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Zdenek Dvorak
Comparable box dimension
Commits
c9a0340c
Commit
c9a0340c
authored
Sep 16, 2021
by
Zdenek Dvorak
Browse files
Added the consequence with regards to the existence of sublinear separators.
parent
4278f250
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comparableboxdimension.tex
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c9a0340c
...
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@@ 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 treewidthfragility. 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
{
thmtwfrag
}
.
\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.
...
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