1-Fan-Bundle-Planar Drawings of Graphs Patrizio Angelini Michael - - PowerPoint PPT Presentation

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1 fan bundle planar drawings of graphs
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1-Fan-Bundle-Planar Drawings of Graphs Patrizio Angelini Michael - - PowerPoint PPT Presentation

Dec 21, 2022 563 likes •2.2k views

1-Fan-Bundle-Planar Drawings of Graphs Patrizio Angelini Michael A. Bekos Michael Kaufmann Philipp Kindermann Thomas Schneck Beyond Planarity Planar Beyond Planarity 1-planar Beyond Planarity 2-planar Beyond Planarity k -planar Beyond

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SLIDE 1

1-Fan-Bundle-Planar Drawings of Graphs

Patrizio Angelini Michael A. Bekos Michael Kaufmann Philipp Kindermann Thomas Schneck

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SLIDE 2

Beyond Planarity

Planar

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SLIDE 3

Beyond Planarity

1-planar

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SLIDE 4

Beyond Planarity

2-planar

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SLIDE 5

Beyond Planarity

k-planar

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SLIDE 6

Beyond Planarity

Fan-planar k-planar

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SLIDE 7

Beyond Planarity

Fan-planar k-planar

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SLIDE 8

Beyond Planarity

Fan-planar k-planar

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SLIDE 9

Beyond Planarity

Fan-planar k-planar

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SLIDE 10

Bundled Edge Drawings

[Holten & van Wijk ’09]

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SLIDE 11

Bundled Edge Drawings

[Lambert, Bourqui &Auber ’10]

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SLIDE 12

Bundled Edge Drawings

[Wu, Zeng, Zhu & Yu ’17]

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SLIDE 13

Bundled Edge Drawings

[Ye ’17]

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SLIDE 14

k-Fan-Bundle-Planarity

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SLIDE 15

k-Fan-Bundle-Planarity

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SLIDE 16

k-Fan-Bundle-Planarity

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SLIDE 17

k-Fan-Bundle-Planarity

1-sided

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SLIDE 18

k-Fan-Bundle-Planarity

1-sided 2-sided

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SLIDE 19

k-Fan-Bundle-Planarity

1-sided 2-sided k crossings per bundle

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SLIDE 20

k-Fan-Bundle-Planarity

1-sided 2-sided k crossings per bundle 1-Fan-Bundle-Planarity

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SLIDE 21

k-Fan-Bundle-Planarity

1-sided 2-sided k crossings per bundle Density 1-Fan-Bundle-Planarity

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SLIDE 22

k-Fan-Bundle-Planarity

1-sided 2-sided k crossings per bundle Density Relationships 1-Fan-Bundle-Planarity

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SLIDE 23

k-Fan-Bundle-Planarity

1-sided 2-sided k crossings per bundle Density Relationships Recognition 1-Fan-Bundle-Planarity

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SLIDE 24

Density

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SLIDE 25

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 26

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 27

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 28

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 29

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 30

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 31

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 32

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 33

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 34

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 35

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 36

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 37

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar

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SLIDE 38

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 39

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 40

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 41

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 42

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 43

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 44

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces

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SLIDE 45

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces Remove 1 edge, create 2 faces

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SLIDE 46

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces Remove 1 edge, create 2 faces

⇒ Planar graph G′, m′ ≤ 3n − 6, f ′ ≤ 2n − 4

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SLIDE 47

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces Remove 1 edge, create 2 faces

⇒ Planar graph G′, m′ ≤ 3n − 6, f ′ ≤ 2n − 4 ⇒ m ≤ m′ + 2f ′/3

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SLIDE 48

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces Remove 1 edge, create 2 faces

⇒ Planar graph G′, m′ ≤ 3n − 6, f ′ ≤ 2n − 4 ⇒ m ≤ m′ + 2f ′/3 ≤ 3n − 6 + 2 · (2n − 4)/3

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SLIDE 49

Density: 1-sided

Upper bound: Take maximally dense graph G, make it planar Remove 2 edges, create 3 faces Remove 1 edge, create 2 faces

⇒ Planar graph G′, m′ ≤ 3n − 6, f ′ ≤ 2n − 4 ⇒ m ≤ m′ + 2f ′/3 ≤ 3n − 6 + 2 · (2n − 4)/3 ≤ (13n − 26)/3

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SLIDE 50

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 51

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 52

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 53

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 54

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 55

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 56

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 57

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

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SLIDE 58

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

(5n − 10)/3

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SLIDE 59

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

(5n − 10)/3 +4 · (2n − 4)/3

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SLIDE 60

Density: 1-sided

Lower bound: Upper bound: (13n − 26)/3

(5n − 10)/3 +4 · (2n − 4)/3= (13n − 26)/3

Tight

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SLIDE 61

Density: 1-sided

2-layer

  • uter

general 1-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

LB UB LB UB LB UB

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SLIDE 62

Density: 2-sided

Flower Drawing:

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SLIDE 63

Density: 2-sided

Flower Drawing: – Vertices on circle

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SLIDE 64

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle

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SLIDE 65

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle

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SLIDE 66

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle

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SLIDE 67

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross

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SLIDE 68

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross

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SLIDE 69

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing

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SLIDE 70

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets

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SLIDE 71

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets

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SLIDE 72

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets

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SLIDE 73

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets

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SLIDE 74

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets

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SLIDE 75

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals

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SLIDE 76

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals

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SLIDE 77

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals – Zigzag pattern in each partition

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SLIDE 78

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals – Zigzag pattern in each partition

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SLIDE 79

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals – Zigzag pattern in each partition

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SLIDE 80

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals – Zigzag pattern in each partition

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SLIDE 81

Density: 2-sided

Flower Drawing: – Vertices on circle – Every vertex has left and right bundle – Consecutive bundles cross Water Lily Drawing: – Flower Drawing – Terminals partitioned into 3 sets – Cycle through terminals – Zigzag pattern in each partition

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SLIDE 82

Density: 2-sided

Water Lily Drawings have 4n − 9 edges

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SLIDE 83

Density: 2-sided

Water Lily Drawings have 4n − 9 edges

⇒ LB for outer-2-sided: 4n − 9

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SLIDE 84

Density: 2-sided

Water Lily Drawings have 4n − 9 edges

⇒ LB for outer-2-sided:

Merge 2 Water Lilys

⇒ LB for 2-sided:

4n − 9 6n − 18

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SLIDE 85

Density: 2-sided

Water Lily Drawings have 4n − 9 edges

⇒ LB for outer-2-sided:

Merge 2 Water Lilys

⇒ LB for 2-sided:

4n − 9 6n − 18

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SLIDE 86

Density: 2-sided

Water Lily Drawings have 4n − 9 edges

⇒ LB for outer-2-sided:

Merge 2 Water Lilys

⇒ LB for 2-sided: ⇒ LB for 2-layer 2-sided:

4n − 9 6n − 18 2n − 4

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SLIDE 87

Density

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 4n − 9 6n − 18 LB UB LB UB LB UB

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SLIDE 88

Density

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 6n − 18 LB UB LB UB LB UB

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SLIDE 89

Density

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 4n − 9 6n − 18 LB UB LB UB LB UB

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SLIDE 90

Density

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 4n − 9 6n − 18 8.6n − 15.6 LB UB LB UB LB UB

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SLIDE 91

Relationships

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SLIDE 92

Relationships

PLANAR

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SLIDE 93

Relationships

K4

PLANAR

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SLIDE 94

Relationships

K4

PLANAR

1-PLANAR

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SLIDE 95

Relationships

K4

PLANAR

K6 1-PLANAR

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SLIDE 96

Relationships

2-PLANAR

K4

PLANAR

K6 1-PLANAR

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SLIDE 97

Relationships

2-PLANAR

K4

PLANAR

K6 1-PLANAR K3,10

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SLIDE 98

Relationships

2-PLANAR

FAN-PLANAR

K4

PLANAR

K6 1-PLANAR K3,10

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SLIDE 99

Relationships

2-PLANAR

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

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SLIDE 100

Relationships

2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

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SLIDE 101

Relationships

1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

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SLIDE 102

Relationships

1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

? ? ? ?

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SLIDE 103

K3,14

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SLIDE 104

K3,14

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SLIDE 105

K3,14

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SLIDE 106

K3,14

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SLIDE 107

K3,14

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SLIDE 108

K3,14

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SLIDE 109

K3,14

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SLIDE 110

K3,14

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SLIDE 111

K3,14

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SLIDE 112

Relationships

1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

? ? ? ?

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SLIDE 113

Relationships

1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

? ? ?

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SLIDE 114

Relationships

1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

? ?

K3,14

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SLIDE 115

Relationships

1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

?

K3,14 D12

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SLIDE 116

Relationships

K4,12

2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

?

K3,14 D12

?

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SLIDE 117

K9

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SLIDE 118

K9

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SLIDE 119

K9

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SLIDE 120

Relationships

K4,12

2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

?

K3,14 D12

?

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SLIDE 121

Relationships

K4,12

2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4,n−4 K4

PLANAR

K6 1-PLANAR K3,10

?

K3,14 D12 K9

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SLIDE 122

Relationships

K4,12

2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR

[Binucci et al.]

FAN-PLANAR

K4

PLANAR

K6 1-PLANAR K3,10 K3,14 D12 K9 K4,567

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SLIDE 123

Recognition

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SLIDE 124

Recognition: general

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SLIDE 125

Recognition: 2-layer 1-sided

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SLIDE 126

Recognition: 2-layer 1-sided

K2,3 is drawable

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SLIDE 127

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not!

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SLIDE 128

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

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SLIDE 129

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

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SLIDE 130

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

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SLIDE 131

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

  • Max. bicon. fan-planar:
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SLIDE 132

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

  • Max. bicon. fan-planar:
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SLIDE 133

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

  • Max. bicon. fan-planar:
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SLIDE 134

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

  • Max. bicon. fan-planar:
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SLIDE 135

Recognition: 2-layer 1-sided

K2,3 is drawable ... but K2,4 is not! [Binucci et al.]

  • Max. bicon. fan-planar:
  • Max. bicon. 1-sided 1-fbp:
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SLIDE 136

Recognition: 2-layer 1-sided

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SLIDE 137

Recognition: 2-layer 1-sided

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SLIDE 138

Recognition: 2-layer 1-sided

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SLIDE 139

Recognition: 2-layer 1-sided

slide-140
SLIDE 140

Recognition: 2-layer 1-sided

slide-141
SLIDE 141

Recognition: 2-layer 1-sided

slide-142
SLIDE 142

Recognition: 2-layer 1-sided

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SLIDE 143

Recognition: 2-layer 1-sided

  • Max. fan-planar
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SLIDE 144

Recognition: 2-layer 1-sided

  • Max. fan-planar
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SLIDE 145

Recognition: 2-layer 1-sided

  • Max. fan-planar
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SLIDE 146

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp
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SLIDE 147

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp
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SLIDE 148

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp
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SLIDE 149

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp
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SLIDE 150

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

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SLIDE 151

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

slide-152
SLIDE 152

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

slide-153
SLIDE 153

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs Small Legs

slide-154
SLIDE 154

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs Small Legs

slide-155
SLIDE 155

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

slide-156
SLIDE 156

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

slide-157
SLIDE 157

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

slide-158
SLIDE 158

Recognition: 2-layer 1-sided

  • Max. fan-planar
  • Max. 1-sided 1-fbp

Big Legs

  • Max. 1-sided 1-fbp
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SLIDE 159

Summary

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 4n − 9 6n − 18 8.6n − 15.6 LB UB LB UB LB UB

slide-160
SLIDE 160

Summary

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 4n − 9 6n − 18 8.6n − 15.6 LB UB LB UB LB UB K4,12 2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR [Binucci et al.]

FAN-PLANAR

K4 PLANAR K6 1-PLANAR K3,10 K3,14 D12 K9 K4,567

slide-161
SLIDE 161

Summary

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 4n − 9 6n − 18 8.6n − 15.6 LB UB LB UB LB UB K4,12 2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR [Binucci et al.]

FAN-PLANAR

K4 PLANAR K6 1-PLANAR K3,10 K3,14 D12 K9 K4,567 General:

slide-162
SLIDE 162

Summary

2-layer

  • uter

general 1-sided 2-sided

5n−7 3 5n−7 3 8n−13 3 8n−13 3 13n−26 3 13n−26 3

2n − 4 3n − 7 4n − 9 4n − 9 6n − 18 8.6n − 15.6 LB UB LB UB LB UB K4,12 2-SIDED 1-FBP 1-SIDED 1-FBP 2-PLANAR [Binucci et al.]

FAN-PLANAR

K4 PLANAR K6 1-PLANAR K3,10 K3,14 D12 K9 K4,567 General: 2-layer 1-sided: