GD 2014September 26,

2014

Drawing Planar Graphs with Reduced Height

Department of Computer ScienceUniversity of Manitoba

Stephane Durocher Debajyoti Mondal

3

Straight-line Drawings (Fixed Vs. Variable)

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b c

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GD 2014 September 26, 2014

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A straight-line drawing of G on a 5×5 grid

A planar graph GA straight-line drawing

of G with height 4

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eg

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Straight-line Drawings (Fixed Vs. Variable)

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b c

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A planar graph G

Via visibility representationBiedl, 2014

Here we allow variable embedding.Sometimes we use edge-bends.

Focus on height.

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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

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)

The Big Picture

w w'

v4

v7

v6v1

v5

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v5

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v6

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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

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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Technical Details (Draw Go and Gi)

Gi Go

GD 2014September 26,

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w

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Technical Details (Draw Go and Gi)

Gi Go

v4

v8 w

wv7

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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

Technical Details (Draw Go and Gi)

GiGo

v4

v8 w

w

v7

v6

v1

v5

Di

w’

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v5

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v8

v6

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Towards w’

Do

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Technical Details (Merge Do and Di)

v4

v8

v7

v6

v1

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Di

Do

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v3

Technical Details (Merge Do and Di)

v4

v8

v7

v6

v1

v5

Di

Do

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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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g

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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

Plane 3-Trees

f f

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The representative tree

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b c

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fa

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f eA planar 3-tree

Plane 3-Treesr

The representative tree T of G

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cA planar 3-tree G

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18GD 2014September 26,

2014

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Plane 3-Trees

A planar 3-tree G

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The representative tree T of G b

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Each component with at most n/2

vertices

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cA planar 3-tree G

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Choosing a suitable embedding

v wG1

G3

G2F

F

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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

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w

w1

w2

F

F

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wt

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4n/9 + O(1)

21GD 2014September 26,

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Plane 3-Trees

A planar 3-tree Gb

v w

Choosing a suitable embedding

v wG1

G3

G2

w1w2

w3

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w

F

F

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y

wt

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4n/9 + O(1)

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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

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

!