Skip to content
GitLab
Menu
Projects
Groups
Snippets
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in
Toggle navigation
Menu
Open sidebar
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
}
...
...
Write
Preview
Supports
Markdown
0%
Try again
or
attach a new file
.
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment