Merge branch 'main' of gitlab.kam.mff.cuni.cz:rakdver/comparable-box-dimension into main

comparable-box-dimension.tex

@@ -34,7 +34,7 @@
 \title{On comparable box dimension}
 \author{Zden\v{e}k Dvo\v{r}\'ak\thanks{Computer Science Institute, Charles University, Prague, Czech Republic. E-mail: {\tt rakdver@iuuk.mff.cuni.cz}.
 Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}\and
-Daniel Gon\c{c}alves\thanks{...}
+Daniel Gon\c{c}alves\thanks{...}\and 
 Abhiruk Lahiri\thanks{...}\and
 Jane Tan\thanks{...}\and
 Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology.  E-mail: {\tt torsten.ueckerdt@kit.edu}}}
@@ -53,20 +53,20 @@ comparable box dimension and explore further properties of this notion.
 \section{Introduction}
 
 For a system $\OO$ of subsets of $\mathbb{R}^d$, we say that a graph $G$ is a \emph{touching graph of objects from $\OO$}
-if there exists a function $f:V(G)\to \OO$ (called the \emph{touching representation by objects from $\OO$})
+if there exists a function $f:V(G)\to \OO$ (called a \emph{touching representation by objects from $\OO$})
 such that for distinct $u,v\in V(G)$, the interiors of $f(u)$ and $f(v)$ are disjoint
 and $f(u)\cap f(v)\neq\emptyset$ if and only if $uv\in E(G)$.
 Famously, Koebe~\cite{koebe} proved that a graph is planar if and only if it is a touching graph of balls in $\mathbb{R}^2$.
-This result motivated a number of strenthenings and variations~\cite{...}; most relevantly for us, every planar graph is
+This result motivated a number of strengthenings and variations~\cite{...}; most relevantly for us, every planar graph is
 a touching graph of cubes in $\mathbb{R}^3$~\cite{felsner2011contact}.
 
 An attractive feature of touching representation is that it makes it possible to represent graph classes that are sparse
 (e.g., planar graphs, or more generally, graph classes with bounded expansion theory~\cite{nesbook}),
 whereas in a general intersection representation, the represented class always includes arbitrarily large cliques.
 Of course, whether the class of touching graph of objects from $\OO$ is sparse or not depends on the system $\OO$.
-For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis-aligned boxes in $\mathbb{R}^d$, where the vertices in
+For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis-aligned boxes in $\mathbb{R}^3$, where the vertices in
 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
 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 $B_2\sqsubseteq B_1$.
@@ -82,7 +82,7 @@ numbers, which implies that $\GG$ has strongly sublinear separators.  They also
 that for any function $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
 comparable box dimension.  Dvo\v{r}\'ak et al.~\cite{wcolig}
 proved that graphs of comparable box dimension $3$ have exponential weak coloring number, giving the
-first natural graph class with olynomial strong coloring numbers and superpolynomial weak coloring numbers
+first natural graph class with polynomial strong coloring numbers and superpolynomial weak coloring numbers
 (the previous example is obtained by subdividing edges of every graph suitably many times~\cite{covcol}).
 
 We show that the comparable box dimension behaves well under the operations of addition of apex vertices,
@@ -94,13 +94,13 @@ The comparable box dimension of every proper minor-closed class of graphs is fin
 \end{theorem}
 
 Additionally, we show that classes of graphs with finite comparable box dimension are fractionally treewidth-fragile.
-This gives arbitrarily precise approximation algorithms for alll monotone maximization problems that are
+This gives arbitrarily precise approximation algorithms for all monotone maximization problems that are
 expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx}
 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.
+Let us start with a simple lemma saying that the 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 $\cbdim(G)\le \cbdim(G-v)+1$.
@@ -114,12 +114,15 @@ Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^
 \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 $B[i]=I_i$.
+For a box $B=I_1\times \cdots \times I_d$ 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}
-...
+ For any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty intersection.
+ Since axis-aligned boxes have the Helly property, there is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap f(a_w)$.
+ As each box is full-dimensional, its interior intersects at least one of the $2^d$ orthants at $p$.
+ Since $f$ is a touching representation, $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors and hence $w \leq 2^d$.
 \end{proof}
 
 A \emph{tree decomposition} of a graph $G$
@@ -130,11 +133,11 @@ such that
 \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 $T$.
+For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ and $x$ is 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?)
+of the widths of its tree decompositions.  We will need to know that graphs of bounded treewidth have bounded comparable box dimension.
+In fact, we will prove the following stronger fact (TODO: Was this published somewhere before? I am only aware of the upper bound of $t+2$ on the boxicity of $G$~\cite{box-treewidth}.)
 
 \begin{lemma}\label{lemma-tw}
 Let $(T,\beta)$ be a tree decomposition of a graph $G$ of width $t$.
@@ -219,13 +222,13 @@ implying that the boxes $h(x_i)$ and $h(w)$ are disjoint.  Therefore, $h$ is a t
 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
+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 arranged 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.
+even after an 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 $G$.
+Consider a tree decomposition $(T,\beta)$ of a graph $G$.
 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.
@@ -235,7 +238,7 @@ A graph $G$ is obtained from graphs in a class $\GG$ by repeated clique-sums if
 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 $a$.
+If $(T,\beta)$ is a tree decomposition 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}

data.bib

@@ -5321,3 +5321,16 @@ note = {In Press}
   volume    = {2103.08698},
   year      = {2021}
 }
+
+@article{box-treewidth,
+  author    = {L. Sunil Chandran and Naveen Sivadasan},
+  title     = {Boxicity and treewidth},
+  journal   = {Journal of Combinatorial Theory, Series B},
+  volume    = {97},
+  number    = {5},
+  pages     = {733--744},
+  year      = {2007},
+  issn      = {0095-8956},
+  doi       = {https://doi.org/10.1016/j.jctb.2006.12.004},
+  url       = {https://www.sciencedirect.com/science/article/pii/S0095895607000111},
+}