WALCOM 2012February 16, 2012 Stephane Durocher Debajyoti Mondal Department of Computer Science University of Manitoba

a b c d e f g h i A plane graph G A point set P 1 WALCOM 2012February 16, 2012

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

Reference Graph Class Time complexity 3 WALCOM 2012February 16, 2012

WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) A chain 4 y c1c1 x y x c1c1 c2c2 c|S|c|S| S

WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 5 y x c1c1 c2c2 c|S|c|S| S y x

WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 6 y x c1c1 c2c2 c|S|c|S| S y x G

WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 7 y x G B B B B P

WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 8 y x G P x y

WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 9 y x G P x y

WALCOM 2012February 16, Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G admits a 1-bend PSE on P ? Does G admits a 1-bend PSE on P ? P

WALCOM 2012February 16, Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v4v4 v5v5 G P v1v1 v3v3 v6v6 v2v2 v1v1 v5v5 v3v3 v2v2 v4v4 v6v6 v1v1 v5v5 v2v2 v3v3 v4v4 v6v6 If G contains a Hamiltonian Cycle, then G admits a 1-bend PSE on P

WALCOM 2012February 16, Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v4v4 v5v5 G P v1v1 v3v3 v6v6 v2v2 v1v1 v5v5 v2v2 v3v3 v4v4 v6v6 If G admits a 1-bend PSE on P, then G contains a Hamiltonian Cycle e How to get rid off bends? How to get rid off bends?

WALCOM 2012February 16, Point-Set Embeddability is NP-hard for 3-Connected Graphs G / G

WALCOM 2012February 16, v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G / admits a PSE on P ? Does G / admits a PSE on P ? P G / ? ? ?

WALCOM 2012February 16, P G / P ?

WALCOM 2012February 16, h a d c b e f g a d c b e f g h G h a d c b e f g G /

WALCOM a d c b e f g h h a d c b e f g G / G A B P

WALCOM a d c b e f g h h a d c b e f g G / G a dcbh g e f A B P

WALCOM 2012 A B A B 19 a dcbh g e f G /

WALCOM 2012 A B 20 G / h a d c b e f g h a d c b e f g 12 points

WALCOM a d c b e f g h h a d c b e f g G / G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G / admit a PSE on some point set among P 1, P 2, …, P k ? Does G / admit a PSE on some point set among P 1, P 2, …, P k ? February 16, 2012

WALCOM 2012February 16, 2012 A plane 3-tree G f g h k m a b c d e A construction for G 22 a b c d e f g h k m k e c g m Nishat et al. (2010), Durocher et al. (2011), Moosa and Rahman (2011) Point-set embeddability can be tested for plane 3-trees in O(n 4/3 + ɛ ) time.

WALCOM 2012February 16, a b c d a b c e f g a b c e f h i j a b c f h i j k l m

WALCOM 2012February 16, a b c d a b c e f g a b c e f h i j a b c f h i j k l m a b c d a c b g f e a c b f e h j i a c b f i j l h k m

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

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

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

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

WALCOM 2012February 16, 2012 A plane 3-tree G 4 v1v1 v2v2 v3v3 v4v4 v8v8 v5v5 v6v6 v7v7 v9v9 v 10 v1v1 v2v2 v3v3 v4v4 v8v8 v5v5 v6v6 v7v7 v9v9

WALCOM 2012February 16, 2012 A plane 3-tree G 4 f1f1 f2f2 f3f3 f4f4 f1f1 f3f3 f4f4 e f d f1f1 f2f2 f4f4 q o p f2f2 f3f3 n m l k j i g h f4f4 r

WALCOM 2012February 16, 2012 A plane 3-tree G 4 b f1f1 f2f2 f3f3 f4f4 a c q o p n m l k j i g h e f d f1f1 f3f3 f4f4 e f d f1f1 f2f2 f4f4 q o p f2f2 f3f3 n m l k j i g h f4f4 r r

y = 0 y = -1 y = 1 (c) h a d c b e f g M / (a) A B (d) (b) a dcbh g e f L PiPi

WALCOM 2012February 16, Tricks… a d c b e f g h h a d c b e f g Instead of the blue vertices use blue cycles. Replace black vertices with some special graph structure. Use simpler input instances. h a d c b e f g

WALCOM 2012February 16,

WALCOM 2012February 16, Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G admits a 1-bend PSE on P ? Does G admits a 1-bend PSE on P ? P

WALCOM 2012February 16, Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v5v5 v6v6 v8v8 Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? v2v2 v3v3 v1v1 v7v7 v4v4 v1v1 v2v2 v8v8 v3v3 v4v4 v7v7 v5v5 v6v6 G

WALCOM 2012February 16, 2012 a b c d e f g h i j k l m n o A plane 3-tree G f g h i j k l m n o a b c d e A construction for G 4

WALCOM 2012February 16, 2012 a b c d e f g h i j k l m n o A plane 3-tree G f g h i j k l m n o a b c d e The representative vertex of G k l e A plane 3-tree A construction for G c o c g m n d A plane 3-tree 5

h a d c b e f g a d c b e f g h M M / (a) (b) (c) a / b / c / r st