### 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 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). 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 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 $\mathbb{R}^d$. ... ... @@ -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}$. \end{proof} %We will need the fact that the chromatic number is at most exponential in the comparable box dimension; %this follows from~\cite{subconvex} and we include the argument to make the dependence clear. %\begin{lemma}\label{lemma-chrom} %If$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. 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} ... ... @@ -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$v\in V(G)$, the set$\{x\in V(T):v\in\beta(x)\}$is non-empty and induces a connected subtree of$T$. \end{itemize} 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$T$. 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 ... ... @@ -151,8 +139,8 @@ In fact, we will prove the following stronger fact (TODO: Was this published som \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$\vol(h(u))>\vol(h(v))$. for$u,v\in V(G)$, if$p(u)\preceq p(v)$, then$h(u)\sqsubseteq h(v)$. Moreover, the representation can be chosen so that no two hypercubes have the same size. \end{lemma} \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 $xy\in E(T_\beta)$ 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 $a$. Moreover, $\pi(x)$ is a clique in $T_\beta$ and $p(x)=x$ for each $x\in V(T)$. \end{lemma} \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. ... ... @@ -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$, then$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$. Suppose now that$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$. \end{proof} We are now ready to deal with the clique-sums. ... ... @@ -202,37 +198,56 @@$G\subseteq G'$and$\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$. \begin{proof} Let$(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$.$T_\beta$has a touching representation$h$by hypercubes in$\mathbb{R}^{a+1}$such that$h(x)\sqsubseteq h(y)$whenever$x\preceq y$. Since$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\}$, For every$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$, where$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)$such that \begin{itemize} \item whenever$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)$. \item[(a)] for distinct$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)$, \item[(b)] if$x\prec y$, then$E_i(x)\subseteq E_i(y)$, \item[(c)] if$i=\varphi(x)$, then$f_x(v)\subseteq E_i(x)$for every$v\in\beta(x)$, and \item[(d)] if$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$, the box$E_i(x)$contains a point belonging to$\bigcap_{v\in K} f_y(v)$. \end{itemize} Some explanation is in order for the last point: Firstly, since$\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)$, where$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)$have disjoint$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)$, then this is the case because$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$. Let$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$. Moreover,$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$. Moreover,$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)$, (b) and (c) if$\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$. \end{proof} %We will need the fact that the chromatic number is at most exponential in the comparable box dimension; %this follows from~\cite{subconvex} and we include the argument to make the dependence clear. %\begin{lemma}\label{lemma-chrom} %If$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} ... ...
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