1. n u 1 u 2 u Canonical Decomposition

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

3. 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

4. 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 ．

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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 ．

20. 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 ．

21. 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 ．

22. 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 ．

23. 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 ．

24. 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 ．

25. 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 ．

26. 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 ．

27. 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 ．

28. 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 ．

29. 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 ．

30. 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 ．

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

32. 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

33. 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)

34. 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

35. 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)

36. 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

37. 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)

38. 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

39. 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)

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

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