Update arxiv_cbd.tex

arxiv_cbd.tex

@@ -670,6 +670,12 @@ of $\cbdim(G)$ and $\chi(G)$.
   Therefore, (c2) holds.
 \end{proof}
 
+A touching representation of axis-aligned boxes in $\mathbb{R}^d$ is said \emph{fully touching} if any two intersecting boxes intersect on a $(d-1)$-dimensional box. Note that the construction above is fully touching.
+Indeed, two intersecting boxes corresponding to vertices $u,v$ of colors $c(u) < c(v)$, only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} lead to the following.
+\begin{corollary}
+\label{cor-fully-touching}
+Any graph $G$ has a fully touching representation of comparable axis-aligned boxes in $\mathbb{R}^d$, where $d= \cbdim(G) + 3^{\cbdim(G)}$.
+\end{corollary}
 
 Together, the lemmas from this section show that comparable box dimension is almost preserved by
 full clique-sums.
@@ -778,7 +784,7 @@ $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for suffici
 \end{proof}
 The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree.
 Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree}
-extends to graphs of treewidth at most $k$ (see the proof in the appendix).
+extends to graphs of treewidth at most $k$.
 \begin{corollary}\label{cor-tw}
 Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
 \end{corollary}
@@ -791,8 +797,8 @@ a representation $h$ of $T-V(C^\star)$ in $\mathbb{R}^{k+1}$ such that
 \item the vertices are represented by hypercubes of pairwise different sizes,
 \item if $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$, then $h(u)\cap h(v)$ is a facet of $h(u)$ incident
 with its point with minimum coordinates, and
-\item for each vertex $u$ and each facet of $h(u)$ incident with its point with minimum coordinates, there exists
-at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.
+%\item for each vertex $u$ and each facet of $h(u)$ incident with its point with minimum coordinates, 
+%there exists at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.
 \end{itemize}
 If for some $u,v\in V(G)$, we have $uv\in E(T)\setminus E(G)$, where without loss of generality $h(u)\sqsubseteq h(v)$,
 we now alter the representation by shrinking $h(u)$ slightly away from $h(v)$ (so that all other touchings are preserved).
@@ -848,10 +854,44 @@ 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.
 
+
+Before going further, 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 $T$.
+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.  Let us remark that the value of treewidth obtained via this definition coincides
+with the one via $k$-trees which we used in the previous 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}.
+by Erlebach et al.~\cite{erlebach2005polynomial}. Before introducing this more general setting, and as a warm-up, let us outline
+how to prove that disk graphs of thickness $t$ are fractionally treewidth-fragile. Consider first unit disk graphs. 
+By partitionning the plane with a random grid $\HH$, having squared cells of side-length $2k$, any unit disk has probability $1/2k$ 
+to intersect a vertical (resp. horizontal) line of the grid. By union bound, any disk has probability at most $1/k$ to intersect 
+the grid. Considering this probability distribution, let us now show that removing the disks intersected by the grid leads to a 
+unit disk graph of bounded treewidth. Indeed, in such a graph any connected component corresponds to unit disks contained in the 
+same cell of the grid. Such cell having area bounded by $4k^2$, there are at most $16tk^2/\pi$ disks contained in a cell. 
+The size of the connected components being bounded, so is the treewidth. Note that this distribution also works if we are given
+disks whose diameter lie in a certain range. If any diameter $\delta$ is such that $1/c \le \delta \le 1$, then the same process
+with a random grid of $2k\times 2k$ cells, ensures that any disk is deleted with probability at most $1/k$, while now the
+connected components have size at most $4tc^2k^2/\pi$. Dealing with arbitrary disk graphs (with any diameter $\delta$ being in the range 
+$0< \delta \le 1$) requires to delete more disks. This is why each $(2k\times 2k)$-cell is now partitionned in a quadtree-like manner. 
+Now a disk with diameter between $\ell /2$ and $\ell$ (with $\ell =1/2^i$ for some integer $i\ge 0$) is deleted if it is not contained 
+in a $(2k\ell \times 2k\ell)$-cell of a quadtree. It is not hard to see that a disk is deleted with probability at most $1/k$.
+To prove that the remaining graph has bounded treewidth one should consider the following tree decomposition $(T,\beta)$. The 
+tree $T$ is obtained by linking the roots of the quadtrees we used (as trees) to a new common root. 
+Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all the disks of diameter at least $\ell/2$ intersecting $C$.
+To see that such bag is bounded consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any
+disk in $\beta(C)$ intersects $C'$ on an area at least $\pi\ell^2/16$. This implies that $|\beta(C)| \le 16t(2k+1)^2 / \pi$.
 
 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 $A\sqsubseteq_s B$
@@ -876,20 +916,6 @@ 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}
 
-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 $T$.
-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.  Let us remark that the value of treewidth obtained via this definition coincides
-with the one via $k$-trees which we used in the previous section.
 
 \begin{theorem}\label{thm-twfrag}
 For positive integers $t$, $s$, and $d$, the class of graphs