1. Vertex orderings

2. Vertex ordering 16 13 14 12 11 15 10 4 8 9 5 6 3 7 2 1

3. st-numbering t = 16 s = 1 has two neighbors j, k 13 14 12 15 11 10 4 87 8 9 10 11 12 13 14 15
s = 1
t = 16
has two neighbors j, k

4. st-numbering t = 16 s = 1 has two neighbors j, k 13 14 12 15 11 10 4 89 5 6 3 7 2
s = 1
has two neighbors j, k

5. st-numbering t = 16 s = 1 and For any i, both vertices13 14 12 11 15 10 4 8 9 5 6 3 7 2
s = 1
and
For any i, both vertices
induce connected subgraphs.

6. Application of st-numbersing
Planarity testing
Visibility drawing
Internet routing

10. Canonical Ordering 16 13 14 15 10 12 11 6 5 9 4 8 7 3 1 2
Triangulated plane graph

11. Canonical Ordering 16 13 14 15 10 12 11 6 5 9 4 8 7 3 1 2
Gk : subgraph of G induced by vertices

12. Canonical Ordering 16 13 14 15 10 12 11 6 G9 5 9 4 8 7 3 1 2
Gk : subgraph of G induced by vertices

13. Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2
(co1)
Gk is biconnected and internally triangulated 9 4 8 7 3 1 2
(co2) vertices 1 and 2 are on the outer face of Gk
(co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

14. Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G3
(co1)
Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G3
(co2) vertices 1 and 2 are on the outer face of Gk
(co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

15. Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G4
(co1)
Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G4
(co2) vertices 1 and 2 are on the outer face of Gk
(co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

16. Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G10
(co1)
Gk is biconnected and internally triangulated 9 4 8 7 3 1 2
G10
(co2) vertices 1 and 2 are on the outer face of Gk
(co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

17. Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G10
(co1)
Gk is biconnected and internally triangulated 9 4 8 7 3 1 2
G10
(co2) vertices 1 and 2 are on the outer face of Gk
(co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

18. Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G10
(co1)
Gk is biconnected and internally triangulated 9 4 8 7 3 1 2
G10
(co2) vertices 1 and 2 are on the outer face of Gk
(co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

19. Every triangulated plane graph has a canonical ordering.

20. Canonical Ordering
Chord: For a cycle C in a graph, an edge joining two non-consecutive vertices in C is called a chord of C.

21. Lemma 4.2.1 Every triangulated plane graph G has a canonical ordering.
Proof. Using reverse induction
Basis: Since G = Gn, clearly (co1)-(co3) hold for k=n.
Inductive hypothesis: The vertices vn, vn-1, …, vk+1, k+1≥ 4 have been appropriately chosen.
Induction step: Now we have to choose vk. If one can choose as vk a vertex w ≠ v1, v2 on the cycle Co(Gk) which is not an end of a chord of Co(Gk) then clearly (co1)-(co3) hold for k-1 since Gk-1 = Gk – vk.

22. Now we have to proof that there is such a vertex.
w = vk
Gk-1 v2 v1
Now we have to proof that there is such a vertex.
We have to consider two cases – (i) Co(Gk) has no chord (ii) Co(Gk) has chord.
Let Co(Gk) = w1, w2, …., wt, where w1= v1 and wt = v2.

23. In Case (i), any of the vertices w2, w3, …, wt-1 is such a vertex w.
wq-1 wq
wp+1 wp w2
w1=v1
wt=v2
In case (ii), find a minimal chord (wp, wq), p+2≤q, and any of the vertices wp+1, wp+2,…,wq-1 is such a vertex w.

24. Algorithm: Canonical-Ordering
In the animation,
Red number means ordering;
Outer vertex are colored red;
Blue number indicates number of chords of the outer cycles end with the associated vertex

25. 16 13 14
C0(G14) 15 10 12 11
Co(G15) 9 6
C0(G13) 8 5 7 4 3 2 1 v2 v1
Choose a vertex x such that chords(x) = 0 and x ≠ v1, v2

26. Straight Line Drawing
Straight line drawing
Plane graph
Each vertex is drawn as a point.
Each edge is drawn as a single straight line segment.

27. Every plane graph has a straight line drawing.
Wagner ’ Fary ’48
Straight Line Drawing
Polynomial-time algorithm
Straight line drawing
Plane graph
Each vertex is drawn as a point.
Each edge is drawn as a single straight line segment.

28. Straight Line Grid Drawing
Plane graph
In a straight line grid drawing each vertex is drawn on a grid point.

29. Every plane graph has a straight line drawing.
Wagner ’ Fary ’48
Straight Line Grid Drawing
Grid-size is not polynomial of the number of vertices n
Straight line grid drawing.
Plane graph

30. Straight Line Grid Drawing
Plane graph
de Fraysseix ‘90

31. Schnyder ‘90 H
n-2
n-2 W

32. What is the minimum size of a grid required for a straight line drawing?

33. de Fraysseix ‘90
Shift Method
Schnyder ‘90
Realizer Method

34. Shift Method Canonically ordered input graph G 16 13 14 15 10 12 11 98 5 7 4 3 2 1
Canonically ordered input graph G

39. Straight-line grid drawing of G using shift method16 13 14 15 1 3 10 12 11 2 9 6 8 5 7 4 3 1 2 1
(0, 0) 2
(2, 0) 3
(1, 1) 3 1 2
Straight-line grid drawing of G using shift method

40. Straight-line grid drawing of G using shift method16 13 14 15 1 3 10 12 11 2 9 6 8 5 7 4 3 1 2 1
(0, 0) 2
(2, 0) 3
(1, 1) 3 1 2
Straight-line grid drawing of G using shift method

41. Straight-line grid drawing of G using shift method16 13 14 15 1 3 10 12 11 2 9 6 8 5 7 4 3 1 2 1
(0, 0) 2
(2, 0) 3
(1, 1) 3 1 2
Straight-line grid drawing of G using shift method

42. Straight-line grid drawing of G using shift method16 13 14 15 1 10 3 12 11 4 9 6 2 8 5 7 4 3 1 2 1
(0, 0) 2
(4, 0) 3
(1, 1) 4
(2, 2) 4 3 1 2
Straight-line grid drawing of G using shift method

43. Straight-line grid drawing of G using shift method16 13 14 15 1 10 3 12 11 4 9 6 2 8 5 7 4 3 1 2 1
(0, 0) 2
(4, 0) 3
(1, 1) 4
(2, 2) 4 3 1 2
Straight-line grid drawing of G using shift method

44. Straight-line grid drawing of G using shift method16 13 14 15 1 10 3 12 11 4 9 6 2 8 5 7 4 3 1 2 1
(0, 0) 2
(4, 0) 3
(1, 1) 4
(2, 2) 4 3 1 2
Straight-line grid drawing of G using shift method

45. Straight-line grid drawing of G using shift method16 13 14 15 1 10 3 12 11 4 9 6 5 8 5 2 7 4 3 1 2 1
(0, 0) 2
(6, 0) 3
(1, 1) 4
(2, 2) 5 5
(3, 3) 4 3 2 1
Straight-line grid drawing of G using shift method

46. Straight-line grid drawing of G using shift method16 13 14 15 1 10 3 12 11 4 9 6 5 8 5 2 7 4 3 1 2 1
(0, 0) 2
(6, 0) 3
(1, 1) 4
(2, 2) 5 5
(3, 3) 4 3 2 1
Straight-line grid drawing of G using shift method

47. Straight-line grid drawing of G using shift method16 13 14 15 1 10 3 12 11 4 9 6 5 8 5 2 7 4 3 1 2 1
(0, 0) 2
(6, 0) 3
(1, 1) 4
(2, 2) 5 5
(3, 3) 4 3 2 1
Straight-line grid drawing of G using shift method

48. Straight-line grid drawing of G using shift method16 13 14 1 15 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(8, 0) 3
(1, 1) 4
(2, 2) 6 5 5
(3, 3) 4 6
(4, 4) 3 2 1
Straight-line grid drawing of G using shift method

49. Straight-line grid drawing of G using shift method16 13 14 1 15 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(8, 0) 3
(1, 1) 4
(2, 2) 6 5 5
(3, 3) 4 6
(4, 4) 3 2 1
Straight-line grid drawing of G using shift method

50. Straight-line grid drawing of G using shift method16 13 14 1 15 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(8, 0) 3
(1, 1) 4
(2, 2) 6 5 5
(3, 3) 4 6
(4, 4) 3 2 1
Straight-line grid drawing of G using shift method

51. Straight-line grid drawing of G using shift method16 13 1 14 15 7 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(10, 0) 3
(3, 1) 4
(4, 2) 6 5 5
(5, 3) 4 7 6
(6, 4) 3 7
(2, 2) 2 1
Straight-line grid drawing of G using shift method

52. Straight-line grid drawing of G using shift method16 13 1 14 15 7 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(10, 0) 3
(3, 1) 4
(4, 2) 6 5 5
(5, 3) 4 7 6
(6, 4) 3 7
(2, 2) 2 1
Straight-line grid drawing of G using shift method

53. Straight-line grid drawing of G using shift method16 13 1 14 15 7 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(10, 0) 3
(3, 1) 4
(4, 2) 6 5 5
(5, 3) 4 7 6
(6, 4) 3 7
(2, 2) 2 1
Straight-line grid drawing of G using shift method

54. Straight-line grid drawing of G using shift method16 1 13 8 14 15 7 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(12, 0) 3
(5, 1) 4
(6, 2) 6 8 5 5
(7, 3) 7 4 6
(8, 4) 3 7
(4, 2) 2 1 8
(3, 3)
Straight-line grid drawing of G using shift method

55. Straight-line grid drawing of G using shift method16 1 13 8 14 15 7 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(12, 0) 3
(5, 1) 4
(6, 2) 6 8 5 5
(7, 3) 7 4 6
(8, 4) 3 7
(4, 2) 2 1 8
(3, 3)
Straight-line grid drawing of G using shift method

56. Straight-line grid drawing of G using shift method16 1 13 8 14 15 7 10 3 12 11 4 9 6 5 8 5 6 7 4 3 2 1 2 1
(0, 0) 2
(12, 0) 3
(5, 1) 4
(6, 2) 6 8 5 5
(7, 3) 7 4 6
(8, 4) 3 7
(4, 2) 2 1 8
(3, 3)
Straight-line grid drawing of G using shift method

57. Straight-line grid drawing of G using shift method16 1 9 13 14 8 15 10 7 12 11 3 9 4 6 8 5 5 7 4 6 3 2 1 2 9 8 6 5 7 4 3 2 1
Straight-line grid drawing of G using shift method

58. Straight-line grid drawing of G using shift method16 1 9 13 14 8 15 10 7 12 11 3 9 4 6 8 5 5 7 4 6 3 2 1 2 9 8 6 5 7 4 3 2 1
Straight-line grid drawing of G using shift method

59. Straight-line grid drawing of G using shift method16 1 9 13 14 8 15 10 7 12 11 3 9 4 6 8 5 5 7 4 6 3 2 1 2 9 8 6 5 7 4 3 2 1
Straight-line grid drawing of G using shift method

60. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 5 6 9 7 6 8 3 2 5 7 4 3 1 2 10 9 6 8 5 7 4 3 1 2
Straight-line grid drawing of G using shift method

61. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 5 6 9 7 6 8 3 2 5 7 4 3 1 2 10 9 6 8 5 7 4 3 1 2
Straight-line grid drawing of G using shift method

62. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 5 6 9 7 6 8 3 2 5 7 4 3 1 2 10 9 6 8 5 7 4 3 1 2
Straight-line grid drawing of G using shift method

63. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 11 5 9 7 6 6 8 3 2 5 7 4 3 1 2 10 11 9 6 8 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

64. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 11 5 6 9 7 6 8 3 2 5 7 4 3 1 2 10 11 9 6 8 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

65. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 11 5 6 9 7 6 8 3 2 5 7 4 3 1 2 10 11 9 6 8 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

66. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 12 9 7 11 6 8 3 5 6 5 7 4 2 3 1 2 10 12 11 9 6 8 7 4 5 3 2 1
Straight-line grid drawing of G using shift method

67. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 12 9 7 11 6 8 3 5 6 5 7 4 2 3 1 2 10 12 11 9 6 8 7 4 5 3 2 1
Straight-line grid drawing of G using shift method

68. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 4 12 11 12 9 7 11 6 8 3 5 6 5 7 4 2 3 1 2 10 12 11 9 6 8 7 4 5 3 2 1
Straight-line grid drawing of G using shift method

69. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 11 8 13 12 4 12 9 6 7 11 8 5 5 3 6 7 4 3 2 1 2 13 10 12 11 9 6 8 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

70. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 13 12 11 4 12 9 6 7 11 8 5 3 5 6 7 4 3 2 1 2 13 10 12 11 9 6 8 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

71. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 8 13 12 11 4 12 9 6 7 11 8 5 3 5 6 7 4 3 2 1 2 13 10 12 11 9 6 8 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

72. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 13 12 11 8 4 14 12 9 6 6 7 8 11 5 3 5 2 7 4 3 1 2 14 13 10 12 9 11 8 6 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

73. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 11 8 13 12 4 14 12 9 6 6 7 8 11 5 3 5 2 7 4 3 1 2 14 13 10 12 9 11 8 6 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

74. Straight-line grid drawing of G using shift method16 1 9 13 14 10 15 10 11 8 13 12 4 14 12 9 6 6 7 8 11 5 3 5 2 7 4 3 1 2 14 13 10 12 9 11 8 6 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

75. Straight-line grid drawing of G using shift method16 1 15 13 14 13 15 9 4 10 14 12 11 12 10 6 8 11 9 5 2 6 8 5 7 4 7 3 3 1 2 15 14 13 12 10 11 8 6 9 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

76. Straight-line grid drawing of G using shift method16 1 15 13 14 13 15 9 4 10 14 12 11 12 10 6 8 11 9 5 2 6 8 5 7 4 7 3 3 1 2 15 14 13 12 10 11 8 6 9 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

77. Straight-line grid drawing of G using shift method16 1 15 13 14 13 15 9 4 10 14 12 11 12 10 6 8 11 9 5 2 6 8 5 7 4 7 3 3 1 2 15 14 13 12 10 11 8 6 9 7 4 3 5 2 1
Straight-line grid drawing of G using shift method

78. Straight-line grid drawing of G using shift method16 1 15 13 14 13 15 9 4 10 14 12 11 12 10 6 8 11 9 5 2 6 8 5 7 4 7 3 3 1 2 16 15 14 13 12 10 11 8 6 9 7 4 3 5 1 2
Straight-line grid drawing of G using shift method

81. Some other vertex orsering
Antibandwidth labeling
Graceful labeling

82. Thank You