1. Minimum-Segment Convex Drawings of 3-Connected Cubic Plane Graphs Sudip Biswas Debajyoti Mondal Rahnuma Islam Nishat Md. Saidur Rahman Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering Bangladesh University of Engineering and Technology (BUET) Dhaka – 1000, Bangladesh COCOON 2010July 19, 2010

2. 1 2 3 4 6 7 9 8 5 1 2 3 4 6 7 9 8 5 1 2 3 4 6 7 9 8 5 Minimum-Segment Convex Drawings Convex Drawing

3. 9 segments 8 segments 6 segments Minimum-Segment Convex Drawings Minimum-Segment 7 8 9 4 56 3 2 1

4. Previous Results M. Chrobak et al. [1997] Straight-line convex grid drawings of 3-connected plane graphs (n-2) x (n-2) area G. Kant [1994] Orthogonal grid drawings of 3-connected cubic plane graphs ( n / 2 +1)x( n / 2 +1) area Dujmovic et al. [2006] Straight-line drawings of cubic graphs with few segments (n-2) segments Keszegh et al. [2008] Straight-line drawings with few slopes 5 slopes and at most 3 bends

5. Our Results Straight-line convex grid-drawings of cubic graphs ( n / 2 +1) x ( n / 2 +1) area Minimum segment 6 slopes, no bend

6. 8 1 23 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 10 9 11 12 13 14 Straight-line convex grid-drawings of cubic graphs Input: 3-Connected Plane Cubic Graph G Output: Minimum- Segment Drawing of G

7. 8 1 2 3 4 5 6 7 10 9 11 12 13 14 Intuitive Idea A Minimum-Segment Drawing Vertices on the same segment have straight corners

8. 8 1 2 3 4 5 6 7 10 9 11 12 13 14 Intuitive Idea A Minimum-Segment Drawing number of segment decreases after ensuring a straight corner at a vertex Lets try to ensure a straight corner at each vertex in the drawing

9. 1 2345 6 7 8 4 1 235 6 An Example 9 10 7 8 4 1 235 6 1 9 7 8 4235 6 11 9 10 7 8 4 1 235 6 11 12 13 14 … 6 1 234 5 7 8 9 10 11 12 13 14 How do we choose the set of vertices at each step? The number of straight corners is (n-3) and this is the maximum The number of segments is the minimum.

10. An Example Canonical Decomposition 6 1 234 5 7 8 9 10 11 12 13 14 G. Kant: Every 3-connected plane graph has a canonical decomposition which can be obtained in linear time. 1 2345 6 7 8 4 1 235 6 9 10 7 8 4 1 235 6 … Choose a partition at each step such that the resulting graph is 2-connected

11. 1 2345 6 7 8 4 1 235 6 Let’s Impose some rules 9 10 7 8 4 1 235 6 1 9 7 8 4235 6 11 6 1 234 5 7 8 9 10 11 12 13 14 Chain 3 is the left-end of the chain {7,8} 4 is the right-end of the chain {7,8} (3,7) is the left-edge of {7,8} (4,8) is the right-edge of {7,8}

12. 1 2345 6 7 8 4 1 235 6 Let’s Impose some rules 9 10 7 8 4 1 235 6 1 9 7 8 4235 6 11 6 1 234 5 7 8 9 10 11 12 13 14 If the left-end of the chain has a straight corner, use slope +1

13. 1 2345 6 7 8 4 1 235 6 Let’s Impose some rules 9 10 7 8 4 1 235 6 1 9 7 8 4235 6 11 6 1 234 5 7 8 9 10 11 12 13 14 If the right-end of the chain has a straight corner, use slope -1

14. 1 2345 6 7 8 4 1 235 6 Let’s Impose some rules 9 10 7 8 4 1 235 6 1 9 7 8 4235 6 11 6 1 234 5 7 8 9 10 11 12 13 14 If the right-end is at the rightmost position of the drawing, use the slope 

15. 1 2345 6 7 8 4 1 235 6 Let’s Impose some rules 9 10 7 8 4 1 235 6 1 9 7 8 4235 6 11 6 1 234 5 7 8 9 10 11 12 13 14 In all other cases, use the slope of the outer-edges. Slope of (7,8) = Slope of (8,11)

16. If the right-end of the chain has a straight corner, use slope -1 If the left-end of the chain has a straight corner, use slope +1 If the right-end is at the rightmost position of the drawing, use the slope  In all other cases, use the slope of the outer-edges. These four rules works for minimum-segment convex drawings! Minimum-Segment Convex Drawings

17. How can we obtain a grid drawing? Minimum-Segment Convex Drawings 9 10 7 8 4 1 235 6 11 12 13 14

18. Minimum-Segment Convex Grid Drawings 9 10 7 8 4 1 235 6 11 12 13 14 9 10 7 8 4 1 235 6 11 12 13 14 Now the rules of placing the partitions are not so simple!

19. 1 2345 6 7 8 4 1 235 6 An Example 6 1 234 5 7 8 9 10 11 12 13 14 11 1 2345 6 7 8 9 10 … 1 2345 6 7 89 11 12 13 14 9 10 1 2345 6 7 8

20. 1 2345 6 7 8 4 1 235 6 Calculation of Grid Size 6 1 234 5 7 8 9 10 11 12 13 14 11 1 2345 6 7 8 9 10 … 1 2345 6 7 89 11 12 13 14 9 10 1 2345 6 7 8 |V 1 | = 6 Width= 6 |V 2 | = 2 Width= 6+1= 7 Width= |V 1 | + (|V 2 |-1) |V 3 | = 2 Width= 7+1=8 Width= |V 1 | + (|V 2 |-1) + (|V 3 |-1) Width = |V 1 | + ∑ (|V k |-1) = |V 1 | + ∑ (|V k |-1) = n -∑ k 1 = n / 2 +1

21. 1 2345 6 7 8 4 1 235 6 Calculation of Grid Size 6 1 234 5 7 8 9 10 11 12 13 14 11 1 2345 6 7 8 9 10 … 1 2345 6 7 89 11 12 13 14 9 10 1 2345 6 7 8 n/2n/2 n/2n/2 Area of the drawing = ( n / 2 +1) x ( n / 2 +1)

22. 1 2345 6 7 8 4 1 235 6 The number of slopes is six 6 1 234 5 7 8 9 10 11 12 13 14 11 1 2345 6 7 8 9 10 … 1 2345 6 7 89 11 12 13 14 9 10 1 2345 6 7 8 0o0o 45 o  (1,14) (1, 6) (5, 6)

23. Thank You

24. 7 8 4 1 235 6 6 1 234 5 7 8 9 10 11 12 13 14 An Example 9 10 1 2345 6 7 8

25. 6 1 234 5 7 8 9 11 12 13 14 An Example 9 10 1 2345 6 7 811 1 2345 6 7 8 9 10

26. 6 1 234 5 7 8 9 11 12 13 14 An Example 11 1 2345 6 7 8 9 10 12 1 2345 6 7 8 9 1011

27. 6 1 234 5 7 8 9 10 11 12 13 14 An Example 12 1 2345 6 7 8 9 1011 1 2345 6 7 8 9 1011 12 13

28. 6 1 234 5 7 8 9 10 11 12 13 14 An Example 1 2345 6 7 8 9 1011 12 13 1 2345 6 7 89 10 11 12 13 14

29. 1 2 3 4 15 Canonical decomposition of a 3-Connected Cubic graph G. Kant: Every 3-connected plane graph has a canonical decomposition which can be obtained in linear time. 5 6 7 8 9 10 18 13 14 17 16 G1G1 G2G2 G3G3 G4G4 G5G5 G k-1 GkGk

30. Canonical Ordering of a 3-Connected Cubic graph We call a partition V k = (z 1, z 2,…, z l ) The leftmost and the rightmost neighbor of V k on G k-1 are w p and w q 2 18 1 3 4 5 6 7 8 9 10 11 12 13 14 17 15 16 7 1 3 4 5 6 8 9 10 2 wpwp wqwq z1z1 z2z2 z3z3 z4z4 G k-1 GkGk

31. Minimum-Segment Convex Drawing G 1 is drawn as a tricorner. (z 1, z 2,…, z l ) form a segment of slope 0. 2 1 3 4 A Vertex with a Straight Corner (180 o ) w1w1 w2w2 w3w3 w4w4 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

32. V k = (6, 5), w p =3, w q =2. 2 1 3 4 5 6 If x(w q ) is the maximum, then slope(w q, z l ) = ∞ If x(w p ) is not the minimum, then slope(w p, w p-1 ) = slope(w p, z 1 ) w p-1 wpwp z1z1 wqwq zlzl Minimum-Segment Convex Drawing 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

33. 2 1 3 4 5 6 9 8 7 10 V k = (10,9,8,7), w p =4, w q =6. If w q has a straight corner, then slope(w q, z 1 ) = -1 If w p has a straight corner, then slope(w p, z 1 ) = +1 w p-1 z1z1 zlzl wpwp wqwq Minimum-Segment Convex Drawing 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

34. 2 1 3 4 56 11 9 8 7 10 V k = (11), w p =1, w q =10. If x(w q ) is not the minimum, then slope(w q, w q+1 ) = slope(w q, z l ) If x(w p ) is the minimum, then slope(w p, z 1 ) = +1 z l = z l wpwp wqwq w q+1 Minimum-Segment Convex Drawing 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

35. 2 1 3 4 56 11 9 8 7 10 V k = (11), w p =1, w q =10. If x(w q ) is not the minimum, then slope(w q, w q+1 ) = slope(w q, z l ) If x(w p ) is the minimum, then slope(w p, z 1 ) = +1 z l = z l wpwp wqwq w q+1 Minimum-Segment Convex Drawing 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

36. 2 1 3 4 5 6 11 9 8 7 12 13 10 V k = (13,12), w p =7 w q =5. 2 1 3 4 5 6 11987 12 13 10 1514 V k = (15,14), w p =9, w q =8. 2 1 3 4 5 6 11987 12 13 10 1514 16 V k = (16), w p =11, w q =15. 2 1 3 4 5 6 11987 12 13 10 1514 16 17 V k = (17), w p =13, w q =12. 2 1 3 4 5 6 11 987 12 13 10 1514 16 17 18 V k = (18), w p =16, w q =17. Minimum-Segment Convex Drawing 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

37. 12 2 1 3 4 5 6 11 987 12 13 10 15 14 16 17 18 The vertices 1, 2, 14 and 18 do not have any straight corner Minimum-Segment Convex Drawing 14 The vertices 1, 2 and 18 do not have any straight corner 2 18 1 3 4 5 6 7 8 9 10 12 13 14 17 16

38. 13 A segment S has |S| edges and |S|-1 straight corners. Minimum-Segment Convex Drawing 2 1 3 4 5 6 15 14 16 17 18 11 987 12 13 10 14 11 98 7 12 10 S |S| = 6 Number of Straight Corners =|S| - 1

39. A segment S has |S| edges and |S|-1 straight corners. Minimum-Segment Convex Drawing 2 1 3 4 5 6 15 16 17 18 11 987 12 13 10 14 Γ Let, Γ has n>4 vertices and x segments. Denote the segments by S 1, S 2, …, S x x is the minimum (|S 1 |-1) + (|S 2 |-1)+ … + (|S x |-1) = (n-3) (|S 1 | + |S 2 |+ … + |S x |) - x = (n-3) Constant - x = (n-3) x = Constant - (n-3)