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Zdenek Dvorak
Comparable box dimension
Commits
0b81ca9b
Commit
0b81ca9b
authored
Sep 07, 2021
by
Zdenek Dvorak
Browse files
Improved readability of the cliquesum argument.
parent
e743a8cd
Changes
1
Hide whitespace changes
Inline
Sidebyside
comparableboxdimension.tex
View file @
0b81ca9b
...
...
@@ 68,7 +68,8 @@ For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis
one part are represented by
$
m
\times
1
\times
1
$
boxes and the vertices of the other part are represented by
$
1
\times
n
\times
1
$
boxes (a
\emph
{
box
}
is the cartesian product of intervals of nonzero length).
Dvo
\v
{
r
}
\'
ak, McCarty and Norin~
\cite
{
subconvex
}
noticed that this issue disappears if we forbid such a combination of
long and wide boxes: Two boxes are
\emph
{
comparable
}
if a translation of one of them is a subset of the other one.
long and wide boxes: For two boxes
$
B
_
1
$
and
$
B
_
2
$
, we write
$
B
_
1
\sqsubseteq
B
_
2
$
if a translation of
$
B
_
1
$
is a subset of
$
B
_
2
$
.
We say that
$
B
_
1
$
and
$
B
_
2
$
are
\emph
{
comparable
}
if
$
B
_
1
\sqsubseteq
B
_
2
$
or
$
B
_
2
\sqsubseteq
B
_
1
$
.
A
\emph
{
touching representation by comparable boxes
}
of a graph
$
G
$
is a touching representation
$
f
$
by boxes
such that for every
$
u,v
\in
V
(
G
)
$
, the boxes
$
f
(
u
)
$
and
$
f
(
v
)
$
are comparable. For a graph
$
G
$
, let the
\emph
{
comparable box dimension
}
$
\cbdim
(
G
)
$
of
$
G
$
be the smallest integer
$
d
$
such that
$
G
$
has a touching representation by comparable boxes in
$
\mathbb
{
R
}^
d
$
.
...
...
@@ 112,20 +113,6 @@ where $M$ is chosen large enough so that $f(u)\subseteq [M,M] \times \cdots \ti
Then
$
h
$
is a touching representation of
$
G
$
by comparable boxes in
$
\mathbb
{
R
}^{
d
+
1
}$
.
\end{proof}
%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.
%\begin{lemma}\label{lemmachrom}
%If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$colorable.
%\end{lemma}
%\begin{proof}
%We actually show that $G$ is $(3^d1)$degenerate. Since every induced subgraph of $G$ also
%has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$
%is less than $3^d$. Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume. For every neighbor $u$ of $v$,
%there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$.
%Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in
%a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+N(v)\le 3^d$.
%\end{proof}
We need a bound on the clique number in terms of the comparable box dimension.
For a box
$
B
=
I
_
1
\times
\cdots
I
_
d
$
and
$
i
\in\{
1
,
\ldots
,d
\}
$
, let
$
B
[
i
]=
I
_
i
$
.
\begin{lemma}
\label
{
lemmacliq
}
...
...
@@ 142,6 +129,7 @@ such that
\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 nonempty 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
)
$
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
\emph
{
width
}
is the maximum of the sizes of the bags minus
$
1
$
. The
\emph
{
treewidth
}
of a graph is the minimum
...
...
@@ 151,8 +139,8 @@ 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
)
\
n
eq
p
(
v
)
$
and
$
p
(
u
)
$
is an ancestor of
$
p
(
v
)
$
in
$
T
$
,
then
$
\vol
(
h
(
u
))
>
\vol
(
h
(
v
))
$
.
for
$
u,v
\in
V
(
G
)
$
, if
$
p
(
u
)
\
prec
eq
p
(
v
)
$
, then
$
h
(
u
)
\sqsubseteq
h
(
v
)
$
.
Moreover, the representation can be chosen so that no two hypercubes have the same size
.
\end{lemma}
\begin{proof}
...
...
...
@@ 178,6 +166,7 @@ For each note $x\in V(T)$, let $\pi(x)=\{x\}\cup \{p(v):v\in\beta(x)\}$. Let $T
$
xy
\in
E
(
T
_
\beta
)
$
if and only if
$
x
\in\pi
(
y
)
$
or
$
y
\in\pi
(
x
)
$
.
\begin{lemma}
\label
{
lemmalegraf
}
If
$
(
T,
\beta
)
$
is a tree decompositon of
$
G
$
of adhesion
$
a
$
, then
$
(
T,
\pi
)
$
is a tree decomposition of
$
T
_
\beta
$
of width at most
$
a
$
.
Moreover,
$
\pi
(
x
)
$
is a clique in
$
T
_
\beta
$
and
$
p
(
x
)=
x
$
for each
$
x
\in
V
(
T
)
$
.
\end{lemma}
\begin{proof}
For each edge
$
xy
\in
E
(
T
_
\beta
)
$
, we have
$
x,y
\in
\pi
(
x
)
$
or
$
x,y
\in
\pi
(
y
)
$
by definition.
...
...
@@ 191,6 +180,13 @@ Consider a node $x\in V(T)$. Note that for each $v\in \beta(x)$, the vertex $p(
In particular, if
$
x
$
is the root of
$
T
$
, then
$
\pi
(
x
)=
\{
x
\}
$
. Otherwise, if
$
y
$
is the parent of
$
x
$
in
$
T
$
, then
$
p
(
v
)=
x
$
for every
$
v
\in
\beta
(
x
)
\setminus\beta
(
y
)
$
, and thus
$

\pi
(
x
)

\le

\beta
(
x
)
\cap
\beta
(
y
)

+
1
\le
a
+
1
$
.
Hence, the width of
$
(
T,
\pi
)
$
is at most
$
a
$
.
Suppose now that
$
y
$
and
$
z
$
are distinct vertices in
$
\pi
(
x
)
$
. Then both
$
y
$
and
$
z
$
are ancestors of
$
x
$
in
$
T
$
,
and thus without loss of generality, we can assume that
$
y
\preceq
z
$
. If
$
y
=
x
$
, then
$
yz
\in
E
(
T
_
\beta
)
$
by definition.
Otherwise, there exist vertices
$
u,v
\in
\beta
(
x
)
$
such that
$
p
(
u
)=
y
$
and
$
p
(
v
)=
z
$
. Since
$
v
\in
\beta
(
x
)
\cap\beta
(
z
)
$
and
$
y
$
is on the path in
$
T
$
from
$
x
$
to
$
z
$
, we also have
$
v
\in\beta
(
y
)
$
. This implies
$
z
\in\pi
(
y
)
$
and
$
yz
\in
E
(
T
_
\beta
)
$
.
Hence,
$
\pi
(
x
)
$
is a clique in
$
T
_
\beta
$
. Moreover, note that
$
x
\not\in
\pi
(
w
)
$
for any ancestor
$
w
\neq
x
$
of
$
x
$
in
$
T
$
,
and thus
$
p
(
x
)=
x
$
.
\end{proof}
We are now ready to deal with the cliquesums.
...
...
@@ 202,37 +198,56 @@ $G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)(\omega(\GG)+1)$.
\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
}$
. Moreover,
letting
$
\prec
$
be a linear ordering on
$
V
(
T
)
$
in a nondecreasing order according to the volume of the hypercubes assigned by
$
h
$
,
we have that
$
x
\prec
y
$
whenever
$
x
$
is a descendant of
$
y
$
in
$
T
$
.
$
T
_
\beta
$
has a touching representation
$
h
$
by 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 of
$
x
$
by comparable boxes in
$
\mathbb
{
R
}^
d
$
for
$
d
=
\cbdim
(
\GG
)
$
. We scale and translate the representations so that for every
$
x
\in
V
(
T
)
$
and
$
i
\in\{
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
$
,
where
$
d
=
\cbdim
(
\GG
)
$
. We scale and translate the representations so that for every
$
x
\in
V
(
T
)
$
and
$
i
\in\{
0
,
\ldots
,a
\}
$
,
there exists a box
$
E
_
i
(
x
)
$
such that
\begin{itemize}
\item
whenever
$
x
\prec
y
$
, we have
$
E
_
i
(
x
)
\subseteq
E
_
i
(
y
)
$
and a translation of
$
E
_
i
(
x
)
$
is a subset of every box of
the representation
$
f
_
y
$
whenever
$
x
\prec
y
$
,
\item
if
$
i
=
\varphi
(
x
)
$
, then all boxes of
$
f
_
x
$
are subsets of
$
E
_
i
(
x
)
$
, and
\item
if
$
i
\neq
p
(
v
)
$
and
$
K
=
\{
v
\in\beta
(
x
)
:
\varphi
(
p
(
v
))=
i
\}
$
, then letting
$
y
=
p
(
v
)
$
for
$
v
\in
K
$
(and noting that this
$
y
$
is unique, since
$
\varphi
$
is a proper coloring of
$
T
_
\beta
$
and that
$
K
$
is a clique in
$
G
_
y
$
), the box
$
E
_
i
(
x
)
$
contains a point belonging to
$
\bigcap
_{
v
\in
K
}
f
_
y
(
v
)
$
.
\item
[(a)]
for distinct
$
x,y
\in
V
(
T
)
$
, if
$
h
(
x
)
\sqsubset
h
(
y
)
$
, then
$
E
_
i
(
x
)
\sqsubset
E
_
i
(
y
)
$
and
$
E
_
i
(
x
)
\sqsubset
f
_
y
(
v
)
$
for every
$
v
\in
\beta
(
y
)
$
,
\item
[(b)]
if
$
x
\prec
y
$
, then
$
E
_
i
(
x
)
\subseteq
E
_
i
(
y
)
$
,
\item
[(c)]
if
$
i
=
\varphi
(
x
)
$
, then
$
f
_
x
(
v
)
\subseteq
E
_
i
(
x
)
$
for every
$
v
\in\beta
(
x
)
$
, and
\item
[(d)]
if
$
i
\neq
\varphi
(
x
)
$
and
$
K
=
\{
v
\in\beta
(
x
)
:
\varphi
(
p
(
v
))=
i
\}
$
is nonempty, then letting
$
y
=
p
(
v
)
$
for
$
v
\in
K
$
,
the box
$
E
_
i
(
x
)
$
contains a point belonging to
$
\bigcap
_{
v
\in
K
}
f
_
y
(
v
)
$
.
\end{itemize}
Some explanation is in order for the last point: Firstly, since
$
\pi
(
x
)
$
is a clique in
$
T
_
\beta
$
, there
exists only one vertex
$
y
\in
\pi
(
x
)
$
of color
$
i
$
, and thus
$
y
=
p
(
v
)
$
for all
$
v
\in
K
$
.
Moreover,
$
K
$
is a clique in
$
G
_
y
$
, and thus
$
\bigcap
_{
v
\in
K
}
f
_
y
(
v
)
$
is nonempty. Lastly,
note that if
$
x
\prec
z
\prec
y
$
, then
$
K
=
\beta
(
x
)
\cap\beta
(
y
)
\subseteq
\beta
(
z
)
\cap
\beta
(
y
)
$
,
and thus
$
E
_
i
(
z
)
$
was also chosen to contain a point of
$
\bigcap
_{
v
\in
K
}
f
_
y
(
v
)
$
;
hence, a choice of
$
E
_
i
(
x
)
$
satisfying
$
E
_
i
(
x
)
\subseteq
E
_
i
(
z
)
$
as required by (b) is possible.
Let us now define
$
f
(
v
)=
h
(
p
(
v
))
\times
E
_
0
(
v
)
\times
\cdots\times
E
_
a
(
v
)
$
for each
$
v
\in
V
(
G
)
$
,
where
$
E
_
i
(
v
)=
f
_{
p
(
v
)
}
(
v
)
$
if
$
i
=
\varphi
(
p
(
v
))
$
and
$
E
_
i
(
v
)=
E
_
i
(
p
(
v
))
$
otherwise. We claim this gives a touching representation
of a supergraph of
$
G
$
by comparable boxes in
$
\mathbb
{
R
}^{
(
d
+
1
)(
a
+
1
)
}$
. First, note that the boxes are indeed comparable;
if
$
p
(
u
)=
p
(
v
)
$
, then this is the case since
$
f
_{
p
(
v
)
}$
is a representation by comparable boxes, and if say
$
p
(
u
)
\prec
p
(
v
)
$
, then this is due to
the scaling of
$
f
_{
p
(
u
)
}$
. Next, let us argue
$
f
(
u
)
$
and
$
f
(
v
)
$
have disjoint
$
p
(
u
)
\prec
p
(
v
)
$
, then this is due to
(a) and (c)
. Next, let us argue
$
f
(
u
)
$
and
$
f
(
v
)
$
have disjoint
interiors. If
$
p
(
u
)=
p
(
v
)
$
, this is the case since
$
f
_{
p
(
v
)
}$
is a touching representation, and if
$
p
(
u
)
\neq
p
(
v
)
$
,
then this is the case because
$
h
$
is a touching representation. Finally, suppose that
$
uv
\in
E
(
G
)
$
. Let
$
x
$
be the node of
$
T
$
nearest to the root such that
$
u,v
\in
\beta
(
x
)
$
. Without loss of generality,
$
p
(
u
)=
x
$
.
Let
$
y
=
p
(
v
)
$
. If
$
x
=
y
$
, then
$
f
(
u
)
\cap
f
(
v
)
\neq\emptyset
$
, since
$
f
_
x
$
is a touching representation of
$
G
_
x
$
.
Let
$
y
=
p
(
v
)
$
. If
$
x
=
y
$
, then
$
f
(
u
)
\cap
f
(
v
)
\neq\emptyset
$
, since
$
f
_
x
$
is a touching representation of
$
G
_
x
$
and
$
uv
\in
E
(
G
_
x
)
$
.
If
$
x
\neq
y
$
, then
$
y
\in\pi
(
x
)
$
and
$
xy
\in
E
(
T
_
\beta
)
$
, implying that
$
h
(
x
)
\cap
h
(
y
)
\neq\emptyset
$
.
Moreover,
$
x
\prec
y
$
, implying that
$
E
_
i
(
u
)
\cap
E
_
i
(
v
)
\neq\emptyset
$
for
$
i
=
0
,
\ldots
,a
$
.
Hence, again we have
$
f
(
u
)
\cap
f
(
v
)
\neq\emptyset
$
.
Moreover,
$
x
\prec
y
$
, implying that
$
E
_
i
(
u
)
\cap
E
_
i
(
v
)
\neq\emptyset
$
for
$
i
=
0
,
\ldots
,a
$
by (b) if
$
\varphi
(
x
)
\neq
i
\neq
\varphi
(
y
)
$
,
(b) and (c) if
$
\varphi
(
x
)=
i
\neq\varphi
(
y
)
$
, and (d) if
$
\varphi
(
x
)
\neq
i
=
\varphi
(
y
)
$
(we cannot have
$
\varphi
(
x
)=
i
=
\varphi
(
y
)
$
, since
$
xy
\in
E
(
T
_
\beta
)
$
. Hence, again we have
$
f
(
u
)
\cap
f
(
v
)
\neq\emptyset
$
.
\end{proof}
%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.
%\begin{lemma}\label{lemmachrom}
%If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$colorable.
%\end{lemma}
%\begin{proof}
%We actually show that $G$ is $(3^d1)$degenerate. Since every induced subgraph of $G$ also
%has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$
%is less than $3^d$. Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume. For every neighbor $u$ of $v$,
%there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$.
%Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in
%a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+N(v)\le 3^d$.
%\end{proof}
\section
{
Exploiting the product structure
}
...
...
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