Stephane Durocher amp Debajyoti Mondal University of Manitoba Contact Graph Each vertex is represented by a closed region The interiors of every pair of vertices are disjoint Two vertices are joined by an edge ID: 805028
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Slide1
On Balanced +-Contact Representations
Stephane
Durocher &
Debajyoti
Mondal
University of Manitoba
Slide2Contact GraphEach vertex is represented by a closed region.The interiors of every pair of vertices are disjoint.
Two vertices are joined by an edge
iff
the boundaries of their regions touch.
Theorem [Koebe 1936] Every planar graph has a circle contact representation.
Cover Contact Graph (Circle Contact Representation)
a
b
c
d
e
a
b
e
d
c
Graph Drawing, Bordeaux.
2
September 23-25, 2013
Slide3b
g
Other Shapes
Graph Drawing, Bordeaux.
3
September 23-25, 2013
Point-contact of disks
(Every Planar Graph)
[Koebe 1936]
Point-contact of triangles
(Every Planar Graph)
[
de
Fraysseix
et al. 1994
]
a
b
c
d
e
g
f
f
a
c
d
e
a
b
c
d
e
f
g
Rectangle contact representation
(Complete Characterization)
[Kozminski & Kinnen 1985,
Kant & He 2003]
A node-link diagram
Side-contact of polygons
(octagon
hexagon
)
[He 1999, Liao et al. 2003,
Duncan et al. 2011]
a
b
g
c
f
d
e
b
a
d
c
e
f
g
g
point contact
side contact
Slide4+-Contact Representations
Each vertex is represented by an axis-aligned + .
Two + shapes never cross.
Two + shapes
touch iff the corresponding vertices are adjacent. Graph Drawing, Bordeaux, France.
4
September 23-25, 2013
a
d
f
g
e
c
b
g
e
f
d
c
b
a
Allowed
Allowed
Not Allowed
Not Allowed
Slide5c-Balanced +
-
Contact
Representations
Each arm can touch at most ⌈c∆⌉ other arms.Graph Drawing, Bordeaux.
5
September 23-25, 2013
a
d
f
g
e
c
b
g
e
f
d
c
b
a
A plane graph
G
with
maximum degree ∆ = 5
A (1/2)-balanced
+
-contact
representation of
G
Slide6c-Balanced +-Contact vs.
T
- and
L
-Contact
Every planar graph
admits
a
T-contact representation [de Fraysseix
et al. 1994]. Several recent attempts to characterize L
-contact graphs [Kobourov et al. 2013
, Chaplick et al. 2013].
T
- and L
-contact representations may be unbalanced, but our goal with +-contact is to construct balanced representations.
Graph Drawing, Bordeaux.
6September 23-25, 2013
a
b
c
d
f
e
g
a
d
f
g
e
c
b
g
e
f
d
c
b
a
A plane graph
G
with ∆ = 5
A (1/2)-balanced
+
-contact representation
A
T
-contact
r
epresentation
Slide7c-Balanced Representations: Applications
Graph Drawing, Bordeaux.
7
September 23-25, 2013
a
d
f
g
e
c
b
g
e
f
d
c
b
a
a
d
g
e
f
b
c
g
e
f
d
c
b
a
An 1-bend
o
rthogonal
d
rawing
with boxes of size
⌈
c
∆
⌉×⌈
c
∆
⌉
A transformation into an 1-bend polyline drawing with 2
⌈
c
∆
⌉
slopes
[
Keszegh
et al, 2000]
Slide8Results Graph Drawing, Bordeaux.
8
September 23-25, 2013
2-trees have (1/3)-balanced +-contact representations, but not necessarily a (1/4-
ϵ)-balanced +-contact representation.Plane 3-trees have (1/2)-balanced +-contact
representations,
but not necessarily a (
1/3)-balanced +-contact representation.
Strengthens the result that 2-trees
with ∆ = 4, have 1-bend orthogonal (equivalent to (1/4)-balanced +-contact) [Tayu
et al., 2009].
Implies 1-bend polyline drawings of
2-trees
with 2⌈
∆/3⌉
-slopes, and for
plane
3-trees with 2⌈
∆/2⌉
slopes, which is significantly
smaller than the upper bound of 2∆ for general planar graphs [
Keszegh et al. 2010].
It is interesting that with 1-bend per edge, we use roughly
2∆/3 slopes for 2-trees
, where the planar slope number of 2-trees is in [∆-3, 2∆] [Lenhart
et al., 2013].
Slide92-Trees
Graph Drawing, Bordeaux.
9
September 23-25, 2013
A 2-tree (series-parallel graph)
G
with
n ≥
2
vertices is constructed as follows.
Base Case:
Series combination :
Parallel combination:
G
1
G
2
G
1
G
2
s
1
t
1
s
2
t
2
s
1
t
1
s
2
t
2
s
1
s
1
= s
2
t
2
t
1
= t
2
t
1
= s
2
poles
poles
Slide10Series-Parallel DecompositionsGraph Drawing, Bordeaux.
10
September 23-25, 2013
S
P
S
P
S
S
P
P
a
b
c
d
e
f
g
a
b
c
d
e
f
g
Slide11Series-Parallel DecompositionsGraph Drawing, Bordeaux.
11
September 23-25, 2013
S
P
S
P
S
S
P
P
a
b
c
d
e
f
g
Slide12(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
12
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Let f(
a) denote the free points of the arm a
. Initially, f(
a) = ⌈∆
/2⌉ or
f
(a) = ⌈
∆/2⌉
-1 (if there is an edge (
s,t) in G
).Claim
: Given a rectangle
R=abcd
such that degree(s,H
) ≤ f(
ad) + f
(ar)
and degree(t
,H) ≤ f(c
l) + f(c
u),
one can construct a (1/2)-balanced representation of H inside R
such that s and
t lie on a and
c, respectively.
a
r
G
H
R
a
b
c
d
s
s
t
t
a
d
c
l
c
u
Slide13(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
13
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Claim:
Given a rectangle R=
abcd such that degree(s,H
) ≤ f(a
d) + f
(
ar)
and degree(t
,H) ≤ f
(cl) +
f(c
u), one can construct a (1/2)-balanced representation of
H inside R
such that s and
t lie on a
and c, respectively.
H
R
a
b
c
d
R
a
(
= s
)
b
c (
= t)
d
Base Case:
H
consists of two isolated vertices:
s
and
t.
s
t
Slide14(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
14
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Claim: Given a rectangle
R=abcd
such that degree(s,H) ≤
f(a
d) + f
(a
r)
and degree(t
,H) ≤ f
(cl
) + f
(cu),
one can construct a (1/2)-balanced representation
of H inside
R such that s
and t lie on a
and c, respectively.
H
R
a
b
c
d
Series Combination
Induction
: Draw
H
1\ (s
1,t
1)
and
H2
\ (
s
2
,
t
2
)
inside
R
1
and
R
2
,
respectively.
s = s
1
t = t
2
t
1
= s
2
H1H2
a (= s)bc (=
t)d
m ( = t1)
R1R
2f(ad)-1⌈∆/2
⌉ -1a1⌈∆/2⌉-1
⌈∆/2⌉⌈∆/2⌉m1f(a
r)m2f(cl)
f(cu)-1c2
Slide15(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
15
September 23-25, 2013
a
(= s
)
b
c
(=
t)
d
m
( =
t
1
)
R
1
R
2
f
(
a
d
)-1
⌈
∆
/2
⌉
-1
a
1
⌈
∆
/2
⌉
-1
⌈∆/2⌉
⌈
∆
/2
⌉
m
1
f
(
a
r
)
m
2
f
(
c
l
)-1
f
(
c
u)
c2
a
(
= s)
bc (= t)d
m ( = t1)
R
1R2f(ad)-1
⌈∆/2⌉ -1a1⌈∆/2
⌉⌈∆/2⌉⌈∆/2⌉-1m1
f(ar)m2f(cl
)f(cu)-1c2
Series CombinationInduction: Draw H1\ (s1,
t1) and H2 \ (s2,t2) inside
R1 and R2 , respectively.
Slide16(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
16
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Claim: Given a rectangle
R=abcd
such that degree(s,H) ≤
f(a
d) + f
(a
r)
and degree(t
,H) ≤ f
(cl
) + f
(cu),
one can construct a (1/2)-balanced representation
of H inside
R such that s
and t lie on a
and c, respectively.
H
R
a
b
c
d
a
1
b
c
1
d
Parallel Combination
Distribute the free points of
R
among
R
1
and
R
2
.
H
1
H
2
s
1
= s
2
t
1
= t
2
R
1
a
2
b
c
(= t1)
d
R2
Slide17(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
17
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Claim: Given a rectangle
R=abcd
such that degree(s,H) ≤
f(a
d) + f
(a
r)
and degree(t
,H) ≤ f
(cl
) + f
(cu),
one can construct a (1/2)-balanced representation
of H inside
R such that s
and t lie on a
and c, respectively.
H
R
a
b
c
d
10
b
1
c
1
d1
H1
H
2
s
1
= s
2
t
1
= t
2
R
1
a
2
b
2
c
2
d
2
R
2
5
0
21
10
5
0
3
0
16
02degree (s1,H
1) = 15degree (t1,H1) = 3a1Parallel Combination
Distribute the free points of R among R1 and R2.
Slide18(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
18
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Claim: Given a rectangle
R=abcd
such that degree(s,H) ≤
f(a
d) + f
(a
r)
and degree(t
,H) ≤ f
(cl
) + f
(cu),
one can construct a (1/2)-balanced representation
of H inside
R such that s
and t lie on a
and c, respectively.
H
R
a
b
c
d
Parallel Combination
Draw
H
1 and H
2 using induction, and merge them avoiding edge crossing.
H
1H
2
s
1
= s
2
t
1
= t
2
Slide19(1/2)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
19
September 23-25, 2013
Let
G be a 2-tree, and H=G \ (s
,
t
).Claim: Given a rectangle
R=abcd
such that degree(s,H) ≤
f(a
d) + f(
a
r)
and degree(t
,H) ≤ f
(cl
) + f
(cu),
one can construct a (1/2)-balanced representation
of H inside
R such that s
and t lie on a
and c, respectively.
We started with
G
and proved that H admits a (1/2)-balanced +-contact representation inside
R. If the poles of
G are adjacent, we initialize f
(a
d)=⌈∆
/2⌉ -
1 and f
(c
l)=⌈∆
/2⌉ -
1, then draw H. Finally,
draw (s
,
t
) along
abc
.
a
r
G
H
R
a
b
c
d
s
s
t
t
a
d
c
l
c
u
G
a
b
c
d
Hab
cd
Slide20Refinement: (1/3)-Balanced Representation Graph Drawing, Bordeaux.
20
September 23-25, 2013
Why was the previous construction (1/2)-balanced?
While adding a new arm, we assigned at most ⌈∆/2⌉ free points to it. Since ⌈∆/2⌉ + ⌈∆/2
⌉ ≥ ∆, we could find a ‘nice’ rectangle partition, i.e., using at most two arms.
Recall series combination.
a
(
= s
)
b
c
(
=
t
)
d
m
R
1
R
2
⌈
∆
/2
⌉
-1
a
1
⌈
∆
/2
⌉
-1
⌈∆/2⌉
⌈
∆
/2
⌉
m
1
m
2
c
2
Slide21For (1/3)-balanced we assign at most ⌈∆/3⌉ free points to any arm.
Sometimes we need at least three of the arms of
m
to lie in the same rectangle. E.g., if degree(
m,H1) > 2⌈∆/3⌉ .Sometimes we need to share an arm among the rectangles. E.g., assume degree(m,H1
) > ⌈∆/3⌉ and degree(
m,H
2) > ⌈∆/3⌉ in the following.
Refinement: (1/3)-Balanced Representation
Graph Drawing, Bordeaux. 21
September 23-25, 2013
a
(
= s
)
b
c
(
=
t)
d
m
⌈
∆
/3
⌉
-1
a
1
⌈
∆
/3
⌉
-1
⌈∆
/3⌉
0
m
1
m
2
c
2
⌈∆
/3⌉
a
(
= s
)
b
c
(
=
t
)
d
m
⌈
∆
/3
⌉
-1
a
1
⌈
∆
/3⌉-1m1m2c2
deg(m1,H1) – (⌈∆/3⌉-1)
deg(m2,H2) – (⌈∆/3⌉-1)0
Slide22Refinement: (1/3)-Balanced Representation Graph Drawing, Bordeaux.
22
September 23-25, 2013
a
(
= s
)
b
c
(=
t)
d
m
⌈
∆
/3
⌉
-1
a
1
⌈
∆
/3
⌉
-1
⌈∆
/3⌉
⌈
∆
/3
⌉
m
1
m
2
c
2
⌈∆
/3⌉
a
(
= s
)
b
c
(
=
t
)
d
m
⌈
∆
/3
⌉
-1
a
1
⌈
∆
/3
⌉
-1
m
1
m2c2deg
(m1) – (⌈∆/3⌉-1)outnDeg(m2) – (⌈∆/3⌉-1)0
Some poles do not lie at the corners.More case Analysis!
Slide23(1/3)-Balanced Representation for 2-treesGraph Drawing, Bordeaux.
23
September 23-25, 2013
a
(
= s
)
b
c
(=
t)
d
m
⌈
∆
/3
⌉
-1
a
1
⌈
∆
/3
⌉
-1
⌈∆
/3⌉
⌈
∆
/3
⌉
m
1
m
2
c
2
⌈∆
/3⌉
a
(
= s
)
b
c
(
=
t
)
d
m
⌈
∆
/3
⌉
-1
m
2
c
2
outnDeg
(
m
2
) – (⌈∆/3⌉-1)
0⌈∆/3⌉ -1
a1m1deg(m1) – (⌈∆/3⌉-1)
Sometimes flip sub-problems to apply induction.a1
m1Some poles do not lie at the corners.More case Analysis!
Slide24Plane 3-trees: (1/2)-BalancedGraph Drawing, Bordeaux.
24
September 23-25, 2013
a
b
c
p
a
b
p
b
c
p
a
c
p
Slide25ConclusionGraph Drawing, Bordeaux.
25
September 23-25, 2013
Summary
2-trees have (1/3)-balanced +-contact representations, but not necessarily a (1/4-ϵ)-balanced +-contact representation.Plane 3-trees have (1/2)-
balanced +-contact
representations,
but not necessarily a (1/3)-balanced +-contact representation.
Open Questions
Although our representations for planar 3-trees preserve the input
embedding, our
representations for 2-trees do not have this property. Do there exist algorithms for (1/3)-balanced representations of 2-trees that preserve input embedding?
Close the gap between the lower and upper bounds.
Characterize planar graphs that admit
c-balanced +-contact representations, for small fixed values
c.
Slide26Thank YouSeptember 23-25, 201326Graph Drawing, Bordeaux.