1. CCCG 2014 August 11, 2014 Department of Computer Science University of Manitoba Stephane Durocher Debajyoti Mondal

2. a b c d e f g h i (a) (b) a b c d e f g h i (c) a b c d e f g h i CCCG 2014August 11, 2014 2

3. a b c d e f g h i (a) (b) a b c d e f g h i (c) a b c d e f g h i CCCG 2014August 11, 2014 3 a b c d e f g h i A segment is maximal path P such that the vertices on P are collinear in the drawing.

4. a b c d e f g h i (a) (b) a b c d e f g h i (c) a b c d e f g h i CCCG 2014August 11, 2014 4 A 8-segment drawingA 10-segment drawing

5. a b c d e f g h i (a) (b) a b c d e f g h i (c) a b c d e f g h i CCCG 2014August 11, 2014 5 A 8-segment drawingA 10-segment drawing Minimization is NP-complete [Durocher, Mondal, Nishat, and Whitesides, CCCG 2011]

6. CCCG 2014August 11, 2014 6 Graph ClassLower BoundsUpper BoundsReferences Trees| {v: deg(v) is odd} | / 2 Dujmović, Eppstein, Suderman and Wood, CGTA 2007 Maximal Outerplanarnn Plane 2-Trees and 3-Trees2n2n2n2n 3-Connected Cubic Planen/2+3n/2+4 Mondal, Nishat, Biswas, and Rahman, JOCO 2010 3-Connected Plane2n2n5n/2 (= 2.50n) Dujmović, Eppstein, Suderman and Wood, CGTA 2007 Triangulations2n2n7n/3 (= 2.33n)This Presentation 4-Conneted Triangulations 2n2n9n/4 (= 2.25n)This Presentation

7. CCCG 2014August 11, 2014 7 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

8. CCCG 2014August 11, 2014 8 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. a b c Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

9. CCCG 2014August 11, 2014 9 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. a b c d Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

10. CCCG 2014August 11, 2014 10 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. a b c d e Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

11. CCCG 2014August 11, 2014 11 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. a b c d e h f g Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

12. CCCG 2014August 11, 2014 12 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. a b c d e h f g i j Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

13. CCCG 2014August 11, 2014 13 a b c d e f g h i j k l Every rooted tree T has an upward drawing with leaf (T) segments. a b c d e h f g i j k l Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

14. CCCG 2014August 11, 2014 14 a b c d e f g h i j k l Whenever we create a new segment, we ensure that this new segment starts at a non-leaf vertex T and ends at a leaf of T  The drawing has leaf (T) segments. a b c d e h f g i j k l Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

15. CCCG 2014August 11, 2014 15 Whenever we create a new segment, we ensure that this new segment starts at a non-leaf vertex T and ends at a leaf of T  The drawing has leaf (T) segments. a b c d e h f g i j k l a b c d h f g i j k l e Divergence: downward extension of the segments does not create edge crossings. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

16. v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 G3G3 G6G6 G7G7 G8G8 G4G4 G5G5 A Canonical Ordering of G [De Fraysseix, Pach, and Pollack 1988]

17. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 G3G3 G6G6 G7G7 G8G8 G4G4 G5G5 A Canonical Ordering of G [De Fraysseix, Pach, and Pollack 1988]

18. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 A Canonical Ordering of G [De Fraysseix et al. 1988] v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 A Schnyder realizer of G [Schnyder 1990]

19. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 A Canonical Ordering of G [De Fraysseix et al. 1988] v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 A Schnyder realizer of G [Schnyder 1990] leaf (T l ) =3 v7v7 leaf (T r ) =3 TmTm

20. CCCG 2014August 11, 2014 20 Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8

21. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 Incremental construction in canonical order while maintaining nice drawings of the subtrees v1v1 v2v2 v3v3

22. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 Incremental construction in canonical order while maintaining nice drawings of the subtrees

23. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 Incremental construction in canonical order while maintaining nice drawings of the subtrees

24. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 Incremental construction in canonical order while maintaining nice drawings of the subtrees v1v1 v2v2 v3v3 v4v4 v5v5 v6v6

25. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 Incremental construction in canonical order while maintaining nice drawings of the subtrees v1v1 v6v6 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7

26. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 Incremental construction in canonical order while maintaining nice drawings of the subtrees v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 leaf (T l ) + leaf (T r ) + 3 segments

27. Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees Draw G with at most leaf (T l ) + leaf (T r ) + n segments v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 Incremental construction in canonical order while maintaining nice drawings of the subtrees v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 leaf (T l ) + leaf (T r ) + 3 segments + at most (n-3) segments

28. CCCG 2014August 11, 2014 28 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 Question 1. What makes it possible to maintain nice drawings of the subtrees? - We can always create a new segment satisfying the ‘divergence’ property. A triangulation G and A drawing of G with at most leaf (T l ) + leaf (T r ) + n segments v p q

29. CCCG 2014August 11, 2014 29 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 A triangulation G and A drawing of G with at most leaf (T l ) + leaf (T r ) + n segments v Question 1. What makes it possible to maintain nice drawings of the subtrees? - We can always create a new segment satisfying the ‘divergence’ property. p q p q

30. v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 A triangulation G and A drawing of G with at most leaf (T l ) + leaf (T r ) + n segments v p q p q v Question 2. Why the drawing of the edges in T m does not create any edge crossing? - The slopes of the l-edges incident to the outerface are smaller than the slope of edge (v, p). - The slopes of the r-edges incident to the outerface are larger than the slope of edge (v, q).

31. CCCG 2014August 11, 2014 31 Graph Class Lower Bounds Upper BoundsReferences Trees | {v: deg(v) is odd} | / 2 Dujmović, Eppstein, Suderman and Wood, CGTA 2007 Maximal Outerplanarnn Plane 2-Trees and 3-Trees 2n2n2n2n Samee, Alam, Adnan and Rahman, GD 2008 3-Connected Cubic Plane Graphs n/2 Mondal, Nishat, Biswas, and Rahman, JOCO 2010 3-Connected Plane Graphs 2n2n5n/2 (= 2.50n) Dujmović, Eppstein, Suderman and Wood, CGTA 2007 Triangulations2n2nleaf (T l ) + leaf (T r ) + n <= 2.33nThis Presentation 4-Conneted Triangulations 2n2nleaf (T l ) + leaf (T r ) + n <= 2.25nThis Presentation Combine the upper bounds on the number of leaves [Bonichon, Saëc and Mosbah, ICALP 2002] [Zhang and He, DCG 2005]

32. CCCG 2014August 11, 2014 32 Tight Bounds: What is the smallest constant c such that every n vertex planar graph admits a (cn)-segment drawing? Can we improve the bound in the variable embedding setting? Generalization: Does the upper bound of 7n/3 segments hold also for 3-connected planar graphs? Optimization: Is there a polynomial-time algorithm for computing minimum-segmentdrawings of triangulations, or simpler classes of graphs such as plane 3-trees or outerplanar graphs?

33. v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 a b c d h f g i j k l e Tight Bounds? Generalization? Optimization?