1. Canonical Decomposition, Realizer, Schnyder Labeling and Orderly Spanning Trees of Plane Graphs Kazuyuki Miura, Machiko Azuma and Takao Nishizeki title

2. Convex grid drawing convex drawing 1:all vertices are put on grid points 2:all edges are drawn as straight line segments 3:all faces are drawn as convex polygons

3. 1. canonical decomposition 2. realizer 3. Schnyder labeling drawing methods Convex grid drawing

4. method (cd) V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition [CK97] [CK97] convex grid drawing How to construct a convex grid drawing outer triangular ordered partition convex grid drawing but not outer triangular convex grid drawing

5. method (re) Realizer [DTV99] How to construct a convex grid drawing [Fe01] outer triangular convex grid drawing

6. method (sl) Schnyder labeling [Sc90,Fe01] 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 How to construct a convex grid drawing outer triangular convex grid drawing [Fe01]

7. relations V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Convex Grid Drawing Are there any relations between these concepts? Are there any relations between these concepts? Question 1

8. known result [BTV99] Realizer [Fe01] Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 [Fe01] Known results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 3-connectivity is a sufficient condition

9. [BTV99] Realizer [Fe01] Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 [Fe01] Known results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing What is the necessary and sufficient condition? What is the necessary and sufficient condition? Question 2

10. [BTV99] Realizer [Fe01] Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Known results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition [Fe01] Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree [CLL01]

11. application Applications of a canonical decomposition a realizer a Schnyder labeling an orderly spanning tree convex grid drawing floor-planning graph encoding 2-visibility drawing etc.

12. Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree our result Are there any relations between these concepts? Are there any relations between these concepts? Question 1 What is the necessary and sufficient condition? What is the necessary and sufficient condition? Question 2

13. Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree these five notions are equivalent with each other What is the necessary and sufficient condition? What is the necessary and sufficient condition? Question 2

14. Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree these five notions are equivalent with each other a necessary and sufficient condition for plane graphs for their existence

15. theorem Theorem G : plane graph with each degree ≥ 3 (a) - (f) are equivalent with each other. (a) G has a canonical decomposition. (b) G has a realizer. (c ) G has a Schnyder labeling. (d) G has an outer triangular convex grid drawing. (e) G has an orderly spanning tree. ( f ) necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v } such that both u and v are on the same P i (1 ≤ i ≤ 3).

16. each degree ≥ 3 necessary and sufficient condition (f) Our necessary and sufficient condition P 1 P 2 P 3 G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3).

17. each degree ≥ 3 P 1 P 2 P 3 -1. internally 3-connected (f) Our necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3). For any separation pair {u,v} of G, 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex. For any separation pair {u,v} of G, 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex. G separation pair u v G - {u,v}

18. each degree ≥ 3 P 1 P 2 P 3 (f) Our necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3). For any separation pair {u,v} of G, 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex. For any separation pair {u,v} of G, 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex. G separation pair u v G - {u,v}

19. each degree ≥ 3 P 1 P 2 P 3 (f) Our necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3). For any separation pair {u,v} of G, 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex. For any separation pair {u,v} of G, 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

20. each degree ≥ 3 P 1 P 2 P 3 (f) Our necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3). -2. separation pair on Pi v u

21. each degree ≥ 3 P 1 P 2 P 3 (f) Our necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3). v u

22. each degree ≥ 3 P 1 P 2 P 3 (f) Our necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same path P i (1 ≤ i ≤ 3). P 3 P 1 P 2 u v

23. theorem Theorem G : plane graph with each degree ≥ 3 (a) - (f) are equivalent with each other. (a) G has a canonical decomposition. (b) G has a realizer. (c ) G has a Schnyder labeling. (d) G has an outer triangular convex grid drawing. (e) G has an orderly spanning tree. ( f ) necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v } such that both u and v are on the same P i (1 ≤ i ≤ 3).

24. (outline : convexdraw=>ns) Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition [Fe01] Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree our result necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3). [BTV99] [Fe01]

25. G has an outer triangular convex grid drawing G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) G has no outer triangular convex grid drawing G is not internally 3-connected, or G has a separation pair {u,v} such that both u and v are on the same P i (1 ≤ i ≤ 3) if These faces cannot be simultaneously drawn as convex polygons. G has no convex drawing.

26. G has an outer triangular convex grid drawing G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) G has no outer triangular convex grid drawing G is not internally 3-connected, or G has a separation pair {u,v} such that both u and v are on the same P i (1 ≤ i ≤ 3) if u v This face cannot be drawn as a convex polygon. G has no outer triangular convex grid drawing.

27. (outline : ns =>cd ) Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree our result necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3). [BTV99] [Fe01]

28. P 3 P 2 G has a canonical decomposition G G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 P 1

29. V 1 P 3 P 2 G has a canonical decomposition G G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) V 8 V 7 V 6 V 5 V 4 V 3 V 2 P 1 (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b).

30. V 1 V h G G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b).

31. V 1 V h G G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). V 7 V 6 V 5 V 4 V 3 V 2

32. V 1 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G V 5 V 4 V 3 V 2 5

33. V 1 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G 1

34. V 1 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G V 2 2

35. V 1 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd1) V 1 consists of all vertices on the inner face containing, and V h ={ }. (cd2) Each G k (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G V 3 V 2 3

36. h (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G=G V h G k-1 (a) VkVk G k-1 (b) VkVk G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3)

37. V h-1 G (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) G k-1 (a) VkVk G k-1 (b) VkVk

38. G h-2 (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). V h-2 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) G k-1 (a) VkVk G k-1 (b) VkVk

39. G h-3 (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). V h-3 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) G k-1 (a) VkVk G k-1 (b) VkVk

40. (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) G k-1 (a) VkVk G k-1 (b) VkVk G V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1

41. (outline : ost  realizer) Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3). [BTV99] [Fe01]

42. Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Convex Grid Drawing [Fe01] Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3). [BTV99] [Fe01] 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree Realizer

43. ost Orderly Spanning Tree 2 3 4 5 6 1 preorder 7 8 9 10 11 12 n =13 parent smaller children larger v

44. realizer Realizer for each vertex

45. (outline : re =>ost ) Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} such that both u and v are on the same path P i (1 ≤ i ≤ 3). [BTV99] [Fe01]

46. Realizer Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree parent smaller children larger v

47. (outline : ost =>re ) Realizer Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Our results V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} such that both u and v are on the same path P i (1 ≤ i ≤ 3). [BTV99] [Fe01]

48. Realizer Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 v

49. Realizer Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 v v

50. Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 Realizer v v

51. Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 Realizer v v

52. Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 Realizer v v

53. Orderly Spanning Tree 1 2 3 4 5 6 7 8 9 10 11 12 13 Realizer v v

54. Schnyder labeling 2 1 3 1 1 1 2 2 3 3 1 2 3 3 3 3 1 23 3 3 2 2 2 1 1 1 1 2 1 2 3 1 2 3 12 3 1 2 3 2 1 2 3 Conclusion V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 Canonical Decomposition Convex Grid Drawing 1 2 3 4 5 6 7 8 9 10 11 12 13 Orderly Spanning Tree necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v} such that both u and v are on the same path P i (1 ≤ i ≤ 3). [BTV99] [Fe01]

55. theorem Theorem G : plane graph with each degree ≥ 3 (a) - (f) are equivalent with each other. (a) G has a canonical decomposition. (b) G has a realizer. (c ) G has a Schnyder labeling. (d) G has an outer triangular convex grid drawing. (e) G has an orderly spanning tree. ( f ) necessary and sufficient condition G is internally 3-connected. G has no separation pair {u,v } such that both u and v are on the same P i (1 ≤ i ≤ 3).

56. If a plane graph G satisfies the necessary and sufficient condition, then one can find the followings in linear time (a) canonical decomposition, (b) realizer, (c) Schnyder labeling, (d) orderly spanning tree, and (e) outer triangular convex grid drawing of G having the size (n -1)×(n -1). Corollary G : plane graph with each degree ≥ 3 3-connected not 3-connected

57. remaining problem The remaining problem is to characterize the class of plane graphs having convex grid drawings such that the size is (n -1)×(n -1) and the outer face is not always a triangle. triangle pentagon

58. E N D

59. ---------------------

61. Orderly Spanning Tree ost-def 1 2 3 4 5 6 7 8 9 10 11 12 13 r uu uu parent smaller children larger (each subset may be empty) N1N1 N2N2 N3N3 N4N4 (ost2) For each leaf other than u  and u , neither N 2 nor N 4 is empty. (ost1) For each edge not in the tree T, none of the endpoints is an ancestor of the other in T.

63. 1 u 2 u G i F F P choose subset G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

64. 1 u 2 u G i F P Case 1 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

65. 1 u 2 u G i P Case 1 ViVi G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

66. 1 u 2 u G i P Case 1 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

67. 1 u 2 u G i Case 2 F P G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

68. 1 u 2 u G i Case 2 F P u v P2P2 G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

69. G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) 2 u G i Case 2 F P 1 u u v P3P3 G has a canonical decomposition (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

70. 1 u 2 u G i Case 2 F P G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

71. 2 u G i Case 2 F P 1 u ViVi G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

72. 2 u G i Case 2 F P 1 u G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) (cd2) Each G k (1≤k≤h) is internally 3-connected. (cd3) All the vertices in each V k (2 ≤ k ≤ h -1) are outer vertices of G k, and either (a) or (b). G k-1 (a) VkVk G k-1 (b) VkVk

74. chord-path chord-path P 1. P connects two outer vertices u and v. 2. u,v is a separation pair of G. 3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v. u v G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3)

75. chord-path P u v u v u v u v G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) 1. P connects two outer vertices u and v. 2. u,v is a separation pair of G. 3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

76. chord-path P w1w1 w2w2 wtwt minimal G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) 1. P connects two outer vertices u and v. 2. u,v is a separation pair of G. 3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

77. chord-path P w1w1 w2w2 wtwt minimal find a minimal chord-path P, then choose such a vertex set. plan of proof : G has a canonical decomposition G is internally 3-connected G has no separation pair {u,v} s.t. both u and v are on the same P i (1 ≤ i ≤ 3) 1. P connects two outer vertices u and v. 2. u,v is a separation pair of G. 3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

79. 3-connected plane graph