Clustered Planarity = Flat Clustered Planarity 537-dht-03.ppt 039-dht-06537-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

Clustered graph

Inclusion tree

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]

All leaves of the inclusion tree have depth two 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]

039-dht-06537-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

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

Reduction strategy

Reduction strategy “flat” subtree

Reduction strategy

Reduction strategy “flat” subtree

Reduction strategy “flat” subtree

Reduction strategy

Reduction strategy “flat” subtree

Reduction strategy

Reduction strategy

Reduction strategy “flat” subtree

Reduction strategy

Reduction strategy

Reduction strategy

Reduction strategy “flat” subtree

Reduction strategy

Reduction strategy

Reduction strategy flat inclusion tree

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

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

 * 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

 * 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

   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

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

  

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

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

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

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

   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

   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

   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

   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

   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

   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

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

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

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

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

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

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

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

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

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

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

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

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

Consequences of the proof of Theorem 1 039-dht-06537-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

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

Small faces Theorem [Da Lozzo, Eppstein, Goodrich, Gupta, 18] 039-dht-06537-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)

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

Iterating the transformation

Iterating the transformation

Iterating the transformation

Iterating the transformation

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

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?

Thanks! Any question?