Te Planar Split Tickness of Graphs Philipp Kindermann FernUniversit - - PowerPoint PPT Presentation

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Te Planar Split Tickness of Graphs Philipp Kindermann FernUniversit - - PowerPoint PPT Presentation

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Te Planar Split Tickness of Graphs Philipp Kindermann FernUniversit at in Hagen Joint work with David Eppstein, Stephen Kobourov, Giuseppe Liotta, Anna Lubiw, Aude Maignan, Debajyoti Mondal, Hamideh Vosoughpour, Sue Whitesides, and Stephen

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Te Planar Split Tickness of Graphs

Philipp Kindermann FernUniversit¨ at in Hagen

Joint work with David Eppstein, Stephen Kobourov, Giuseppe Liotta, Anna Lubiw, Aude Maignan, Debajyoti Mondal, Hamideh Vosoughpour, Sue Whitesides, and Stephen Wismath

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SLIDE 2

Split Tickness

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SLIDE 3

Split Tickness

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SLIDE 4

Split Tickness

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SLIDE 5

Split Tickness

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Split Tickness

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SLIDE 7

Split Tickness

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SLIDE 8

Split Tickness

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SLIDE 9

Split Tickness

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Split Tickness

2-split

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Split Tickness

2-split G has P split thickness k: there is a k-split of G with property P

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SLIDE 12

Split Tickness

2-split G has P split thickness k: there is a k-split of G with property P planar which is planar

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Split Tickness

2-split G has P split thickness k: there is a k-split of G with property P planar which is planar ⇒ G is k-splittable

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SLIDE 14

Maps of clustered social networks

k-split of cluster graph

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Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components

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Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need?

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SLIDE 17

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors

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Heawood’s empire problem [1889]

1 6 6 1 2 2 3 3 4 5 5 7 7 8 9 9 10 10 11 12 12 M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 4 11 8

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SLIDE 19

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 3-pire: 18 colors [Taylor ’81]

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SLIDE 20

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 3-pire: 18 colors [Taylor ’81] 4-pire: 24 colors [Taylor ’81]

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SLIDE 21

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 3-pire: 18 colors [Taylor ’81] 4-pire: 24 colors [Taylor ’81] 5-pire: 30 colors [Jackson & Ringel ’82]

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SLIDE 22

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 3-pire: 18 colors [Taylor ’81] 4-pire: 24 colors [Taylor ’81] 5-pire: 30 colors [Jackson & Ringel ’82]

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SLIDE 23

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 3-pire: 18 colors [Taylor ’81] 4-pire: 24 colors [Taylor ’81] 5-pire: 30 colors [Jackson & Ringel ’82] M-pire: 6M colors [Jackson & Ringel ’82]

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SLIDE 24

Heawood’s empire problem [1889]

M-pire map: n empires, each at most M components How many colors do you need? 1-pire: 4 colors 2-pire: 12 colors [Kim ’??] 3-pire: 18 colors [Taylor ’81] 4-pire: 24 colors [Taylor ’81] 5-pire: 30 colors [Jackson & Ringel ’82] M-pire: 6M colors [Jackson & Ringel ’82] Optimal k-splittability for Kn (n > 6) is k = ⌈n/6⌉

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SLIDE 25

Known Results

Kn ⌈n/6⌉ planar

[Jackson & Ringel ’82]

Input #-split Output

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SLIDE 26

Known Results

Kn ⌈n/6⌉ planar

  • uterplanar

2 interval

[Jackson & Ringel ’82] [Scheinermann & West ’83]

Input #-split Output

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Known Results

Kn ⌈n/6⌉ planar planar interval 3

  • uterplanar

2 interval

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83]

Input #-split Output

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Known Results

Kn ⌈n/6⌉ planar planar interval 3 planar 4 caterpillar forest

  • uterplanar

2 interval

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83] [Scheinermann & West ’83]

Input #-split Output

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SLIDE 29

Known Results

Kn ⌈n/6⌉ planar planar interval 3 planar 4 caterpillar forest

  • uterplanar

2 interval planar 4 star forest

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83] [Scheinermann & West ’83] [Knauer & Ueckerdt ’16]

Input #-split Output

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Known Results

Kn ⌈n/6⌉ planar planar interval 3 planar 4 caterpillar forest

  • uterplanar

2 interval planar 4 star forest planar bipartite 3 star forest

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83] [Scheinermann & West ’83] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16]

Input #-split Output

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Known Results

Kn ⌈n/6⌉ planar planar interval 3 planar 4 caterpillar forest

  • uterplanar

2 interval planar 4 star forest planar bipartite 3 star forest

  • uterplanar

3 star forest

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83] [Scheinermann & West ’83] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16]

Input #-split Output

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Known Results

Kn ⌈n/6⌉ planar planar interval 3 planar 4 caterpillar forest

  • uterplanar

2 interval planar 4 star forest planar bipartite 3 star forest

  • uterplanar

3 star forest

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83] [Scheinermann & West ’83] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16]

Input #-split Output

≤ thickness planar

anything

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Known Results

Kn ⌈n/6⌉ planar planar interval 3 planar 4 caterpillar forest

  • uterplanar

2 interval planar 4 star forest planar bipartite 3 star forest

  • uterplanar

3 star forest

[Jackson & Ringel ’82] [Scheinermann & West ’83] [Scheinermann & West ’83] [Scheinermann & West ’83] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16] [Knauer & Ueckerdt ’16]

Input #-split Output

≤ thickness planar ≤ arboricity forest

anything anything

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2-Splits of Complete Bipartite Graphs

K2,n ?

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2-Splits of Complete Bipartite Graphs

K2,n ?

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2-Splits of Complete Bipartite Graphs

K2,n ?✓

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?

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SLIDE 39

2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓

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SLIDE 41

2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 10

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 11

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 12

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 13

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 14

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 15

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ?

n = 16

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ?

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ? n ≤ 10

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ? n ≤ 10 K7,n ?

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SLIDE 53

2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ? n ≤ 10 K7,n ? n ≤ 8

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ? n ≤ 10 K7,n ? n ≤ 8 Ka,b is 2-splittable if and only if ab ≤ 4(a + b) − 4

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ? n ≤ 10 K7,n ? n ≤ 8 Ka,b is 2-splittable if and only if ab ≤ 4(a + b) − 4 Proof: ab ≤ 4(a + b) − 4 ⇒ G ⊆ K4,b, K5,16, K6,10, or K7,8

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2-Splits of Complete Bipartite Graphs

K2,n ?✓ K3,n ?✓ K4,n ?✓ K5,n ? n ≤ 16 K6,n ? n ≤ 10 K7,n ? n ≤ 8 Ka,b is 2-splittable if and only if ab ≤ 4(a + b) − 4 Proof: ab ≤ 4(a + b) − 4 ⇒ G ⊆ K4,b, K5,16, K6,10, or K7,8 ab > 4(a + b) − 4 ⇒ too many edges (Euler)

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Max-Degree-∆ Graphs

Every max-degree-∆ graph is ⌈∆/2⌉-splittable

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Max-Degree-∆ Graphs

Every max-degree-∆ graph is ⌈∆/2⌉-splittable

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Max-Degree-∆ Graphs

Every max-degree-∆ graph is ⌈∆/2⌉-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 Every max-degree-∆ graph is ⌈∆/2⌉-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar Every max-degree-∆ graph is ⌈∆/2⌉-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar Every max-degree-∆ graph is ⌈∆/2⌉-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar Every max-degree-∆ graph is ⌈∆/2⌉-splittable Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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SLIDE 64

Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)

Every max-degree-∆ graph is ⌈∆/2⌉-splittable Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)

length of smallest cycle Every max-degree-∆ graph is ⌈∆/2⌉-splittable Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)
  • 2. Splitting a graph cannot decrease its girth.

length of smallest cycle Every max-degree-∆ graph is ⌈∆/2⌉-splittable Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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SLIDE 67

Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)
  • 2. Splitting a graph cannot decrease its girth.
  • 3. High-girth planar graphs have ≤ (1 + o(1))n edges

length of smallest cycle Every max-degree-∆ graph is ⌈∆/2⌉-splittable Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)
  • 2. Splitting a graph cannot decrease its girth.
  • 3. High-girth planar graphs have ≤ (1 + o(1))n edges
  • 4. Any ⌊∆/2⌋-split would have (1 +

1 ∆−1)n edges

length of smallest cycle Every max-degree-∆ graph is ⌈∆/2⌉-splittable Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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SLIDE 69

Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)
  • 2. Splitting a graph cannot decrease its girth.
  • 3. High-girth planar graphs have ≤ (1 + o(1))n edges
  • 4. Any ⌊∆/2⌋-split would have (1 +

1 ∆−1)n edges

length of smallest cycle Every max-degree-∆ graph is ⌈∆/2⌉-splittable

≥

Not every max-degree-∆ graph is ⌊∆/2⌋-splittable

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SLIDE 70

Max-Degree-∆ Graphs

⇒ max-degree 2 ⇒ planar

  • 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω(log n)
  • 2. Splitting a graph cannot decrease its girth.
  • 3. High-girth planar graphs have ≤ (1 + o(1))n edges
  • 4. Any ⌊∆/2⌋-split would have (1 +

1 ∆−1)n edges

length of smallest cycle Every max-degree-∆ graph is ⌈∆/2⌉-splittable

≥

Not every max-degree-∆ graph is ⌊∆/2⌋-splittable Lower bound holds for every minor-free graph class

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SLIDE 71

Genus-1-Planar Graphs

projective plane

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Genus-1-Planar Graphs

projective plane torus

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SLIDE 73

Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus

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SLIDE 77

Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus

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Genus-1-Planar Graphs

projective plane torus Projective-planar and toroidal graphs are 2-splittable

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NP-hardness of 2-Splittability

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

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SLIDE 83

NP-hardness of 2-Splittability

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SLIDE 84

NP-hardness of 2-Splittability

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SLIDE 85

NP-hardness of 2-Splittability

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SLIDE 86

NP-hardness of 2-Splittability

Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991]

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SLIDE 87

NP-hardness of 2-Splittability

Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] Variable:

vi vi

vi = true

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SLIDE 88

NP-hardness of 2-Splittability

Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] Variable:

vi vi

vi = true

vi vi

vi = false

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SLIDE 89

NP-hardness of 2-Splittability

Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] Variable:

vi vi

vi = true

vi vi

vi = false Clause:

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SLIDE 90

NP-hardness of 2-Splittability

Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] Variable:

vi vi

vi = true

vi vi

vi = false Clause:

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SLIDE 91

NP-hardness of 2-Splittability

v1 v1 v2 v2 v3 v3 v4 v4

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SLIDE 92

NP-hardness of 2-Splittability

v1 v1 v2 v2 v3 v3 v4 v4

(v1 ∨ v2 ∨ v3)

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SLIDE 93

NP-hardness of 2-Splittability

v1 v1 v2 v2 v3 v3 v4 v4

(v1 ∨ v2 ∨ v3) (v1 ∨ v2 ∨ v4) ∧

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SLIDE 94

NP-hardness of 2-Splittability

v1 v1 v2 v2 v3 v3 v4 v4

(v1 ∨ v2 ∨ v3) (v1 ∨ v2 ∨ v4) (v2 ∨ v3 ∨ v4) ∧ ∧

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SLIDE 95

NP-hardness of 2-Splittability

v2 v2 v1 v1 v3 v3 v4 v4

(v1 ∨ v2 ∨ v3) (v1 ∨ v2 ∨ v4) (v2 ∨ v3 ∨ v4) ∧ ∧ v1 ∧ v2 ∧ v3 ∧ v4

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SLIDE 96

NP-hardness of 2-Splittability

v2 v2 v1 v1 v3 v3 v4 v4

(v1 ∨ v2 ∨ v3) (v1 ∨ v2 ∨ v4) (v2 ∨ v3 ∨ v4) ∧ ∧ 2-splittability is NP-complete v1 ∧ v2 ∧ v3 ∧ v4

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SLIDE 97

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph

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SLIDE 98

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph

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SLIDE 99

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph Every graph is pa(G)-splittable

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SLIDE 100

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph Every graph is pa(G)-splittable Every n-vertex k-splittable graph G has ≤ 3kn − 6 edges

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SLIDE 101

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph Every graph is pa(G)-splittable Every n-vertex k-splittable graph G has ≤ 3kn − 6 edges = 3k(n − 1) + 3k − 6

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SLIDE 102

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph Every graph is pa(G)-splittable Every n-vertex k-splittable graph G has ≤ 3kn − 6 edges = 3k(n − 1) + 3k − 6 Nash-Williams add ⇒ 3k trees

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SLIDE 103

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph Every graph is pa(G)-splittable Every n-vertex k-splittable graph G has ≤ 3kn − 6 edges = 3k(n − 1) + 3k − 6 Nash-Williams add ⇒ pa(G) ≤ 3k ⇒ 3k trees

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SLIDE 104

Approximation

Pseudoarboricity pa(G): minimum # pseudotrees whose union is the given graph Every graph is pa(G)-splittable Every n-vertex k-splittable graph G has ≤ 3kn − 6 edges = 3k(n − 1) + 3k − 6 Nash-Williams add ⇒ pa(G) ≤ 3k Pseudoarboricity approximates splittability with factor 3 ⇒ 3k trees

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SLIDE 105

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈

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SLIDE 106

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . .

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SLIDE 107

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

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SLIDE 108

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

  • 1. create DFS tree → directed edges

∃T ⊆ E ∶ ∃r ∈ V ∶

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SLIDE 109

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

  • 1. create DFS tree → directed edges

∃T ⊆ E ∶ ∃r ∈ V ∶

∀S1 , S2 ⊂ E ∶ (∀e ∈ E ∶ (e ∈ S1 ↔ ¬(e ∈ S2)) → ∃(v, w) ∈ E ∶ v ∈ S1 ∧ w ∈ S2 ∀S ⊆ V ∶ ¬(S ≡ ∅) → ∃v ∈ S ∶ ¬(w1 , w2 ∈ S ∶ ¬(w1 ≡ w2) → ¬((v, w1), (v, w2) ∈ E)) ∀(v, w) ∈ E ∶ ¬((v, w) ∈ T) → ∃S ⊂ T ∶ ∃y ∈ V ∶ ∃(y, ∗) ∈ S ∧ (¬(y ≡ r) → ∀z ∈ V ∶ ¬(z ≡ r, y) → ¬(∃(a, z) ∈ T) ∨ (∃(a, z), (b, z) ∈ T)) ∧ ∃(v, ∗) ∈ S ∧ ∃(w, ∗) ∈ S)

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SLIDE 110

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

  • 1. create DFS tree → directed edges

∃T ⊆ E ∶ ∃r ∈ V ∶

  • 2. create k2 edge sets S1,1, . . . , Sk,k to partition edges
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SLIDE 111

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

  • 1. create DFS tree → directed edges

∃T ⊆ E ∶ ∃r ∈ V ∶

  • 2. create k2 edge sets S1,1, . . . , Sk,k to partition edges
  • 3. Simulate the MSO formula on the split graph
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SLIDE 112

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

  • 1. create DFS tree → directed edges

∃T ⊆ E ∶ ∃r ∈ V ∶

  • 2. create k2 edge sets S1,1, . . . , Sk,k to partition edges
  • 3. Simulate the MSO formula on the split graph

Courcelle’s Teorem Every graph property definable in the monadic second-order logic of graphs can be decided in linear time on graphs of bounded treewidth

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SLIDE 113

Fixed-Parameter Tractability

Monadic second-order logic: ¬, ∧, ∨, →, ↔, (, ), ∀, ∃, ≡, ⊂, ∈ quantified formulae over vertex and edge sets: ∀S ⊂ E ∶ ∃T ⊂ V ∶ . . . can test for absence of minors → planar = (K5, K3,3)-free

  • 1. create DFS tree → directed edges

∃T ⊆ E ∶ ∃r ∈ V ∶

  • 2. create k2 edge sets S1,1, . . . , Sk,k to partition edges
  • 3. Simulate the MSO formula on the split graph

Courcelle’s Teorem Every graph property definable in the monadic second-order logic of graphs can be decided in linear time on graphs of bounded treewidth Can test k-splittability of graphs of treewidth ≤ w in time O(f (k, w) ⋅ n)

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SLIDE 114

Conclusion

New concept of k-splittability → draw nonplanar graphs in a planar way

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SLIDE 115

Conclusion

New concept of k-splittability → draw nonplanar graphs in a planar way Tight bounds for complete graphs complete bipartite graphs graphs of bounded maximum degree

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SLIDE 116

Conclusion

New concept of k-splittability → draw nonplanar graphs in a planar way Tight bounds for complete graphs complete bipartite graphs graphs of bounded maximum degree NP-complete but 3-approximable

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SLIDE 117

Conclusion

New concept of k-splittability → draw nonplanar graphs in a planar way Tight bounds for complete graphs complete bipartite graphs graphs of bounded maximum degree NP-complete but 3-approximable FPT for bounded treewidth

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SLIDE 118

Conclusion

New concept of k-splittability → draw nonplanar graphs in a planar way Tight bounds for complete graphs complete bipartite graphs graphs of bounded maximum degree NP-complete but 3-approximable FPT for bounded treewidth Open Problems: anything you want! k-split