The treewidth-fragility part.

comparable-box-dimension.tex

@@ -7,12 +7,11 @@
 \usepackage{epsfig}
 \usepackage{url}
 \newcommand{\GG}{{\cal G}}
+\newcommand{\HH}{{\cal H}}
 \newcommand{\CC}{{\cal C}}
 \newcommand{\OO}{{\cal O}}
 \newcommand{\PP}{{\cal P}}
 \newcommand{\RR}{{\cal R}}
-\newcommand{\col}{\text{col}}
-\newcommand{\vol}{\text{vol}}
 
 \newcommand{\eps}{\varepsilon}
 \newcommand{\mc}[1]{\mathcal{#1}}
@@ -20,6 +19,8 @@
 \newcommand{\bb}[1]{\mathbb{#1}}
 \newcommand{\brm}[1]{\operatorname{#1}}
 \newcommand{\cbdim}{\brm{dim}_{cb}}
+\newcommand{\tw}{\brm{tw}}
+\newcommand{\vol}{\brm{vol}}
 %%%%%
 
 
@@ -449,6 +450,126 @@ certainly can be improved).  The dependence of the comparable box dimension on t
 established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}.
 It would be interesting to prove this part of Corollary~\ref{cor-minor} without using the structure theorem.
 
+\section{Fractional treewidth-fragility}
+
+Suppose $G$ is a connected planar graph and $v$ is a vertex of $G$.  For an integer $k\ge 2$,
+give each vertex at distance $d$ from $v$ the color $d\bmod k$.  Then deleting the vertices of any of the $k$ colors
+results in a graph of treewidth at most $3k$. This fact (which follows from the result of Robertson and Seymour~\cite{rs3}
+on treewidth of planar graphs of bounded radius) is (in the modern terms) the basis of Baker's technique~\cite{baker1994approximation}
+for design of approximation algorithms.  However, even quite simple graph classes (e.g., strong products of three paths~\cite{gridtw})
+do not admit such a coloring (where the removal of any color class results in a graph of bounded treewidth).
+However, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx}
+and applies to much more general graph classes, including all graph classes with strongly sublinear separators and bounded maximum degree~\cite{twd}.
+
+We say that a class of graphs $\GG$ is \emph{fractionally treewidth-fragile} if there exists a function $f$ such that
+for every graph $G\in\GG$ and integer $k\ge 2$, there exist sets $X_1, \ldots, X_m\subseteq V(G)$ such that
+each vertex belongs to at most $m/k$ of them and $\tw(G-X_i)\le f(k)$ for every $i$
+(equivalently, there exists a probability distribution on the set $\{X\subseteq V(G):\tw(G-X)\le f(k)\}$
+such that $\text{Pr}[v\in X]\le 1/k$ for each $v\in V(G)$).
+For example, the class of planar graphs is (fractionally) treewidth-fragile, since we can let $X_i$ consist of the
+vertices of color $i-1$ in the coloring described at the beginning of the section.
+
+Our main result is that all graph classes of bounded comparable box dimension are fractionally treewidth-fragile.
+We will show the result in a more general setting, motivated by concepts from~\cite{subconvex} and by applications to related
+representations. The argument is motivated by the idea used in the approximation algorithms for disk graphs
+by Erlebach et al.~\cite{erlebach2005polynomial}.
+
+For a measurable set $A\subseteq \mathbb{R}^d$, let $\vol(A)$ denote the Lebesgue measure of $A$.
+For two measurable subsets $A$ and $B$ of $\mathbb{R}^d$ and a positive integer $s$, we write $A\sqsubseteq_s B$
+if for every $x\in B$, there exists a translation $A'$ of $A$ such that $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$.
+Note that for two boxes $A$ and $B$, we have $A\sqsubseteq_1 B$ if and only if $A\sqsubseteq B$.
+An \emph{$s$-comparable envelope representation} $(\iota,\omega)$ of a graph $G$ in $\mathbb{R}^d$ consists of 
+two functions $\iota,\omega:V(G)\to 2^{\mathbb{R}^d}$ such that for some ordering $v_1$, \ldots, $v_n$ of vertices of $G$,
+\begin{itemize}
+\item for each $i$, $\omega(v_i)$ is a box, $\iota(v_i)$ is a measurable set, and $\iota(v_i)\subseteq \omega(v_i)$,
+\item if $i<j$, then $\omega(v_j)\sqsubseteq_s \iota(v_i)$, and
+\item if $i<j$ and $v_iv_j\in E(G)$, then $\omega(v_j)\cap \iota(v_i)\neq\emptyset$.
+\end{itemize}
+We say that the representation has \emph{thickness at most $t$} if for every point $x\in \mathbb{R}^d$, there
+exist at most $t$ vertices $v\in V(G)$ such that $x\in\iota(v)$.
+
+For example:
+\begin{itemize}
+\item If $f$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^d$, then
+$(f,f)$ is a $1$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2^d$.
+\item If $f$ is a touching representation of $G$ by balls in $\mathbb{R}^d$ and letting $\omega(v)$ be
+the smallest axis-aligned hypercube containing $f(v)$, then there exists a positive integer $s_d$ depending only on $d$ such that
+$(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2$.
+\end{itemize}
+
+\begin{theorem}\label{thm-twfrag}
+For positive integers $t$, $s$, and $d$, the class of graphs
+with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at most $t$
+is fractionally treewidth-fragile.
+\end{theorem}
+\begin{proof}
+For a positive integer $k$, let $f(k)=(2ksd+2)^dst$.
+Let $(\iota,\omega)$ be an $s$-comparable envelope representation of a graph $G$
+in $\mathbb{R}^d$ of thickness at most $t$, and let $v_1$, \ldots, $v_n$ be the corresponding ordering of the vertices of $G$.
+Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as follows.
+\begin{itemize}
+\item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$.
+\item For $i=2,\ldots, n$, let $\ell_{i,j} = \ell_{i-1,j} / b_{i,j}$, where
+$b_{i,j}=\max\bigl(1,\lfloor \tfrac{\ell_{i-1,j}}{ksd|\omega(v_i)[j]|} \bigr)$.
+\end{itemize}
+Let $x_j\in [0,\ell_{1,j}]$ be chosen uniformly at random, and let $\HH_j$ be the set of hyperplanes in $\mathbb{R}^d$
+consisting of the points whose $j$-th coordinate is equal to $x_j+m\ell_{i,j}$ for some $m\in\mathbb{Z}$ and $i\in\{1,\ldots, n\}$.
+For each hyperplane $H\in \HH_j$, let $i(H)$ denote the minimum integer $i$ such that $H$ can be expressed in this form.
+
+Let $\HH=\bigcup_{j=1}^d \HH_j$.
+We let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such that the box $\omega(v_a)$ intersects some hyperplane $H\in \HH$
+such that $i(H)\le a$.  First, let us argue that $\text{Pr}[v_a\in X]\le 1/k$.  Indeed, since $\ell{i,j}$ is an integer multiple
+of $\ell_{a,j}$ for every $i<a$, we conclude that $v\in X$ if and only if for some $j\in\{1,\ldots,d\}$ and $m\in\mathbb{Z}$,
+we have $x_j+m\ell_{a,j}\in \omega(v_a)[j]$.  The set $[0,\ell_{1,j}]\cap \bigcup_{m\in\mathbb{Z}} (\omega(v_a)[j]-m\ell_{a,j})$
+has measure $\tfrac{\ell_{1,j}}{\ell_{a,j}}\cdot |\omega(v_a)[j]|$, implying that for fixed $j$, this happens with probability
+$|\omega(v_a)[j]|/\ell_{a,j}$.  Let $a'\le a$ be the largest integer such that $b_{a',j}\neq 1$ if such an index exists,
+and $a'=1$ otherwise; note that $\ell_{a,j}=\ell_{a',j}\ge ksd|\omega(v_{a'})[j]|$.  Moreover, since
+$\omega(v_a)\sqsubseteq_s\iota(v_{a'})\subseteq \omega(v_{a'})$, we have $\omega(v_a)[j]\le s\omega(v_{a'})[j]$.
+Combining these inequalities,
+$$\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_{a'})[j]|}=\frac{1}{kd}.$$
+By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$.
+
+Let us now bound the treewidth of $G-X$.  For $a\in\{1,\ldots,n\}$, the \emph{$a$-grid} is $F_a=\bigcup_{H\in \HH:i(h)\le a} H$, and we let
+the $0$-grid $F_0=\emptyset$.  For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus F_a$.
+A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge 0$.
+A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$.
+Note that there exists a rooted tree $T$ whose vertices are
+the non-empty cells and such that for $x,y\in V(T)$, we have $x\preceq y$ if and only if $x\subseteq y$.
+For each non-empty cell $C$, let us define $\beta(C)$ as the set of vertices $v_i\in V(G-X)$ such that
+$\iota(v)\cap C\neq\emptyset$ and $C$ is an $a$-cell for some $a\ge i$.
+
+Let us show that $(T,\beta)$ is a tree decomposition of $G-X$.  For each $v_j\in V(G-X)$, the $j$-grid is disjoint from $\omega(v_j)$,
+and thus $\iota(v_j)\subseteq \omega(v_j)\subset C$ for some $j$-cell $C\in V(T)$ and $v_j\in \beta(C)$.  Consider now an edge $v_iv_j\in E(G-X)$, where $i<j$.
+We have $\omega(v_j)\cap \iota(v_i)\neq\emptyset$, and thus $\iota(v_i)\cap C\neq\emptyset$ and $v_i\in \beta(C)$.
+Finally, suppose that $v_j\in C'$ for some $C'\in V(T)$.  Then $C'$ is an $a$-cell for some $a\ge j$, and since
+$\iota(v_j)\cap C'\neq\emptyset$ and $\iota(v_j)\subset C$, we conclude that $C'\subseteq C$, and consequently $C'\preceq C$.
+Moreover, any cell $C''$ such that $C'\preceq C''\preceq C$ (and thus $C'\subseteq C''\subseteq C$) is an $a'$-cell
+for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq \iota(v_j)\cap C'\neq\emptyset$, implying $v_j\in\beta(C'')$.
+It follows that $\{C':v_j\in\beta(C')\}$ induces a connected subtree of $T$.
+
+Finally, let us bound the width of the decomposition $(T,\beta)$.  Let $C$ be a non-empty cell and let $a$ be maximum such that $C$
+is an $a$-cell.  Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ldots, $\ell_{a,d}$.  Consider $j\in\{1,\ldots,d\}$:
+\begin{itemize}
+\item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$.
+\item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $b_{a,j}=1$, implying $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$.
+\item If $a>1$ and $\ell_{a,j}>\ell_{a-1,j}$, then $\ell_{a-1,j}\ge 2ksd|\omega(v_a)[j]|$ and
+$$\ell_{a,j}=\frac{\ell_{a-1,j}}{\lfloor \frac{\ell_{a-1,j}}{ksd|\omega(v_a)[j]|}\rfloor}<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$
+\end{itemize}
+Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$.  Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
+For any $v_i\in \beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
+there exists a translation $B_i\subseteq C'$ of $\omega(v_a)$ such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$.
+Since the representation has thickness at most $t$,
+\begin{align*}
+\vol(C')&\ge \vol\left(C'\cap \bigcup_{v_i\in \beta(C)\setminus\{v_a\}} \iota(v_i)\right)\\
+&\ge \vol\left(\bigcup_{v_i\in \beta(C)\setminus\{v_a\}} B_i\cap\iota(v_i)\right)\\
+&\ge \frac{1}{t}\sum_{v_i\in \beta(C)\setminus\{v_a\}} \vol(B_i\cap\iota(v_i))\\
+&\ge \frac{\vol(\omega(v_a))(|\beta(C)|-1)}{st}.
+\end{align*}
+Since $\vol(C')=(2ksd+2)^d\vol(\omega(v_a))$, it follows that
+$$|\beta(C)|-1\le (2ksd+2)^dst=f(k),$$
+as required.
+\end{proof}
+
 \subsection*{Acknowledgments}
 This research was carried out at the workshop on Geometric Graphs and Hypergraphs organized by Yelena Yuditsky and Torsten Ueckerdt
 in September 2021.  We would like to thank the organizers and all participants for creating a friendly and productive environment.