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SPQR Tree

SPQR Tree. v. u. skeleton of S. SPQR-tree. v. Q. u. S. P. Q. R. Q. S. Q. Q. Q. Q. Q. Q. Q. Decomposition of biconected graph with respected to its triconnected components. Applications. SPQR tree represents all embeddings of a graph.

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SPQR Tree

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  1. SPQR Tree

  2. v u skeleton of S SPQR-tree v Q u S P Q R Q S Q Q Q Q Q Q Q Decomposition of biconected graph with respected to its triconnected components.

  3. Applications SPQR tree represents all embeddings of a graph. • Online problems connected with planarity and triconnectivity. • To maintain the minimum spanning tree and transitive closure of a graph. • Orthogonal drawings of planar graphs with minimum number of bends. • Symmetric drawings of graphs in three dimension. • For cluster planarity testing. • Minimum segment drawings of planar graphs. • Upward planarity • Monotone drawing of graphs.

  4. biconnected graphs • a cut-vertex is a vertex such that its removal produces a disconnected graph • a biconnected graph does not have cut-vertices • a separation pair of a biconnected graph is a pair of vertices whose removal produces a disconnected graph • a split pair is either a separation pair or a pair of adjacent vertices

  5. Split pair v G1 A split pair v u G2 u A split pair A split pair {u, v} of G v G u A pair of adjacent vertices

  6. Maximal split pair v v’ u u’ A split pair b G a split pair {u, v} a For any other split pair {u’, v’}, a, b, u, v are in the same split component

  7. A split pair v b v’ v’ a u u’ u’ For any other split pair {u’, v’}, a, b, u, v are in the same split component

  8. v A maximal split pair with respect to a reference edge (a, b) u A split pair b G a split pair {u, v} a For any other split pair {u’, v’}, a, b, u, v are in the same split component

  9. v’ (= u’) A split pair b v G a split pair {u, v} a u

  10. A maximal split pair with respect to a reference edge (a, b) A split pair v’ v’ b a split pair {u, v} v a u (= u’) u (= u’) For other split pair {u’, v’}, a, b, u, v are not in the same split component

  11. SPQR-tree v With a reference edge u

  12. SPQR-tree v Q u

  13. SPQR-tree v Q u

  14. SPQR-tree v Q u S

  15. SPQR-tree v Q u S Q

  16. SPQR-tree Q S P Q

  17. SPQR-tree Q S P Q S Q Q Q

  18. SPQR-tree Q S P Q R Q S Q Q Q Q Q Q Q

  19. v u skeleton of S SPQR-tree v • each internal node of the tree is associated with a skeleton representing its configuration • the graph represented by node  into its parent  is called the pertinent of  Q u S P Q R Q S Q Q Q Q Q Q Q

  20. c s s e t t One or more cut vertices. An SPQR-tree two split components (One of them consist of e.) s e t biconnected series- parallel graph G

  21. s S x t x1 x2 pertinent graph of x An SPQR-tree two split components (One of them consist of e.) c G1 s s s e e G2 t t t Series case biconnected series- parallel graph G pole

  22. c G1 s e c e G2 t An SPQR-tree c G1 s e G2 t S x x1 x2

  23. c two parallel edge s e S c x s Q x2 pertinent graph of x1 An SPQR-tree c s G1 e Trivial case S x x1 x2 pole

  24. c S x P x2 t xa xb pertinent graph of x2 An SPQR-tree three or more split components c c e e Ga Gb G2 t t Parallel case S pole x x2

  25. None of the case above t Skeleton of R node s R-node

  26. Q S R Q P R Q Q S S R Q Q Q R Q Q Q Q Q Q Q Q

  27. Series-parallel graphs G1 series connection G1 sink source G2 G2 A series-parallel graphs a single edge source sink

  28. G1 parallel connection G1 sink source G2 G2 A series-parallel graphs a single edge source sink

  29. SPQ-tree P S S S P S S An SPQ-tree of G : an edge : series connection S : parallel connection P An SPQ-tree c b j d k a i e l h g f biconnected series- parallel graph G

  30. An SPQ-tree P c S S S b j d P k c g e d h k l f a i e S S l h g f j b a i biconnected series- parallel graph G An SPQ-tree of G : an edge : series connection S : parallel connection P

  31. An SPQ-tree P c S S S b j d P k c g e d h k l f a i e S S l h g f j b a i biconnected series- parallel graph G An SPQ-tree of G : an edge : series connection S : parallel connection P

  32. An SPQ-tree P c S S S b j d P k c g e d h k l f a i e S S l h g f j b a i biconnected series- parallel graph G An SPQ-tree of G : an edge : series connection S : parallel connection P

  33. An SPQ-tree P c S S S b j d P k c g e d h k l f a i e S S l h g f j b a i biconnected series- parallel graph G An SPQ-tree of G : 0 The number of children : 2 or more S : 2(non-root) or 3(root) P

  34. Thank You

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