1. GD 2014 September 26, 2014 Department of Computer Science University of Manitoba Stephane Durocher Debajyoti Mondal

2. a b c d f GD 2014September 26, 2014 e g A straight-line drawing of G on a 5×5 grid A planar graph G A straight-line drawing of G with height 4 a b c a b c 2

3. a b c d f GD 2014September 26, 2014 e g A straight-line drawing of G on a 5×5 grid A planar graph G A straight-line drawing of G with height 4 a b c f g e e g f a b c 3

4. a b c d f e g A planar graph G Via visibility representation Biedl, 2014 Here we allow variable embedding. Sometimes we use edge-bends. Focus on height. Here we allow variable embedding. Sometimes we use edge-bends. Focus on height. a b c f g e

5. Nested Triangles Graph Area Height 0.44n 2 +O(1) 0.66n [ Dolev et al. 1984 ] A class of planar 3-trees Area Height 0.44n 2 + O(1) 0.66n [ Frati and Patrignani 2008, Mondal et al. 2010 ] Triangulations Area Height 2n 2 + O(1) n − 2 [de Fraysseix et al. 1990] n 2 + O(1) n − 2 [Schnyder 1990] 1.78n 2 + O(1) 0.66n [Chrobak and Nakano 1998] 0.88n 2 + O(1) 0.66n [Brandenburg 2008] 0.44n 2 + O(1) 0.66n ( polyline ) [Bonichon et al. 2003] Upper BoundsLower Bounds Fixed Embedding Triangulations Polyline drawing with height 4n/9+O(λ∆) ≈ 0.44n+O(λ∆) This is 0.44n+o(n) when ∆ is o(n) Planar 3-trees Straight-line drawing with height 4n/9+O(1) ≈ 0.44n+O(1) Triangulations Area Height 0.88n 2 + O(1) 0.66n [Brandenburg 2008] 0.44n 2 + O(1) 0.66n ( polyline ) [Bonichon et al. 2003] Planar 3-trees Area Height 0.88n 2 + O(1) 0.5n [Brandenburg 2008, Hossain et al. 2013] Nested Triangles Graph Area Height 0.22n 2 + O(1) 0.33n [Frati and Patrignani 2008] Upper Bounds Variable Embedding Improved Upper Bounds

6. A separator of size O(√n) An n-vertex planar graph GA simple cycle separator of G G o with 2n/3+O(√n) vertices G i with 2n/3+O(√n) vertices [Djidjev and Venkatesan, 1997] Every planar triangulation has a simple cycle separator of size O(√n)

7. w w' v4v4 v7v7 v6v6 v1v1 v5v5 v4v4 v5v5 v1v1 v8v8 v6v6 v2v2 v3v3 Towards w ’ a v8v8 wb GiGi GoGo Choose an Embedding Decomposition Drawing and Merge

8. Choose a face which is incident to some edge of the cycle separator as the new outer face. 8 GD 2014September 26, 2014

9. Construct G o and G i GiGi GoGo Choose a face which is incident to some edge of the cycle separator as the new outer face. GD 2014September 26, 2014

10. GiGi GoGo 10 GD 2014September 26, 2014 w

11. GiGi GoGo v4v4 v8v8 w w v7v7 v6v6 v1v1 v5v5 11 GD 2014September 26, 2014

12. GiGi GoGo v4v4 v8v8 w w v7v7 v6v6 v1v1 v5v5 w' 12 GD 2014September 26, 2014

13. GiGi GoGo v4v4 v8v8 w w v7v7 v6v6 v1v1 v5v5 DiDi w’w’ v4v4 v5v5 v1v1 v8v8 v6v6 v2v2 v3v3 Towards w ’ DoDo b a

14. v4v4 v8v8 v7v7 v6v6 v1v1 v5v5 DiDi DoDo v2v2 v3v3

15. v4v4 v8v8 v7v7 v6v6 v1v1 v5v5 DiDi DoDo v2v2 v3v3 Height of D i is (2/3)×|D i | = 4n/9+O(λ) Height of D o is (2/3)×|D o | = 4n/9+O(λ∆) Height of the final drawing is 4n/9+O(λ∆) At most 6 bends per edge - two bends to enter D o from D i - two bends on separator - two bends to return to D i from D o Height of D i is (2/3)×|D i | = 4n/9+O(λ) Height of D o is (2/3)×|D o | = 4n/9+O(λ∆) Height of the final drawing is 4n/9+O(λ∆) At most 6 bends per edge - two bends to enter D o from D i - two bends on separator - two bends to return to D i from D o Improve to 4 bends per edge using the transformation via visibility representation [Biedl 2014]

16. a b c d f e g a b c a b c f a b c d f a b c d f e A planar 3-tree Start with a triangle, then repeatedly add a vertex and triangulate the resulting graph. GD 2014September 26, 2014

17. f f d f d e f d e g The representative tree a b c d f e g a b c a b c f a b c d f a b c d f e A planar 3-tree

18. r The representative tree T of G a c A planar 3-tree G b 18 GD 2014September 26, 2014

19. a c A planar 3-tree G r v The representative tree T of G b v r Each component with at most n/2 vertices a c A planar 3-tree G b v w Choosing a suitable embedding v w G1G1 G3G3 G2G2 F F

20. a c A planar 3-tree G b a c b r G1G1 G2G2 G3G3 G4G4 v w Choosing a suitable embedding v w G1G1 G3G3 G2G2 F F

21. a c A planar 3-tree G b v w Plane 3 trees inside each of these triangles has n/2+O(1) vertices v w G1G1 G3G3 G2G2 w1w1 w2w2 w3w3 v w w1w1 w2w2 F F x y wtwt y x 4n/9 + O(1) 21 GD 2014September 26, 2014

22. a c A planar 3-tree G b v w Choosing a suitable embedding v w G1G1 G3G3 G2G2 w1w1 w2w2 w3w3 v w F F x y wtwt y x 4n/9 + O(1) v w w1w1 w2w2 The main challenge here is to show that the number of lines that are intersecting each triangle is sufficient to draw the corresponding plane 3-tree 22 GD 2014September 26, 2014

23. v4v4 v8v8 v7v7 v6v6 v1v1 v5v5 v2v2 v3v3 Triangulations Polyline drawing with height 4n/9+O(λ∆) ≈ 0.44n+O(λ∆) This is 0.44n+o(n) when ∆ is o(n) Planar 3-trees Straight-line drawing with height 4n/9+O(1) ≈ 0.44n+O(1) Triangulations Area Height 0.88n 2 + O(1) 0.66n [Brandenburg 2008] 0.44n 2 + O(1) 0.66n ( polyline ) [Bonichon et al. 2003] Planar 3-trees Area Height 0.88n 2 + O(1) 0.5n [Brandenburg 2008, Hossain et al. 2013] Upper Bounds Improved Upper Bounds a c r v b v r Thank you OPEN: Close the gap!

24. Better Trade-offs? Lower Bounds? Different Aesthetics and Styles? v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Thank You..

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26. GD 2014September 26, 2014 Nested Triangles Graph [Dolev et al. 1984] (2n/3 − 1) × (2n/3 − 1) ≈ 0.44n 2 A class of planar 3-trees [Frati and Patrignani 2008, 2 ⌈ n/3 ⌉ × ⌊ 2n/3−1 ⌋ ≈ 0.44n 2 Mondal et al. 2010] Triangulations (2n − 4) × (n − 2) ≈ 2n 2 [de Fraysseix et al. 1990] (n − 2) × (n − 2) ≈ n 2 [Schnyder 1990] 4 ⌊ 2(n-1)/3 ⌋ × ⌊ 2(n-1)/3 ⌋ ≈ 1.78n 2 [Chrobak and Nakano 1998] 4n/3 × 2n/3 ≈ 0.88n 2 [Brandenburg 2008] Upper BoundsLower Bounds Nested Triangles Graph [Frati and Patrignani 2008] (n/3 − 1) × (n/3 − 1) ≈ 0.11n 2 Triangulations Same as fixed embedding setting Nested Triangles Graph (2n/3+6) × (n/3+3) ≈ 0.22n 2 [Frati and Patrignani 2008] planar 3-trees Drawing with height n/2 [Hossain et al. 2013] Upper BoundsLower Bounds 26

27. a b c d f e g a b c a b c f a b c d f a b c d f e A planar 3-tree f f d f d e f d e g The representative tree

28. Nested Triangles Graph Area Height 0.44n 2 +O(1) 0.66n [ Dolev et al. 1984 ] A class of planar 3-trees Area Height 0.44n 2 + O(1) 0.66n [ Frati and Patrignani 2008, Mondal et al. 2010 ] Triangulations Area Height 2n 2 + O(1) n − 2 [de Fraysseix et al. 1990] n 2 + O(1) n − 2 [Schnyder 1990] 1.78n 2 + O(1) 0.66n [Chrobak and Nakano 1998] 0.88n 2 + O(1) 0.66n [Brandenburg 2008] 0.44n 2 + O(1) 0.66n (polyline) [Brandenburg 2008] Upper BoundsLower Bounds Nested Triangles Graph Area Height 0.11n 2 +O(1) n/3 [Frati and Patrignani 2008] Triangulations Area Height 0.88n 2 + O(1) 0.66n [Brandenburg 2008] Nested Triangles Graph Area Height 0.22n 2 + O(1) n/3 [Frati and Patrignani 2008] Planar 3-trees Area Height 0.88n 2 + O(1) n/2 [Brandenburg 2008, Hossain et al. 2013] Upper BoundsLower Bounds 28

29. G o 2n/3+O(√n) vertices G i 2n/3+O(√n) vertices A simple cycle separator of G Drawing Algorithm - Compute a suitable embedding of G. - Draw G o and G i with small height using Chrobak and Nakano’s Algorihtm. - Merge the drawings using edge-bends (ensure the height remains small). Drawing Algorithm - Compute a suitable embedding of G. - Draw G o and G i with small height using Chrobak and Nakano’s Algorihtm. - Merge the drawings using edge-bends (ensure the height remains small). 29 GD 2014September 26, 2014

30. a b c d f GD 2014September 26, 2014 e g A straight-line drawing of G on a 5×5 grid A planar graph G A straight-line drawing of G with height 4 f ge c d a b a b c f g e e g f a b c 30

31. a b c d f e g a b c a b c f a b c d f a b c d f e a b c d f e A planar 3-tree NOT a planar 3-tree Every vertex is of degree four.

32. v4v4 v8v8 v7v7 v6v6 v1v1 v5v5 DiDi DoDo v2v2 v3v3 Height of D i is (2/3)×|D i | = 4n/9+O(λ) Height of D o is (2/3)×|D o | = 4n/9+O(λ∆) Height of the final drawing is 4n/9+O(λ∆) At most 6 bends per edge - two bends to enter D o from D i - two bends on separator - two bends to return to D i from D o Height of D i is (2/3)×|D i | = 4n/9+O(λ) Height of D o is (2/3)×|D o | = 4n/9+O(λ∆) Height of the final drawing is 4n/9+O(λ∆) At most 6 bends per edge - two bends to enter D o from D i - two bends on separator - two bends to return to D i from D o