### Fixed the clique-sum argument.

parent 889b062f
 ... ... @@ -100,63 +100,140 @@ or expressible in the first-order logic~\cite{logapx}. \section{Operations} Let us start with a simple lemma saying that addition of a vertex increases the comparable box dimension by at most one. In particular, this implies that $\cbdim(G)\le |V(G)$. \begin{lemma}\label{lemma-apex} For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$. \end{lemma} \begin{proof} Let $f$ be a comparable box representation of $G-v$ in $\mathbb{R}^d$, where $d=\cbdim(G-v)$. Let $f$ be a touching representation of $G-v$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G-v)$. For each $u\in V(G)\setminus\{v\}$, let $h(u)=[0,1]\times f(u)$ if $uv\in E(G)$ and $h(u)=[1/2,3/2]\times f(u)$ if $uv\not\in E(G)$. Let $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$, where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \times [-M,M]$ for every $u\in V(G)\setminus\{v\}$. Then $h$ is a comparable box representation of $G$ in $\mathbb{R}^{d+1}$. Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$. \end{proof} Next, let us deal with clique-sums. A \emph{clique-sum} of two graphs $G_1$ and $G_2$ is obtained from their disjoint union by identifying vertices of a clique in $G_1$ and a clique of the same size in $G_2$ and possibly deleting some of the edges of the resulting clique. The main issue to overcome in obtaining a representation for a clique-sum is that the representations of $G_1$ and $G_2$ can be degenerate''. Consider e.g. the case that $G_1$ is represented by unit squares arrangef in a grid; in this case, there is no space to attach $G_2$ at the cliques formed by four squares intersecting in a single corner. This can be avoided by increasing the dimension, but we need to be careful so that the dimension stays bounded even after, motivating the following definition. %We will need the fact that the chromatic number is at most exponential in the comparable box dimension; %this follows from~\cite{subconvex} and we include the argument to make the dependence clear. %\begin{lemma}\label{lemma-chrom} %If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$-colorable. %\end{lemma} %\begin{proof} %We actually show that $G$ is $(3^d-1)$-degenerate. Since every induced subgraph of $G$ also %has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$ %is less than $3^d$. Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume. For every neighbor $u$ of $v$, %there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$. %Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in %a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+|N(v)|\le 3^d$. %\end{proof} We need a bound on the clique number in terms of the comparable box dimension. For a box $B=I_1\times \cdots I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_i$. \begin{lemma}\label{lemma-cliq} If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$. \end{lemma} \begin{proof} ... \end{proof} For a box $B=I_1\times \cdot\times I_d$ and $i\in\{1,\ldots,d\}$ let $B[i]$ denote the interval $I_i$. Let $B_1$, \ldots, $B_k$ be pairwise touching boxes in $\mathbb{R}^d$. A box $B$ \emph{touches $B_1$, \ldots, $B_k$ generically} if there exist distinct $i_1,\ldots, i_k\in\{1,\ldots, d\}$ such that for $j=1,\ldots, k$, \begin{itemize} \item $B[i_j]\cap B_j[i_j]$ consists of a single point, and \item for every $i\in\{1,\ldots,d\}\setminus \{i_j\}$, $B[i]\cap B_j[i]$ is a non-empty interval of non-zero length. \end{itemize} A clique $K$ in a touching box representation $f$ of a graph $G$ in $\mathbb{R}^d$ is \emph{exposed} if there exists a box $B$ A \emph{tree decomposition} of a graph $G$ is a pair $(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$ assigns a \emph{bag} to each of its nodes, such that \begin{itemize} \item the interior of $B$ is disjoint from $f(v)$ for every $v\in V(G)$, \item $B$ touches the boxes $\{f(u):u\in K\}$ generically, and \item there exists $i\in\{1,\ldots,d\}$ such that $B[i]\subseteq f(u)[i]$ for every $u\in K$. \item for each $uv\in E(G)$, there exists $x\in V(T)$ such that $u,v\in\beta(x)$, and \item for each $v\in V(G)$, the set $\{x\in V(T):v\in\beta(x)\}$ is non-empty and induces a connected subtree of $T$. \end{itemize} The representation is \emph{exposed} if all cliques are exposed. \begin{lemma}\label{lemma-expose} Every graph $G$ has an exposed comparable box representation in $\mathbb{R}^{\chi(G)}$. For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ nearest to the root of $T$. The \emph{adhesion} of the tree decomposition is the maximum of $|\beta(x)\cap\beta(y)|$ over distinct $x,y\in V(T)$, and its \emph{width} is the maximum of the sizes of the bags minus $1$. The \emph{treewidth} of a graph is the minimum of the widths of its tree decompositions. We will need to know that graphs of bounded treewidth have bounded dimension. In fact, we will prove the following stronger fact (TODO: Was this published somehere before?) \begin{lemma}\label{lemma-tw} Let $(T,\beta)$ be a tree decomposition of a graph $G$ of width $t$. Then $G$ has a touching representation $h$ by hypercubes in $R^{t+1}$ such that for $u,v\in V(G)$, if $p(u)\neq p(v)$ and $p(u)$ is an ancestor of $p(v)$ in $T$, then $\vol(h(u))>\vol(h(v))$. \end{lemma} \begin{proof} ... \end{proof} Note that the chromatic number of $G$ is at most exponential in the comparable box dimension; this follows from~\cite{subconvex} and we include the argument to make the dependence clear. \begin{lemma}\label{lemma-chrom} If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$-colorable. Next, let us deal with clique-sums. A \emph{clique-sum} of two graphs $G_1$ and $G_2$ is obtained from their disjoint union by identifying vertices of a clique in $G_1$ and a clique of the same size in $G_2$ and possibly deleting some of the edges of the resulting clique. The main issue to overcome in obtaining a representation for a clique-sum is that the representations of $G_1$ and $G_2$ can be degenerate''. Consider e.g. the case that $G_1$ is represented by unit squares arrangef in a grid; in this case, there is no space to attach $G_2$ at the cliques formed by four squares intersecting in a single corner. This can be avoided by increasing the dimension, but we need to be careful so that the dimension stays bounded even after arbitrary number of clique-sums. It will be convenient to work in the setting of tree decompositions. Consider a tree decompostion $(T,\beta)$ of a graphs $G$. For each $x\in V(T)$, the \emph{torso} $G_x$ of $x$ is the graph obtained from $G[\beta(x)]$ by adding a clique on $\beta(x)\cap\beta(y)$ for each $y\in V(T)$. For a class of graphs $\GG$, we say that the tree decomposition is \emph{over $\GG$} if all the torsos belong to $\GG$. We use the following well-known fact. \begin{observation} A graph $G$ is obtained from graphs in a class $\GG$ by repeated clique-sums if and only if $G$ has a tree decomposition over $\GG$. \end{observation} For each note $x\in V(T)$, let $\pi(x)=\{x\}\cup \{p(v):v\in\beta(x)\}$. Let $T_\beta$ be the graph with vertex set $V(T)$ such that $xy\in E(T_\beta)$ if and only if $x\in\pi(y)$ or $y\in\pi(x)$. \begin{lemma}\label{lemma-legraf} If $(T,\beta)$ is a tree decompositon of $G$ of adhesion $a$, then $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$. \end{lemma} \begin{proof} We actually show that $G$ is $(3^d-1)$-degenerate. Since every induced subgraph of $G$ also has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$ is less than $3^d$. Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume. For every neighbor $u$ of $v$, there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$. Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+|N(v)|\le 3^d$. For each edge $xy\in E(T_\beta)$, we have $x,y\in \pi(x)$ or $x,y\in \pi(y)$ by definition. Moreover, for each $x\in V(T_\beta)$, we have $$\{y:x\in\pi(y)\}=\{x\}\cup \bigcup_{v\in V(G):p(v)=x} \{y:v\in\beta(y)\},$$ and all the sets on the right-hand size induce connected subtrees containing $x$, implying that $\{y:x\in\pi(y)\}$ also induces a connected subtree containing $x$. Hence, $(T,\pi)$ is a tree decomposition of $T_\beta$. Consider a node $x\in V(T)$. Note that for each $v\in \beta(x)$, the vertex $p(v)$ is an ancestor of $x$ in $T$. In particular, if $x$ is the root of $T$, then $\pi(x)=\{x\}$. Otherwise, if $y$ is the parent of $x$ in $T$, then $p(v)=x$ for every $v\in \beta(x)\setminus\beta(y)$, and thus $|\pi(x)|\le |\beta(x)\cap \beta(y)|+1\le a+1$. Hence, the width of $(T,\pi)$ is at most $a$. \end{proof} We are now ready to deal with the clique-sums. \begin{theorem} If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then there exists a graph $G'$ such that $G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$. \end{theorem} \begin{proof} Let $(T,\beta)$ be a tree decomposition of $G$ over $\GG$; the adhesion $a$ of $(T,\beta)$ is at most $\omega(\GG)$. By Lemma~\ref{lemma-legraf}, $(T,\pi)$ is a tree decomposition of $T_\beta$ of width at most $a$. By Lemma~\ref{lemma-tw}, $T_\beta$ has a touching representation $h$ by hypercubes in $\mathbb{R}^{a+1}$. Moreover, letting $\prec$ be a linear ordering on $V(T)$ in a non-decreasing order according to the volume of the hypercubes assigned by $h$, we have that $x\prec y$ whenever $x$ is a descendant of $y$ in $T$. Since $T_\beta$ has treewidth at most $a$, it has a proper coloring $\varphi$ by colors $\{0,\ldots,a\}$. For every $x\in V(T)$, let $f_x$ be a touching representation of the torso of $x$ by comparable boxes in $\mathbb{R}^d$ for $d=\cbdim(\GG)$. We scale and translate the representations so that for every $x\in V(T)$ and $i\in\{0,\ldots,a\}$, there exists a box $E_i(x)$ such that \begin{itemize} \item whenever $x\prec y$, we have $E_i(x)\subseteq E_i(y)$ and a translation of $E_i(x)$ is a subset of every box of the representation $f_y$ whenever $x\prec y$, \item if $i=\varphi(x)$, then all boxes of $f_x$ are subsets of $E_i(x)$, and \item if $i\neq p(v)$ and $K=\{v\in\beta(x):\varphi(p(v))=i\}$, then letting $y=p(v)$ for $v\in K$ (and noting that this $y$ is unique, since $\varphi$ is a proper coloring of $T_\beta$ and that $K$ is a clique in $G_y$), the box $E_i(x)$ contains a point belonging to $\bigcap_{v\in K} f_y(v)$. \end{itemize} Let us now define $f(v)=h(p(v))\times E_0(v)\times \cdots\times E_a(v)$ for each $v\in V(G)$, where $E_i(v)=f_{p(v)}(v)$ if $i=\varphi(p(v))$ and $E_i(v)=E_i(p(v))$ otherwise. We claim this gives a touching representation of a supergraph of $G$ by comparable boxes in $\mathbb{R}^{(d+1)(a+1)}$. First, note that the boxes are indeed comparable; if $p(u)=p(v)$, then this is the case since $f_{p(v)}$ is a representation by comparable boxes, and if say $p(u)\prec p(v)$, then this is due to the scaling of $f_{p(u)}$. Next, let us argue $f(u)$ and $f(v)$ have disjoint interiors. If $p(u)=p(v)$, this is the case since $f_{p(v)}$ is a touching representation, and if $p(u)\neq p(v)$, then this is the case because $h$ is a touching representation. Finally, suppose that $uv\in E(G)$. Let $x$ be the node of $T$ nearest to the root such that $u,v\in \beta(x)$. Without loss of generality, $p(u)=x$. Let $y=p(v)$. If $x=y$, then $f(u)\cap f(v)\neq\emptyset$, since $f_x$ is a touching representation of $G_x$. If $x\neq y$, then $y\in\pi(x)$ and $xy\in E(T_\beta)$, implying that $h(x)\cap h(y)\neq\emptyset$. Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\ldots,a$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$. \end{proof} \section{Exploiting the product structure} \subsection*{Acknowledgments} ... ...
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