Clustered Planarity = Flat Clustered Planarity

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

Clustered graph

Inclusion tree

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

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]

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

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]

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

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

Reduction strategy

Reduction strategy “flat” subtree

Reduction strategy flat inclusion tree

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

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

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

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

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

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  *

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   

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

  

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

 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

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

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

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

Clustered Planarity = Flat Clustered Planarity