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WALCOM 2012

On the Hardness of Point-Set Embeddabiltiy. Stephane Durocher Debajyoti Mondal. Department of Computer Science University of Manitoba. WALCOM 2012. February 16, 2012. Point-Set Embeddings. b. c. h. a. i. g. f. e. d. A point set P. A plane graph G.

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WALCOM 2012

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  1. On the Hardness of Point-Set Embeddabiltiy Stephane Durocher DebajyotiMondal Department of Computer Science University of Manitoba WALCOM 2012 February 16, 2012

  2. Point-Set Embeddings b c h a i g f e d A point set P A plane graph G WALCOM 2012 February 16, 2012 1

  3. Point-Set Embeddings b b c c h h a a i i g g f f e e d d An embedding of G on P A plane graph G WALCOM 2012 February 16, 2012 2

  4. Previous Results Reference Graph Class Time complexity WALCOM 2012 February 16, 2012 3

  5. Sketch of Cabello’s Proof S c1 c1 9 c2 10 c|S| 11 3-Partition  Point-Set Embeddability (2-connected graphs) S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x x y y A chain S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} WALCOM 2012 February 16, 2012 4

  6. Sketch of Cabello’s Proof S c1 9 c2 10 c|S| 11 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x y S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} WALCOM 2012 February 16, 2012 5

  7. Sketch of Cabello’s Proof S c1 9 c2 10 c|S| 11 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x y G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} WALCOM 2012 February 16, 2012 6

  8. Sketch of Cabello’s Proof B B B B 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} P WALCOM 2012 February 16, 2012 7

  9. Sketch of Cabello’s Proof y 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} P WALCOM 2012 February 16, 2012 8

  10. Sketch of Cabello’s Proof y 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} P WALCOM 2012 February 16, 2012 9

  11. Sketch of Kaufmann and Wiese’s Proof Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v3 v4 v6 v5 v1 v2 P G Does G contain a Hamiltonian Cycle? Does G admits a 1-bend PSE on P ? WALCOM 2012 February 16, 2012 10

  12. Sketch of Kaufmann and Wiese’s Proof Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v3 v3 v2 v5 v4 v6 v4 v6 v1 v6 v1 v2 v5 v4 v3 v5 v1 v2 P G If G contains a Hamiltonian Cycle, then G admits a 1-bend PSE on P WALCOM 2012 February 16, 2012 11

  13. Sketch of Kaufmann and Wiese’s Proof Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v3 How to get rid off bends? v4 e v6 v6 v1 v2 v5 v4 v3 v5 v1 v2 G P If G admits a 1-bend PSE on P, then G contains a Hamiltonian Cycle WALCOM 2012 February 16, 2012 12

  14. Our Result Point-Set Embeddability is NP-hard for 3-Connected Graphs G G / WALCOM 2012 February 16, 2012 13

  15. Lets Try... v3 Does G contain a Hamiltonian Cycle? v4 v6 ? ? v5 ? v1 v2 G Does G /admits a PSE on P ? P G / WALCOM 2012 February 16, 2012 14

  16. Problems… ? P P G / WALCOM 2012 February 16, 2012 15

  17. Tricks… c c a c a a G d d d b h h b h g b g g f f f e e e G / WALCOM 2012 February 16, 2012 16

  18. Idea of Reduction A c a d h b g f G e c a d b h g P f e G / B WALCOM 2012 17

  19. Idea of Reduction A c a d h b g f G e g f d c b h e a c a d b h g P f e G / B WALCOM 2012 18

  20. Idea of Reduction A A g f d c b h e a B B G / WALCOM 2012 19

  21. Multiple Point-Sets A 12 12 points 12 c c a a 14 d d b b h h g g f f 12 points e e 10 G / B G / WALCOM 2012 20

  22. PSE is NP-hard for 3-Connected Graphs c a Does G contain a Hamiltonian Cycle? d h b g f G e c a Does G / admit a PSE on some point set among P1, P2, … , Pk? d b h g f e G / WALCOM 2012 February 16, 2012 21

  23. Positive Results b Nishat et al. (2010), Durocher et al. (2011), Moosa and Rahman (2011) Point-set embeddability can be tested for plane 3-trees in O(n4/3 + ɛ) time. b k k k e d e e h d h a f f g g g c a m c m m c A plane 3-tree G A construction for G WALCOM 2012 February 16, 2012 22

  24. Graphs with Plane 3-Trees as Weak Dual c c c c i i g j j h h d l f f f k e e m a a a a b b b b WALCOM 2012 February 16, 2012 23

  25. Graphs with Plane 3-Trees as Weak Dual c c c c i i g j j h h d l f f f k e e m a a a a b b b b f f m f d e e h h l g k j j i i b b b b a a a a c c c c WALCOM 2012 February 16, 2012 24

  26. Convex PSE of Klee Graphs a q p n o m l k j f r d i c h e c b g r P A klee graph G m h l g n j i k p e d Does G admits a convex point-set embedding on P ? o f q a b WALCOM 2012 February 16, 2012 25

  27. Convex PSE of Klee Graphs a q p n o m l k j f r d i h e c b g P A klee graph G Does G admits a convex point-set embedding on P ? WALCOM 2012 February 16, 2012 26

  28. Dynamic Programming q p o n k m a l j f a r d i q e g q h p n p n o o k m l k m f l j r j d f r i d e g i h h c b e c b g WALCOM 2012 February 16, 2012 27

  29. Future Works • Is PSE NP-hard for 4-connected graphs? • Convex PSE algorithms for general kleegraphs . • PSE algorithms for klee graphs without convexity constraint. WALCOM 2012 February 16, 2012 28

  30. Thank You..

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