[PPT] - Drawing Graphs on Few Circles and Few Spheres Alexander Ravsky PowerPoint Presentation

SLIDE 1

Drawing Graphs

n Few Circles and Few Spheres

Julius-Maximilians-Universit¨ at W¨ urzburg, Germany Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

Myroslav Kryven Alexander Ravsky Alexander Wolff

SLIDE 2

Motivation

Given a planar graph,...

SLIDE 3

Motivation

Given a planar graph,... 10 lines ...find a straight-line drawing with as few lines as possible that together cover the drawing.

[Chaplick et al., 2016]

SLIDE 4

Motivation

Given a planar graph,... 10 lines 7 lines ...find a straight-line drawing with as few lines as possible that together cover the drawing.

[Chaplick et al., 2016]

SLIDE 5

Motivation

Given a planar graph,... 10 lines 7 lines 4 circles 6 arcs ...find a straight-line drawing with as few lines as possible that together cover the drawing. ...find a circular-arc drawing with as few circles as possible that together cover the drawing.

[Chaplick et al., 2016]

SLIDE 6

Motivation

Given a planar graph,... 10 lines 7 lines 4 circles 6 arcs 4 circles 4 arcs ...find a straight-line drawing with as few lines as possible that together cover the drawing. ...find a circular-arc drawing with as few circles as possible that together cover the drawing.

[Chaplick et al., 2016]

SLIDE 7

Motivation

Given a planar graph,... 10 lines 7 lines 4 circles 6 arcs 4 circles 4 arcs ...find a straight-line drawing with as few lines as possible that together cover the drawing. ...find a circular-arc drawing with as few circles as possible that together cover the drawing.

[Chaplick et al., 2016]

Smaller visual complexity

Advantages:

SLIDE 8

Motivation

Given a planar graph,... 10 lines 7 lines 4 circles 6 arcs 4 circles 4 arcs ...find a straight-line drawing with as few lines as possible that together cover the drawing. ...find a circular-arc drawing with as few circles as possible that together cover the drawing.

[Chaplick et al., 2016]

Smaller visual complexity
Better reflects symmetry

Advantages:

SLIDE 9

Outline

Platonic solids

affine cover number
segment number
spherical cover number
arc number

Lower Bounds for σ1

d w.r.t. Other Parameters

Open Problem Motivation A Combinatorial Lover Bound Formal Definitions

SLIDE 10

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

SLIDE 11

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ1

2(cube) =

SLIDE 12

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ1

2(cube) = 7

SLIDE 13

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ1

2(cube) = 7

The spherical cover number σm

d (G) is

the minimum number of m-dimensional spheres in Rd such that G has a crossing-free circular-arc drawing that is contained in these spheres. Def.

SLIDE 14

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ1

2(cube) = 7

σ1

2(cube) =

The spherical cover number σm

d (G) is

the minimum number of m-dimensional spheres in Rd such that G has a crossing-free circular-arc drawing that is contained in these spheres. Def.

SLIDE 15

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ1

2(cube) = 7

σ1

2(cube) = 4

The spherical cover number σm

d (G) is

the minimum number of m-dimensional spheres in Rd such that G has a crossing-free circular-arc drawing that is contained in these spheres. Def.

SLIDE 16

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ2

3(K5) =

The spherical cover number σm

d (G) is

the minimum number of m-dimensional spheres in Rd such that G has a crossing-free circular-arc drawing that is contained in these spheres. Def.

SLIDE 17

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ2

3(K5) =

The spherical cover number σm

d (G) is

the minimum number of m-dimensional spheres in Rd such that G has a crossing-free circular-arc drawing that is contained in these spheres. Def.

3

SLIDE 18

Affine Covers & Spherical Covers

Let G be a graph, and let 1 ≤ m < d. The affine cover number ρm

d (G) is

the minimum number of m-dimensional hyperplanes in Rd such that G has a crossing-free straight-line drawing that is contained in these planes. Def.

[⋆ Chaplick et al., 2016]

⋆

ρ2

3(K5) =

The spherical cover number σm

d (G) is

the minimum number of m-dimensional spheres in Rd such that G has a crossing-free circular-arc drawing that is contained in these spheres. Def.

3 σ2

3(K5) =2.

SLIDE 19

Segment Number and Arc Number

Def. The segment number of G, seg(G), is the minimum number of line segments formed by the edges of G in a straight-line drawing.

[Dujmovi´ c, Eppstein, Suderman, Wood CGTA’07]

1 line, 2 segments

SLIDE 20

Segment Number and Arc Number

Def. The segment number of G, seg(G), is the minimum number of line segments formed by the edges of G in a straight-line drawing.

[Dujmovi´ c, Eppstein, Suderman, Wood CGTA’07] [Schulz JGAA’15]

1 line, 2 segments Def. The arc number of G, arc(G), is the minimum number

f arcs formed by the edges of G in a circular-arc

drawing.

SLIDE 21

Outline

Formal Definitions

affine cover number
segment number
spherical cover number
arc number

Lower Bounds for σ1

d w.r.t. Other Parameters

Open Problem Motivation A Combinatorial Lover Bound Platonic solids

SLIDE 22

Combinatorial Lower Bounds on ρ1

2 and σ1 2 Let G be a graph. v

Obs. 1

Any vertex v of G lies on ≥ ⌈deg(v)/2⌉ lines.

[Chaplick et al., 2016 ]

SLIDE 23

Combinatorial Lower Bounds on ρ1

2 and σ1 2 Let G be a graph. v

Obs. 1

Any vertex v of G lies on ≥ ⌈deg(v)/2⌉ lines.

[Chaplick et al., 2016 ]

= ⇒ ρ1

2(G)

2

≥
v∈V (G)
deg v

2

2

SLIDE 24

Combinatorial Lower Bounds on ρ1

2 and σ1 2 Let G be a graph. v

Obs. 1

Any vertex v of G lies on ≥ ⌈deg(v)/2⌉ lines. = ⇒ ρ1

2(G) ≥ 1

2   1 +

1 + 8
v∈V (G)
deg v

2

2


 

[Chaplick et al., 2016 ]

= ⇒ ρ1

2(G)

2

≥
v∈V (G)
deg v

2

2

SLIDE 25

Combinatorial Lower Bounds on ρ1

2 and σ1 2 Let G be a graph. Any vertex v of G lies on ≥ ⌈deg(v)/2⌉ circles. v = ⇒ σ1

2(G) ≥ 1

2   1 +

1 + 4
v∈V (G)
deg v

2

2


 

Obs. 2

= ⇒ 2 σ1

2(G)

2

≥
v∈V (G)
deg v

2

2

SLIDE 26

Outline

Formal Definitions

affine cover number
segment number
spherical cover number
arc number

Lower Bounds for σ1

d w.r.t. Other Parameters

Open Problem Motivation A Combinatorial Lover Bound Platonic solids

SLIDE 27

Platonic Solids: Affine Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4

ctahedron

6 12 8 cube 8 12 6 dodecahedron 20 30 12 icosahedron 12 30 20

SLIDE 28

Platonic Solids: Affine Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4

ctahedron

6 12 8 cube 8 12 6 dodecahedron 20 30 12 icosahedron 12 30 20

SLIDE 29

Platonic Solids: Affine Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4

ctahedron

6 12 8 cube 8 12 6 dodecahedron 20 30 12 icosahedron 12 30 20

Recall Obs. 1: ρ1

2(G) ≥ 1

2   1 +

1 + 8
v∈V (G)
deg v

2

2


 

SLIDE 30

Platonic Solids: Affine Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 ≥ 4

ctahedron

6 12 8 cube 8 12 6 dodecahedron 20 30 12 icosahedron 12 30 20

Recall Obs. 1: ρ1

2(tetrahedron) ≥ 1

2  1 +

1 + 8 · 4

3

2

2

  ≥ 3.37

SLIDE 31

Platonic Solids: Affine Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 ≥ 4

ctahedron

6 12 8 ≥ 4 cube 8 12 6 ≥ 5 dodecahedron 20 30 12 ≥ 7 icosahedron 12 30 20 ≥ 9

Recall Obs. 1: = ⇒ ρ1

2(G) ≥ 1

2   1 +

1 + 8
v∈V (G)
deg v

2

2


 

SLIDE 32

Platonic Solids: Affine Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6

ctahedron

6 12 8 9 cube 8 12 6 7 dodecahedron 20 30 12 9 . . . 10 icosahedron 12 30 20 13 . . . 15

[ S c h e r m , B . T h . ’ 1 7 ]

Arguments: We use the number of nested cycles and the internal degree of the outer face.

SLIDE 33

Platonic Solids: Segment Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6

ctahedron

6 12 8 9 cube 8 12 6 7 dodecahedron 20 30 12 9 . . . 10 icosahedron 12 30 20 13 . . . 15

SLIDE 34

Platonic Solids: Segment Numbers

ρ2

1(G) ≤ seg(G)

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 ≥ 6

ctahedron

6 12 8 9 ≥ 9 cube 8 12 6 7 ≥ 7 dodecahedron 20 30 12 9 . . . 10 ≥ 9 icosahedron 12 30 20 13 . . . 15 ≥ 13

Trivial bound:

SLIDE 35

Platonic Solids: Segment Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6

ctahedron

6 12 8 9 9 cube 8 12 6 7 7 dodecahedron 20 30 12 9 . . . 10 ≥ 9 icosahedron 12 30 20 13 . . . 15 ≥ 13

SLIDE 36

Platonic Solids: Segment Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6

ctahedron

6 12 8 9 9 cube 8 12 6 7 7 dodecahedron 20 30 12 9 . . . 10 ≥ 9 icosahedron 12 30 20 13 . . . 15 ≥ 13 (For fixed embedding.) Find locally consistent angle assignment with maximum number of π-angles.

ILP:

SLIDE 37

Platonic Solids: Segment Numbers

(For fixed embedding.) Find locally consistent angle assignment with maximum number of π-angles.

ILP:

⇒ Lower bounds for the minimum number

f segments in the

corresponding drawing. G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6

ctahedron

6 12 8 9 9 cube 8 12 6 7 7 dodecahedron 20 30 12 9 . . . 10 ≥ 13 icosahedron 12 30 20 13 . . . 15 ≥ 15

SLIDE 38

Platonic Solids: Segment Numbers

(For fixed embedding.) Find locally consistent angle assignment with maximum number of π-angles.

ILP:

⇒ Lower bounds for the minimum number

f segments in the

corresponding drawing.

13 segments 15 segments

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6

ctahedron

6 12 8 9 9 cube 8 12 6 7 7 dodecahedron 20 30 12 9 . . . 10 13 icosahedron 12 30 20 13 . . . 15 15

SLIDE 39

Platonic Solids: Spherical Cover Numbers

SLIDE 40

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 ≥ 3

ctahedron

6 12 8 9 9 ≥ 3 cube 8 12 6 7 7 ≥ 4 dodecahedron 20 30 12 9 . . . 10 13 ≥ 5 icosahedron 12 30 20 13 . . . 15 15 ≥ 7

Recall Obs. 2: = ⇒ σ1

2(G) ≥ 1

2   1 +

1 + 4
v∈V (G)
deg v

2

2


 

SLIDE 41

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 ≥ 3 cube 8 12 6 7 7 ≥ 4 dodecahedron 20 30 12 9 . . . 10 13 ≥ 5 icosahedron 12 30 20 13 . . . 15 15 ≥ 7

SLIDE 42

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 3 cube 8 12 6 7 7 ≥ 4 dodecahedron 20 30 12 9 . . . 10 13 ≥ 5 icosahedron 12 30 20 13 . . . 15 15 ≥ 7

SLIDE 43

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 3 cube 8 12 6 7 7 4 dodecahedron 20 30 12 9 . . . 10 13 ≥ 5 icosahedron 12 30 20 13 . . . 15 15 ≥ 7

SLIDE 44

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 3 cube 8 12 6 7 7 4 dodecahedron 20 30 12 9 . . . 10 13 5 icosahedron 12 30 20 13 . . . 15 15 ≥ 7

[Andr´ e Schulz, JGAA’15]

SLIDE 45

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 3 cube 8 12 6 7 7 4 dodecahedron 20 30 12 9 . . . 10 13 5 icosahedron 12 30 20 13 . . . 15 15 7

SLIDE 46

Platonic Solids: Spherical Cover Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 3 cube 8 12 6 7 7 4 dodecahedron 20 30 12 9 . . . 10 13 5 icosahedron 12 30 20 13 . . . 15 15 7

7 circles / 10 arcs

SLIDE 47

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3

ctahedron

6 12 8 9 9 3 cube 8 12 6 7 7 4 dodecahedron 20 30 12 9 . . . 10 13 5 icosahedron 12 30 20 13 . . . 15 15 7

SLIDE 48

Platonic Solids: Arc Numbers

σ2

1(G) ≤ arc(G)

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 ≥ 3

ctahedron

6 12 8 9 9 3 ≥ 3 cube 8 12 6 7 7 4 ≥ 4 dodecahedron 20 30 12 9 . . . 10 13 5 ≥ 5 icosahedron 12 30 20 13 . . . 15 15 7 ≥ 7

Trivial bound:

SLIDE 49

Platonic Solids: Arc Numbers

σ2

1(G) ≤ arc(G)

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 ≥ 3

ctahedron

6 12 8 9 9 3 ≥ 3 cube 8 12 6 7 7 4 ≥ 4 dodecahedron 20 30 12 9 . . . 10 13 5 ≥ 10 icosahedron 12 30 20 13 . . . 15 15 7 ≥ 7

Trivial bound: Obs: For any graph G, arc(G) ≥ #

dd-deg. vtc. of G
2

[Dujmovi´ c, Eppstein, Suderman, Wood CGTA’07]

SLIDE 50

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 3

ctahedron

6 12 8 9 9 3 3 cube 8 12 6 7 7 4 4 dodecahedron 20 30 12 9 . . . 10 13 5 10 icosahedron 12 30 20 13 . . . 15 15 7 ≥ 7

SLIDE 51

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 3

ctahedron

6 12 8 9 9 3 3 cube 8 12 6 7 7 4 4 dodecahedron 20 30 12 9 . . . 10 13 5 10 icosahedron 12 30 20 13 . . . 15 15 7 7

7 arcs :-)

SLIDE 52

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 3

ctahedron

6 12 8 9 9 3 3 cube 8 12 6 7 7 4 4 dodecahedron 20 30 12 9 . . . 10 13 5 10 icosahedron 12 30 20 13 . . . 15 15 7 7

7 arcs :-)

SLIDE 53

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 3

ctahedron

6 12 8 9 9 3 3 cube 8 12 6 7 7 4 4 dodecahedron 20 30 12 9 . . . 10 13 5 10 icosahedron 12 30 20 13 . . . 15 15 7 7

7 arcs :-) How to draw?

SLIDE 54

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 3

ctahedron

6 12 8 9 9 3 3 cube 8 12 6 7 7 4 4 dodecahedron 20 30 12 9 . . . 10 13 5 10 icosahedron 12 30 20 13 . . . 15 15 7 7

7 arcs :-) How to draw? Solve a system

f quadratic

equations.

SLIDE 55

Platonic Solids: Arc Numbers

G = (V , E) |V | |E| |F| ρ1

2(G)

seg(G) σ1

2(G)

arc(G) tetrahedron 4 6 4 6 6 3 3

ctahedron

6 12 8 9 9 3 3 cube 8 12 6 7 7 4 4 dodecahedron 20 30 12 9 . . . 10 13 5 10 icosahedron 12 30 20 13 . . . 15 15 7 7

7 arcs :-) How to draw? Solve a system

f quadratic

equations. Solution exists!

SLIDE 56

Outline

Formal Definitions

affine cover number
segment number
spherical cover number
arc number

Lower Bounds for σ1

d w.r.t. Other Parameters

Open Problem Motivation A Combinatorial Lover Bound Platonic solids

SLIDE 57

Lower Bounds for σ1

d w.r.t. Other Parameters σ1

d(G) ≥ χe(G)/3,

Edge-chromatic # Edge-chromatic # For any d ≥ 1 and any graph G, the following bounds hold:

SLIDE 58

Lower Bounds for σ1

d w.r.t. Other Parameters σ1

d(G) ≥ χe(G)/3,

Edge-chromatic # Edge-chromatic # For any d ≥ 1 and any graph G, the following bounds hold:

SLIDE 59

Lower Bounds for σ1

d w.r.t. Other Parameters σ1

d(G) ≥ χe(G)/3,

σ1

d(G) ≥ bw(G)/2,

bisection width ⌈n/2⌉ ⌊n/2⌋ Edge-chromatic # For any d ≥ 1 and any graph G, the following bounds hold: bisection width

SLIDE 60

Lower Bounds for σ1

d w.r.t. Other Parameters σ1

d(G) ≥ χe(G)/3,

σ1

d(G) ≥ 2 3 la(G),

σ1

d(G) ≥ bw(G)/2,

σ1

d(G) > n/10,

σ1

d(G) ≥ tw(G)/6.

σ1

d(G) ≥ sepW (G)/2,

bisection width for almost all G cubic balanced separator Edge-chromatic # treewidth For any d ≥ 1 and any graph G, the following bounds hold: linear arboricity

SLIDE 61

Outline

Formal Definitions

affine cover number
segment number
spherical cover number
arc number

Lower Bounds for σ1

d w.r.t. Other Parameters

Open Problem Motivation A Combinatorial Lover Bound Platonic solids

SLIDE 62

Open Problems

Is there a family of planar graphs whose circle cover number grows asymptotically more slowly than their line cover number? Problem 1: vs.

SLIDE 63

Open Problems

Is there a family of planar graphs whose circle cover number grows asymptotically more slowly than their line cover number? Problem 1: Problem 2: 9...10 13...15 Determine the line cover number for the dodecahedron and icosahedron. vs.