Recognizing and Drawing IC-planar Graphs Philipp Kindermann - - PowerPoint PPT Presentation

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Recognizing and Drawing IC-planar Graphs Philipp Kindermann - - PowerPoint PPT Presentation

Aug 01, 2023 360 likes •1.82k views

Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg, Walter Didimo, William S. Evans, Giuseppe Liotta & Fabrizio Montecchiani

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SLIDE 1

Recognizing and Drawing IC-planar Graphs

Philipp Kindermann Universit¨ at W¨ urzburg / FernUniversit¨ at in Hagen

Joint work with Franz J. Brandenburg, Walter Didimo, William S. Evans, Giuseppe Liotta & Fabrizio Montecchiani

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SLIDE 2

1-planar Graphs

Planar graphs: Can be drawn without crossings.

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SLIDE 3

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

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SLIDE 4

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

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SLIDE 5

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

  • ≤ 4n − 8 edges
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SLIDE 6

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

  • ≤ 4n − 8 edges
  • straight-line: ≤ 4n − 9 edges
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SLIDE 7

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

  • ≤ 4n − 8 edges
  • straight-line: ≤ 4n − 9 edges
  • Recognition: NP-hard

[Grigoriev & Bodlander ALG’07]

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SLIDE 8

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

  • ≤ 4n − 8 edges
  • straight-line: ≤ 4n − 9 edges
  • Recognition: NP-hard

[Grigoriev & Bodlander ALG’07]

  • for planar graphs + 1 edge

[Korzhik & Mohar JGT’13]

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SLIDE 9

1-planar Graphs

1-planar graphs: Each edge is crossed at most once. Planar graphs: Can be drawn without crossings.

  • ≤ 4n − 8 edges
  • straight-line: ≤ 4n − 9 edges
  • Recognition: NP-hard

[Grigoriev & Bodlander ALG’07]

  • for planar graphs + 1 edge

[Korzhik & Mohar JGT’13]

  • with given rotation system

[Auer et al. JGAA’15]

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SLIDE 10

RAC Graphs

RAC graphs: Can be drawn straight-line with only right-angle crossings.

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SLIDE 11

RAC Graphs

RAC graphs: Can be drawn straight-line with only right-angle crossings.

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SLIDE 12

RAC Graphs

  • Increases readability

RAC graphs:

[Huang et al. PacificVis’08]

Can be drawn straight-line with only right-angle crossings.

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SLIDE 13

RAC Graphs

  • Increases readability

RAC graphs:

[Huang et al. PacificVis’08]

... even for planar graphs

[van Krefeld GD’11]

Can be drawn straight-line with only right-angle crossings.

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SLIDE 14

RAC Graphs

  • Increases readability

RAC graphs:

[Huang et al. PacificVis’08]

... even for planar graphs

[van Krefeld GD’11]

  • ≤ 4n − 10 edges

[Didimo et al. WADS’09]

Can be drawn straight-line with only right-angle crossings.

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SLIDE 15

RAC Graphs

  • Increases readability

RAC graphs:

[Huang et al. PacificVis’08]

... even for planar graphs

[van Krefeld GD’11]

  • ≤ 4n − 10 edges

[Didimo et al. WADS’09]

Can be drawn straight-line with only right-angle crossings.

  • Recognition: NP-hard

[Argyriou et al. JGAA’12]

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SLIDE 16

1-planar RAC graphs

1-planar

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SLIDE 17

1-planar RAC graphs

1-planar

  • 1-planar = RAC

[Eades & Liotta DMA’13]

RAC

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SLIDE 18

1-planar RAC graphs

1-planar

  • 1-planar = RAC

[Eades & Liotta DMA’13]

?

RAC

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SLIDE 19

1-planar RAC graphs

1-planar

  • 1-planar = RAC

[Eades & Liotta DMA’13]

RAC

  • uter-1-planar
  • outer-1-planar ⊂ RAC

[Dehkordi & Eades IJCGA’12]

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SLIDE 20

1-planar RAC graphs

1-planar

  • 1-planar = RAC

[Eades & Liotta DMA’13]

RAC

  • uter-1-planar
  • outer-1-planar ⊂ RAC

[Dehkordi & Eades IJCGA’12] perfect RAC

  • perfect RAC ⊂ 1-planar

[Eades & Liotta DMA’13]

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SLIDE 21

IC-planar Graphs

IC-planar graphs: Each edge is crossed at most once

independent crossings

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SLIDE 22

IC-planar Graphs

IC-planar graphs: Each edge is crossed at most once and each vertex is incident to at most one crossing edge.

independent crossings

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SLIDE 23

IC-planar Graphs

IC-planar graphs: Each edge is crossed at most once and each vertex is incident to at most one crossing edge.

independent crossings

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SLIDE 24

IC-planar Graphs

  • ≤ 13n/4 − 6 edges

IC-planar graphs: Each edge is crossed at most once and each vertex is incident to at most one crossing edge.

independent crossings

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SLIDE 25

Recognition

Reduction from 1-planarity testing.

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SLIDE 26

Recognition

Reduction from 1-planarity testing. u v

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SLIDE 27

Recognition

Reduction from 1-planarity testing. u v

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SLIDE 28

Recognition

Reduction from 1-planarity testing. u v

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SLIDE 29

Recognition

Reduction from 1-planarity testing. u v u

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SLIDE 30

Recognition

Reduction from 1-planarity testing. u v u

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SLIDE 31

Recognition

Reduction from 1-planarity testing. u v u

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SLIDE 32

Recognition

Testing IC-planarity is NP-hard

Theorem.

Reduction from 1-planarity testing. u v u

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SLIDE 33

Recognition

Testing IC-planarity is NP-hard

Theorem.

Reduction from 1-planarity testing. u v Reduction from planar-3SAT

Recognition

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SLIDE 34

Recognition

Testing IC-planarity is NP-hard

Theorem.

Reduction from 1-planarity testing. u v Reduction from planar-3SAT

Recognition

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SLIDE 35

Recognition

Testing IC-planarity is NP-hard

Theorem.

even if the rotation system is given. Reduction from 1-planarity testing. u v Reduction from planar-3SAT

Recognition

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SLIDE 36

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar?

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SLIDE 37

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge!

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SLIDE 38

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! Compute extended dual T ∗ of T.

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SLIDE 39

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T.

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SLIDE 40

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T.

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SLIDE 41

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T.

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SLIDE 42

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. T ∗ :

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SLIDE 43

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM

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SLIDE 44

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 45

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 46

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 47

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 48

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 49

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 50

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 51

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v

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SLIDE 52

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? Task: Find a valid routing for each matching edge! T : Compute extended dual T ∗ of T. u v T ∗ : (u, v) ∈ EM u v Routing in T ˆ = path of length 3 in T ∗

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SLIDE 53

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv

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SLIDE 54

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv Interior I(u, v)

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SLIDE 55

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv Interior I(u, v) The boundaries of two interiors may not intersect.

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SLIDE 56

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv Interior I(u, v) The boundaries of two interiors may not intersect.

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SLIDE 57

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv Interior I(u, v) The boundaries of two interiors may not intersect.

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SLIDE 58

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv Interior I(u, v) The boundaries of two interiors may not intersect.

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SLIDE 59

Triangulation + Matching

Given a triconnected plane graph T = (V, ET ) and a matching M = (V, EM), is G = (V, ET ∪ EM) IC-planar? u v luv ruv Interior I(u, v) The boundaries of two interiors may not intersect.

  • ×
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SLIDE 60

Triangulation + Matching

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SLIDE 61

Triangulation + Matching

u v

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SLIDE 62

Triangulation + Matching

u v

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SLIDE 63

Triangulation + Matching

u v

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SLIDE 64

Triangulation + Matching

a b u v

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SLIDE 65

Triangulation + Matching

a b u v

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SLIDE 66

Triangulation + Matching

a b u v

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SLIDE 67

Triangulation + Matching

a b c d u v

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SLIDE 68

Triangulation + Matching

a b c d u v

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SLIDE 69

Triangulation + Matching

a b c d u v

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SLIDE 70

Triangulation + Matching

a b c d u v Hierarchical structure: Tree H = (VH, EH)

H:

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SLIDE 71

Triangulation + Matching

a b c d u v Icd Iab Iuv Hierarchical structure: Tree H = (VH, EH) VH = {Iuv | (u, v) ∈ M}

H:

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SLIDE 72

Triangulation + Matching

a b c d u v Icd Iab Iuv G Hierarchical structure: Tree H = (VH, EH) VH = {Iuv | (u, v) ∈ M} ∪ {G}

H:

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SLIDE 73

Triangulation + Matching

a b c d u v Icd Iab Iuv G Hierarchical structure: Tree H = (VH, EH) (Iuv, Iab) ∈ EH ⇔ Iuv ⊂ Iab VH = {Iuv | (u, v) ∈ M} ∪ {G}

H:

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SLIDE 74

Triangulation + Matching

a b c d u v Icd Iab Iuv G Hierarchical structure: Tree H = (VH, EH) (Iuv, Iab) ∈ EH ⇔ Iuv ⊂ Iab

  • utdeg(Iuv) = 0 ⇒ (Iuv, G) ∈ EH

VH = {Iuv | (u, v) ∈ M} ∪ {G}

H:

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SLIDE 75

Triangulation + Matching

a b c d u v Icd Iab Iuv G

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SLIDE 76

Triangulation + Matching

a b c d Icd Iab Iuv G

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SLIDE 77

Triangulation + Matching

a b c d Icd Iab Iuv G

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SLIDE 78

Triangulation + Matching

a b c d Icd Iab Iuv G

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SLIDE 79

Triangulation + Matching

a b c d Icd Iab Iuv G

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SLIDE 80

Triangulation + Matching

a b c d Icd Iab Iuv G

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SLIDE 81

Triangulation + Matching

a b c d Icd Iab Iuv G

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SLIDE 82

Triangulation + Matching

  • Always pick “middle” routing

a b c d Icd Iab Iuv G

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SLIDE 83

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d Icd Iab Iuv G

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SLIDE 84

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d u v Icd Iab Iuv G

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SLIDE 85

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d u v Icd Iab Iuv G

  • Recursively check which routings are valid
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SLIDE 86

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d u v Icd Iab Iuv G

  • Recursively check which routings are valid
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SLIDE 87

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d u v Icd Iab Iuv G

  • Recursively check which routings are valid
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SLIDE 88

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d u v Icd Iab Iuv G

  • Recursively check which routings are valid
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SLIDE 89

Triangulation + Matching

  • Always pick “middle” routing

a b

  • Solve rest with 2SAT

c d u v Icd Iab Iuv G

  • Recursively check which routings are valid

Theorem.

IC-planarity can be tested efficiently if the input graph is a triangulated planar graph and a matching

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SLIDE 90

Straight-Line Drawings

IC-plane graphs can be drawn straight-line

  • n the O(n) × O(n) grid in O(n) time.

Theorem.

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SLIDE 91

Straight-Line Drawings

IC-plane graphs can be drawn straight-line

  • n the O(n) × O(n) grid in O(n) time.

Theorem.

Using a special 1-planar drawing... [Alam et al. GD’13]

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SLIDE 92

Straight-Line Drawings

IC-plane graphs can be drawn straight-line

  • n the O(n) × O(n) grid in O(n) time.

Theorem.

Using a special 1-planar drawing...

RAC?

[Alam et al. GD’13]

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SLIDE 93

Straight-Line Drawings

IC-plane graphs can be drawn straight-line

  • n the O(n) × O(n) grid in O(n) time.

Theorem.

Using a special 1-planar drawing...

RAC?

[Alam et al. GD’13]

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SLIDE 94

Straight-Line Drawings

IC-plane graphs can be drawn straight-line

  • n the O(n) × O(n) grid in O(n) time.

Theorem.

Straight-line RAC drawings of IC-planar graphs may require exponential area.

Theorem.

Using a special 1-planar drawing...

RAC?

[Alam et al. GD’13]

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SLIDE 95

Straight-Line Drawings

IC-plane graphs can be drawn straight-line

  • n the O(n) × O(n) grid in O(n) time.

Theorem.

Straight-line RAC drawings of IC-planar graphs may require exponential area.

Theorem.

Using a special 1-planar drawing...

RAC?

[Alam et al. GD’13]

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SLIDE 96

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90]

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SLIDE 97

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph

[de Fraysseix, Pach & Pollack Comb’90]

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SLIDE 98

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

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SLIDE 99

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-100
SLIDE 100

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-101
SLIDE 101

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-102
SLIDE 102

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-103
SLIDE 103

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-104
SLIDE 104

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-105
SLIDE 105

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-106
SLIDE 106

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Augment to 3-connected planar graph
  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

slide-107
SLIDE 107

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

  • Augment to planar-maximal IC-planar graph
slide-108
SLIDE 108

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs

  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90] d c b a

  • Augment to planar-maximal IC-planar graph
  • Each crossing → Kite K = (a, b, c, d)
slide-109
SLIDE 109

Straight-Line RAC Drawings

  • Remove one edge per crossing

Adjust Shift-Algorithm for planar graphs

  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

  • Augment to planar-maximal IC-planar graph
  • Each crossing → Kite K = (a, b, c, d)

d c b a

slide-110
SLIDE 110

Straight-Line RAC Drawings

  • Adjust step in which d is placed
  • Remove one edge per crossing

Adjust Shift-Algorithm for planar graphs

  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

  • Augment to planar-maximal IC-planar graph
  • Each crossing → Kite K = (a, b, c, d)

d c b a

slide-111
SLIDE 111

Straight-Line RAC Drawings

  • Adjust step in which d is placed
  • Remove one edge per crossing

Adjust Shift-Algorithm for planar graphs

  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

  • Augment to planar-maximal IC-planar graph
  • Each crossing → Kite K = (a, b, c, d)

d c b a Highest number in canonical order

slide-112
SLIDE 112

Straight-Line RAC Drawings

  • Adjust step in which d is placed
  • Remove one edge per crossing

Adjust Shift-Algorithm for planar graphs

  • Contour only has slopes ±1
  • Insert vertices in canonical order

[de Fraysseix, Pach & Pollack Comb’90]

  • Augment to planar-maximal IC-planar graph
  • Each crossing → Kite K = (a, b, c, d)

d c b a c a Al(b) b

slide-113
SLIDE 113

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c a Al(b) b

slide-114
SLIDE 114

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c a Al(b) b

slide-115
SLIDE 115

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c a Al(b) b

slide-116
SLIDE 116

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c a Al(b) b

slide-117
SLIDE 117

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c a

slide-118
SLIDE 118

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b c a

slide-119
SLIDE 119

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b c a

slide-120
SLIDE 120

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b c a d

slide-121
SLIDE 121

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b c a d

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SLIDE 122

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) c a d b

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SLIDE 123

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b a c

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SLIDE 124

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b u d a c

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SLIDE 125

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b u d a c

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SLIDE 126

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b u d a c

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SLIDE 127

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b u d a c

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SLIDE 128

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a Al(b) b u d a c

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SLIDE 129

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a b u d a c

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SLIDE 130

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a b u d a c

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SLIDE 131

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a b u c a

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SLIDE 132

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a u c a b

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SLIDE 133

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a u c a b

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SLIDE 134

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a u c a Al(b) b

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SLIDE 135

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a u c a Al(b) b

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SLIDE 136

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a u c a Al(b) b

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SLIDE 137

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c Al(b) b a u

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SLIDE 138

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c Al(b) b d a u

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SLIDE 139

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c Al(b) b d a u

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SLIDE 140

Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] d c b a c Al(b) b d a u IC-planar graphs can be drawn straight-line RAC in exponential area in O(n3) time.

Theorem.

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SLIDE 141

Conclusion

1-planar RAC

  • uter-1-planar

perfect RAC

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SLIDE 142

Conclusion

1-planar RAC

  • uter-1-planar

perfect RAC IC-planar

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SLIDE 143

Conclusion

1-planar RAC

  • uter-1-planar

perfect RAC IC-planar ?

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SLIDE 144

Conclusion

1-planar RAC

  • uter-1-planar

perfect RAC IC-planar ?

Draw in polynomial area with good crossing resolution?

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SLIDE 145

Conclusion

1-planar RAC

  • uter-1-planar

perfect RAC IC-planar ?

Draw in polynomial area with good crossing resolution? What about maximal IC-planar graphs?