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Planar Embeddings of Graphs with Specified Edge Lengths Sergio Cabello GIVE, University Utrecht Erik D. Demaine MIT G unter Rote FU Berlin Problem and motivation Given a graph G Sergio Cabello, sergio@cs.uu.nl, 2/11 Problem and

  1. Planar Embeddings of Graphs with Specified Edge Lengths Sergio Cabello GIVE, University Utrecht Erik D. Demaine MIT G¨ unter Rote FU Berlin

  2. Problem and motivation Given a graph G Sergio Cabello, sergio@cs.uu.nl, 2/11

  3. Problem and motivation Given a graph G 3 3 1 2 3 2 1 1 1 1 1 2 2 2 1 1 3 2 and specified edge lengths Question: Can we draw G with these edge lengths? Sergio Cabello, sergio@cs.uu.nl, 2/11

  4. Problem and motivation Applications: • Sensor networks • Structural analysis of molecules • Linear cartograms Sergio Cabello, sergio@cs.uu.nl, 3/11

  5. Our results Restriction to planar embeddings (so planar graphs): • Triangulated graphs ⇒ linear time decision   3-connected ( → fixed topology)         unit edge lengths,     • ⇒ NP-hard. bounded face degree, and           generically rigid     2-connected + unit length, or   Improves Eades and Wormald ’90 3-connected   also in simplicity Sergio Cabello, sergio@cs.uu.nl, 4/11

  6. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. Sergio Cabello, sergio@cs.uu.nl, 5/11

  7. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. Sergio Cabello, sergio@cs.uu.nl, 5/11

  8. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  9. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  10. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  11. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  12. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  13. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  14. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  15. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Sergio Cabello, sergio@cs.uu.nl, 5/11

  16. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] ⇒ In R 2 : fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. 1 3 2 8 4 5 7 6 Longest edge in the outermost face ⇒ Two outer face candidates. Sergio Cabello, sergio@cs.uu.nl, 5/11

  17. Triangulated graphs Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S 2 [Whitney] . e ⇒ In R 2 : fixing outer face, m the topology is completely fixed. i t r ⇒ Fixed outer face, incremental treatment of triangles in O ( n ) time. a e n 1 3 2 8 i 4 L 5 7 6 Longest edge in the outermost face ⇒ Two outer face candidates. Sergio Cabello, sergio@cs.uu.nl, 5/11

  18. NP-hardness Th: It is NP-hard for planar 3-connected graphs. Reduction from planar 3-SAT v 1 v 2 v 3 v 4 v 5 v n . . . Sergio Cabello, sergio@cs.uu.nl, 6/11

  19. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  20. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  21. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  22. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  23. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  24. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  25. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  26. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  27. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  28. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  29. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  30. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  31. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  32. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  33. NP-hardness v 3 v 4 v 5 The holder gadget. Sergio Cabello, sergio@cs.uu.nl, 7/11

  34. NP-hardness The variable gadget. v 3 v 4 v 5 (true literal ≡ pushing towards variable) ¬ v i v i v i v i ≡ true ¬ v i v i v i Sergio Cabello, sergio@cs.uu.nl, 8/11

  35. NP-hardness The variable gadget. v 3 v 4 v 5 (true literal ≡ pushing towards variable) ¬ v i v i v i v i ≡ false ¬ v i v i v i Sergio Cabello, sergio@cs.uu.nl, 8/11

  36. NP-hardness The inverter gadget. pushing towards variable v 3 v 4 v 5 � pushing towards clause Sergio Cabello, sergio@cs.uu.nl, 9/11

  37. NP-hardness The inverter gadget. pushing towards variable v 3 v 4 v 5 � pushing towards clause Sergio Cabello, sergio@cs.uu.nl, 9/11

  38. NP-hardness The inverter gadget. pushing towards variable v 3 v 4 v 5 � pushing towards clause Sergio Cabello, sergio@cs.uu.nl, 9/11

  39. NP-hardness The clause gadget. v 3 v 4 v 5 (realizable ⇔ ∃ literal pushing towards clause) l i ≡ false l j ≡ false the same l k ≡ false Sergio Cabello, sergio@cs.uu.nl, 10/11

  40. NP-hardness The clause gadget. v 3 v 4 v 5 (realizable ⇔ ∃ literal pushing towards clause) l i ≡ true l j ≡ false l k ≡ false Sergio Cabello, sergio@cs.uu.nl, 10/11

  41. NP-hardness The clause gadget. v 3 v 4 v 5 (realizable ⇔ ∃ literal pushing towards clause) l i ≡ true l j ≡ true l k ≡ true Sergio Cabello, sergio@cs.uu.nl, 10/11

  42. NP-hardness The clause gadget. v 3 v 4 v 5 (realizable ⇔ ∃ literal pushing towards clause) l i ≡ false l j ≡ false l k ≡ true Sergio Cabello, sergio@cs.uu.nl, 10/11

  43. What did I explain? Planar drawing of graphs with specified edge lengths. • ⇒ linear time decision Triangulated graphs   3-connected ( → fixed topology)         unit edge lengths,     • ⇒ NP-hard. bounded face degree, and           generically rigid   Sergio Cabello, sergio@cs.uu.nl, 11/11

  44. Contents • Problem and motivation • Our results • Triangulated graphs • NP-hardness • What did I explain? Sergio Cabello, sergio@cs.uu.nl, 12/11

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