On the maximum number of crossings in star-simple drawings of K n - - PowerPoint PPT Presentation

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On the maximum number of crossings in star-simple drawings of K n - - PowerPoint PPT Presentation

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On the maximum number of crossings in star-simple drawings of K n with no empty lens Stefan Felsner, Michael Hoffmann, Kristin Knorr , Irene Parada EuroCG2020 18.03.2020, EuroCG 2020 Introduction Structure of Star-simple Drawings Lower and

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SLIDE 1

On the maximum number of crossings in star-simple drawings of Kn with no empty lens

Stefan Felsner, Michael Hoffmann, Kristin Knorr, Irene Parada

EuroCG2020

18.03.2020, EuroCG 2020

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SLIDE 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges Not adjacent edges

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

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SLIDE 3

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

slide-4
SLIDE 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

slide-5
SLIDE 5

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

slide-6
SLIDE 6

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

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SLIDE 7

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges at most 1

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

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SLIDE 8

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges at most 1

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

slide-9
SLIDE 9

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges at most 1

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

slide-10
SLIDE 10

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges at most 1

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

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SLIDE 11

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges at most 1

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

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SLIDE 12

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Simple Drawings Star-simple Drawings

Adjacent edges no crossings no crossings Not adjacent edges at most 1 arbitrary many

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 2

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SLIDE 13

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Definition (Maximum Crossing Number)

max-cr(G) = max

D(G)∈ Simple Drawings

(# Crossings in D(G))

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 3

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SLIDE 14

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Definition (Maximum Crossing Number)

max-cr(G) = max

D(G)∈ Simple Drawings

(# Crossings in D(G))

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗(Kn)

Lower Bound Upper Bound

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 3

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SLIDE 15

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Definition (Maximum Crossing Number)

max-cr(G) = max

D(G)∈ Simple Drawings

(# Crossings in D(G))

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗(Kn)

Lower Bound n

4

† Upper Bound n

4

†

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 3

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SLIDE 16

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Simple Drawings vs. Star-simple Drawings

Definition (Maximum Crossing Number)

max-cr(G) = max

D(G)∈ Simple Drawings

(# Crossings in D(G))

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗(Kn)

Lower Bound n

4

† unbounded? Upper Bound n

4

† unbounded?

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 3

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SLIDE 17

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 18

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 19

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 20

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

slide-21
SLIDE 21

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 22

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 23

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 24

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

slide-25
SLIDE 25

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

slide-26
SLIDE 26

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 27

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

slide-28
SLIDE 28

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

slide-29
SLIDE 29

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Restrictions

twisting empty lens no empty lens spiraling

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 4

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SLIDE 30

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-31
SLIDE 31

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-32
SLIDE 32

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-33
SLIDE 33

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-34
SLIDE 34

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-35
SLIDE 35

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-36
SLIDE 36

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-37
SLIDE 37

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-38
SLIDE 38

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-39
SLIDE 39

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-40
SLIDE 40

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-41
SLIDE 41

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-42
SLIDE 42

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-43
SLIDE 43

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-44
SLIDE 44

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Crossing Patterns

Lemma

The four vertices incident to two crossing edges belong to the same region of the Drawing. deadlocks spirals

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 5

slide-45
SLIDE 45

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-46
SLIDE 46

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-47
SLIDE 47

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-48
SLIDE 48

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-49
SLIDE 49

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-50
SLIDE 50

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-51
SLIDE 51

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-52
SLIDE 52

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Lower Bound of maximum crossing number in Kn

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† unbounded?

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 6

slide-53
SLIDE 53

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-54
SLIDE 54

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-55
SLIDE 55

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-56
SLIDE 56

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-57
SLIDE 57

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-58
SLIDE 58

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-59
SLIDE 59

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-60
SLIDE 60

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-61
SLIDE 61

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-62
SLIDE 62

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-63
SLIDE 63

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-64
SLIDE 64

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Slight Improvement of Lower Bound

2n−4

5 4 · 2n−4

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 7

slide-65
SLIDE 65

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Theorem

Consider D∗

neL(Kn):

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 8

slide-66
SLIDE 66

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Theorem

Consider D∗

neL(Kn):

If C(k) is the maximum number of crossings of a pair of edges that

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 8

slide-67
SLIDE 67

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Theorem

Consider D∗

neL(Kn):

If C(k) is the maximum number of crossings of a pair of edges that (a) form no deadlock/no spiral

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 8

slide-68
SLIDE 68

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Theorem

Consider D∗

neL(Kn):

If C(k) is the maximum number of crossings of a pair of edges that (a) form no deadlock/no spiral (b) all lenses formed by the two edges can be hit by k points

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 8

slide-69
SLIDE 69

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Theorem

Consider D∗

neL(Kn):

If C(k) is the maximum number of crossings of a pair of edges that (a) form no deadlock/no spiral (b) all lenses formed by the two edges can be hit by k points then C(k) ≤ e · k!.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 8

slide-70
SLIDE 70

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Setting

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 9

slide-71
SLIDE 71

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Setting

  • crossing edges: e, e′
  • vertices of e, e′ on outer face
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 9

slide-72
SLIDE 72

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Setting

  • crossing edges: e, e′
  • vertices of e, e′ on outer face
  • e′ vertical straight line
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 9

slide-73
SLIDE 73

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Setting

  • crossing edges: e, e′
  • vertices of e, e′ on outer face
  • e′ vertical straight line
  • e = {u, v} meander edge
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 9

slide-74
SLIDE 74

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Setting

  • crossing edges: e, e′
  • vertices of e, e′ on outer face
  • e′ vertical straight line
  • e = {u, v} meander edge
  • pi lens node
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 9

slide-75
SLIDE 75

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Setting

  • crossing edges: e, e′
  • vertices of e, e′ on outer face
  • e′ vertical straight line
  • e = {u, v} meander edge
  • pi lens node
  • ei = {pi, u}
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 9

slide-76
SLIDE 76

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-77
SLIDE 77

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-78
SLIDE 78

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-79
SLIDE 79

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-80
SLIDE 80

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-81
SLIDE 81

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-82
SLIDE 82

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-83
SLIDE 83

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-84
SLIDE 84

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei
  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-85
SLIDE 85

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(b) Edges ei and e′ form no deadlock/no spiral

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-86
SLIDE 86

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(b) Edges ei and e′ form no deadlock/no spiral

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-87
SLIDE 87

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(b) Edges ei and e′ form no deadlock/no spiral

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 10

slide-88
SLIDE 88

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei

(b) Edges ei and e′ form no deadlock/no spiral (c) All the lenses of ei and e′ are hit by k − 1 points p1, . . . , pi−1, pi+1, . . . , pk

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 11

slide-89
SLIDE 89

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei

(b) Edges ei and e′ form no deadlock/no spiral (c) All the lenses of ei and e′ are hit by k − 1 points p1, . . . , pi−1, pi+1, . . . , pk

Result

C(k) ≤ k · C(k − 1) + 1

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 11

slide-90
SLIDE 90

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei

(b) Edges ei and e′ form no deadlock/no spiral (c) All the lenses of ei and e′ are hit by k − 1 points p1, . . . , pi−1, pi+1, . . . , pk

Result

C(k) ≤ k · C(k − 1) + 1 ≤ k! · k

s=0 1 s! ≤ e · k!

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 11

slide-91
SLIDE 91

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei

(b) Edges ei and e′ form no deadlock/no spiral (c) All the lenses of ei and e′ are hit by k − 1 points p1, . . . , pi−1, pi+1, . . . , pk

Result

C(k) ≤ k · C(k − 1) + 1 ≤ k! · k

s=0 1 s! ≤ e · k!

k ≤ (n − 4)

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 11

slide-92
SLIDE 92

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei

(b) Edges ei and e′ form no deadlock/no spiral (c) All the lenses of ei and e′ are hit by k − 1 points p1, . . . , pi−1, pi+1, . . . , pk

Result

C(k) ≤ k · C(k − 1) + 1 ≤ k! · k

s=0 1 s! ≤ e · k!

k ≤ (n − 4) ⇒ e · (n − 4)!

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 11

slide-93
SLIDE 93

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Upper Bound for maximum crossing number

Properties

(a) Between any two crossings of e and e’ from left to right, ≤ one crossing of e’ with

  • ne of the ei

(b) Edges ei and e′ form no deadlock/no spiral (c) All the lenses of ei and e′ are hit by k − 1 points p1, . . . , pi−1, pi+1, . . . , pk

Result

C(k) ≤ k · C(k − 1) + 1 ≤ k! · k

s=0 1 s! ≤ e · k!

k ≤ (n − 4) ⇒ e · (n − 4)! ⇒ max-cr∗

neL(Kn) ≤ n!

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 11

slide-94
SLIDE 94

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Summary

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

†

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 12

slide-95
SLIDE 95

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Summary

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† n!

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 12

slide-96
SLIDE 96

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Summary

Simple Drawings max-cr(Kn) Star-simple Drawings max-cr ∗

neL(Kn)

Lower Bound n

4

† 2n ‡ Upper Bound n

4

† n!

Open Problem

  • Big gap between upper bound and lower bound
  • Further graph classes

†Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol.

  • 8590. 1964.

‡Aichholzer, Oswin, et al. ”On semi-simple drawings of the complete graph.” XVII Spanish Meeting on Computational Geometry. 2017.

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 12

slide-97
SLIDE 97

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary

Thank you! Questions?

  • S. Felsner, M. Hoffmann, K. Knorr, I. Parada

Maximum crossing number in star-simple drawings of Kn EuroCG2020 13