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Commit e743a8cd authored by Zdenek Dvorak's avatar Zdenek Dvorak
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Fixed the clique-sum argument.

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...@@ -100,63 +100,140 @@ or expressible in the first-order logic~\cite{logapx}. ...@@ -100,63 +100,140 @@ or expressible in the first-order logic~\cite{logapx}.
\section{Operations} \section{Operations}
Let us start with a simple lemma saying that addition of a vertex increases the comparable box dimension by at most one. 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} \begin{lemma}\label{lemma-apex}
For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$. For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$.
\end{lemma} \end{lemma}
\begin{proof} \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 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 $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$, $h(u)=[1/2,3/2]\times f(u)$ 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 $u\in V(G)\setminus\{v\}$. 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} \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 %We will need the fact that the chromatic number is at most exponential in the comparable box dimension;
by identifying vertices of a clique in $G_1$ and a clique of the same size in $G_2$ and possibly %this follows from~\cite{subconvex} and we include the argument to make the dependence clear.
deleting some of the edges of the resulting clique. The main issue to overcome in obtaining a representation for a clique-sum %\begin{lemma}\label{lemma-chrom}
is that the representations of $G_1$ and $G_2$ can be ``degenerate''. Consider e.g. the case that $G_1$ is represented %If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$-colorable.
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 %\end{lemma}
in a single corner. This can be avoided by increasing the dimension, but we need to be careful so that the dimension stays bounded %\begin{proof}
even after, motivating the following definition. %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 $I_i$. A \emph{tree decomposition} of a graph $G$
Let $B_1$, \ldots, $B_k$ be pairwise touching boxes in $\mathbb{R}^d$. 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,
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 $B$
such that such that
\begin{itemize} \begin{itemize}
\item the interior of $B$ is disjoint from $f(v)$ for every $v\in V(G)$, \item for each $uv\in E(G)$, there exists $x\in V(T)$ such that $u,v\in\beta(x)$, and
\item $B$ touches the boxes $\{f(u):u\in K\}$ generically, 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 $T$.
\item there exists $i\in\{1,\ldots,d\}$ such that $B[i]\subseteq f(u)[i]$ for every $u\in K$.
\end{itemize} \end{itemize}
The representation is \emph{exposed} if all cliques are exposed. 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 $x,y\in V(T)$,
\begin{lemma}\label{lemma-expose} and its \emph{width} is the maximum of the sizes of the bags minus $1$. The \emph{treewidth} of a graph is the minimum
Every graph $G$ has an exposed comparable box representation in $\mathbb{R}^{\chi(G)}$. 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} \end{lemma}
\begin{proof} \begin{proof}
... ...
\end{proof} \end{proof}
Note that the chromatic number of $G$ is at most exponential in the comparable box dimension; 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
this follows from~\cite{subconvex} and we include the argument to make the dependence clear. by identifying vertices of a clique in $G_1$ and a clique of the same size in $G_2$ and possibly
\begin{lemma}\label{lemma-chrom} deleting some of the edges of the resulting clique. The main issue to overcome in obtaining a representation for a clique-sum
If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$-colorable. 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} \end{lemma}
\begin{proof} \begin{proof}
We actually show that $G$ is $(3^d-1)$-degenerate. Since every induced subgraph of $G$ also For each edge $xy\in E(T_\beta)$, we have $x,y\in \pi(x)$ or $x,y\in \pi(y)$ by definition.
has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$ Moreover, for each $x\in V(T_\beta)$, we have
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 $v$, $$\{y:x\in\pi(y)\}=\{x\}\cup \bigcup_{v\in V(G):p(v)=x} \{y:v\in\beta(y)\},$$
there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$. and all the sets on the right-hand size induce connected subtrees containing $x$,
Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in implying that $\{y:x\in\pi(y)\}$ also induces a connected subtree containing $x$.
a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+|N(v)|\le 3^d$. 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} \end{proof}
\section{Exploiting the product structure} \section{Exploiting the product structure}
\subsection*{Acknowledgments} \subsection*{Acknowledgments}
......
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