From c9a0340cf3429dba677dfd594db2129440399273 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Zden=C4=9Bk=20Dvo=C5=99=C3=A1k?= <rakdver@iuuk.mff.cuni.cz>
Date: Thu, 16 Sep 2021 14:14:08 +0200
Subject: [PATCH] Added the consequence with regards to the existence of
sublinear separators.
---
comparable-box-dimension.tex | 14 ++++++++++++++
1 file changed, 14 insertions(+)
diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index dd48772..94bf604 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -570,6 +570,20 @@ $$|\beta(C)|-1\le (2ksd+2)^dst=f(k),$$
as required.
\end{proof}
+The proof that (generalizations of) graphs with bounded comparable box dimensions have sublinear separators in~\cite{subconvex}
+is indirect; it is established that these graphs have polynomial coloring numbers, which in turn implies they have polynomial
+expansion, which then gives sublinear separators using the algorithm of Plotkin, Rao, and Smith~\cite{plotkin}.
+The existence of sublinear separators is known to follow more directly from fractional treewidth-fragility. Indeed, since $\text{Pr}[v\in X]\le 1/k$,
+there exists $X\subseteq V(G)$ such that $\tw(G-X)\le f(k)$ and $|X|\le |V(G)|/k$. The graph $G-X$ has a balanced separator of size
+at most $\tw(G-X)+1$, which combines with $X$ to a balanced separator of size at most $V(G)|/k+f(k)+1$ in $G$.
+Optimizing the value of $k$ (choosing it so that $V(G)|/k=f(k)$), we obtain the following corollary of Theorem~\ref{thm-twfrag}.
+
+\begin{corollary}
+For positive integers $t$, $s$, and $d$, every graph $G$
+with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at most $t$
+has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).$
+\end{corollary}
+
\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.
--
GitLab