1. Various Orders and Drawings of Plane Graphs Takao Nishizeki Tohoku University

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

3. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 16 Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

4. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

5. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s = 1 t = 16 st-numbering has two neighbors j, k

6. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s = 1 t = 16 st-numbering has two neighbors j, k

7. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s = 1 t = 16 st-numbering and induce connected subgraphs. For any i, both vertices

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

9. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

10. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 Triangulated plane graph

11. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 G k : subgraph of G induced by vertices

12. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 G k : subgraph of G induced by vertices G9G9

13. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ).

14. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G3G3

15. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G4G4

16. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G 10

17. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G 10

18. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G 10

19. Plane graph Straight line grid drawing. de Fraysseix et al. ’90 Straight Line Grid Drawing

20. Initial drawing of G 3

21. Install k + 1 GkGk

22. Shift and install k + 1 Shift method

23. n-2 Schnyder ’90 W H Upper bound

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

25. Lower Bound

26. A restricted class of plane graphs may have more compact grid drawing. Triangulated plane graph 3-connected graph

27. 4-connected ? not 4-connected disconnected

28. How much area is required for 4-connected plane graphs?

29. W < n/2 H < n/2 = = Straight line grid drawing Miura et al. ’01 Input: 4-connected plane graph G Output: a straight line grid drawing Grid Size : Area:

30. n-2 Area < 1/4 = Area=n 2 Area<n /4 2 W < n/2 H < n/2 = =.. = plane graph G 4 -connected plane graph G Schnyder ’90 Miura et al. ’01

31. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

32. |slope|>1 Main idea G Step2: Divide G into two halves G’ and G” n/2=9 G” G’ 1 2 n-1=17 n=18 3 4 7 8 5 6 14 10 15 16 13 11 12 W < n/2 -1 = W/2 1 G’ G” Step3 and 4 : Draw G’ and G” in isosceles right-angled triangles G n/2 n/2 -1 Step5: Combine the drawings of G’ and G” Triangulate all inner faces 1 2 n-1=17 n=18 3 4 7 8 5 6 14 9 10 15 16 13 11 12 Step1: find a 4-canonical ordering |slope|<1 =

33. 4-canonical ordering [KH97](4-connected graph) (1) Edges (1,2) and (n,n-1) are on the outer face (2) For each vertex k, 3 < k < n-2, at least two neighbors have lower number and at least two neighbors have higher neighbor. 1 2 n-1=17 n=18 3 47 8 5 6 14 10 15 16 13 11 12 9

34. 4-canonical ordering [KH97](4-connected graph) 1 2 n-1=17 n=18 3 47 8 5 6 14 10 15 16 13 11 12 9 and induce 2-connected subgraphs. Both vertices Generalization of an st-numbering

35. Shift and install k + 1 Shift method Only one shift

36. Shift and install k + 1 Shift method

37. Graph is not triangulated Is there any ordering?

38. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

39. n u 1 u 2 u Canonical Decomposition

40. V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u

41. V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ． Canonical Decomposition

42. V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

43. separation pair u v Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

44. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

45. v 2 separation pair u v Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

46. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

47. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

48. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

49. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

50. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

51. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

52. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

53. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

54. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

55. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

56. Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

57. V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

58. V 1 V 8 V 7 V 6 V 5 V 4 V 3 V 2 n u 1 u 2 u G 1 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

59. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 2 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

60. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 3 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

61. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 4 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

62. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 5 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

63. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 6 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

64. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 7 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

65. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 8 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

66. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G k-1 (a) G k-1 (b) VkVk VkVk Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

67. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G k-1 (a) G k-1 (b) VkVk VkVk G 2 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

68. V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G k-1 (a) G k-1 (b) VkVk VkVk G 6 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

69. n-2 Convex Grid Drawing Chrobak and Kant ’97 Input: 3-connected graph Output: convex grid drawing Area Grid Size

70. 12 3 456 7 8 9 10 11 12 p q Shift method

71. 12 3 456 7 8 9 10 11 12 p q Shift method

72. 12 3 456 7 8 9 10 11 12 p q Shift method

73. 12 3 456 7 8 9 10 11 12 p q Shift method

74. 12 3 456 7 8 9 10 11 12 p q Shift method

75. Area < 1/4 = 3 -connected graph 4 -connected graph Area n-2 W H Area Chrobak and Kant ’97 Miura et al. 2000

76. W H The algorithm of Miura et al. is best possible

77. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

78. 1: 4-canonical decomposition 12 3 45 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 13 14 15 1617 1819 20 21 11 12 9 10 78 6 3 4 5 4: Decide y-coordinates 1 2 Time complexity: O(n) 9 16 1 2 3 4 5 6 7 8 14 21 12 13 20 10 19 11 17 15 18 O(n)[NRN97] 2: Find paths 3: Decide x-coordinates Main idea

79. U3U3 G k G k -1 (a) G k G k -1 G k G (b) (c) face U k U k U k U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering) G

80. G k G k -1 (a) G k G k -1 G k G (b) (c) face U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering)

81. U3U3 4-canonical decomposition[NRN97] (a generalization of st-numbering) G k -1 G k G k G (a) G k G k -1 G k G (b) (c) face

82. G k G k -1 (a) G k G k -1 G k G (b) (c) face U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering)

83. U9U9 4-canonical decomposition[NRN97] (a generalization of st-numbering) G k -1 G k face G k G k -1 (a) G k G k -1 G k G (b) (c) face

84. G k G k -1 (a) G k G k -1 G k G (b) (c) face U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering)

85. U5U5 4-canonical decomposition[NRN97] (a generalization of st-numbering) G k -1 G k face G k G k -1 (a) G k G k -1 G k G (b) (c) face

86. U1U1 U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 n = 21 (a generalization of st-numbering) G k G k -1 (a) G k G k -1 G k G (b) (c) face U k U k U k G O(n)O(n)

87. Conclusions st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

88. Conclusions Recent Development Orderly spanning treesChiang et al., 2001 Miura et al. 2004 Canonical decomposition, realizer and orderly Spanning tree are equivalent notions.