On Balanced + -Contact Representations
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On Balanced +-Contact Representations
Stephane Durocher & Debajyoti MondalUniversity of Manitoba
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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 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
de
a
b
ed
c
September 23-25, 2013
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b g
Other Shapes
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
de
gf
f
a
c
d
e
a
bc
d
ef
g
Rectangle contact representation (Complete Characterization)
[Kozminski & Kinnen 1985, Kant & He 2003]
A node-link diagram
Side-contact of polygons(octagonhexagon)
[He 1999, Liao et al. 2003, Duncan et al. 2011]
abg
cfd e
b
ad
ce
f
g
g
poin
t conta
ctsid
e co
nta
ct
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+-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
df g
e
c b
ge
fd
c
b
a
Allowed Allowed Not Allowed Not Allowed
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c-Balanced +-Contact Representations
Each arm can touch at most ⌈c∆⌉ other arms.
September 23-25, 2013
a
df g
e
c b
ge
fd
c
b
a
A plane graph G with maximum degree ∆ = 5
A (1/2)-balanced +-contactrepresentation of G
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c-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.
September 23-25, 2013
ab
cd
f
e
g
a
df g
e
c b
ge
fd
c
b
a
A plane graph G with ∆ = 5
A (1/2)-balanced +-contact representation
A T-contact representation
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c-Balanced Representations: Applications
September 23-25, 2013
a
df g
e
c b
ge
fd
c
b
a
ad
ge
f
bc
ge
f
d
c
b
a
An 1-bend orthogonal drawingwith boxes of size ⌈c∆⌉×⌈c∆⌉
A transformation into an 1-bend polyline drawing with 2⌈c∆⌉ slopes
[Keszegh et al, 2000]
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Results
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].
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2-Trees
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:
G1
G2
G1
G2
s1 t1
s2 t2
s1 t1
s2 t2
s1
s1= s2
t2
t1= t2
t1= s2
poles
poles
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Series-Parallel Decompositions
September 23-25, 2013
SP
S
P
SS
P
P
a
b
cd e
f
g
a
b
cd e
f
g
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Series-Parallel Decompositions
September 23-25, 2013
SP
S
P
SS
P
P
a
b
cd e
f
g
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(1/2)-Balanced Representation for 2-trees
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(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.
ar
G H
R
a
b c
ds s
t t
ad
cl
cu
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(1/2)-Balanced Representation for 2-trees
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(ad) + f(ar) 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
R
a (= s)
b c (= t)
d
Base Case: H consists of two isolated vertices: s and t.
s
t
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(1/2)-Balanced Representation for 2-trees
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(ad) + f(ar) 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 H1\ (s1,t1) and H2 \ (s2,t2)
inside R1 and R2 , respectively.
s =
s1
t = t 2
t 1= s
2
H1
H2
a (= s)
b c (= t)
dm ( = t1)
R1
R2
f(a d)
-1
⌈∆/2⌉ -1
a1
⌈∆/2⌉-1
⌈∆/2⌉⌈∆/2⌉
m1
f(ar)
m2
f(cl)
f(cu)-1
c2
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(1/2)-Balanced Representation for 2-trees
September 23-25, 2013
a (= s)
b c (= t)
dm ( = t1)
R1
R2
f(a d)
-1
⌈∆/2⌉ -1
a1
⌈∆/2⌉-1
⌈∆/2⌉⌈∆/2⌉
m1
f(ar)
m2
f(cl)-1
f(cu)
c2
a (= s)
b c (= t)
dm ( = t1)
R1
R2
f(a d)
-1
⌈∆/2⌉ -1
a1
⌈∆/2⌉
⌈∆/2⌉ ⌈∆/2⌉-1m1
f(ar)
m2
f(cl)
f(cu)-1
c2
Series Combination
Induction: Draw H1\ (s1,t1) and H2 \ (s2,t2)
inside R1 and R2 , respectively.
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(1/2)-Balanced Representation for 2-trees
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(ad) + f(ar) 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
a1
b c1
d
Parallel Combination
Distribute the free points of R among R1 and R2.
H1
H2
s 1= s
2 t1 = t2
R1
a2
b c (= t1)
d
R2
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(1/2)-Balanced Representation for 2-trees
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(ad) + f(ar) 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 b1 c1
d1
H1
H2
s 1= s
2 t1 = t2
R1
a2
b2c2
d2
R2
5
0
2110
5
0
3 0
16
0
2
degree (s1,H1) = 15
degree (t1,H1) = 3
a1
Parallel Combination
Distribute the free points of R among R1 and R2.
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(1/2)-Balanced Representation for 2-trees
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(ad) + f(ar) 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 H1 and H2 using induction, and merge them avoiding edge crossing.
H1
H2
s 1= s
2 t1 = t2
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(1/2)-Balanced Representation for 2-trees
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(ad) + f(ar) 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(ad)=⌈∆/2⌉ -1 and f(cl)=⌈∆/2⌉ -1, then draw H. Finally, draw (s,t) along abc.
ar
G H
R
a
b c
ds s
t t
ad
cl
cuG
a
b c
d
H
a
b c
d
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Refinement: (1/3)-Balanced Representation
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
mR1
R2⌈∆/2⌉ -1
a1
⌈∆/2⌉-1
⌈∆/2⌉⌈∆/2⌉
m1
m2
c2
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For (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,H2) > ∆/3 in the following.⌈ ⌉
Refinement: (1/3)-Balanced Representation
September 23-25, 2013
a (= s)
b c (= t)
dm
⌈∆/3⌉ -1
a1
⌈∆/3⌉-1⌈∆/3⌉
0
m1
m2
c2
⌈∆/3⌉
a (= s)
b c (= t)
dm
⌈∆/3⌉ -1
a1
⌈∆/3⌉-1
m1
m2
c2
deg(m1,H1) – (⌈∆/3⌉-1)
deg(m2,H2) – (⌈∆/3⌉-1)
0
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Refinement: (1/3)-Balanced Representation
September 23-25, 2013
a (= s)
b c (= t)
dm
⌈∆/3⌉ -1
a1
⌈∆/3⌉-1⌈∆/3⌉
⌈∆/3⌉
m1
m2
c2
⌈∆/3⌉
a (= s)
b c (= t)
dm
⌈∆/3⌉ -1
a1
⌈∆/3⌉-1
m1
m2
c2
deg(m1) – (⌈∆/3⌉-1)
outnDeg(m2) – (⌈∆/3⌉-1)
0
Some poles do not lie at the corners.
More case Analysis!
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(1/3)-Balanced Representation for 2-trees
September 23-25, 2013
a (= s)
b c (= t)
dm
⌈∆/3⌉ -1
a1
⌈∆/3⌉-1⌈∆/3⌉
⌈∆/3⌉
m1
m2
c2
⌈∆/3⌉
a (= s)
b c (= t)
dm
⌈∆/3⌉-1
m2
c2
outnDeg(m2) – (⌈∆/3⌉-1)
0⌈∆/3⌉ -1
a1
m1 deg(m1) – (⌈∆/3⌉-1)
Sometimes flip sub-problems to apply
induction.
a1
m1
Some poles do not lie at the corners.
More case Analysis!
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Plane 3-trees: (1/2)-Balanced
September 23-25, 2013
a
b c
p
a
b
p
b c
p
a
c
p
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Conclusion
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.
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Thank You
September 23-25, 201326