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

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

Max-Degree-∆ Graphs Every max-degree-∆ graph is ⌈ ∆ / 2 ⌉ -splittable ⇒ max-degree 2 ⇒ planar Not every max-degree-∆ graph is ⌊ ∆ / 2 ⌋ -splittable 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω ( log n )

Max-Degree-∆ Graphs Every max-degree-∆ graph is ⌈ ∆ / 2 ⌉ -splittable ⇒ max-degree 2 ⇒ planar length of smallest cycle Not every max-degree-∆ graph is ⌊ ∆ / 2 ⌋ -splittable 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω ( log n )

Max-Degree-∆ Graphs Every max-degree-∆ graph is ⌈ ∆ / 2 ⌉ -splittable ⇒ max-degree 2 ⇒ planar length of smallest cycle Not every max-degree-∆ graph is ⌊ ∆ / 2 ⌋ -splittable 1. ∆ ≥ 5 ⇒ ∃ ∆-regular graphs of size n with girth Ω ( log n ) 2. Splitting a graph cannot decrease its girth.

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

Max-Degree-∆ Graphs Every max-degree-∆ graph is ⌈ ∆ / 2 ⌉ -splittable ⇒ max-degree 2 ⇒ planar length of smallest cycle Not every max-degree-∆ graph is ⌊ ∆ / 2 ⌋ -splittable 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 1 4. Any ⌊ ∆ / 2 ⌋ -split would have ( 1 + ∆ − 1 ) n edges

Max-Degree-∆ Graphs Every max-degree-∆ graph is ⌈ ∆ / 2 ⌉ -splittable ⇒ max-degree 2 ⇒ planar length of smallest cycle Not every max-degree-∆ graph is ⌊ ∆ / 2 ⌋ -splittable 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 ≥ 1 4. Any ⌊ ∆ / 2 ⌋ -split would have ( 1 + ∆ − 1 ) n edges

Max-Degree-∆ Graphs Every max-degree-∆ graph is ⌈ ∆ / 2 ⌉ -splittable ⇒ max-degree 2 ⇒ planar length of smallest cycle Not every max-degree-∆ graph is ⌊ ∆ / 2 ⌋ -splittable 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 ≥ 1 4. Any ⌊ ∆ / 2 ⌋ -split would have ( 1 + ∆ − 1 ) n edges Lower bound holds for every minor-free graph class

Genus-1-Planar Graphs projective plane

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus

Genus-1-Planar Graphs projective plane torus Projective-planar and toroidal graphs are 2-splittable

NP-hardness of 2-Splittability 4 11 1 8 5 2 3 7 3 10 6 6 9 12 4 11 1 8 7 10 9 5 12 2

NP-hardness of 2-Splittability

NP-hardness of 2-Splittability

NP-hardness of 2-Splittability

NP-hardness of 2-Splittability Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991]

NP-hardness of 2-Splittability Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] v i Variable: v i v i = true

NP-hardness of 2-Splittability Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] v i v i Variable: v i v i v i = true v i = false

NP-hardness of 2-Splittability Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] v i v i Variable: v i v i v i = true v i = false Clause:

NP-hardness of 2-Splittability Reduction from planar 3-SAT with a cycle through clause vertices [Kratochv´ ıl, Lubiw & Neˇ setˇ ril 1991] v i v i Variable: v i v i v i = true v i = false Clause:

NP-hardness of 2-Splittability v 1 v 2 v 4 v 3 v 1 v 2 v 3 v 4

NP-hardness of 2-Splittability ( v 1 ∨ v 2 ∨ v 3 ) v 1 v 2 v 4 v 3 v 1 v 2 v 3 v 4

NP-hardness of 2-Splittability ( v 1 ∨ v 2 ∨ v 3 ) ( v 1 ∨ v 2 ∨ v 4 ) ∧ v 1 v 2 v 4 v 3 v 1 v 2 v 3 v 4

NP-hardness of 2-Splittability ( v 1 ∨ v 2 ∨ v 3 ) ( v 1 ∨ v 2 ∨ v 4 ) ( v 2 ∨ v 3 ∨ v 4 ) ∧ ∧ v 1 v 2 v 4 v 3 v 1 v 2 v 3 v 4

NP-hardness of 2-Splittability ( v 1 ∨ v 2 ∨ v 3 ) ( v 1 ∨ v 2 ∨ v 4 ) ( v 2 ∨ v 3 ∨ v 4 ) ∧ ∧ v 2 v 4 v 1 v 3 v 1 v 2 v 3 v 4 v 1 ∧ v 2 ∧ v 3 ∧ v 4

NP-hardness of 2-Splittability ( v 1 ∨ v 2 ∨ v 3 ) ( v 1 ∨ v 2 ∨ v 4 ) ( v 2 ∨ v 3 ∨ v 4 ) ∧ ∧ v 2 v 4 v 1 v 3 v 1 v 2 v 3 v 4 v 1 ∧ v 2 ∧ v 3 ∧ v 4 2-splittability is NP-complete

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

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

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

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 ≤ 3 kn − 6 edges