [0, \frac12]$ if $i= k+1$) and $h^\epsilon(C)[i] = [\frac14,\frac14+\epsilon]$ (resp. $h^\epsilon(C)[i] = [\frac34,\frac34+\epsilon]$). Finally, if $v_i\in V(C)$ we have that $h(v_i)[i] \cap h^\epsilon(C)[i] = \{p(C)[i]\}$ and that $h(v_i)[j] \cap h^\epsilon(C)[j] = [p(C)[j],p(C)[j]+\epsilon]$ for any $j\neq i$ and any $\epsilon <\frac14$. This concludes the proof of the theorem. \end{proof} As every planar graph $G$ has a touching representation with cubes in Note that actually the bound on the comparable boxes dimension of Theorem~\ref{thm-ktree} extends to graphs of treewidth $k$. For this, note that the construction in this proof can provide us with a representation $h$ of any $k$-tree $G$ with hypercubes of distinct sizes. Note also that this representation is such that for any two adjacent vertices $u$ and $v$, with $h(u) \sqsubset h(v)$ say, the intersection $I = h(u) \cap h(v)$ is a facet of $h(u)$. Actually $I[i] = h(u)[i]$ for every dimension, except one that we denote $j$. For this dimension we have that $I[j]=\{c\}$ for some $c$, and that $h(u)[j]=[c,c+s]$, where $s$ is the length of the sides of $h(u)$. In that context to delete an edge $uv$ one can simply replace $h(u)[j]=[c,c+s]$ with $[c+\epsilon,c+s]$, for a sufficiently small $\epsilon$. One can proceed similarly for any subset of edges, and note that as the hypercubes in $h$ have distinct sizes these small perturbations give rise to boxes that are still comparable. Thus for any treewidth $k$ graph $H$ (that is a subgraph of a $k$-tree $G$) we have $\cbdim(H)\le k+1$. As every planar graph $G$ has a touching representation by cubes in $\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le 3$. For the graphs with higher Euler genus we can also derive upper bounds. Indeed, combining the previous observation on the ... ...
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