. Convex Grid Drawing: Canonical Decomposition & Realizer

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

2. convex drawing Convex grid 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. drawing methods • canonical decomposition • realizer • Schnyder labeling Convex grid drawing

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

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

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

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

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

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

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

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

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

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

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 Pi (1 ≤ i ≤ 3).

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

17. -1. internally 3-connected each degree ≥ 3 P P 3 2 P 1 separation pair u G G-{u,v} v (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 Pi (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.

18. each degree ≥ 3 P P 3 2 P 1 u separation pair G G-{u,v} v (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 Pi (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.

19. each degree ≥ 3 P P 3 2 P 1 (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 Pi (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.

20. -2. separation pair on Pi each degree ≥ 3 u P P 3 2 v P 1 (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 Pi (1 ≤ i ≤ 3).

21. each degree ≥ 3 P P 3 2 v P u 1 (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 Pi (1 ≤ i ≤ 3).

22. each degree ≥ 3 P P 3 2 P P 3 1 P u 2 v P 1 (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 Pi (1 ≤ i ≤ 3).

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 Pi (1 ≤ i ≤ 3).

24. (outline : convexdraw=>ns) our result V 8 V V 7 6 V 5 1 V V 4 V 2 3 6 7 5 V 8 1 2 4 Convex Grid Drawing Canonical Decomposition 10 9 [Fe01] [BTV99] 11 13 3 12 3 Orderly Spanning Tree 2 1 1 1 2 3 2 3 3 2 1 1 1 3 2 3 3 2 1 3 [Fe01] 1 2 2 3 1 2 2 1 3 3 1 1 2 3 3 2 3 2 3 2 2 1 1 1 Realizer Schnyder labeling Our results • 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 Pi (1 ≤ i ≤ 3).

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

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

27. (outline : ns =>cd ) our result V 8 V V 7 6 V 5 1 V V 4 V 2 3 6 7 5 V 8 1 2 4 Convex Grid Drawing Canonical Decomposition 10 9 [Fe01] [BTV99] 11 13 3 12 3 Orderly Spanning Tree 2 1 1 1 2 3 2 3 3 2 1 1 1 3 2 3 3 2 1 3 [Fe01] 1 2 2 3 1 2 2 1 3 3 1 1 2 3 3 2 3 2 3 2 2 1 1 1 Realizer Schnyder labeling Our results • 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 Pi (1 ≤ i ≤ 3).

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

29. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. V P 8 2 P V 3 (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. 7 V 5 V (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). 6 V 4 V V 2 3 P 1 V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G

30. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. V h (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G

31. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. V h V (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. 7 V 5 V (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). 6 V 4 V V 2 3 V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G

32. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. V 5 (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). V 4 V V 2 3 V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G 5

33. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G 1

34. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). V 2 V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G 2

35. (cd1) V1 consists of all vertices on the inner face containing , and Vh={}. (cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected. (cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b). V V 2 3 V 1 • G is internally 3-connected • G has no separation pair {u,v} • s.t. both u and v are • on the same Pi (1 ≤ i ≤ 3) G has a canonical decomposition G 3

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

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

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

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

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

41. (outline : ostrealizer) V 8 V V 7 6 V 5 1 V V 4 V 2 3 6 7 5 V 8 1 2 4 Convex Grid Drawing Canonical Decomposition 10 9 [Fe01] [BTV99] 11 13 3 12 3 Orderly Spanning Tree 2 1 1 1 2 3 2 3 3 2 1 1 1 3 2 3 3 2 1 3 [Fe01] 1 2 2 3 1 2 2 1 3 3 1 1 2 3 3 2 3 2 3 2 2 1 1 1 Realizer Schnyder labeling Our results • 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 Pi (1 ≤ i ≤ 3).

42. V 8 V V 7 6 V 5 1 V V 4 V 2 3 6 7 5 V 8 1 2 4 Convex Grid Drawing Canonical Decomposition 10 9 [Fe01] [BTV99] 11 13 3 12 3 2 1 1 1 2 3 2 3 3 2 1 1 1 3 2 3 3 2 1 3 [Fe01] 1 2 2 3 1 2 2 1 3 3 1 1 2 3 3 2 3 2 3 2 2 1 1 1 Schnyder labeling Realizer Our results • 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 Pi (1 ≤ i ≤ 3). Orderly Spanning Tree

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

44. realizer for each vertex Realizer

45. (outline : re =>ost ) V 8 V V 7 6 V 5 1 V V 4 V 2 3 6 7 5 V 8 1 2 4 Convex Grid Drawing Canonical Decomposition 10 9 [Fe01] [BTV99] 11 13 3 12 3 Orderly Spanning Tree 2 1 1 1 2 3 2 3 3 2 1 1 1 3 2 3 3 2 1 3 [Fe01] 1 2 2 3 1 2 2 1 3 3 1 1 2 3 3 2 3 2 3 2 2 1 1 1 Realizer Schnyder labeling Our results • 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 Pi (1 ≤ i ≤ 3).

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

47. (outline : ost =>re ) V 8 V V 7 6 V 5 1 V V 4 V 2 3 6 7 5 V 8 1 2 4 Convex Grid Drawing Canonical Decomposition 10 9 [Fe01] [BTV99] 11 13 3 12 3 Orderly Spanning Tree 2 1 1 1 2 3 2 3 3 2 1 1 1 3 2 3 3 2 1 3 [Fe01] 1 2 2 3 1 2 2 1 3 3 1 1 2 3 3 2 3 2 3 2 2 1 1 1 Realizer Schnyder labeling Our results • 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 Pi (1 ≤ i ≤ 3).

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

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

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