Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann - - PowerPoint PPT Presentation

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Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann - - PowerPoint PPT Presentation

Jul 03, 2023 179 likes •933 views

Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann Universit at W urzburg joint work with Markus Chimani Fabrizio Montecchiani Pavel Valtr Crossing ratio Crossing number cr ( G ) : Min. # crossings over all drawings of G

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SLIDE 1

Crossing Numbers of Beyond-Planar Graphs

Philipp Kindermann Universit¨ at W¨ urzburg joint work with Markus Chimani Fabrizio Montecchiani Pavel Valtr

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SLIDE 2

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G

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SLIDE 3

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G?

  • ✗

✗ ✗

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SLIDE 4

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G

  • ✗

✗ ✗

  • ✗
  • ✗
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SLIDE 5

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G (k-)quasi-planar cr. number cr(k-)qp(G):

  • Min. # crossings over

all (k-)quasi-planar drawings of G

  • ✗

✗ ✗

  • ✗
  • ✗
  • ✗
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SLIDE 6

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G (k-)quasi-planar cr. number cr(k-)qp(G):

  • Min. # crossings over

all (k-)quasi-planar drawings of G

  • ✗

✗ ✗

  • ✗
  • ✗
  • ✗
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SLIDE 7

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G (k-)quasi-planar cr. number cr(k-)qp(G):

  • Min. # crossings over

all (k-)quasi-planar drawings of G Crossing ratio:

  • ✗

✗ ✗

  • ✗
  • ✗
  • ✗
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SLIDE 8

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G (k-)quasi-planar cr. number cr(k-)qp(G):

  • Min. # crossings over

all (k-)quasi-planar drawings of G Crossing ratio: ̺1-pl(G): supremum of cr1-pl(G)/cr(G) over all 1-planar graphs

  • ✗

✗ ✗

  • ✗
  • ✗
  • ✗
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SLIDE 9

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G (k-)quasi-planar cr. number cr(k-)qp(G):

  • Min. # crossings over

all (k-)quasi-planar drawings of G Crossing ratio: ̺1-pl(G): supremum of cr1-pl(G)/cr(G) over all 1-planar graphs ̺fan(G) crfan(G)/cr(G) fan-planar

  • ✗

✗ ✗

  • ✗
  • ✗
  • ✗
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SLIDE 10

Crossing ratio

Crossing number cr(G):

  • Min. # crossings over

all drawings of G 1-planar cr. number cr1-pl(G):

  • Min. # crossings over

all 1-planar drawings of G? Fan-planar cr. number crfan(G):

  • Min. # crossings over

all fan-planar drawings of G (k-)quasi-planar cr. number cr(k-)qp(G):

  • Min. # crossings over

all (k-)quasi-planar drawings of G Crossing ratio: ̺1-pl(G): supremum of cr1-pl(G)/cr(G) over all 1-planar graphs ̺fan(G) crfan(G)/cr(G) fan-planar ̺(k-)qp(G) cr(k-)qp(G)/cr(G) (k-)quasi-planar

  • ✗

✗ ✗

  • ✗
  • ✗
  • ✗
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SLIDE 11

1-Planar Graphs

at most 4n − 8 edges, n − 2 crossings

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SLIDE 12

1-Planar Graphs

at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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1-Planar Graphs

Graph G at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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1-Planar Graphs

Graph G at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 15

1-Planar Graphs

Graph G at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 16

1-Planar Graphs

Graph G at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 19

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 20

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 21

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 22

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 23

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 24

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 25

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 26

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges Special edge at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 27

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges Special edge at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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SLIDE 28

1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges Special edge not 1-planar, 2 crossings at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges Special edge not 1-planar, 2 crossings 1-planar, n − 2 crossings at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1

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1-Planar Graphs

Graph G [Korzhik & Mohar ’13] This is the only 1-planar embedding of G Dual G∗ Fixing edges Special edge not 1-planar, 2 crossings 1-planar, n − 2 crossings at most 4n − 8 edges, n − 2 crossings ⇒ ̺1-pl ≤ n/2 − 1 ̺1-pl = n/2 − 1

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Fan-Planar Graphs

at most 5n − 10 edges

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

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SLIDE 34

Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) K3,3

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SLIDE 35

Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) K3,3

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 38

Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 39

Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 40

Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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Fan-Planar Graphs

at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 46

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 47

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 48

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

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SLIDE 49

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

fan-planar, ℓ + 1 ≈ n/12 crossings

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SLIDE 50

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

fan-planar, ℓ + 1 ≈ n/12 crossings

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Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

fan-planar, ℓ + 1 ≈ n/12 crossings Θ(n2) crossings

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Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2)

ℓ

fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

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SLIDE 53

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

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SLIDE 54

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

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SLIDE 55

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

not fan-planar, 16 crossings

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SLIDE 56

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

not fan-planar, 16 crossings

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SLIDE 57

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

not fan-planar, 16 crossings fan-planar, Ω(n2) crossings

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SLIDE 58

Fan-Planar Graphs

not fan-planar, 2 crossings at most 5n − 10 edges ⇒ O(n2) crossings ⇒ ̺fan ∈ O(n2) fan-planar, ℓ + 1 ≈ n/12 crossings ̺fan ∈ Ω(n)

⇒

not fan-planar, 16 crossings fan-planar, Ω(n2) crossings

⇒ ̺fan ∈ Ω(n2)?

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k-Quasi-Planar Graphs

at most f (k)n log n edges

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SLIDE 60

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

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SLIDE 61

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

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SLIDE 62

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel

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SLIDE 63

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel

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SLIDE 64

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel

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SLIDE 65

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel

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SLIDE 66

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel +k edges

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SLIDE 67

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel +k edges not k-quasi-planar, k(k − 1)/2 crossings

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SLIDE 68

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel +k edges not k-quasi-planar, k(k − 1)/2 crossings

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SLIDE 69

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel +k edges not k-quasi-planar, k(k − 1)/2 crossings k-quasi-planar, O(n/k + k2) crossings

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SLIDE 70

k-Quasi-Planar Graphs

at most f (k)n log n edges ⇒ O( f ′(k)n2 log2 n) crossings

⇒ ̺k-pl ∈ O( f ′′(k)n2 log2 n)

2k-wheel +k edges not k-quasi-planar, k(k − 1)/2 crossings k-quasi-planar, O(n/k + k2) crossings

⇒ ̺k-pl ∈ Ω(n/k3)

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Summary

Graph class 1-planar fan-planar quasi-planar k-quasi-planar lower bound upper bound n/2 − 1 Ω(n) Ω(n/k3) Ω(n) n/2 − 1 O(n2) f (k)n2 log2 n O(n2)

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SLIDE 72

Summary

Improve bounds? Graph class 1-planar fan-planar quasi-planar k-quasi-planar lower bound upper bound n/2 − 1 Ω(n) Ω(n/k3) Ω(n) n/2 − 1 O(n2) f (k)n2 log2 n O(n2)

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SLIDE 73

Summary

Improve bounds? Graph class 1-planar fan-planar quasi-planar k-quasi-planar lower bound upper bound n/2 − 1 Ω(n) Ω(n/k3) Ω(n) n/2 − 1 O(n2) f (k)n2 log2 n O(n2) Compare between different drawing styles?

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SLIDE 74

Summary

Improve bounds? Graph class 1-planar fan-planar quasi-planar k-quasi-planar lower bound upper bound n/2 − 1 Ω(n) Ω(n/k3) Ω(n) n/2 − 1 O(n2) f (k)n2 log2 n O(n2) Restrict to straight-line? Compare between different drawing styles?