Strongly Monotone Drawings of Planar Graphs Stefan Felsner - - PowerPoint PPT Presentation

strongly monotone drawings of planar graphs
SMART_READER_LITE
LIVE PREVIEW

Strongly Monotone Drawings of Planar Graphs Stefan Felsner - - PowerPoint PPT Presentation

Strongly Monotone Drawings of Planar Graphs Stefan Felsner Technische Universit at Berlin FernUniversit at in Hagen Alexander Igamberdiev Philipp Kindermann FernUniversit at in Hagen Freie Universit at Berlin Boris Klemz


slide-1
SLIDE 1

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

EuroCG 2016, Lugano Stefan Felsner

Strongly Monotone Drawings of Planar Graphs

Graz University of Technology Alexander Igamberdiev Philipp Kindermann Boris Klemz Tamara Mchedlidze Manfred Scheucher Karlsruhe Institute of Technology Freie Universit¨ at Berlin FernUniversit¨ at in Hagen FernUniversit¨ at in Hagen Technische Universit¨ at Berlin

slide-2
SLIDE 2

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Strongly Monotone Drawings

We want planar straight-line graph drawings that are ... Strongly monotone: between each vertex pair (u, v) exists a path that is monotonically increasing in direction d = − → uv u v strongly monotone d

slide-3
SLIDE 3

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Strongly Monotone Drawings

We want planar straight-line graph drawings that are ... Strongly monotone: between each vertex pair (u, v) exists a path that is monotonically increasing in direction d = − → uv u v strongly monotone u v not strongly monotone w w d d

slide-4
SLIDE 4

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Strongly Monotone Drawings

We want planar straight-line graph drawings that are ... Strongly monotone: between each vertex pair (u, v) exists a path that is monotonically increasing in direction d = − → uv u v strongly monotone u v not strongly monotone Motivation: Path-finding tasks become easy! w w d d

slide-5
SLIDE 5

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Related Work & Results

Strongly monotone drawings: do not exist for every planar graph

[Kindermann et al. ’14]

exist for every tree

[Kindermann et al. ’14]

exist for every 2-connected outerplanar graph

[Kindermann et al. ’14]

area required can be exponential

[N¨

  • llenburg et al. ’14]
slide-6
SLIDE 6

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Related Work & Results

Strongly monotone drawings: do not exist for every planar graph

[Kindermann et al. ’14]

exist for every tree

[Kindermann et al. ’14]

exist for every 2-connected outerplanar graph

[Kindermann et al. ’14]

area required can be exponential

[N¨

  • llenburg et al. ’14]

exist for every planar 3-connectd graph exist for every outerplanar graph exist for every 2-tree

}

  • ur results
slide-7
SLIDE 7

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

3-connected Graphs

Theorem: Every 3-connected planar graph has a strongly monotone drawing. Proof idea: Every 3-connected planar graph G admits a primal-dual circle packing P. [Brightwell, Scheinerman 1993] Drawing induced by P is strongly monotone.

slide-8
SLIDE 8

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

primal dual circle packing P of G

slide-9
SLIDE 9

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G primal dual circle packing P of G

slide-10
SLIDE 10

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G dual graph of G primal dual circle packing P of G

slide-11
SLIDE 11

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe

slide-12
SLIDE 12

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe

slide-13
SLIDE 13

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe

slide-14
SLIDE 14

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe

face circle = inscribed circle of face

slide-15
SLIDE 15

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings

circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe vertex v has ’star-shaped’ Region Rv v can see all of Rv!

face circle = inscribed circle of face

v

slide-16
SLIDE 16

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Constructing Strongly Monotone Paths

Consider vertices v1, vk where w.l.o.g. s = v1vk horizontal. v1 vk

slide-17
SLIDE 17

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Constructing Strongly Monotone Paths

f1 f2 f3 f4 R1 R2 R3 R4 Rk General position: s does not pass through circle centers or edge points. Consider vertices v1, vk where w.l.o.g. s = v1vk horizontal. v1 vk ⇒ s intersects alternating sequence of vertex regions and face circles:

slide-18
SLIDE 18

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Constructing Strongly Monotone Paths

f1 f2 f3 f4 R1 R2 R3 R4 Rk General position: s does not pass through circle centers or edge points. Concatenation yields strongly monotone (v1, vk)-path! Consider vertices v1, vk where w.l.o.g. s = v1vk horizontal. v1 vk ⇒ s intersects alternating sequence of vertex regions and face circles: Each region Ri has some vertex vi in its center. Idea: Find path Pi from vi of Ri to vi+1 of Ri+1, 1 ≤ i < k.

slide-19
SLIDE 19

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) ci x

slide-20
SLIDE 20

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci x

slide-21
SLIDE 21

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci x

slide-22
SLIDE 22

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci x

slide-23
SLIDE 23

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er x

slide-24
SLIDE 24

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. x Edge points p1, . . . , pr are above s: p1 p2 pr

slide-25
SLIDE 25

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x Edge points p1, . . . , pr are above s: p1 p2 pr

slide-26
SLIDE 26

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x Same for edge point of er. Edge points p1, . . . , pr are above s: p1 p2 pr

slide-27
SLIDE 27

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x ⇒ all edge points p1, . . . pr above ci. p1, . . . , pr in this order around fi. Same for edge point of er. Edge points p1, . . . , pr are above s: p1 p2 pr

slide-28
SLIDE 28

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle

s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x ⇒ all edge points p1, . . . pr above ci. p1, . . . , pr in this order around fi.

}⇒ x-coordinates increase

⇒ strongly monotone Same for edge point of er. Edge points p1, . . . , pr are above s: p1 p2 pr

slide-29
SLIDE 29

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges.

slide-30
SLIDE 30

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges.

slide-31
SLIDE 31

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges.

slide-32
SLIDE 32

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges.

slide-33
SLIDE 33

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently:

slide-34
SLIDE 34

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: an expanded edge

slide-35
SLIDE 35

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: an expanded edge

slide-36
SLIDE 36

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: an expanded edge

slide-37
SLIDE 37

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

2-Trees

Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: Theorem: Every 2-tree has a strongly monotone drawing. an expanded edge

slide-38
SLIDE 38

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Definitions

A drawing with bubbles (DWB) is a straight-line drawing of G = (V, E) where every edge in some E′ ⊆ E is associated with a bubble.

slide-39
SLIDE 39

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Definitions

A drawing with bubbles (DWB) is a straight-line drawing of G = (V, E) where every edge in some E′ ⊆ E is associated with a bubble. An extension of a DWB is a drawing obtained by stacking one vertex into each bubble of some E′′ ⊆ E′.

slide-40
SLIDE 40

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Definitions

A drawing with bubbles (DWB) is a straight-line drawing of G = (V, E) where every edge in some E′ ⊆ E is associated with a bubble. An extension of a DWB is a drawing obtained by stacking one vertex into each bubble of some E′′ ⊆ E′. A DWB is strongly monotone if every extension is strongly monotone. i.e. the drawing without bubbles is strongly monotone stacking into bubbles maintains strong monotonicty

slide-41
SLIDE 41

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles

slide-42
SLIDE 42

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Base case: e

slide-43
SLIDE 43

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step:

slide-44
SLIDE 44

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge e B

slide-45
SLIDE 45

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u e e B C u

slide-46
SLIDE 46

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble e e B C u

slide-47
SLIDE 47

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble e e B C u

slide-48
SLIDE 48

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble e e B C u

slide-49
SLIDE 49

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Drawing a 2-Tree

Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble How to add bubbles for new edges? e e B C u Not done yet!

slide-50
SLIDE 50

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Adding Bubbles

A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α

slide-51
SLIDE 51

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Adding Bubbles

A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α Idea: Find α s.t. between any vertex / bubble pair exists an α-safe strongly monotone path.

slide-52
SLIDE 52

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Adding Bubbles

A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α Idea: Find α s.t. between any vertex / bubble pair exists an α-safe strongly monotone path. Place bubble for e = (u, v) close to v s.t. prolonging paths leading to v changes the direction by < α. v u w

slide-53
SLIDE 53

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Adding Bubbles

A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α Idea: Find α s.t. between any vertex / bubble pair exists an α-safe strongly monotone path. Place bubble for e = (u, v) close to v s.t. prolonging paths leading to v changes the direction by < α. v u v u x w w

slide-54
SLIDE 54

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Related Work & Results & Open Problems

Strongly monotone drawings: do not exist for every planar graph

[Kindermann et al. ’14]

exist for every tree

[Kindermann et al. ’14]

exist for every 2-connected outerplanar graph

[Kindermann et al. ’14]

area required can be exponential

[N¨

  • llenburg et al. ’14]

exist for every planar 3-connectd graph exist for every outerplanar graph exist for every 2-tree

}

  • ur results

exist for every planar 2-connected graph?

  • pen problem!
slide-55
SLIDE 55

Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Related Work & Results & Open Problems

Strongly monotone drawings: do not exist for every planar graph

[Kindermann et al. ’14]

exist for every tree

[Kindermann et al. ’14]

exist for every 2-connected outerplanar graph

[Kindermann et al. ’14]

area required can be exponential

[N¨

  • llenburg et al. ’14]

exist for every planar 3-connectd graph exist for every outerplanar graph exist for every 2-tree

}

  • ur results

exist for every planar 2-connected graph?

  • pen problem!

Thank you!