Commit d42d6198 authored by Jane Tan's avatar Jane Tan
Browse files

Some minor fixes and rewordings

parent c9a0340c
...@@ -5,6 +5,7 @@ ...@@ -5,6 +5,7 @@
\usepackage{amssymb} \usepackage{amssymb}
\usepackage{colonequals} \usepackage{colonequals}
\usepackage{epsfig} \usepackage{epsfig}
\usepackage{tikz}
\usepackage{url} \usepackage{url}
\newcommand{\GG}{{\cal G}} \newcommand{\GG}{{\cal G}}
\newcommand{\HH}{{\cal H}} \newcommand{\HH}{{\cal H}}
...@@ -23,6 +24,7 @@ ...@@ -23,6 +24,7 @@
\newcommand{\vol}{\brm{vol}} \newcommand{\vol}{\brm{vol}}
%%%%% %%%%%
\newcommand{\note}[1]{\textcolor{blue}{#1}}
\newtheorem{theorem}{Theorem} \newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary} \newtheorem{corollary}[theorem]{Corollary}
...@@ -37,7 +39,7 @@ ...@@ -37,7 +39,7 @@
Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}\and 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{...}\and Daniel Gon\c{c}alves\thanks{...}\and
Abhiruk Lahiri\thanks{...}\and Abhiruk Lahiri\thanks{...}\and
Jane Tan\thanks{...}\and Jane Tan\thanks{Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK. E-mail: {\tt jane.tan@maths.ox.ac.uk}}\and
Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten.ueckerdt@kit.edu}}} Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten.ueckerdt@kit.edu}}}
\date{} \date{}
...@@ -45,9 +47,7 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten ...@@ -45,9 +47,7 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten
\maketitle \maketitle
\begin{abstract} \begin{abstract}
The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded
as a touching graph of comparable boxes in $\mathbb{R}^d$ (two boxes are comparable if one of them is
a subset of a translation of the other one). We show that proper minor-closed classes have bounded
comparable box dimension and explore further properties of this notion. comparable box dimension and explore further properties of this notion.
\end{abstract} \end{abstract}
...@@ -58,18 +58,17 @@ if there exists a function $f:V(G)\to \OO$ (called a \emph{touching representati ...@@ -58,18 +58,17 @@ if there exists a function $f:V(G)\to \OO$ (called a \emph{touching representati
such that for distinct $u,v\in V(G)$, the interiors of $f(u)$ and $f(v)$ are disjoint 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 $uv\in E(G)$. 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 $\mathbb{R}^2$. 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 strengthenings and variations~\cite{...}; most relevantly for us, every planar graph is This result motivated a number of strengthenings and variations (see \cite{graphsandgeom, sachs94} for some classical examples); most relevantly for us, every planar graph is a touching graph of cubes in $\mathbb{R}^3$~\cite{felsner2011contact}.
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 An attractive feature of touching representations is that it is possible to represent graph classes that are sparse
(e.g., planar graphs, or more generally, graph classes with bounded expansion theory~\cite{nesbook}), (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. This is in contrast to general intersection representations where 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 $\OO$. 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 $\mathbb{R}^3$, 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 $1\times n\times 1$ 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, in particular axis-aligned).
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: 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 $B_2$. long and wide boxes: For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.
We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$. 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} $\cbdim(G)$
...@@ -102,12 +101,13 @@ or expressible in the first-order logic~\cite{logapx}. ...@@ -102,12 +101,13 @@ or expressible in the first-order logic~\cite{logapx}.
\section{Operations} \section{Operations}
Let us start with a simple lemma saying that the 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 $\cbdim(G)\le |V(G)$. 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 touching representation of $G-v$ by comparable boxes 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 $d=\cbdim(G-v)$.
We define a representation $h$ of $G$ as follows.
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 $u\in V(G)\setminus\{v\}$.
...@@ -120,7 +120,7 @@ For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_ ...@@ -120,7 +120,7 @@ For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_
If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$. If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$.
\end{lemma} \end{lemma}
\begin{proof} \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. To represent 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 $f(a_1) \cap \cdots \cap f(a_w)$. 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 $p$. 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 $w \leq 2^d$. Since $f$ is a touching representation, $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors and hence $w \leq 2^d$.
...@@ -138,11 +138,11 @@ For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in ...@@ -138,11 +138,11 @@ For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in
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
of the widths of its tree decompositions. We will need to know that graphs of bounded treewidth have bounded comparable box dimension. 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}.) In fact, we will prove the following stronger fact. \note{(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} \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 axis-aligned hypercubes in $R^{t+1}$ such that Then $G$ has a touching representation $h$ by axis-aligned hypercubes in $\mathbb{R}^{t+1}$ such that
for $u,v\in V(G)$, if $p(u)\prec p(v)$, then $h(u)\sqsubset h(v)$. for $u,v\in V(G)$, if $p(u)\prec p(v)$, then $h(u)\sqsubset h(v)$.
Moreover, the representation can be chosen so that no two hypercubes have the same size. Moreover, the representation can be chosen so that no two hypercubes have the same size.
\end{lemma} \end{lemma}
...@@ -153,7 +153,7 @@ and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta( ...@@ -153,7 +153,7 @@ and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta(
we can subdivide the edge $xy$, introducing $|\beta(x)\setminus\beta(y)|-1$ new vertices, we can subdivide the edge $xy$, introducing $|\beta(x)\setminus\beta(y)|-1$ new vertices,
and set their bags appropriately). It is now natural to relabel the vertices of $G$ and set their bags appropriately). It is now natural to relabel the vertices of $G$
so that $V(G)=V(T)$, by giving the unique vertex in $\beta(x)\setminus\beta(y)$ so that $V(G)=V(T)$, by giving the unique vertex in $\beta(x)\setminus\beta(y)$
the label $x$. In particular, $p(x)=x$. Furthermore, we can assume that $y\in\beta(x)$. the label $x$. In particular, $p(x)=x$. Furthermore, switching to this new labeling, we can assume that $y\in\beta(x)$.
Otherwise, if $y$ is not the root of $T$, we can replace the edge $xy$ by the edge from $x$ Otherwise, if $y$ is not the root of $T$, we can replace the edge $xy$ by the edge from $x$
to the parent of $y$. If $y$ is the root of $T$, then the subtree rooted in $x$ induces a to the parent of $y$. If $y$ is the root of $T$, then the subtree rooted in $x$ induces a
union of connected components in $G$, and we can process this subtree separately from the union of connected components in $G$, and we can process this subtree separately from the
...@@ -211,7 +211,7 @@ Hence, $\min(h(w)[j])-2\epsilon\le \min(h''(w_1)[j])\le \max(h''(w_1)[j])\le \ma ...@@ -211,7 +211,7 @@ Hence, $\min(h(w)[j])-2\epsilon\le \min(h''(w_1)[j])\le \max(h''(w_1)[j])\le \ma
Since $h(x_i)[j]\subseteq h''(w_1)[j]$ by (a) and the length of the interval $h(x_i)[j]$ is greater than $2\varepsilon$, Since $h(x_i)[j]\subseteq h''(w_1)[j]$ by (a) and the length of the interval $h(x_i)[j]$ is greater than $2\varepsilon$,
we conclude that $h(x_i)[j]\cap h(w)[j]\neq\emptyset$. Therefore, the boxes $h(w)$ and $h(x_i)$ touch. we conclude that $h(x_i)[j]\cap h(w)[j]\neq\emptyset$. Therefore, the boxes $h(w)$ and $h(x_i)$ touch.
Consider now two non-adjacent vertices of $G$, say $x_i$ and $w$. As we noted before, if $x_i$ and $w$ are Finally, consider two non-adjacent vertices of $G$, say $x_i$ and $w$. As we noted before, if $x_i$ and $w$ are
incomparable in $\prec$, then (a) and (b) implies that the boxes $h(x_i)$ and $h(w)$ are disjoint. incomparable in $\prec$, then (a) and (b) implies that the boxes $h(x_i)$ and $h(w)$ are disjoint.
Suppose now that say $x_i\prec w$, and let $j=\varphi(w)$. If $w\in\beta(x_i)$, then $h(x_i)[j]$ and $h(w)[j]$ are disjoint by (i). Suppose now that say $x_i\prec w$, and let $j=\varphi(w)$. If $w\in\beta(x_i)$, then $h(x_i)[j]$ and $h(w)[j]$ are disjoint by (i).
Otherwise, let $y$ be the last vertex on the path from $w$ to $x_i$ in $T$ such that $w\in\beta(y)$ and let $z$ be the child of $y$ Otherwise, let $y$ be the last vertex on the path from $w$ to $x_i$ in $T$ such that $w\in\beta(y)$ and let $z$ be the child of $y$
...@@ -310,7 +310,7 @@ $xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$. ...@@ -310,7 +310,7 @@ $xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$.
We can now combine Theorem~\ref{thm-cs} with Lemma~\ref{lemma-cliq}. We can now combine Theorem~\ref{thm-cs} with Lemma~\ref{lemma-cliq}.
\begin{corollary}\label{cor-cs} \begin{corollary}\label{cor-cs}
If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then there exists a graph $G'$ such that 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 $\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}$. $G\subseteq G'$ and \[\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}.\]
\end{corollary} \end{corollary}
Note that only bound the comparable box dimension of a supergraph Note that only bound the comparable box dimension of a supergraph
...@@ -407,7 +407,7 @@ Any graph $G$ is a subgraph of the strong product of a path, a graph of threewid ...@@ -407,7 +407,7 @@ Any graph $G$ is a subgraph of the strong product of a path, a graph of threewid
\item $t=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$. \item $t=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.
\end{itemize} \end{itemize}
Moreover, for every $k$, there exists $t$ such that if $K_k\not\preceq_m G$, then $G$ is a subgraph of a clique-sum Moreover, for every $k$, there exists $t$ such that if $K_k\not\preceq_m G$, then $G$ is a subgraph of a clique-sum
for graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding of graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding
at most $t$ apex vertices. at most $t$ apex vertices.
\end{theorem} \end{theorem}
...@@ -420,13 +420,10 @@ comparable box dimension at most $t+2+\lceil \log_2 m\rceil$. ...@@ -420,13 +420,10 @@ comparable box dimension at most $t+2+\lceil \log_2 m\rceil$.
Without loss of generality, we can assume that $k=\log_2 m$ is an integer and the Without loss of generality, we can assume that $k=\log_2 m$ is an integer and the
vertices of $K_m$ are elements of $\{0,1\}^k$. Moreover, we can assume that $V(P)=\{1,\ldots,n\}$ vertices of $K_m$ are elements of $\{0,1\}^k$. Moreover, we can assume that $V(P)=\{1,\ldots,n\}$
with $ij\in E(P)$ iff $|i-j|=1$. Let $h$ be the touching representation of $T$ by with $ij\in E(P)$ iff $|i-j|=1$. Let $h$ be the touching representation of $T$ by
hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}. hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}. Then an explicit representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$ can be obtained as follows. For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,
The representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$
is obtained as follows: For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,
we set we set
$$f(p,v,x_1, \ldots, x_k)[j]=\begin{cases} $$f(p,v,x_1, \ldots, x_k)[j]=\begin{cases}
f(v)[j]&\text{ for $j=1,\ldots, t+1$}\\ h(v)[j]&\text{ for $j=1,\ldots, t+1$}\\
[p,p+1]&\text{ if $j=t+2$}\\ [p,p+1]&\text{ if $j=t+2$}\\
[x_{j-t-2},x_{j-t-2}+1]&\text{ if $j>t+2$.} [x_{j-t-2},x_{j-t-2}+1]&\text{ if $j>t+2$.}
\end{cases}$$ \end{cases}$$
...@@ -444,7 +441,7 @@ Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, the ...@@ -444,7 +441,7 @@ Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, the
$\cbdim(G)\le d$. $\cbdim(G)\le d$.
\end{corollary} \end{corollary}
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$ Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
and comparable box dimension $n$, showing that the dependence of the comparable box dimension cannot be and comparable box dimension $n$. It follows that the dependence of the comparable box dimension cannot be
subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-minor} subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-minor}
certainly can be improved). The dependence of the comparable box dimension on the size of the forbidden minor that we certainly can be improved). The dependence of the comparable box dimension on the size of the forbidden minor that we
established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}. established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}.
......
...@@ -5254,7 +5254,7 @@ note="In Press" ...@@ -5254,7 +5254,7 @@ note="In Press"
@article{mohar2002coloring, @article{mohar2002coloring,
title={Coloring Eulerian triangulations of the projective plane}, title={Coloring Eulerian triangulations of the projective plane},
author={Mohar, Bojan}, author={Mohar, Bojan},
journal={Discrete mathematics}, journal={Discrete Mathematics},
volume=244, volume=244,
pages={339--344}, pages={339--344},
year=2002 year=2002
...@@ -5334,3 +5334,19 @@ note = {In Press} ...@@ -5334,3 +5334,19 @@ note = {In Press}
doi = {https://doi.org/10.1016/j.jctb.2006.12.004}, doi = {https://doi.org/10.1016/j.jctb.2006.12.004},
url = {https://www.sciencedirect.com/science/article/pii/S0095895607000111}, url = {https://www.sciencedirect.com/science/article/pii/S0095895607000111},
} }
@book{graphsandgeom,
author= {L. Lov\'asz},
title={Graphs and Geometry},
publisher={American Mathematical Society},
year=2019,
address = "Providence"}
@article{sachs94,
author = {Horst Sachs},
title = {Coin graphs, polyhedra, and conformal mapping },
journal = {Discrete Mathematics},
volume = {134},
year=1994,
pages={133--138}}
}
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