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![Page 1: GD 2014 September 26, 2014 Department of Computer Science University of Manitoba Stephane Durocher Debajyoti Mondal.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ec65503460f94bd1f1d/html5/thumbnails/1.jpg)
GD 2014September 26,
2014
Drawing Planar Graphs with Reduced Height
Department of Computer ScienceUniversity of Manitoba
Stephane Durocher Debajyoti Mondal
![Page 2: GD 2014 September 26, 2014 Department of Computer Science University of Manitoba Stephane Durocher Debajyoti Mondal.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ec65503460f94bd1f1d/html5/thumbnails/2.jpg)
3
Straight-line Drawings (Fixed Vs. Variable)
a
b c
d
f
GD 2014 September 26, 2014
e
g
A straight-line drawing of G on a 5×5 grid
A planar graph GA straight-line drawing
of G with height 4
a
b
c
f
g
e
eg
f
a
b
c
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Straight-line Drawings (Fixed Vs. Variable)
a
b c
d
f e
g
A planar graph G
Via visibility representationBiedl, 2014
Here we allow variable embedding.Sometimes we use edge-bends.
Focus on height.
a
b
c
f
g e
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Nested Triangles GraphArea Height 0.44n2 +O(1) 0.66n [Dolev et al. 1984]
A class of planar 3-trees Area Height 0.44n2 + O(1) 0.66n [Frati and
Patrignani 2008, Mondal et al. 2010]
Triangulations
Area Height2n2 + O(1) n − 2 [de Fraysseix et al. 1990]n2 + O(1) n − 2 [Schnyder 1990]1.78n2 + O(1) 0.66n [Chrobak and Nakano 1998]0.88n2 + O(1) 0.66n [Brandenburg 2008]0.44n2 + O(1) 0.66n (polyline) [Bonichon et al. 2003]
Upper Bounds Lower BoundsFixed Embedding
TriangulationsPolyline drawing with height 4n/9+O(λ∆) ≈ 0.44n+O(λ∆)This is 0.44n+o(n) when ∆ is o(n)
Planar 3-treesStraight-line drawing with height 4n/9+O(1) ≈ 0.44n+O(1)
TriangulationsArea Height0.88n2 + O(1) 0.66n [Brandenburg 2008]0.44n2 + O(1) 0.66n (polyline) [Bonichon et al. 2003]Planar 3-treesArea Height0.88n2 + O(1) 0.5n [Brandenburg 2008,
Hossain et al. 2013]Nested Triangles GraphArea Height0.22n2 + O(1) 0.33n [Frati and Patrignani 2008]
Upper BoundsVariable Embedding
Improved Upper Bounds
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Idea: Use the Simple Cycle Separator
A separator of size O(√n)
An n-vertex planar graph G A simple cycle separator of G
Go with 2n/3+O(√n) vertices Gi with 2n/3+O(√n) vertices
[Djidjev and Venkatesan, 1997] Every planar triangulation has a simple cycle separator of size O(√n)
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The Big Picture
w w'
v4
v7
v6v1
v5
v4
v5
v1
v8
v6
v2
v3
Towards w’
a
v8 w b
Gi Go
Choose an Embedding
Decomposition
Drawing and Merge
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Technical Details (Choose an Embedding)
Choose a face which is incident to some edge of the cycle separator as the
new outer face.
GD 2014September 26,
2014
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Technical Details (Construct Go and Gi)
Construct Go and Gi
Gi Go
Choose a face which is incident to some edge of the cycle separator as the
new outer face.
GD 2014September 26,
2014
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10
Technical Details (Draw Go and Gi)
Gi Go
GD 2014September 26,
2014
w
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11
Technical Details (Draw Go and Gi)
Gi Go
v4
v8 w
wv7
v6
v1
v5
GD 2014September 26,
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Technical Details (Draw Go and Gi)
Gi Go
v4
v8 w
wv7
v6
v1
v5
w'
GD 2014September 26,
2014
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Technical Details (Draw Go and Gi)
GiGo
v4
v8 w
w
v7
v6
v1
v5
Di
w’
v4
v5
v1
v8
v6
v2
v3
Towards w’
Do
b
a
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Technical Details (Merge Do and Di)
v4
v8
v7
v6
v1
v5
Di
Do
v2
v3
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Technical Details (Merge Do and Di)
v4
v8
v7
v6
v1
v5
Di
Do
v2
v3
Height of Di is (2/3)×|Di| = 4n/9+O(λ)
Height of Do is (2/3)×|Do| = 4n/9+O(λ∆)
Height of the final drawing is 4n/9+O(λ∆)
At most 6 bends per edge - two bends to enter Do from Di
- two bends on separator- two bends to return to Di from Do
Improve to 4 bends per edge using the transformation via
visibility representation [Biedl 2014]
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a
b c
d
f e
g
a
b c
a
b c
f
a
b c
d
fa
b c
d
f e
Plane 3-Trees
A planar 3-tree
Start with a triangle, then repeatedly add a vertex and triangulate the resulting graph.
GD 2014 September 26, 2014
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Plane 3-Trees
f f
d
f
de
fd
eg
The representative tree
a
b c
d
f e
g
a
b c
a
b c
f
a
b c
d
fa
b c
d
f eA planar 3-tree
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Plane 3-Treesr
The representative tree T of G
a
cA planar 3-tree G
b
18GD 2014September 26,
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a
c
Plane 3-Trees
A planar 3-tree G
r
v
The representative tree T of G b
v
r
Each component with at most n/2
vertices
a
cA planar 3-tree G
b
v w
Choosing a suitable embedding
v wG1
G3
G2F
F
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a
c
Plane 3-Trees
A planar 3-tree Gb
v w
Plane 3 trees inside each of these triangles has n/2+O(1) vertices
v wG1
G3
G2
w1w2
w3
v
w
w1
w2
F
F
x
y
wt
y
x
4n/9 + O(1)
21GD 2014September 26,
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a
c
Plane 3-Trees
A planar 3-tree Gb
v w
Choosing a suitable embedding
v wG1
G3
G2
w1w2
w3
v
w
F
F
x
y
wt
y
x
4n/9 + O(1)
v
w
w1
w2
The main challenge here is to show that the number of lines that are intersecting each triangle is sufficient to
draw the corresponding plane 3-tree
22GD 2014September 26,
2014
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v4
v8
v7
v6
v1v5
v2
v3
TriangulationsPolyline drawing with height 4n/9+O(λ∆) ≈ 0.44n+O(λ∆)This is 0.44n+o(n) when ∆ is o(n)
Planar 3-treesStraight-line drawing with height 4n/9+O(1) ≈ 0.44n+O(1)
TriangulationsArea Height0.88n2 + O(1) 0.66n [Brandenburg 2008]0.44n2 + O(1) 0.66n (polyline) [Bonichon et al. 2003]
Planar 3-treesArea Height0.88n2 + O(1) 0.5n [Brandenburg 2008,
Hossain et al. 2013]
Upper Bounds Improved Upper Bounds
a
c
r
v
b
v
r
Thank
you
OPEN: Clo
se th
e gap
!