Integral Unit Bar-Visibility Graphs Therese Biedl Ahmad Biniaz - - PowerPoint PPT Presentation

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Integral Unit Bar-Visibility Graphs Therese Biedl Ahmad Biniaz - - PowerPoint PPT Presentation

Apr 11, 2023 104 likes •1.75k views

Integral Unit Bar-Visibility Graphs Therese Biedl Ahmad Biniaz Veronika Irvine Philipp Kindermann Anurag Murty Naredla Alexi Turcotte Integral Unit Bar-Visibility Integral Unit Bar-Visibility Integral Unit Bar-Visibility Integral Unit

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SLIDE 1

Integral Unit Bar-Visibility Graphs

Therese Biedl Veronika Irvine Anurag Murty Naredla Ahmad Biniaz Philipp Kindermann Alexi Turcotte

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SLIDE 2

Integral Unit Bar-Visibility

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SLIDE 3

Integral Unit Bar-Visibility

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SLIDE 4

Integral Unit Bar-Visibility

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SLIDE 5

Integral Unit Bar-Visibility

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SLIDE 6

Integral Unit Bar-Visibility

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SLIDE 7

Integral Unit Bar-Visibility

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SLIDE 8

Integral Unit Bar-Visibility

Strong

⇔

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SLIDE 9

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

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SLIDE 10

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

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SLIDE 11

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Every planar graph, O(n) time

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SLIDE 12

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Every planar graph, O(n) time 3-connected: NP-hard

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SLIDE 13

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Every planar graph, O(n) time 3-connected: NP-hard Triangulation: O(n) time

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SLIDE 14

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Every planar graph, O(n) time 3-connected: NP-hard Triangulation: O(n) time 4-connected: O(n) time

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SLIDE 15

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

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SLIDE 16

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

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SLIDE 17

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Characterizations for...

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SLIDE 18

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Characterizations for...

  • trees
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SLIDE 19

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Characterizations for...

  • trees
  • complete bipartite
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SLIDE 20

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Characterizations for...

  • trees
  • complete bipartite
  • outerplanar
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SLIDE 21

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

Characterizations for...

  • trees
  • complete bipartite
  • outerplanar
  • triangulations
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SLIDE 22

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

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SLIDE 23

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 24

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 25

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 26

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 27

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 28

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 29

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 30

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

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SLIDE 31

Integral Unit Bar-Visibility

Weak Strong

⇒ ⇔

1 2 3 4 5 6 7 8

max-deg 6

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SLIDE 32

Weak Model

⇒

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SLIDE 33

Weak: Trees, maxdeg 5

v

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SLIDE 34

Weak: Trees, maxdeg 5

v v

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SLIDE 35

Weak: Trees, maxdeg 5

v v

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SLIDE 36

Weak: Trees, maxdeg 5

v v

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SLIDE 37

Weak: Trees, maxdeg 5

v v

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SLIDE 38

Weak: Trees, maxdeg 5

v v

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SLIDE 39

Weak: Trees, maxdeg 5

v v

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SLIDE 40

Weak: Trees, maxdeg 5

v v

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SLIDE 41

Weak: Trees, maxdeg 5

v v

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SLIDE 42

Weak: Trees, maxdeg 5

v v Tree with maxdeg 5 ✗

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SLIDE 43

Weak: Trees, maxdeg 4

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SLIDE 44

Weak: Trees, maxdeg 4

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SLIDE 45

Weak: Trees, maxdeg 4

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SLIDE 46

Weak: Trees, maxdeg 4

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SLIDE 47

Weak: Trees, maxdeg 4

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SLIDE 48

Weak: Trees, maxdeg 4

Tree with maxdeg 4✓

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SLIDE 49

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 50

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 51

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 52

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 53

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 54

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 55

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 56

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 57

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 58

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 59

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 60

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 61

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 62

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 63

Weak: Biconnected Outerplanar, maxdeg 3

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SLIDE 64

Weak: Biconnected Outerplanar, maxdeg 3

Biconnected Outerplanar with maxdeg 3

✓

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SLIDE 65

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 66

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 67

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 68

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 69

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 70

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 71

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 72

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 73

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 74

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 75

Weak: Biconnected Outerplanar, maxdeg 4

Biconnected Outerplanar with maxdeg 4

✓

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SLIDE 76

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 77

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 78

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 79

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 80

Weak: Biconnected Outerplanar, maxdeg 4

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SLIDE 81

Layered Drawings

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SLIDE 82

Layered Drawings

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SLIDE 83

Layered Drawings

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SLIDE 84

Layered Drawings

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SLIDE 85

Layered Drawings

consecutive consecutive

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SLIDE 86

Weak: NP-hardness

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SLIDE 87

Weak: NP-hardness

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SLIDE 88

Weak: NP-hardness

NP-hard [Heath & Rosenberg]

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SLIDE 89

Weak: NP-hardness

NP-hard [Heath & Rosenberg]

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SLIDE 90

Weak: NP-hardness

NP-hard [Heath & Rosenberg] P

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SLIDE 91

Weak: NP-hardness

Edges:

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SLIDE 92

Weak: NP-hardness

Edges:

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SLIDE 93

Weak: NP-hardness

Edges: Node block:

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SLIDE 94

Weak: NP-hardness

Edges: Node block:

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SLIDE 95

Weak: NP-hardness

Edges: Node block:

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SLIDE 96

Weak: NP-hardness

Edges: Node block:

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SLIDE 97

Weak: NP-hardness

Edges: Node block:

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SLIDE 98

Weak: NP-hardness

Edges: Node block:

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SLIDE 99

Weak: NP-hardness

Edges: Node block:

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SLIDE 100

Weak: NP-hardness

Edges: Node block: Nodes:

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SLIDE 101

Weak: NP-hardness

Edges: Node block: Nodes:

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SLIDE 102

Weak: NP-hardness

Edges: Node block: Nodes:

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SLIDE 103

Weak: NP-hardness

Edges: Node block: Nodes:

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SLIDE 104

Weak: NP-hardness

NP-hard [Heath & Rosenberg] P

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SLIDE 105

Weak: NP-hardness

NP-hard [Heath & Rosenberg] Recognition: NP-hard P

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SLIDE 106

Strong Model

⇔

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SLIDE 107

Strong: Trees

[Dean & Veytsel] Unit Bar-Visibility Representation

⇔ maxdeg 3, subdivision of caterpillar

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SLIDE 108

Strong: Trees

[Dean & Veytsel] Unit Bar-Visibility Representation

⇔ maxdeg 3, subdivision of caterpillar

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SLIDE 109

Strong: Trees

[Dean & Veytsel] Unit Bar-Visibility Representation

⇔ maxdeg 3, subdivision of caterpillar

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SLIDE 110

Strong: Trees

[Dean & Veytsel] Unit Bar-Visibility Representation

⇔ maxdeg 3, subdivision of caterpillar

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SLIDE 111

Strong: Trees

[Dean & Veytsel] Unit Bar-Visibility Representation

⇔ maxdeg 3, subdivision of caterpillar

Integral Unit Bar-Visibility Representation

⇔ maxdeg 3, subdivision of caterpillar

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SLIDE 112

Strong: Biconnected Graphs

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SLIDE 113

Strong: Biconnected Graphs

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SLIDE 114

Strong: Biconnected Graphs

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SLIDE 115

Strong: Biconnected Graphs

Every face spans two slabs

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SLIDE 116

Strong: Biconnected Graphs

Every face spans two slabs

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SLIDE 117

Strong: Biconnected Graphs

Every face spans two slabs

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SLIDE 118

Strong: Biconnected Graphs

Every face spans two slabs

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SLIDE 119

Strong: Biconnected Graphs

Every face spans two slabs

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SLIDE 120

Strong: Biconnected Graphs

Every face spans two slabs

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SLIDE 121

Strong: Biconnected Graphs

Every face spans two slabs and is a trapezoid

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SLIDE 122

Strong: Biconnected Graphs

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SLIDE 123

Strong: Biconnected Graphs

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SLIDE 124

Strong: Biconnected Graphs

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SLIDE 125

Strong: Biconnected Graphs

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SLIDE 126

Strong: Biconnected Graphs

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SLIDE 127

Strong: Biconnected Graphs

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SLIDE 128

Strong: Biconnected Graphs

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SLIDE 129

Strong: Biconnected Graphs

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SLIDE 130

Strong: Biconnected Graphs

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SLIDE 131

Strong: Biconnected Graphs

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SLIDE 132

Strong: Biconnected Graphs

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SLIDE 133

Strong: Biconnected Graphs

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SLIDE 134

Strong: Biconnected Graphs

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SLIDE 135

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab

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SLIDE 136

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab Idea: guess the topmost diagonal

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SLIDE 137

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab Idea: guess the topmost diagonal

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SLIDE 138

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab Idea: guess the topmost diagonal

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SLIDE 139

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab Idea: guess the topmost diagonal

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SLIDE 140

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab Idea: guess the topmost diagonal

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SLIDE 141

Strong: Biconnected Graphs

Fixing one diagonal fixes the whole slab Idea: guess the topmost diagonal

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SLIDE 142

Strong: Biconnected Graphs

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SLIDE 143

Strong: Biconnected Graphs

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SLIDE 144

Strong: Biconnected Graphs

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SLIDE 145

Strong: Biconnected Graphs

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SLIDE 146

Strong: Biconnected Graphs

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SLIDE 147

Strong: Biconnected Graphs

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SLIDE 148

Strong: Biconnected Graphs

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SLIDE 149

Strong: Biconnected Graphs s

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SLIDE 150

Strong: Biconnected Graphs s

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SLIDE 151

Strong: Biconnected Graphs s t

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SLIDE 152

Strong: Biconnected Graphs s t

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SLIDE 153

Strong: Biconnected Graphs s t

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SLIDE 154

Strong: Biconnected Graphs s t

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SLIDE 155

Strong: Biconnected Graphs s t

Biconnected Graph

⇒ Test + draw in O(n3 ) time

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SLIDE 156

Strong: Biconnected Graphs s t

Biconnected Graph

⇒ Test + draw in O(n3 ) time

2

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SLIDE 157

Strong: Biconnected Graphs s t

Biconnected Graph

⇒ Test + draw in O(n3 ) time

2

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SLIDE 158

Summary

Weak Strong

⇒ ⇔

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SLIDE 159

Summary

Weak Strong

⇒ ⇔

Trees, maxdeg 5

✗

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SLIDE 160

Summary

Weak Strong

⇒ ⇔

Trees, maxdeg 5 Trees, maxdeg 4

✗

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SLIDE 161

Summary

Weak Strong

⇒ ⇔

Trees, maxdeg 5 Trees, maxdeg 4

  • Biconn. outerplanar,

maxdeg 4

✗

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SLIDE 162

Summary

Weak Strong

⇒ ⇔

Trees, maxdeg 5 Trees, maxdeg 4

  • Biconn. outerplanar,

maxdeg 4 General: NP-hard

✗

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SLIDE 163

Summary

Weak Strong

⇒ ⇔

Trees, maxdeg 5 Trees, maxdeg 4

  • Biconn. outerplanar,

maxdeg 4 General: NP-hard

✗

  • Trees: maxdeg 3,
  • subdiv. of caterpillar
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SLIDE 164

Summary

Weak Strong

⇒ ⇔

Trees, maxdeg 5 Trees, maxdeg 4

  • Biconn. outerplanar,

maxdeg 4 General: NP-hard

✗

  • Trees: maxdeg 3,
  • subdiv. of caterpillar

General: O(n2) time recognition

s t