Commit 0b81ca9b by Zdenek Dvorak

### Improved readability of the clique-sum argument.

parent e743a8cd
 ... @@ -68,7 +68,8 @@ For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis ... @@ -68,7 +68,8 @@ For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$ one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$ boxes (a \emph{box} is the cartesian product of intervals of non-zero length). boxes (a \emph{box} is the cartesian product of intervals of non-zero length). Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of long and wide boxes: Two boxes are \emph{comparable} if a translation of one of them is a subset of the other one. long and wide boxes: For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if a translation of $B_1$ is a subset of $B_2$. We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$. A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. For a graph $G$, let the \emph{comparable box dimension} $\cbdim(G)$ such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. For a graph $G$, let the \emph{comparable box dimension} $\cbdim(G)$ of $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$. of $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$. ... @@ -112,20 +113,6 @@ where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \ti ... @@ -112,20 +113,6 @@ where$M$is chosen large enough so that$f(u)\subseteq [-M,M] \times \cdots \ti Then $h$ is a touching representation of $G$ by comparable boxes 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} %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. 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 $B[i]=I_i$. 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} \begin{lemma}\label{lemma-cliq} ... @@ -142,6 +129,7 @@ such that ... @@ -142,6 +129,7 @@ such that \item for each $uv\in E(G)$, there exists $x\in V(T)$ such that $u,v\in\beta(x)$, and \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 $T$. \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} \end{itemize} For nodes $x,y\in V(T)$, we write $x\preceq y$ if $x=y$ or $x$ is a descendant of $y$ in $T$. 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 $T$. 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 $x,y\in V(T)$, 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 and its \emph{width} is the maximum of the sizes of the bags minus $1$. The \emph{treewidth} of a graph is the minimum ... @@ -151,8 +139,8 @@ In fact, we will prove the following stronger fact (TODO: Was this published som ... @@ -151,8 +139,8 @@ In fact, we will prove the following stronger fact (TODO: Was this published som \begin{lemma}\label{lemma-tw} \begin{lemma}\label{lemma-tw} Let $(T,\beta)$ be a tree decomposition of a graph $G$ of width $t$. 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 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 $T$, for $u,v\in V(G)$, if $p(u)\preceq p(v)$, then $h(u)\sqsubseteq h(v)$. then $\vol(h(u))>\vol(h(v))$. Moreover, the representation can be chosen so that no two hypercubes have the same size. \end{lemma} \end{lemma} \begin{proof} \begin{proof} ... ... ... @@ -178,6 +166,7 @@ For each note $x\in V(T)$, let $\pi(x)=\{x\}\cup \{p(v):v\in\beta(x)\}$. Let $T ... @@ -178,6 +166,7 @@ For each note$x\in V(T)$, let$\pi(x)=\{x\}\cup \{p(v):v\in\beta(x)\}$. Let$T $xy\in E(T_\beta)$ if and only if $x\in\pi(y)$ or $y\in\pi(x)$. $xy\in E(T_\beta)$ if and only if $x\in\pi(y)$ or $y\in\pi(x)$. \begin{lemma}\label{lemma-legraf} \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 $a$. 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 $a$. Moreover, $\pi(x)$ is a clique in $T_\beta$ and $p(x)=x$ for each $x\in V(T)$. \end{lemma} \end{lemma} \begin{proof} \begin{proof} For each edge $xy\in E(T_\beta)$, we have $x,y\in \pi(x)$ or $x,y\in \pi(y)$ by definition. For each edge $xy\in E(T_\beta)$, we have $x,y\in \pi(x)$ or $x,y\in \pi(y)$ by definition. ... @@ -191,6 +180,13 @@ Consider a node $x\in V(T)$. Note that for each $v\in \beta(x)$, the vertex $p( ... @@ -191,6 +180,13 @@ Consider a node$x\in V(T)$. Note that for each$v\in \beta(x)$, the vertex$p( In particular, if $x$ is the root of $T$, then $\pi(x)=\{x\}$. Otherwise, if $y$ is the parent of $x$ in $T$, then 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 $|\pi(x)|\le |\beta(x)\cap \beta(y)|+1\le a+1$. $p(v)=x$ 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 $a$. Hence, the width of $(T,\pi)$ is at most $a$. Suppose now that $y$ and $z$ are distinct vertices in $\pi(x)$. Then both $y$ and $z$ are ancestors of $x$ in $T$, and thus without loss of generality, we can assume that $y\preceq z$. If $y=x$, then $yz\in E(T_\beta)$ by definition. Otherwise, there exist vertices $u,v\in \beta(x)$ such that $p(u)=y$ and $p(v)=z$. Since $v\in \beta(x)\cap\beta(z)$ and $y$ is on the path in $T$ from $x$ to $z$, we also have $v\in\beta(y)$. This implies $z\in\pi(y)$ and $yz\in E(T_\beta)$. Hence, $\pi(x)$ is a clique in $T_\beta$. Moreover, note that $x\not\in \pi(w)$ for any ancestor $w\neq x$ of $x$ in $T$, and thus $p(x)=x$. \end{proof} \end{proof} We are now ready to deal with the clique-sums. We are now ready to deal with the clique-sums. ... @@ -202,37 +198,56 @@ $G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$. ... @@ -202,37 +198,56 @@ $G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$. \begin{proof} \begin{proof} Let $(T,\beta)$ be a tree decomposition of $G$ over $\GG$; the adhesion $a$ of $(T,\beta)$ is at most $\omega(\GG)$. 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}, 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, $T_\beta$ has a touching representation $h$ by hypercubes in $\mathbb{R}^{a+1}$ such that letting $\prec$ be a linear ordering on $V(T)$ in a non-decreasing order according to the volume of the hypercubes assigned by $h$, $h(x)\sqsubseteq h(y)$ whenever $x\preceq y$. 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 $\{0,\ldots,a\}$. 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 every $x\in V(T)$, let $f_x$ be a touching representation of the torso $G_x$ 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 $i\in\{0,\ldots,a\}$, where $d=\cbdim(\GG)$. 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 there exists a box $E_i(x)$ such that \begin{itemize} \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 \item[(a)] for distinct $x,y\in V(T)$, if $h(x)\sqsubset h(y)$, then $E_i(x) \sqsubset E_i(y)$ and $E_i(x) \sqsubset f_y(v)$ for every $v\in \beta(y)$, the representation $f_y$ whenever $x\prec y$, \item[(b)] if $x\prec y$, then $E_i(x)\subseteq E_i(y)$, \item if $i=\varphi(x)$, then all boxes of $f_x$ are subsets of $E_i(x)$, and \item[(c)] if $i=\varphi(x)$, then $f_x(v)\subseteq E_i(x)$ for every $v\in\beta(x)$, 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 $y$ \item[(d)] if $i\neq \varphi(x)$ and $K=\{v\in\beta(x):\varphi(p(v))=i\}$ is non-empty, then letting $y=p(v)$ for $v\in K$, is unique, since $\varphi$ is a proper coloring of $T_\beta$ and that $K$ is a clique in $G_y$), the box $E_i(x)$ the box $E_i(x)$ contains a point belonging to $\bigcap_{v\in K} f_y(v)$. contains a point belonging to $\bigcap_{v\in K} f_y(v)$. \end{itemize} \end{itemize} Some explanation is in order for the last point: Firstly, since $\pi(x)$ is a clique in $T_\beta$, there exists only one vertex $y\in \pi(x)$ of color $i$, and thus $y=p(v)$ for all $v\in K$. Moreover, $K$ is a clique in $G_y$, and thus $\bigcap_{v\in K} f_y(v)$ is non-empty. Lastly, note that if $x\prec z\prec y$, then $K=\beta(x)\cap\beta(y)\subseteq \beta(z)\cap \beta(y)$, and thus $E_i(z)$ was also chosen to contain a point of $\bigcap_{v\in K} f_y(v)$; hence, a choice of $E_i(x)$ satisfying $E_i(x)\subseteq E_i(z)$ as required by (b) is possible. Let us now define $f(v)=h(p(v))\times E_0(v)\times \cdots\times E_a(v)$ for each $v\in V(G)$, 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 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; 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 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 $p(u)\prec p(v)$, then this is due to (a) and (c). 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 $p(u)\neq p(v)$, 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 $x$ 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, $p(u)=x$. 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 $G_x$. Let $y=p(v)$. If $x=y$, then $f(u)\cap f(v)\neq\emptyset$, since $f_x$ is a touching representation of $G_x$ and $uv\in E(G_x)$. If $x\neq y$, then $y\in\pi(x)$ and $xy\in E(T_\beta)$, implying that $h(x)\cap h(y)\neq\emptyset$. 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 $i=0,\ldots,a$. Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\ldots,a$ by (b) if $\varphi(x)\neq i\neq \varphi(y)$, Hence, again we have $f(u)\cap f(v)\neq\emptyset$. (b) and (c) if $\varphi(x)=i\neq\varphi(y)$, and (d) if $\varphi(x)\neq i=\varphi(y)$ (we cannot have $\varphi(x)=i=\varphi(y)$, since $xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$. \end{proof} \end{proof} %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} \section{Exploiting the product structure} \section{Exploiting the product structure} ... ...
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