1. Clustered Planarity = Flat Clustered Planarity
537-dht-03.ppt
039-dht dht-03.ppt
Clustered Planarity = Flat Clustered Planarity
Pier Francesco Cortese & Maurizio Patrignani
Roma Tre University
26th International Symposium on Graph Drawing and Network Visualization, September 26-28, 2018, Barcelona, Spain 1

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. All leaves of the inclusion tree have depth two
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. 039-dht dht-03.ppt
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
a reduction from FLAT Clustered Planarity to Clustered Planarity is not needed

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 equivalent c-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. A step of the reduction  * 1 2 3 1 2 3

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

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

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

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

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

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

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

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

37.   

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

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

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

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

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

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

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

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

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

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

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

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

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

51.  * 1 10 9 10 9 1 2 3 4 8 3 2 8 5 6 7
4
7 6 5

52. Equivalence    c-planar drawing  c-planar drawing * 1 2 3  1 2 3   
 c-planar drawing
 c-planar drawing 
 c-planar drawing
 c-planar drawing

53. Special case    1 2 7 12 6 3 1 2 4 6 3 5 5
4
7
11
8
9
10

54. Special case  *   1 2 7 12 6 3 1 2 4 6 3 5
4
7 5
12 *
6
3 
4 6
5
7
11
8
9
10

55. Special case    1 2 7 12 6 3 1 2 4 6 3 5 5
4
7 5
12
6
3 
4 6
5
7
11
8
9
10

56. Special case    1 2 7 12 6 3 1 2 4 6 3 5 5
4
7 5
12
6
3 
4 6
5
7
11
8
9
10

57. Special case    1 2 7 12 6 3 1 2 4 6 3 5 5
4
7 5
12
6
3 
4 6
5
7
11
8
9
10

58. Special case  *   1 2 7 12 6 3 1 2 4 6 3 5
4
7 5
12 *
6
3 
4 6
5
7
11
8
9
10

59. Main theorem
Theorem 1
There exists a quadratic-time transformation that maps an instance of Clustered Planarity to an equivalent instance of Flat Clustered Planarity

60. Consequences of the proof of Theorem 1
039-dht dht-03.ppt
Consequences of the proof of Theorem 1
The underlying graph is only subdivided, hence some strong properties of it are preserved
the property of being connected, biconnected, or a subdivision of a triconnected graph
the property of being a tree or a cycle
the property of having treewidth k
the property of having a fixed embedding …
Hence, the complexity of Clustered Planarity restricted to these kinds of graphs can be related to the complexity of Flat Clustered Planarity restricted to the same kinds
of graphs

61. Faces touching few clusters
Theorem [Chimani, Di Battista, Frati, Klein, 14]
Let C(G,T) be an n-vertex flat c-graph where G has a fixed embedding
There exists an O(n3)-time algorithm to test the c-planarity of C if each cluster has at most two vertices on the same face of G
Generalization
Let C(G,T) be an n-vertex c-graph where G has a fixed embedding and T has height h
There exists an O(n3·h3)-time algorithm to test the c-planarity of C if each lower cluster has at most two vertices on the same face of G and each higher cluster has at most two inter-cluster edges on the same face of G

62. Small faces Theorem [Da Lozzo, Eppstein, Goodrich, Gupta, 18]
039-dht dht-03.ppt
Small faces
Theorem [Da Lozzo, Eppstein, Goodrich, Gupta, 18]
Flat Clustered Planarity can be solved in 2O(L·n·log n) time for n-vertex embedded flat c-graphs with maximum face size L
Generalization
Clustered Planarity can be solved in 2O(h·L·n·log(n·h) time for n-vertex embedded c-graphs with maximum face size L and height h of the inclusion tree
Flat Clustered Planarity admits a subexponential-time algorithm when the underlying graph has a fixed embedding and its maximum face size L belongs to o(n).
Clustered Planarity is subexponential whenever L· h2 ∈ o(n)

63. Insight Being enclosed into a cluster…
… is equivalent to being forced to traverse inside-out the cluster boundary…
… which is equivalent to being forced to traverse two clusters

64. Iterating the transformation

65. Iterating the transformation

66. Iterating the transformation

67. Iterating the transformation

68. Further reduction Theorem 2
There exists a linear-time transformation that maps an instance of Flat Clustered Planarity to an equivalent instance of Independent Flat Clustered Planarity

69. Open problems
Can the reduction presented in this paper be used to generalize some other polynomial-time testing algorithm for Flat Clustered Planarity to plain Clustered Planarity?
What is the complexity of independent flat Clustered Planarity when the underlying graph is a cycle?
What is the complexity of independent flat Clustered Planarity when the number of Type 2 clusters is bounded?

70. Thanks!
Any question?