Clustered Planarity = Flat Clustered Planarity

1. Clustered Planarity = Flat Clustered Planarity Roma Tre University Pier Francesco Cortese & Maurizio Patrignani 26th International Symposium on Graph Drawing and Network Visualization, September 26-28, 2018, Barcelona, Spain

2. Clustered graph

3. Inclusion tree

4. Inclusion tree

5. C-planar drawings of c-graphs • Edges do not intersect • Each cluster  is a simple closed region containing exactly the vertices of  • The boundaries of the regions representing clusters do not intersect • Each edge intersects the boundary of a region at most once inter-cluster edge

6. Testing c-planarity • Polynomial, improved to linear, if clusters induce connected subgraphs • [Lengauer 89] [Feng, Cohen, Eades 95] [Dahlhaus 98] [Cornelsen & Wagner 06] [Cortese, Di Battista, Frati, Patrignani, Pizzonia, 08] • Other polynomial cases also based on connectivity • two-component clusters [Jelinek, Jelinkova, Kratochvil, Lidicky, 08] • clusters with few outgoing edges [Jelinek, Suchy, Tesar, Vyskocil, 09] • [Gutwenger, Juenger, Leipert, Mutzel, Percan, Weiskircher, 02] • “extrovert” clusters [Goodrich, Lueker, Sun, 06] • Polynomial for cluster size at most three • [Jelínková, Kára, Kratochvíl, Pergel, Suchý, Vyskocil, 09]

7. Flat clustered graphs All leaves of the inclusion tree have depth two

8. Testing c-planarity of flat c-graphs • Polynomial when the graph of the clusters is embeeded • underlying graph and graph of the clusters is a cycle [Cortese, Di Battista, Patrignani, Pizzonia, 05] • underlying graph is a cycle [Cortese, Di Battista, Patrignani, Pizzonia, 09] • graph of the clusters is a cycle [Fulek, 17] • underlying graph and graph of the clusters are embedded [Fulek, 17] • graph of the clusters is embedded [Fulek, Kyncl, 18][Akitaya, Fulek, Toth, 18] • Polynomial when the underlying graph is embedded and faces touch few clusters • [Di Battista, Frati 09] [Chimani, Di Battista, Frati, Klein, 14]

9. Our results • We show that Clustered Planarity is polynomially equivalent to FLAT Clustered Planarity • we reduce Clustered Planarity to FLAT Clustered Planarity • We show that FLAT Clustered Planarity is polynomially equivalent to INDEPENDENT Flat Clustered Planarity, where clusters induce independent sets

10. Homogeneous inclusion tree • A cluster is homogeneous if either it contains all leaves or it contains all clusters • An inclusion tree is homogeneous is all its clusters are homogeneous • A c-graph with n vertices and c clusters can be transformed in linear time into an equivalentc-graph whose inclusion tree is homogeneous and has height h n-1

11. Reduction strategy

12. Reduction strategy “flat” subtree

13. Reduction strategy

14. Reduction strategy “flat” subtree

15. Reduction strategy “flat” subtree

16. Reduction strategy

17. Reduction strategy “flat” subtree

18. Reduction strategy

19. Reduction strategy

20. Reduction strategy “flat” subtree

21. Reduction strategy

22. Reduction strategy

23. Reduction strategy

24. Reduction strategy “flat” subtree

25. Reduction strategy

26. Reduction strategy

27. Reduction strategy flat inclusion tree

28. 1 2 3 A step of the reduction  * 1 2 3

29. * 1 2 3 A step of the reduction  1 2 3

30. 1 2 3 A step of the reduction  1 2 3

31. 1 2 3 A step of the reduction    1 2 3

32.  1 2 3 * 1 2 3   c-planar drawing Equivalence ( direction)     c-planar drawing

33. 10 9 1 11 3 3 11 1 8 12 10 2 8 2 9 13 5 7 4 7 6 13 6 14 12 15 4 5  *

34. 10 9 1 11 3 3 11 1 8 12 10 2 8 2 9 13 5 7 4 7 6 13 6 14 12 15 4 5  *

35. 10 9 1 11 3 3 11 1 8 12 10 2 8 2 9 13 5 7 4 7 6 13 6 14 12 15 4 5   

36.     1 2 3 * 1 2 3   c-planar drawing Equivalence ( direction)   c-planar drawing  c-planar drawing  c-planar drawing

37.   

38.  1 10 9 11   3 4 2 8 6 5 7 13 12

39.  1 10 9 11   3 4 2 8 6 5 7 13 12

40.  1 10 9 11   3 4 2 8 6 5 7 13 12

41.  1 10 9 11   3 4 2 8 6 5 7 13 12

42.  1 12 10 9 13 10 9 11 11   1 2 3 8 4 3 2 8 6 5 7 4 7 14 13 6 12 15 5

43.  1 12 10 9 13 10 9 11 11   1 2 3 8 4 3 2 8 6 5 7 4 7 14 13 6 12 15 5

44.  1 12 10 9 13 10 9 11 11   1 2 3 8 4 3 2 8 6 5 7 4 7 14 13 6 12 15 5

45. 12 13 11 11 14 13 12 15  1 10 9 10 9   1 2 3 8 4 3 2 8 6 5 7 4 7 6 5

46. 12 13 11 11 14 13 12 15  1 10 9 10 9   1 2 3 8 4 3 2 8 6 5 7 4 7 6 5

47. 12 13 11 11 14 13 12 15  1 10 9 10 9   1 2 3 8 4 3 2 8 6 5 7 4 7 6 5

48.     1 10 9 10 9  1 2 3 8 4 3 2 8 6 5 7 4 7 6 5

49.     1 10 9 10 9 1 2 3 8 4 3 2 8 6 5 7 4 7 6 5

50.     1 10 9 10 9 1 2 3 8 4 3 2 8 6 5 7 4 7 6 5