### Fragility part

- small typos
- def of $l_{i,j}$
- removed $i(H)$,
- reintroduced the definition of tree decomposition (T,b)
parent 86202e3f
 ... ... @@ -708,7 +708,7 @@ 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 $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$. if for every $x\in B$, there exists a translation $A'$ of $A$ such that $x\in A'$ and $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$. 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 $G$, ... ... @@ -729,40 +729,71 @@ the smallest axis-aligned hypercube containing $f(v)$, then there exists a posit $(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2$. \end{itemize} \note{TO REMOVE if we reintroduce tree decomposition earlier !} \note{Let us recall some notions about treewidth. A \emph{tree decomposition} of a graph $G$ is a pair $(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$ assigns a \emph{bag} to each of its nodes, such that \begin{itemize} \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 $v\in \beta(x)$ 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 The \emph{width} of the tree decomposition 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.} \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. is fractionally treewidth-fragile, with a function $f(k) = O_{t,s,d}\bigl(k^{d}\bigr)$. \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. Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $|ksd\omega(v_i)[j]|$ such that $\ell_{i-1,j} / \ell_{i,j}$ is a positive integer. Formally it is defined 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 $b_{i,j}=\max\bigl(1,\lfloor \tfrac{\ell_{i-1,j}}{ksd|\omega(v_i)[j]|} \bigr)$. \item For $i=2,\ldots, n$, let $\ell_{i,j} = \ell_{i-1,j}$, if $\ell_{i-1,j} < ksd|\omega(v_i)[j]|$, and otherwise let $\ell_{i,j}$ be lowest fraction of $\ell_{i-1,j}$ that is greater than $ksd|\omega(v_i)[j]|$, formally $\ell_{i,j} = \min\{\ell_{i-1,j}/b \ |\ b\in \mathbb{N}^+ \text{ and } \ell_{i-1,j}/b \ge ksd|\omega(v_i)[j]|\}$. \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 $i1$ 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 $$\ell_{a,j}=\frac{\ell_{a-1,j}}{\lfloor \frac{\ell_{a-1,j}}{ksd|\omega(v_a)[j]|}\rfloor}<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$ \item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$ (otherwise $\ell_{a,j}=\ell_{a-1,j}/b$ for some integer $b\ge 2$). \item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integers $b\ge 2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that $$\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$ \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 $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$. there exists a translation $B_i$ of $\omega(v_a)$ that intersects $C\cap \iota(v_i)$ and such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$. Note that as $B_i$ intersects $C$, we have that $B_i\subseteq C'$. 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)\\ ... ...
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