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Zdenek Dvorak
Comparable box dimension
Commits
17ff0baa
Commit
17ff0baa
authored
Sep 08, 2021
by
Zdenek Dvorak
Browse files
Added the treewidth argument.
parent
0b81ca9b
Changes
1
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comparableboxdimension.tex
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17ff0baa
...
...
@@ 138,12 +138,82 @@ In fact, we will prove the following stronger fact (TODO: Was this published som
\begin{lemma}
\label
{
lemmatw
}
Let
$
(
T,
\beta
)
$
be a tree decomposition of a graph
$
G
$
of width
$
t
$
.
Then
$
G
$
has a touching representation
$
h
$
by hypercubes in
$
R
^{
t
+
1
}$
such that
for
$
u,v
\in
V
(
G
)
$
, if
$
p
(
u
)
\prec
eq
p
(
v
)
$
, then
$
h
(
u
)
\sqsubset
eq
h
(
v
)
$
.
Then
$
G
$
has a touching representation
$
h
$
by
axisaligned
hypercubes in
$
R
^{
t
+
1
}$
such that
for
$
u,v
\in
V
(
G
)
$
, if
$
p
(
u
)
\prec
p
(
v
)
$
, then
$
h
(
u
)
\sqsubset
h
(
v
)
$
.
Moreover, the representation can be chosen so that no two hypercubes have the same size.
\end{lemma}
\begin{proof}
...
Without loss of generality, we can assume that the root has a bag of size one
and that for each
$
x
\in
V
(
T
)
$
with parent
$
y
$
, we have
$

\beta
(
x
)
\setminus\beta
(
y
)

=
1
$
(if
$
\beta
(
x
)
\subseteq
\beta
(
y
)
$
, we can contract the edge
$
xy
$
; if
$

\beta
(
x
)
\setminus\beta
(
y
)
>
1
$
,
we can subdivide the edge
$
xy
$
, introducing
$

\beta
(
x
)
\setminus\beta
(
y
)


1
$
new vertices,
and set their bags appropriately). It is now natural to relabel the vertices of
$
G
$
so that
$
V
(
G
)=
V
(
T
)
$
, by giving the unique vertex in
$
\beta
(
x
)
\setminus\beta
(
y
)
$
the label
$
x
$
. In particular,
$
p
(
x
)=
x
$
. Furthermore, we can assume that
$
y
\in\beta
(
x
)
$
.
Otherwise, if
$
y
$
is not the root of
$
T
$
, we can replace the edge
$
xy
$
by the edge from
$
x
$
to the parent of
$
y
$
. If
$
y
$
is the root of
$
T
$
, then the subtree rooted in
$
x
$
induces a
union of connected components in
$
G
$
, and we can process this subtree separately from the
rest of the graph (being careful to only use hypercubes smaller than the one representing
$
y
$
and of different sizes from those used on the rest of the graph).
Let us now greedily color
$
G
$
by giving
$
x
$
a color different from the colors of
all other vertices in
$
\beta
(
x
)
$
; such a coloring
$
\varphi
$
uses only colors
$
\{
1
,
\ldots
,t
+
1
\}
$
.
Let
$
D
=
4
\Delta
(
T
)+
1
$
. Let
$
V
(
G
)=
V
(
T
)=
\{
x
_
1
,x
_
2
,
\ldots
, x
_
n
\}
$
, where
for every
$
i<j
$
,
$
x
_
i
$
and
$
x
_
j
$
are either incomparable in
$
\prec
$
or
$
x
_
j
\prec
x
_
i
)
$
; in particular,
$
x
_
1
$
is the root of
$
T
$
. Let
$
\varepsilon
=
D
^{

n

1
}$
.
Let
$
s
_
i
=
D
^{

i
}$
; we will represent
$
x
_
i
$
by a hypercube
$
h
(
x
_
i
)
$
with edges of length
$
s
_
i
$
.
Additionally, we will need to consider larger hypercubes around
$
h
(
x
_
i
)
$
; let
$
h'
(
x
_
i
)
$
be the hypercube with sides of length
$
2
s
_
i
$
and with
$
\min
(
h'
(
x
_
i
)[
j
])=
\min
(
h
(
x
_
i
)[
j
])
$
for
$
j
\in\{
1
,
\ldots
, t
+
1
\}
$
, and
$
h''
(
x
_
i
)
$
the hypercube with sides of length
$
2
s
_
i
+
\epsilon
$
and with
$
\min
(
h''
(
x
_
i
)[
j
])=
\min
(
h
(
x
_
i
)[
j
])
\varepsilon
$
. We will construct the representation
$
h
$
so that the following
invariant is satisfied:
\begin{itemize}
\item
[(a)]
For each
$
x,z
\in
V
(
T
)
$
such that
$
x
\prec
z
$
, we have
$
h'
(
x
)
\subset
h''
(
z
)
$
.
\item
[(b)]
For each
$
y
\in
V
(
T
)
$
and distinct children
$
x
$
and
$
z
$
of
$
y
$
, we have
$
h''
(
x
)
\cap
h''
(
z
)=
\emptyset
$
.
\end{itemize}
Note that this ensures that if
$
x
$
and
$
z
$
are vertices of
$
T
$
and
$
h
(
x
)
\cap
h
(
z
)
\neq\emptyset
$
, then
$
x
\prec
z
$
or
$
z
\prec
x
$
.
We now construct the representation
$
h
$
. For the root
$
x
_
1
$
of
$
T
$
,
$
h
(
r
)
$
is an arbitrary hypercube with sides
of length
$
s
_
1
$
. Assuming now we have already selected
$
h
(
y
)
$
for a vertex
$
y
\in
V
(
T
)
$
, the hypercube
$
h
(
x
_
i
)
$
with sides of length
$
s
_
i
$
for a child
$
x
_
i
$
of
$
y
$
is chosen as follows. For
$
j
\in\{
1
,
\ldots
, t
+
1
\}
$
,
\begin{itemize}
\item
[(i)]
if
$
j
=
\varphi
(
w
)
$
for
$
w
\in\beta
(
x
_
i
)
\setminus\{
x
_
i
\}
$
, we choose
$
h
(
x
_
i
)[
j
]
$
so that
$
\min
(
h
(
x
_
i
)[
j
])=
\max
(
h
(
w
)[
j
])
$
if
$
xw
\in
E
(
G
)
$
and so that
$
\min
(
h
(
x
_
i
)[
j
])=
\max
(
h
(
w
)[
j
])
+
\varepsilon
$
otherwise.
\item
[(ii)]
if
$
j
$
is different from the colors of all vertices in
$
\beta
(
x
_
i
)
\setminus\{
x
_
i
\}
$
,
then we choose
$
h
(
x
_
i
)[
j
]
$
so that
$
h''
(
x
_
i
)[
j
]
$
is a subset of the interior of
$
h
(
y
)[
j
]
$
. The interval
$
h''
(
x
_
i
)[[
j
]
$
is furthermore chosen to be disjoint from
$
h''
(
x
_
m
)[
j
]
$
for any other child
$
x
_
m
$
of
$
y
$
;
this is always possible by the choice of
$
D
$
,
$
s
_
i
$
, and
$
s
_
m
$
.
\end{itemize}
Note that (ii) always applies for
$
j
=
\varphi
(
x
_
i
)
$
and this ensures that the invariant (b) holds.
For the invariant (a), note that in the case (ii), we ensure
$
h''
(
x
_
i
)[[
j
]
\subseteq
h
(
y
)[
j
]
$
and
we have
$
h
(
y
)[
j
]
\subseteq
h''
(
z
)[
j
]
$
by the invariant (a) for
$
y
$
and
$
z
$
. In the case (i),
if
$
z
\prec
w
$
, then we have
$
w
\in\beta
(
z
)
\setminus\{
z
\}
$
and
$
\min
(
h
(
x
_
i
)[
j
])
,
\min
(
h
(
z
)[
j
])
\in\{\max
(
h
(
w
)[
j
])
,
\max
(
h
(
w
)[
j
])+
\varepsilon\}
$
.
If
$
w
\preceq
z
$
, then note we choose
$
h'
(
x
_
i
)[
j
]
\subseteq
h'
(
w
)[
j
]
$
by (i) and that
we have
$
h'
(
w
)[
j
]
\subset
h''
(
z
)[
j
]
$
by (a). This verifies that the invariant (a)
also holds at
$
x
_
i
$
.
Consider now two adjacent vertices of
$
G
$
, say
$
x
_
i
$
and
$
w
$
. Note that any two
adjacent vertices are comparable in
$
\prec
$
, and thus we can assume
$
x
_
i
\prec
w
$
and
$
w
\in\beta
(
x
_
i
)
$
.
By (i), for
$
j
=
\varphi
(
w
)
$
, the intervals
$
h
(
x
_
i
)[
j
]
$
and
$
h
(
w
)[
j
]
$
intersect in a single point. If
$
j
\neq
\varphi
(
w
)
$
, then let
$
w
_
1
$
be the child of
$
w
$
on the path in
$
T
$
from
$
w
$
to
$
x
_
i
$
. If no vertex in
$
\beta
(
w
_
1
)
\setminus\{
w
_
1
\}
$
has color
$
j
$
, then by (ii), we have
$
h''
(
w
_
1
)[
j
]
\subset
h
(
w
)[
j
]
$
,
Otherwise,
$
z
\in
\beta
(
w
_
1
)
\setminus\{
w
_
1
\}
$
such that
$
\varphi
(
z
)=
j
$
; clearly,
$
w
\prec
z
$
and
$
z
\in\beta
(
w
)
\setminus\{
w
\}
$
.
We have
$
\min
(
h
(
w
_
1
)[
j
])
,
\min
(
h
(
w
)[
j
])
\in\{\max
(
h
(
z
)[
j
])
,
\max
(
h
(
z
)[
j
])+
\varepsilon\}
$
by (i).
Hence,
$
\min
(
h
(
w
)[
j
])
2
\epsilon\le
\min
(
h''
(
w
_
1
)[
j
])
\le
\max
(
h''
(
w
_
1
)[
j
])
\le
\max
(
h
(
w
)[
j
])
$
.
Since
$
h
(
x
_
i
)[
j
]
\subseteq
h''
(
w
_
1
)[
j
]
$
by (a) and the length of the interval
$
h
(
x
_
i
)[
j
]
$
is greater than
$
2
\varepsilon
$
,
we conclude that
$
h
(
x
_
i
)[
j
]
\cap
h
(
w
)[
j
]
\neq\emptyset
$
. Therefore, the boxes
$
h
(
w
)
$
and
$
h
(
x
_
i
)
$
touch.
Consider now two nonadjacent vertices of
$
G
$
, say
$
x
_
i
$
and
$
w
$
. As we noted before, if
$
x
_
i
$
and
$
w
$
are
incomparable in
$
\prec
$
, then (a) and (b) implies that the boxes
$
h
(
x
_
i
)
$
and
$
h
(
w
)
$
are disjoint.
Suppose now that say
$
x
_
i
\prec
w
$
, and let
$
j
=
\varphi
(
w
)
$
. If
$
w
\in\beta
(
x
_
i
)
$
, then
$
h
(
x
_
i
)[
j
]
$
and
$
h
(
w
)[
j
]
$
are disjoint by (i).
Otherwise, let
$
y
$
be the last vertex on the path from
$
w
$
to
$
x
_
i
$
in
$
T
$
such that
$
w
\in\beta
(
y
)
$
and let
$
z
$
be the child of
$
y
$
on this path. As we argued in the first paragraph,
$
y
\neq
w
$
, and by (i), the interior of
$
h
(
y
)[
j
]
$
is disjoint from
$
h
(
w
)[
j
]
$
.
By (ii),
$
h''
(
z
)[
j
]
$
is contained in the interior of
$
h
(
y
)[
j
]
$
. By (a), we conclude that
$
h
(
x
_
i
)[
j
]
\subseteq
h''
(
z
)[
j
]
$
,
implying that the boxes
$
h
(
x
_
i
)
$
and
$
h
(
w
)
$
are disjoint. Therefore,
$
h
$
is a touching representation of
$
G
$
.
\end{proof}
Next, let us deal with cliquesums. A
\emph
{
cliquesum
}
of two graphs
$
G
_
1
$
and
$
G
_
2
$
is obtained from their disjoint union
...
...
@@ 191,14 +261,14 @@ and thus $p(x)=x$.
We are now ready to deal with the cliquesums.
\begin{theorem}
\begin{theorem}
\label
{
thmcs
}
If
$
G
$
is obtained from graphs in a class
$
\GG
$
by cliquesums, then there exists a graph
$
G'
$
such that
$
G
\subseteq
G'
$
and
$
\cbdim
(
G'
)
\le
(
\cbdim
(
\GG
)+
1
)(
\omega
(
\GG
)+
1
)
$
.
\end{theorem}
\begin{proof}
Let
$
(
T,
\beta
)
$
be a tree decomposition of
$
G
$
over
$
\GG
$
; the adhesion
$
a
$
of
$
(
T,
\beta
)
$
is at most
$
\omega
(
\GG
)
$
.
By Lemma~
\ref
{
lemmalegraf
}
,
$
(
T,
\pi
)
$
is a tree decomposition of
$
T
_
\beta
$
of width at most
$
a
$
. By Lemma~
\ref
{
lemmatw
}
,
$
T
_
\beta
$
has a touching representation
$
h
$
by hypercubes in
$
\mathbb
{
R
}^{
a
+
1
}$
such that
$
T
_
\beta
$
has a touching representation
$
h
$
by
axisaligned
hypercubes in
$
\mathbb
{
R
}^{
a
+
1
}$
such that
$
h
(
x
)
\sqsubseteq
h
(
y
)
$
whenever
$
x
\preceq
y
$
.
Since
$
T
_
\beta
$
has treewidth at most
$
a
$
, it has a proper coloring
$
\varphi
$
by colors
$
\{
0
,
\ldots
,a
\}
$
.
For every
$
x
\in
V
(
T
)
$
, let
$
f
_
x
$
be a touching representation of the torso
$
G
_
x
$
of
$
x
$
by comparable boxes in
$
\mathbb
{
R
}^
d
$
,
...
...
@@ 233,6 +303,8 @@ Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\l
$
xy
\in
E
(
T
_
\beta
)
$
. Hence, again we have
$
f
(
u
)
\cap
f
(
v
)
\neq\emptyset
$
.
\end{proof}
Note that in Theorem~
\ref
{
thmcs
}
, we only get a representation of a supergraph
of
$
G
$
.
%We will need the fact that the chromatic number is at most exponential in the comparable box dimension;
%this follows from~\cite{subconvex} and we include the argument to make the dependence clear.
...
...
@@ 249,7 +321,7 @@ $xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$.
%\end{proof}
\section
{
Exploiting t
he product structure
}
\section
{
T
he product structure
and minorclosed classes
}
\subsection*
{
Acknowledgments
}
This research was carried out at the workshop on Geometric Graphs and Hypergraphs organized by Yelena Yuditsky and Torsten Ueckerdt
...
...
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