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Zdenek Dvorak
Comparable box dimension
Commits
86202e3f
Commit
86202e3f
authored
Oct 08, 2021
by
Daniel Gonçalves
Browse files
added justifications in the proofs wrt cliquesum extendability
parent
fe97fd5d
Changes
1
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Inline
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comparableboxdimension.tex
View file @
86202e3f
...
...
@@ 162,7 +162,7 @@ vertices. We proceed similarly to bound the chromatic number.
\section
{
Operations
}
It is clear that given a touching representation of a graph
$
G
$
, one
easily obtains a touching representation
with
boxes of an induced
easily obtains a touching representation
by
boxes of an induced
subgraph
$
H
$
of
$
G
$
by simply deleting the boxes corresponding to the
vertices in
$
V
(
G
)
\setminus
V
(
H
)
$
. In this section we are going to
consider other basic operations on graphs.
...
...
@@ 189,8 +189,8 @@ Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^
Let
$
G
\boxtimes
H
$
denote the
\emph
{
strong product
}
of the graphs
$
G
$
and
$
H
$
, i.e., the graph with vertex set
$
V
(
G
)
\times
V
(
H
)
$
and with
distinct vertices
$
(
u
_
1
,v
_
1
)
$
and
$
(
u
_
2
,v
_
2
)
$
adjacent if and only if
either
$
u
_
1
=
u
_
2
$
or
$
u
_
1
u
_
2
\in
E
(
G
)
$
and either
$
v
_
1
=
v
_
2
$
or
distinct vertices
$
(
u
_
1
,v
_
1
)
$
and
$
(
u
_
2
,v
_
2
)
$
adjacent if and only if
,
either
$
u
_
1
=
u
_
2
$
or
$
u
_
1
u
_
2
\in
E
(
G
)
$
,
and either
$
v
_
1
=
v
_
2
$
or
$
v
_
1
v
_
2
\in
E
(
G
)
$
. To obtain a touching representation of
$
G
\boxtimes
H
$
it suffice to take a product of representations of
$
G
$
and
$
H
$
, but
the obtained representation may contain uncomparable boxes. Thus,
...
...
@@ 208,7 +208,7 @@ overcome this issue, by constraining one of the representations.
The proof simply consists in taking a product of the two
representations. Indeed, consider a touching respresentation with
comparable boxes
$
g
$
of
$
G
$
in
$
\mathbb
{
R
}^{
d
_
G
}$
, with
$
d
_
G
=
\cbdim
(
G
)
$
, and the
d
epresentation
$
h
$
of
$
H
$
. Let us define a
$
d
_
G
=
\cbdim
(
G
)
$
, and the
r
epresentation
$
h
$
of
$
H
$
. Let us define a
representation
$
f
$
of
$
G
\boxtimes
H
$
in
$
\mathbb
{
R
}^{
d
_
G
+
d
_
H
}$
as
follows.
$$
f
((
u,v
))[
i
]=
\begin
{
cases
}
...
...
@@ 221,7 +221,7 @@ overcome this issue, by constraining one of the representations.
Now let us observe that for any two vertices
$
u, u'
$
of
$
G
$
, there
is an hyperplane separating the interiors of
$
g
(
u
)
$
and
$
g
(
u'
)
$
, and
similarly for
$
h
$
and
$
H
$
. This implies that the boxes in
$
f
$
are
interior disjoint. Indeed, the same hyperplane that separates
$
g
(
u
)
$
interior
ly
disjoint. Indeed, the same hyperplane that separates
$
g
(
u
)
$
and
$
g
(
u'
)
$
, when extended to
$
\mathbb
{
R
}^{
d
_
G
+
d
_
H
}$
, now separates
any two boxes
$
f
((
u,v
))
$
and
$
f
((
u',v'
))
$
. This implies that
$
f
$
is
a touching representation of a subgraph of
$
G
\boxtimes
H
$
.
...
...
@@ 231,7 +231,7 @@ overcome this issue, by constraining one of the representations.
a point
$
p
_
G
$
in the intersection of
$
g
(
u
)
$
and
$
g
(
u'
)
$
, and a point
$
p
_
H
$
in the intersection of
$
h
(
v
)
$
and
$
h
(
v'
)
$
. Indeed, one can
obtain
$
p
_
G
$
and
$
p
_
H
$
by projecting
$
p
$
in
$
\mathbb
{
R
}^{
d
_
G
}$
or in
$
\mathbb
{
R
}^{
d
_
G
}$
respectively, and conversely
$
p
$
can be obtained
$
\mathbb
{
R
}^{
d
_
H
}$
respectively, and conversely
$
p
$
can be obtained
by taking the product of
$
p
_
G
$
and
$
p
_
H
$
. Thus
$
f
$
is indeed a
touching representation of
$
G
\boxtimes
H
$
.
\end{proof}
...
...
@@ 247,10 +247,6 @@ Corollary~25 in~\cite{subconvex}, but since the setting is somewhat
different and the construction of~
\cite
{
subconvex
}
uses rotated boxes,
we provide details of the argument.
Next, let us show a bound on the comparable box dimension of subgraphs.
\begin{lemma}
\label
{
lemmasubg
}
If
$
G
$
is a subgraph of a graph
$
G'
$
, then
$
\cbdim
(
G
)
\le
\cbdim
(
G'
)+
\chi
^
2
_
s
(
G'
)
$
.
\end{lemma}
...
...
@@ 340,7 +336,7 @@ boxes or not) $h$ of $G$ in $\mathbb{R}^d$ is called
Note that we may consider that the root clique is empty, that is the
empty subgraph with no vertices. In that case the clique is denoted
$
\emptyset
$
. Let
$
\ecbdim
(
G
)
$
be the minimum dimension such that
$
G
$
has a
$
\emptyset
$
cliquesum extendable touching representation
with
has a
$
\emptyset
$
cliquesum extendable touching representation
by
comparable boxes. The following lemma ensures that cliquesum
extendable representations behave well with respect to full
cliquesums.
...
...
@@ 352,12 +348,8 @@ cliquesums.
respectively. Let
$
G
$
be the graph obtained after performing a full
cliquesum of these two graphs on any clique
$
C
_
1
$
of
$
G
_
1
$
, and on
the root clique
$
C
^
*
_
2
$
of
$
G
_
2
$
. Then
$
G
$
admits a
$
C
^
*
_
1
$
clique
sum extendable representation with comparable boxes
$
h
$
in
$
\mathbb
{
R
}^{
\max
(
d
_
1
,d
_
2
)
}$
. Furthermore, we have that the aspect
ratio of
$
h
(
v
)
$
is the same as the one of
$
h
_
1
(
v
)
$
, if
$
v
\in
V
(
G
_
1
)
$
, or the same as
$
h
_
2
(
v
)
$
if
$
v
\in
V
(
G
_
2
)
\setminus
V
(
C
^
*
_
2
)
$
.
\note
{
shall we remove the aspect ratio thing ? only needed for the
hypercubes of the ktrees, but those are not really needed...
}
sum extendable representation by comparable boxes
$
h
$
in
$
\mathbb
{
R
}^{
\max
(
d
_
1
,d
_
2
)
}$
.
\end{lemma}
\begin{proof}
The idea is to translate (allowing also exchanges of dimensions) and
...
...
@@ 394,26 +386,25 @@ cliquesums.
by
$
\epsilon
$
. More formally, for any vertex
$
v
\in
V
(
G
_
2
)
\setminus
\{
v
_
1
,
\ldots
,v
_
k
\}
$
, we set
$
h
(
v
)[
\sigma
(
i
)]
=
p
(
C
_
1
)[
\sigma
(
i
)]
+
\epsilon
h
_
2
(
v
)[
i
]
$
for
$
i
\in
\{
1
,
\ldots
,d
_
2
\}
$
, and
$
h
(
v
)[
j
]
=
p
(
C
_
1
)[
j
]
+
[
0
,
\epsilon
/
2
]
$
, otherwise (for any
$
j
$
not in the
[
p
(
C
_
1
)[
j
]
, p
(
C
_
1
)[
j
]
+
\epsilon
/
2
]
$
, otherwise (for any
$
j
$
not in the
image of
$
\sigma
$
). Note that if we apply the same mapping from
$
h
_
2
$
to
$
h
$
, to the
vertices of
$
\{
v
_
1
,
\ldots
,
v
_
k
\}
$
, then the
image of
$
h
_
2
(
v
_
i
)
$
fit inside the (previously defined) box
$
h
_
2
$
to
$
h
$
, to the
boxes
$
h
_
2
(
v
_
i
)
$
for
$
i
\in
\{
1
,
\ldots
,k
\}
$
, then the
image of
$
h
_
2
(
v
_
i
)
$
fit
s
inside the (previously defined) box
$
h
(
v
_
i
)
$
. Similarly the clique points
$
p
_
2
(
C
)
$
become
$
p
(
C
)
$
by
setting
$
p
(
C
)[
\sigma
(
i
)]
=
p
(
C
_
1
)[
\sigma
(
i
)]
+
\epsilon
p
_
2
(
C
)[
i
]
$
for
$
i
\in
\{
1
,
\ldots
,d
_
2
\}
$
, and
$
p
(
C
)[
j
]
=
p
(
C
_
1
)[
j
]
+
[
0
,
\frac
14
]
$
, otherwise.
for
$
i
\in
\{
1
,
\ldots
,d
_
2
\}
$
, and
$
p
(
C
)[
j
]
=
p
(
C
_
1
)[
j
]
+
\epsilon
/
4
$
, otherwise.
Note that we have defined (differently) both
$
h
^
\epsilon
(
C
_
1
)
$
(resp.
$
p
(
C
_
1
)
$
) and
$
h
^
\epsilon
(
C
^
*
_
2
)
$
(resp.
$
p
(
C
^
*
_
2
)
$
), despite
the fact that those cliques were merged. In the following we use
$
h
^
\epsilon
(
C
_
1
)
$
and
$
p
(
C
_
1
)
$
only for the purpose of the
proof. The point and the box corresponding to th
ese
clique in
$
h
$
is
proof. The point and the box corresponding to th
is
clique in
$
h
$
is
$
p
(
C
^
*
_
2
)
$
and
$
h
^
\epsilon
(
C
^
*
_
2
)
$
.
Let us now check that
$
h
$
is a
$
C
^
*
_
1
$
clique sum extendable
representation
with
comparable boxes. The fact that the boxes are
representation
by
comparable boxes. The fact that the boxes are
comparable follows from the fact that those of
$
V
(
G
_
1
)
$
(resp.
$
V
(
G
_
1
)
$
) are comparable in
$
h
_
1
$
(resp.
$
h
_
2
$
) with the
(resp.
$
V
(
G
_
2
)
$
) are comparable in
$
h
_
1
$
(resp.
$
h
_
2
$
) with the
boxes of
$
V
(
C
^
*
_
1
)
$
(resp.
$
V
(
C
^
*
_
2
)
$
) being hypercubes of side one,
and the other boxes being smaller. Clearly, by construction both
$
h
_
1
(
u
)
\sqsubseteq
h
_
1
(
v
)
$
or
$
h
_
2
(
u
)
\sqsubseteq
h
_
2
(
v
)
$
, imply
...
...
@@ 423,19 +414,27 @@ cliquesums.
We now check that
$
h
$
is a contact representation of
$
G
$
. For
$
u,v
\in
V
(
G
_
1
)
$
(resp.
$
u,v
\in
V
(
G
_
2
)
\setminus
\{
v
_
1
,
\ldots
,v
_
k
\}
$
) it
is clear that
$
h
(
u
)
$
and
$
h
(
v
)
$
are interior disjoint
, and that they
is clear that
$
h
(
u
)
$
and
$
h
(
v
)
$
have disjoint interiors
, and that they
intersect if and only if
$
h
_
1
(
u
)
$
and
$
h
_
1
(
v
)
$
intersect (resp. if
$
h
_
2
(
u
)
$
and
$
h
_
2
(
v
)
$
intersect). Consider now a vertex
$
u
\in
V
(
G
_
1
)
$
and a vertex
$
v
\in
V
(
G
_
2
)
\setminus
\{
v
_
1
,
\ldots
,v
_
k
\}
$
. As
$
h
(
v
)
$
fits inside
$
h
^
\epsilon
(
C
_
1
)
$
, we have that
$
h
(
u
)
$
and
$
h
(
v
)
$
are interior disjoint
. Furthermore, if they intersect then
$
u
\in
have disjoint interiors
. Furthermore, if they intersect then
$
u
\in
V
(
C
_
1
)
$
, say
$
u
=
v
_
1
$
, and
$
h
(
v
)[
1
]
=
[
p
(
C
_
1
)[
1
]
, p
(
C
_
1
)[
1
]+
\alpha
]
$
for some
$
\alpha
>
0
$
. By construction, this implies that
$
h
_
2
(
v
_
1
)
$
and
$
h
_
2
(
v
)
$
intersect.
Finally for the
$
C
^
*
_
1
$
cliquesum extendability, one can easily
check that the (vertex) conditions hold, such as the (clique)
conditions.
check that the (vertices) conditions hold. For the (cliques)
conditions, as the mapping from
$
p
_
2
(
C
)
$
to
$
p
(
C
)
$
(extended to a
mapping from
$
\mathbb
{
R
}^{
d
_
2
}$
to
$
\mathbb
{
R
}^{
d
}$
) is injective,
we have that (c1) clearly holds. For (c2) one has to notice that if
$
d
=
d
_
2
$
, then the mapping from
$
h
_
2
$
to
$
h
$
extended to the clique boxes
would lead to the same clique boxes
$
h
^{
\epsilon
'
}
(
C
)
$
, with the same point
$
p
(
C
)
$
in their lower corner.
If there are extra dimensions, that is if
$
d> d
_
2
$
, then for any such
dimension
$
j
$
that is not in the image of
$
\sigma
$
, we have that
$
h
^{
\epsilon
'
}
(
C
)[
j
]
=
[
p
(
C
_
1
)[
j
]
+
\epsilon
/
4
, p
(
C
_
1
)[
j
]
+
\epsilon
/
4
+
\epsilon
'
]
\subset
[
p
(
C
_
1
)[
j
]
, p
(
C
_
1
)[
j
]
+
\epsilon
/
2
]
=
h
(
v
)[
j
]
$
.
\end{proof}
The following lemma shows that any graphs has a
$
C
^
*
$
cliquesum
...
...
@@ 443,7 +442,7 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
\ecbdim
(
G
)
$
and for any clique
$
C
^
*
$
.
\begin{lemma}
\label
{
lemapexcs
}
For any graph
$
G
$
and any clique
$
C
^
*
$
, we have that
$
G
$
admits a
$
C
^
*
$
cliquesum extendable touching representation
with
comparabe
$
C
^
*
$
cliquesum extendable touching representation
by
comparabe
boxes in
$
\mathbb
{
R
}^
d
$
, for
$
d
=
V
(
C
^
*)

+
\ecbdim
(
G
\setminus
V
(
C
^
*))
$
.
\end{lemma}
...
...
@@ 458,12 +457,40 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
$
d
_{
v
_
i
}
=
i
$
. For each vertex
$
u
\in
V
(
G
)
\setminus
V
(
C
^
*)
$
, if
$
i
\le
k
$
then let
$
h
(
u
)[
i
]
=
[
0
,
1
/
2
]
$
if
$
uv
_
i
\in
E
(
G
)
$
, and
$
h
(
u
)[
i
]
=
[
1
/
4
,
3
/
4
]
$
if
$
uv
_
i
\notin
E
(
G
)
$
. For
$
i>k
$
we have
$
h
(
u
)[
i
]
=
h'
(
u
)[
i

k
]
$
. We proceed similarly for the clique points. For any
\alpha
_
i h'
(
u
)[
i

k
]
$
, for some
$
\alpha
_
i>
0
$
. The values
$
\alpha
_
i>
0
$
are chosen suffciently small so that
$
h
(
u
)[
i
]
\subset
[
0
,
1
)
$
, whenever
$
u
\notin
V
(
C
^
*)
$
.
We proceed similarly for the clique points. For any
clique
$
C
$
of
$
G
$
, if
$
i
\le
k
$
then let
$
p
(
C
)[
i
]
=
0
$
if
$
v
_
i
\in
V
(
C
)
$
, and
$
p
(
C
)[
i
]
=
1
/
4
$
if
$
v
_
i
\notin
V
(
C
)
$
. For
$
i>k
$
we have
to refer to the clique point
$
p'
(
C'
)
$
of
$
C'
=
C
\setminus
\{
v
_
1
,
\ldots
,v
_
k
\}
$
, as we set
$
p
(
C
)[
i
]
=
p'
(
C'
)[
i

k
]
$
. One can
easily check that
$
h
$
is as desired.
\{
v
_
1
,
\ldots
,v
_
k
\}
$
, as we set
$
p
(
C
)[
i
]
=
\alpha
_
i p'
(
C'
)[
i

k
]
$
.
As
$
h'
(
u
)
\sqsubset
h'
(
v
)
$
implies that
$
h
(
u
)
\sqsubset
h
(
v
)
$
, and as
$
h
(
u
)
\sqsubset
h
(
v
_
i
)
$
, for every
$
u
\in
V
(
G
)
\setminus
V
(
C
^
*)
$
and every
$
v
_
i
\in
V
(
C
^
*)
$
, we have that
$
h
$
is a touching representation by comparable boxes.
By the construction, it is clear that
$
h
$
is a representation of
$
G
$
.
For the
$
C
^
*
$
cliquesum extendability, it is clear that the (vertices) conditions hold.
For the (cliques) condition (c1), let us first consider two distinct cliques
$
C
_
1
$
and
$
C
_
2
$
of
$
G
$
such that
$
V
(
C
_
1
)

\ge
V
(
C
_
2
)

$
, and let
$
C'
_
i
=
C
_
i
\setminus
V
(
C
^
*)
$
. If
$
C'
_
1
=
C'
_
2
$
,
there is a vertex
$
v
_
i
\in
V
(
C
_
1
)
\setminus
V
(
C
_
2
)
$
, and
$
p
(
C
_
1
)[
i
]
=
0
\neq
1
/
4
=
p
(
C
_
2
)[
i
]
$
.
Otherwise, if
$
C'
_
1
\neq
C'
_
2
$
, we have that
$
p'
(
C'
_
1
)
\neq
p'
(
C'
_
2
)
$
, which leads to
$
p
(
C
_
1
)
\neq
p
(
C
_
2
)
$
by construction.
For the (cliques) condition (c2), let us first consider a vertex
$
v
\in
V
(
G
)
\setminus
V
(
C
^
*)
$
and a clique
$
C
$
of
$
G
$
containing
$
v
$
.
In the first dimensions
$
i
\le
k
$
, we always have
$
h
^
\epsilon
(
C
)[
i
]
\subseteq
h
(
v
)[
i
]
$
. Indeed, if
$
v
_
i
\in
V
(
C
)
$
we have
$
h
^
\epsilon
(
C
)[
i
]
\subseteq
[
0
,
1
/
2
]
=
h
(
v
)[
i
]
$
(as in that case
$
v
$
and
$
v
_
i
$
are adjacent), and if
$
v
_
i
\notin
V
(
C
)
$
we have
$
h
^
\epsilon
(
C
)[
i
]
\subseteq
[
1
/
4
,
1
/
2
]
\subseteq
h
(
v
)[
i
]
$
. Then for the last
$
d'
$
dimensions, by definition of
$
h'
$
,
we have that
$
h
^
\epsilon
(
C
)[
i
]
\subseteq
h
(
v
)[
i
]
$
for every
$
i>k
$
, except one,
for which
$
h
^
\epsilon
(
C
)[
i
]
\cap
h
(
v
)[
i
]
=
\{
p
(
C
)[
i
]
\}
$
. This completes the first case
and we now consider a vertex
$
v
\in
V
(
G
)
\setminus
V
(
C
^
*)
$
and a clique
$
C
$
of
$
G
$
not containing
$
v
$
.
As
$
v
\notin
V
(
C'
)
$
, there is an hyperplane
${
\mathcal
H
}
'
=
\{
p
\in
\mathbb
{
R
}^{
d'
}
\

\
p
[
i
]
=
c
\}
$
that separates
$
p'
(
C'
)
$
and
$
h'
(
v
)
$
.
This implies that the following hyperplane
${
\mathcal
H
}
=
\{
p
\in
\mathbb
{
R
}^{
d
}
\

\
p
[
k
+
i
]
=
\alpha
_{
k
+
i
}
c
\}
$
separates
$
p
(
C
)
$
and
$
h
(
v
)
$
.
Now we consider a vertex
$
v
_
i
\in
V
(
C
^
*)
$
, and we note that for any clique
$
C
$
containing
$
v
_
i
$
we have that
$
h
^
\epsilon
(
C
)[
i
]
\cap
h
(
v
_
i
)[
i
]
=
[
0
,
\epsilon
]
\cap
[
1
,
0
]
=
\{
0
\}
$
, and
$
h
^
\epsilon
(
C
)[
j
]
\subseteq
[
0
,
1
]
=
h
(
v
_
i
)[
j
]
$
for any
$
j
\neq
i
$
. For a clique
$
C
$
that does not contain
$
v
_
i
$
we have that
$
h
^
\epsilon
(
C
)[
i
]
\cap
h
(
v
_
i
)[
i
]
\subset
(
0
,
1
)
\cap
[
1
,
0
]
=
\emptyset
$
.
Condition (c2) is thus fulfilled and this completes the proof of the lemma.
\end{proof}
The following lemma provides an upper bound on
$
\ecbdim
(
G
)
$
in terms
...
...
@@ 472,7 +499,7 @@ of $\cbdim(G)$ and $\chi(G)$.
For any graph
$
G
$
,
$
\ecbdim
(
G
)
\le
\cbdim
(
G
)
+
\chi
(
G
)
$
.
\end{lemma}
\begin{proof}
Let
$
h
$
be a touching representation
with
comparable boxes of
$
G
$
in
Let
$
h
$
be a touching representation
by
comparable boxes of
$
G
$
in
$
\mathbb
{
R
}^
d
$
, with
$
d
=
\cbdim
(
G
)
$
, and let
$
c
$
be a
$
\chi
(
G
)
$
coloring of
$
G
$
. We start with a slightly modified version
of
$
h
$
. We first scale
$
h
$
to fit in
$
(
0
,
1
)
^
d
$
, and for a
...
...
@@ 495,20 +522,36 @@ of $\cbdim(G)$ and $\chi(G)$.
[
0
,
2
/
5
]
&
\text
{
if $c
(
u
)
=
i

d$
}
\\
[
2
/
5
,
4
/
5
]
&
\text
{
otherwise
(
if $c
(
u
)
> i

d >
0
$
)
}
\end
{
cases
}$$
For any clique
$
C'
$
of
$
G
$
, let us denote
$
c
(
C
)
$
, the set
For any clique
$
C'
$
of
$
G
$
, let us denote
$
c
(
C
'
)
$
, the
color
set
$
\{
c
(
u
)
\

\
u
\in
V
(
C'
)
\}
$
, and let
$
C
$
be one of the maximal cliques
containing
$
C'
$
.We now set
containing
$
C'
$
.
We now set
$$
p
_
2
(
C'
)[
i
]=
\begin
{
cases
}
p
_
1
(
C
)
&
\text
{
if $i
\le
d$
}
\\
2
/
5
&
\text
{
if $i

d
\in
c
(
C'
)
$
}
\\
%\text{if $\exists u\in V(C)$ with $c(u) = id$}\\
1
/
2
&
\text
{
otherwise
}
\end
{
cases
}
$$
One can now check that this is a
$
\emptyset
$
cliquesum
extendable touching representation with comparable boxes. In particular,
one should notice that for a vertex
$
u
$
of a clique
$
C'
$
we have that
$
h
_
2
^{
\epsilon
}
(
C'
)[
i
]
\subset
h
_
2
(
u
)[
i
]
$
for every dimension
$
i
$
,
except if
$
c
(
u
)=
i

d
$
. If
$
c
(
u
)=
i

d
$
, then
$
h
_
2
(
u
)[
i
]
\cap
h
_
2
^{
\epsilon
}
(
C'
)[
i
]
=
\{
2
/
5
\}
$
.
$$
As
$
h
_
2
$
is an extension of
$
h
_
1
$
, and as for all the extra dimensions
$
j
$
(
$
j>d
$
)
we have that
$
2
/
5
\in
h
_
2
(
v
)[
j
]
$
for every vertex
$
v
$
, we have that
$
h
_
2
$
is an
intersection representation of
$
G
$
. To prove that it is touching consider two adjacent
vertices
$
u
$
and
$
v
$
such that
$
c
(
u
)
<c
(
v
)
$
, and let us note that
$
h
_
2
(
u
)[
d
+
c
(
u
)]
=
[
0
,
2
/
5
]
$
and
$
h
_
2
(
v
)[
d
+
c
(
u
)]
=
[
2
/
5
,
4
/
5
]
$
. By construction, the boxes of
$
h
_
1
$
are comparable boxes,
and as
$
h
_
2
$
is an extension of
$
h
_
1
$
such that for all the extra dimensions
$
j
$
(
$
j>d
$
)
the length of
$
h
_
2
(
v
)[
j
]
$
is
$
2
/
5
$
for every vertex
$
v
$
, we have that the boxes in
$
h
_
2
$
are
comparables boxes.
For the
$
\emptyset
$
cliquesum extendability, the (vertices) conditions clearly hold.
For the (cliques) conditions, let us first note that the points
$
p
_
1
(
C
)
$
, defined for
the maximum cliques, are necessarily distinct. This impies that two cliques
$
C
_
1
$
and
$
C
_
2
$
,
which clique points
$
p
_
2
(
C
_
1
)
$
and
$
p
_
2
(
C
_
2
)
$
are based on distinct maximum cliques, necessarily lead to distinct points.
In the case that
$
C
_
1
$
and
$
C
_
2
$
belong to some maximal clique
$
C
$
, we have that
$
c
(
C
_
1
)
\neq
c
(
C
_
2
)
$
and this implies by construction that
$
p
_
2
(
C
_
1
)
$
and
$
p
_
2
(
C
_
2
)
$
are distinct. Thus (c1) holds.
By construction of
$
h
_
1
$
, we have that if
$
h
_
2
^{
\epsilon
}
(
C'
)[
i
]
\cap
h
_
2
(
v
)[
i
]
$
is nonempty for every
$
i
\le
d
$
,
then we have that
$
h
_
2
^{
\epsilon
}
(
C'
)[
i
]
\subset
h
_
2
(
v
)[
i
]
$
for every
$
i
\le
d
$
,
and we have that
$
v
$
belongs to some maximal clique
$
C
$
containing
$
C'
$
. If
$
v
\notin
V
(
C'
)
$
note that
$
p
_
2
(
C'
)[
d
+
c
(
v
)]
=
1
/
2
\notin
[
0
,
2
/
5
]=
h
_
2
(
v
)[
d
+
c
(
v
)]
$
, while if
$
v
\in
V
(
C'
)
$
we have that
$
h
_
2
^{
\epsilon
}
(
C'
)[
i
]
\subset
[
2
/
5
,
1
/
2
+
\epsilon
]
\subset
h
_
2
(
v
)[
i
]
$
for every dimension
$
i>d
$
,
except if
$
c
(
v
)=
i

d
$
, and in that case
$
h
_
2
(
v
)[
i
]
\cap
h
_
2
^{
\epsilon
}
(
C'
)[
i
]
=
[
0
,
2
/
5
]
\cap
[
2
/
5
,
2
/
5
+
\epsilon
]
=
\{
2
/
5
\}
$
. We thus have that (c2) holds, and this concludes the proof of the lemma.
\end{proof}
...
...
@@ 531,7 +574,7 @@ product of a path and a $k$tree, by adding at most $k$ apex vertices.
\end{theorem}
Let us first bound the comparable box dimension of a graph in terms of
its Euler genus. As paths and
$
m
$
clique admit touching
its Euler genus. As paths and
$
m
$
clique
s
admit touching
representations with hypercubes of unit size in
$
\mathbb
{
R
}^{
1
}$
and
in
$
\mathbb
{
R
}^{
\lceil
\log
_
2
m
\rceil
}$
respectively, by
Lemma~
\ref
{
lemmasp
}
it suffice to bound the comparable box
...
...
@@ 552,7 +595,7 @@ containing $C^*$, and by performing successive full cliquesums of
$
K
_{
k
+
1
}$
on a
$
K
_
k
$
subclique. By Lemma~
\ref
{
lemcs
}
, it suffice to
show that
$
K
_{
k
+
1
}$
, the
$
(
k
+
1
)
$
clique with vertex set
$
\{
v
_
1
,
\ldots
, v
_{
k
+
1
}
\}
$
, has a
$
(
K
_{
k
+
1
}

\{
v
_{
k
+
1
}
\}
)
$
cliquesum
extendable touching representation
with
hypercubes. Let us define such
extendable touching representation
by
hypercubes. Let us define such
touching representation
$
h
$
as follows:
\begin{itemize}
\item
$
h
(
v
_
i
)[
i
]
=
[
1
,
0
]
$
if
$
i
\le
k
$
...
...
@@ 580,15 +623,28 @@ For a vertex $v_i$ and a clique $C$, the boxes $h(v_i)$ and
$
h
^
\epsilon
(
C
)
$
intersect if and only if
$
v
_
i
\in
V
(
C
)
$
. Indeed, if
$
v
_
i
\in
V
(
C
)
$
then
$
p
(
C
)
\in
h
(
v
_
i
)
$
and
$
p
(
C
)
\in
h
^
\epsilon
(
C
)
$
, and
if
$
v
_
i
\notin
V
(
C
)
$
then
$
h
(
v
_
i
)[
i
]
=
[
1
,
0
]
$
if
$
i
\le
k
$
(resp.
$
h
(
v
_
i
)[
i
]
=
[
0
\frac
12
]
$
if
$
i
=
k
+
1
$
) and
$
h
^
\epsilon
(
C
)[
i
]
=
(resp.
$
h
(
v
_
i
)[
i
]
=
[
0
,
\frac
12
]
$
if
$
i
=
k
+
1
$
) and
$
h
^
\epsilon
(
C
)[
i
]
=
[
\frac
14
,
\frac
14
+
\epsilon
]
$
(resp.
$
h
^
\epsilon
(
C
)[
i
]
=
[
\frac
34
,
\frac
34
+
\epsilon
]
$
). Finally, if
$
v
_
i
\in
V
(
C
)
$
we have that
$
h
(
v
_
i
)[
i
]
\cap
h
^
\epsilon
(
C
)[
i
]
=
\{
p
(
C
)[
i
]
\}
$
and that
$
h
(
v
_
i
)[
j
]
\cap
h
^
\epsilon
(
C
)[
j
]
=
[
p
(
C
)[
j
]
,p
(
C
)[
j
]+
\epsilon
]
$
for any
$
j
\neq
i
$
and any
$
\epsilon
<
\frac
14
$
. This concludes the proof of the theorem.
\end{proof}
As every planar graph
$
G
$
has a touching representation with cubes in
Note that actually the bound on the comparable boxes dimension of Theorem~
\ref
{
thmktree
}
extends to graphs of treewidth
$
k
$
. For this, note that the construction in this proof can
provide us with a representation
$
h
$
of any
$
k
$
tree
$
G
$
with hypercubes of distinct sizes.
Note also that this representation is such that for any two adjacent vertices
$
u
$
and
$
v
$
,
with
$
h
(
u
)
\sqsubset
h
(
v
)
$
say, the intersection
$
I
=
h
(
u
)
\cap
h
(
v
)
$
is a facet of
$
h
(
u
)
$
.
Actually
$
I
[
i
]
=
h
(
u
)[
i
]
$
for every dimension, except one that we denote
$
j
$
. For this
dimension we have that
$
I
[
j
]=
\{
c
\}
$
for some
$
c
$
, and that
$
h
(
u
)[
j
]=[
c,c
+
s
]
$
,
where
$
s
$
is the length of the sides of
$
h
(
u
)
$
. In that context to delete an edge
$
uv
$
one can simply replace
$
h
(
u
)[
j
]=[
c,c
+
s
]
$
with
$
[
c
+
\epsilon
,c
+
s
]
$
, for a sufficiently small
$
\epsilon
$
.
One can proceed similarly for any subset of edges, and note that as the hypercubes in
$
h
$
have
distinct sizes these small perturbations give rise to boxes that are still comparable.
Thus for any treewidth
$
k
$
graph
$
H
$
(that is a subgraph of a
$
k
$
tree
$
G
$
) we have
$
\cbdim
(
H
)
\le
k
+
1
$
.
As every planar graph
$
G
$
has a touching representation by cubes in
$
\mathbb
{
R
}^
3
$
~
\cite
{
felsner2011contact
}
, we have that
$
\cbdim
(
G
)
\le
3
$
. For the graphs with higher Euler genus we can also derive upper
bounds. Indeed, combining the previous observation on the
...
...
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