[PPT] - Windrose Planarity Embedding Graphs with Direction-Constrained PowerPoint Presentation

SLIDE 1

Windrose Planarity

Embedding Graphs with Direction-Constrained Edges

Published at SODA’16. Joint work with Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Valentino Di Donato, G¨ unter Rote & Ignaz Rutter Philipp Kindermann LG Theoretische Informatik FernUniversit¨ at in Hagen

SLIDE 2

Upward Planarity

An undirected graph is planar: no crossings

SLIDE 3

Upward Planarity

An undirected graph is planar: no crossings

SLIDE 4

Upward Planarity

An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 5

Upward Planarity

An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 6

Upward Planarity

An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 7

Upward Planarity

planar An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 8

Upward Planarity

planar acyclic An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 9

Upward Planarity

planar acyclic An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 10

Upward Planarity

planar acyclic ? An undirected graph is planar: no crossings A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

SLIDE 11

Upward Planarity: Testing

Testing Upward Planarity is...

SLIDE 12

Upward Planarity: Testing

Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

SLIDE 13

Upward Planarity: Testing

Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

poly for single-source graphs

[Di Battista et al. ’98]

SLIDE 14

Upward Planarity: Testing

Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

poly for single-source graphs

[Di Battista et al. ’98]

poly for fixed embedding

[Garg & Tamassia ’95]

SLIDE 15

Upward Planarity: Testing

planar acyclic Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

poly for single-source graphs

[Di Battista et al. ’98]

poly for fixed embedding

[Garg & Tamassia ’95] ?

SLIDE 16

Upward Planarity: Testing

planar acyclic bimodal Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

poly for single-source graphs

[Di Battista et al. ’98]

poly for fixed embedding

[Garg & Tamassia ’95]

SLIDE 17

Upward Planarity: Testing

planar acyclic bimodal Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

poly for single-source graphs

[Di Battista et al. ’98]

poly for fixed embedding

[Garg & Tamassia ’95]

SLIDE 18

Upward Planarity: Testing

planar acyclic bimodal Testing Upward Planarity is...

NP-complete in general

[Garg & Tamassia ’95]

poly for single-source graphs

[Di Battista et al. ’98]

poly for fixed embedding

[Garg & Tamassia ’95]

×

SLIDE 19

Windrose Planarity

q-constrained graph (G, Q):

SLIDE 20

Windrose Planarity

q-constrained graph (G, Q):

G: undirected planar graph

SLIDE 21

Windrose Planarity

q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

SLIDE 22

Windrose Planarity

q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v v

SLIDE 23

Windrose Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v v

SLIDE 24

Windrose Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v A q-constrained graph is windrose planar: v

SLIDE 25

Windrose Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v A q-constrained graph is windrose planar:

no crossings

v

SLIDE 26

Windrose Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v A q-constrained graph is windrose planar:

no crossings
all edges are xy-monotone curves

v

SLIDE 27

Windrose Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v A q-constrained graph is windrose planar:

no crossings
all edges are xy-monotone curves
u ∈
v ⇒ u lies in the ◦-quadrant of v

v

SLIDE 28

Relationship to Upwards Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v,

տ

v,

ւ

v, and

ց

v.

ր

v

տ

v

ց

v

ւ

v A q-constrained graph is windrose planar:

no crossings
all edges are xy-monotone curves
u ∈
v ⇒ u lies in the ◦-quadrant of v

v

SLIDE 29

Relationship to Upwards Planarity

Two directions: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v and

ւ

v v

ր

v

ւ

v A q-constrained graph is windrose planar:

no crossings
all edges are xy-monotone curves
u ∈
v ⇒ u lies in the ◦-quadrant of v

SLIDE 30

Relationship to Upwards Planarity

One direction: q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

ր

v and

ւ

v v

ր

v

ւ

v A q-constrained graph is windrose planar:

no crossings
all edges are xy-monotone curves
u ∈
v ⇒ u lies in the ◦-quadrant of v

SLIDE 31

Relationship to Upwards Planarity

q-constrained graph (G, Q):

G: undirected planar graph
Q: partition of all neighbors of v into

↑

v and

↓

v A q-constrained graph is windrose planar:

no crossings
all edges are xy-monotone curves
u ∈
v ⇒ u lies in the ◦-quadrant of v

v

↑

v

↓

v One direction:

SLIDE 32

Relationship to Upwards Planarity

directed graph One direction: A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

v

↑

v

↓

v

SLIDE 33

Relationship to Upwards Planarity

directed graph One direction: A directed graph is upwards planar:

all edges are y-monotone curves directed upwards
no crossings

v

↑

v

↓

v Testing Windrose Planarity is NP-complete

Theorem.

SLIDE 34

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

SLIDE 35

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

SLIDE 36

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

SLIDE 37

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

SLIDE 38

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

0◦

SLIDE 39

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

0◦

SLIDE 40

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 0◦ 90◦

SLIDE 41

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 0◦ 90◦

SLIDE 42

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 0◦ 90◦ 180◦

SLIDE 43

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 0◦ 90◦ 180◦

SLIDE 44

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 270◦ 0◦ 90◦ 180◦

SLIDE 45

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 270◦ 0◦ 90◦ 180◦

SLIDE 46

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 270◦ 360◦ 0◦ 90◦ 180◦

SLIDE 47

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 360◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦

SLIDE 48

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 360◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦

Labeled graph (G, A): G plane graph, A labeling of angles

SLIDE 49

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 360◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦

Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

SLIDE 50

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 51

Angular Drawing

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 52

Angular Drawing

(G, A) admits angular drawing if: Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 53

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 54

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦

Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 55

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 56

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

SLIDE 57

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

⇒ A is angular labeling

SLIDE 58

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

⇒ A is angular labeling angular labeling A ⇒

unique q-constraints QA

SLIDE 59

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

⇒ A is angular labeling angular labeling A ⇒

unique q-constraints QA

SLIDE 60

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

⇒ A is angular labeling angular labeling A ⇒

unique q-constraints QA

q-constraints Q + large-angle assignment L ⇒

unique angular labeling AQ,L

SLIDE 61

Angular Drawing

(G, A) admits angular drawing if:

Vertex condition: sum of angle cat. at vertex is 360◦
Cycle condition: sum of angle cat. at (int.) face of

length k is k · 180◦ − 360◦ Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦ Labeled graph (G, A): G plane graph, A labeling of angles Angular drawing: end of segments have slopes ≈ ±1

90◦ 90◦ 180◦ 270◦ 0◦ 180◦ 0◦ 270◦ 90◦ 0◦ 270◦ 0◦ 90◦ 180◦ 90◦ 180◦ 90◦ 90◦ 0◦ 360◦

⇒ A is angular labeling angular labeling A ⇒

unique q-constraints QA

q-constraints Q + large-angle assignment L ⇒

unique angular labeling AQ,L

angular drawing ˆ = windrose planar drawing

SLIDE 62

Triangulated Graphs

0 180 0

SLIDE 63

Triangulated Graphs

0 180 0

SLIDE 64

Triangulated Graphs

0 90 90

SLIDE 65

Triangulated Graphs

No (int.) > 180◦ angle categories

0 90 90

SLIDE 66

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90

SLIDE 67

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90

SLIDE 68

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90 G ↑

SLIDE 69

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90 G ↑ G ↑ is acyclic.

Lemma.

Let (G, AQ) be a triangulated angular labeled graph. Then,

SLIDE 70

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90 G ↑ G ↑ is acyclic and has no internal sources or sinks.

Lemma.

Let (G, AQ) be a triangulated angular labeled graph. Then,

270◦ 270◦

SLIDE 71

Triangulated Graphs

G →

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90 G ↑ is acyclic and has no internal sources or sinks.

Lemma.

Let (G, AQ) be a triangulated angular labeled graph. Then,

270◦ 270◦

SLIDE 72

Triangulated Graphs

G →

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90

Lemma.

G ↑ and G → are acyclic and have no internal sources or sinks. Let (G, AQ) be a triangulated angular labeled graph. Then,

SLIDE 73

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90

Lemma.

G ↑ and G → are acyclic and have no internal sources or sinks. Let (G, AQ) be a triangulated angular labeled graph. Then, G

SLIDE 74

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90

Lemma.

G ↑ and G → are acyclic and have no internal sources or sinks. Let (G, AQ) be a triangulated angular labeled graph. Then, G internally

SLIDE 75

Triangulated Graphs

No (int.) > 180◦ angle categories
At least one 0◦ angle category per (int.) face

0 90 90

Lemma.

G ↑ and G → are acyclic and have no internal sources or sinks. Let (G, AQ) be a triangulated angular labeled graph. Then, G internally What if there are no (int.) 180◦ angle categories?

SLIDE 76

Quasi-triangulated Graphs

What if there are no (int.) 180◦ angle categories?

SLIDE 77

Quasi-triangulated Graphs

wN wW wS What if there are no (int.) 180◦ angle categories?

SLIDE 78

Quasi-triangulated Graphs

wN wW wS What if there are no (int.) 180◦ angle categories? wE

SLIDE 79

Quasi-triangulated Graphs

What if there are no (int.) 180◦ angle categories? wN wW wS wE G →

SLIDE 80

Quasi-triangulated Graphs

What if there are no (int.) 180◦ angle categories? wN wW wS wE

topological order on G →:

G → x-coordinates

SLIDE 81

Quasi-triangulated Graphs

What if there are no (int.) 180◦ angle categories? wN wW wS wE

topological order on G →:

G →

1 2 3 4 5 6 7 8 9 10 11 12 13 14

x-coordinates

SLIDE 82

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories? wN wW wS wE

topological order on G →:

G →

1 2 3 4 5 6 7 8 9 10 11 12 13 14

x-coordinates

SLIDE 83

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑ x-coordinates

SLIDE 84

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑

topological order on G ↑:

x-coordinates y-coordinates

SLIDE 85

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑

topological order on G ↑:

2 14 3 10 5 9 8 12 4 7 11 6 1 13

x-coordinates y-coordinates

SLIDE 86

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑

topological order on G ↑:

2 14 3 10 5 9 8 12 4 7 11 6 1 13

x-coordinates y-coordinates

SLIDE 87

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑

topological order on G ↑:

2 14 3 10 5 9 8 12 4 7 11 6 1 13

x-coordinates y-coordinates

SLIDE 88

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑

topological order on G ↑:

2 14 3 10 5 9 8 12 4 7 11 6 1 13

x-coordinates y-coordinates

SLIDE 89

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:

wN wW wS wE G ↑

topological order on G ↑:

2 14 3 10 5 9 8 12 4 7 11 6 1 13

x-coordinates y-coordinates

SLIDE 90

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

x-coordinates y-coordinates

SLIDE 91

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

x-coordinates y-coordinates

SLIDE 92

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

0◦

x-coordinates y-coordinates

SLIDE 93

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

0◦ 90◦ 90◦

x-coordinates y-coordinates

SLIDE 94

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

0◦ 90◦ 90◦

x-coordinates y-coordinates

SLIDE 95

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

0◦ 90◦ 90◦

x-coordinates y-coordinates

SLIDE 96

Quasi-triangulated Graphs

1 3 5 7 9 11 13

What if there are no (int.) 180◦ angle categories?

topological order on G →:
topological order on G ↑:

0◦ 90◦ 90◦

Lemma.

quasi-triangulated angular labeled graph (G, AQ), all internal angles have category 0◦ or 90◦ ⇒ straight-line windrose planar drawing

n n × n grid in O(n) time

x-coordinates y-coordinates

SLIDE 97

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph.

SLIDE 98

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. Task: Augment (G, AQ) to a quasi-triangulated angular labeled graph (G ∗, AQ∗) without internal angle category 180◦.

SLIDE 99

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. Task: Augment (G, AQ) to a quasi-triangulated angular labeled graph (G ∗, AQ∗) without internal angle category 180◦. v

SLIDE 100

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. Task: Augment (G, AQ) to a quasi-triangulated angular labeled graph (G ∗, AQ∗) without internal angle category 180◦. v u

SLIDE 101

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. Task: Augment (G, AQ) to a quasi-triangulated angular labeled graph (G ∗, AQ∗) without internal angle category 180◦. v u w

SLIDE 102

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. Task: Augment (G, AQ) to a quasi-triangulated angular labeled graph (G ∗, AQ∗) without internal angle category 180◦. v u w

SLIDE 103

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

SLIDE 104

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:

SLIDE 105

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:

SLIDE 106

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:

SLIDE 107

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:

SLIDE 108

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

SLIDE 109

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

SLIDE 110

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 111

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 112

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 113

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 114

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 115

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 116

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 117

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 118

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 119

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 120

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 121

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

SLIDE 122

Triangulated graphs

Let (G, AQ) be a triangulated angular labeled graph. v u w

Assume that

ր

u = ∅,

ր

w = ∅ (otherwise, set v = u/w)

if (u, w) not on outer face:
if (u, w) on outer face:

v u w

Add wN, wE, wS, and wW

Theorem.

A triangulated q-constrained graph (G, Q) is windrose planar ⇔ AQ is angular → draw with 1 bend per edge

n an O(n) × O(n) grid in O(n) time

SLIDE 123

Plane Graphs

Lemma.

plane q-constrained graph (G, Q) ⇒ find a large-angle assignment L such that AQ,L is angular (if it exists) in O(n log3 n) time

SLIDE 124

Plane Graphs

Lemma.

plane q-constrained graph (G, Q) ⇒ find a large-angle assignment L such that AQ,L is angular (if it exists) in O(n log3 n) time

Lemma.

Plane angular labeled graph (G, A) ⇒ augment in O(n) time to a triangulated labeled graph (G ′, A′)

SLIDE 125

Plane Graphs

Lemma.

plane q-constrained graph (G, Q) ⇒ find a large-angle assignment L such that AQ,L is angular (if it exists) in O(n log3 n) time

Lemma.

Plane angular labeled graph (G, A) ⇒ augment in O(n) time to a triangulated labeled graph (G ′, A′)

270◦ 90◦ β v1 v2 v3 v4 C

SLIDE 126

Plane Graphs

Lemma.

plane q-constrained graph (G, Q) ⇒ find a large-angle assignment L such that AQ,L is angular (if it exists) in O(n log3 n) time

Lemma.

Plane angular labeled graph (G, A) ⇒ augment in O(n) time to a triangulated labeled graph (G ′, A′)

180◦ 90◦ v1 v3 90◦ 0◦ C β v2 v4

SLIDE 127

Plane Graphs

Lemma.

plane q-constrained graph (G, Q) ⇒ find a large-angle assignment L such that AQ,L is angular (if it exists) in O(n log3 n) time

Lemma.

Plane angular labeled graph (G, A) ⇒ augment in O(n) time to a triangulated labeled graph (G ′, A′)

Theorem.

Plane q-constrained Graph ⇒ test windrose planarity in O(n log3 n) time → draw with 1 bend per edge on O(n) × O(n) grid

SLIDE 128

Further Results

Theorem.

Windrose planar q-constrained graph (G, Q) whose blocks are either edges or planar 3-trees ⇒ straight-line windrose planar drawing

SLIDE 129

Further Results

Theorem.

Windrose planar q-constrained graph (G, Q) whose blocks are either edges or planar 3-trees ⇒ straight-line windrose planar drawing Straight-line windrose planar drawings require exponential area.

Theorem.

SLIDE 130

Further Results

Theorem.

Windrose planar q-constrained graph (G, Q) whose blocks are either edges or planar 3-trees ⇒ straight-line windrose planar drawing Straight-line windrose planar drawings require exponential area.

Theorem.

SLIDE 131

Further Results

Theorem.

Windrose planar q-constrained graph (G, Q) whose blocks are either edges or planar 3-trees ⇒ straight-line windrose planar drawing Straight-line windrose planar drawings require exponential area.

Theorem.

SLIDE 132

Further Results

Theorem.

Windrose planar q-constrained graph (G, Q) whose blocks are either edges or planar 3-trees ⇒ straight-line windrose planar drawing Straight-line windrose planar drawings require exponential area.

Theorem.

SLIDE 133

Open problems

Draw windrose planar graphs straight-line?

SLIDE 134

Open problems

Draw windrose planar graphs straight-line?
Generalizations: each edge has a set of possible directions
r allow more than two directions

SLIDE 135

Open problems

Draw windrose planar graphs straight-line?
Generalizations:
What about bimonotone (relationship between each pair
f vertices)? Not always straight-line!

each edge has a set of possible directions

r allow more than two directions

SLIDE 136

Open problems

Draw windrose planar graphs straight-line?
Generalizations:
What about bimonotone (relationship between each pair
f vertices)? Not always straight-line!

each edge has a set of possible directions

r allow more than two directions