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

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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

  1. Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann Universit¨ at W¨ urzburg joint work with Markus Chimani Fabrizio Montecchiani Pavel Valtr

  2. Crossing ratio Crossing number cr ( G ) : Min. # crossings over all drawings of G

  3. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗

  4. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : Min. # crossings over all fan-planar drawings of G � � ✗ ✗

  5. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : ( k -)quasi-planar cr. number cr ( k -)qp ( G ) : Min. # crossings over Min. # crossings over all fan-planar drawings of G all ( k -)quasi-planar drawings of G � � � � � ✗ ✗ ✗

  6. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : ( k -)quasi-planar cr. number cr ( k -)qp ( G ) : Min. # crossings over Min. # crossings over all fan-planar drawings of G all ( k -)quasi-planar drawings of G � � � � � ✗ ✗ ✗

  7. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : ( k -)quasi-planar cr. number cr ( k -)qp ( G ) : Min. # crossings over Min. # crossings over all fan-planar drawings of G all ( k -)quasi-planar drawings of G � � � � � ✗ ✗ ✗ Crossing ratio:

  8. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : ( k -)quasi-planar cr. number cr ( k -)qp ( G ) : Min. # crossings over Min. # crossings over all fan-planar drawings of G all ( k -)quasi-planar drawings of G � � � � � ✗ ✗ ✗ Crossing ratio: ̺ 1-pl ( G ) : supremum of cr 1-pl ( G ) /cr ( G ) over all 1-planar graphs

  9. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : ( k -)quasi-planar cr. number cr ( k -)qp ( G ) : Min. # crossings over Min. # crossings over all fan-planar drawings of G all ( k -)quasi-planar drawings of G � � � � � ✗ ✗ ✗ Crossing ratio: ̺ 1-pl ( G ) : supremum of cr 1-pl ( G ) /cr ( G ) over all 1-planar graphs ̺ fan ( G ) cr fan ( G ) /cr ( G ) fan-planar

  10. Crossing ratio Crossing number cr ( G ) : 1-planar cr. number cr 1-pl ( G ) : Min. # crossings over Min. # crossings over all drawings of G all 1-planar drawings of G ? � ✗ ✗ ✗ Fan-planar cr. number cr fan ( G ) : ( k -)quasi-planar cr. number cr ( k -)qp ( G ) : Min. # crossings over Min. # crossings over all fan-planar drawings of G all ( k -)quasi-planar drawings of G � � � � � ✗ ✗ ✗ Crossing ratio: ̺ 1-pl ( G ) : supremum of cr 1-pl ( G ) /cr ( G ) over all 1-planar graphs ̺ fan ( G ) cr fan ( G ) /cr ( G ) fan-planar ̺ ( k -)qp ( G ) cr ( k -)qp ( G ) /cr ( G ) ( k -)quasi-planar

  11. 1-Planar Graphs at most 4 n − 8 edges, n − 2 crossings

  12. 1-Planar Graphs at most 4 n − 8 edges, n − 2 crossings ⇒ ̺ 1-pl ≤ n /2 − 1

  13. 1-Planar Graphs at most 4 n − 8 edges, n − 2 crossings ⇒ ̺ 1-pl ≤ n /2 − 1 Graph G

  14. 1-Planar Graphs at most 4 n − 8 edges, n − 2 crossings ⇒ ̺ 1-pl ≤ n /2 − 1 Graph G

  15. 1-Planar Graphs at most 4 n − 8 edges, n − 2 crossings ⇒ ̺ 1-pl ≤ n /2 − 1 Graph G

  16. 1-Planar Graphs at most 4 n − 8 edges, n − 2 crossings ⇒ ̺ 1-pl ≤ n /2 − 1 Graph G

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  31. Fan-Planar Graphs at most 5 n − 10 edges

  32. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings

  33. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 )

  34. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) K 3,3

  35. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) K 3,3

  36. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  37. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  38. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  39. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  40. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  41. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  42. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  43. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) ℓ

  44. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings ℓ

  45. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings ℓ

  46. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings ℓ

  47. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings ℓ

  48. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings ℓ

  49. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings fan-planar, ℓ + 1 ≈ n /12 crossings ℓ

  50. Fan-Planar Graphs at most 5 n − 10 edges ⇒ O ( n 2 ) crossings ⇒ ̺ fan ∈ O ( n 2 ) not fan-planar, 2 crossings fan-planar, ℓ + 1 ≈ n /12 crossings ℓ

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