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

WALCOM 2012February 16,

2012

On the Hardness of Point-Set Embeddabiltiy

Stephane Durocher Debajyoti Mondal

Department of Computer ScienceUniversity of Manitoba

Point-Set Embeddings

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A plane graph G A point set P

1WALCOM 2012February 16,

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Point-Set Embeddings

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A plane graph G An embedding of G on P

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Previous Results

Gritzmann et al. (1991), Castañeda and Urrutia (1996), Bose (2002)

Outerplanar graphs O(n lg 3n)

Nishat et al. (2010), Durocher et al. (2011), Moosa and Rahman(2011)

Plane 3-trees O(n4/3 + ɛ)

Cabello (2006), Nishat et al. (2011) 2-Connected (2-outerplanr graphs), Partial plane 3-trees

NP-complete

Durocher et al.(2011) 3-Connected graphs (Plane 3-trees)

NP-complete (in R3)

This Presentation

3-Connected graphs NP-complete

Klee Graphs (Graphs having plane 3-trees as dual)

O(n8) under convexity

Reference Graph Class Time complexity


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Sketch of Cabello’s Proof

S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32

S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}

3-Partition Point-Set Embeddability (2-connected graphs)

A chain

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y

c1

x yx

c1c2

c|S|

910

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S


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S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32

S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}


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yx

c1c2

c|S|

910

11

S

yx


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S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32

S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}


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yx

c1c2

c|S|

910

11

S

yx

G


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S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32

S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}


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yx

G

B B B B

P


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S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32

S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}


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yx

G P

x

y


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S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32

S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}


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yx

G P

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y


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Sketch of Kaufmann and Wiese’s Proof

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Hamiltonian Cycle 1-Bend P.S.E. (3-connected graphs)

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G

Does G contain a Hamiltonian Cycle?

Does G admits a 1-bend PSE on P ?

P


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v4

v5

G

Pv1

v3

v6

v2

v1

v5

v3

v2

v4v6

v1 v5 v2 v3 v4v6

If G contains a Hamiltonian Cycle, then G admits a 1-bend PSE on P


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v4

v5

G P

v1

v3

v6

v2

v1 v5 v2 v3 v4v6

If G admits a 1-bend PSE on P, then G contains a Hamiltonian Cycle

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How to get rid off bends?


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Our Result

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Point-Set Embeddability is NP-hard for 3-Connected Graphs

G /G


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Lets Try...

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v1 v2

v3

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v5

v6

G


Does G / admits a PSE on P ?

P

G /

???


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Problems…

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P

G /

P

?


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Tricks…

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G /

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Idea of Reduction

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G /

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G /

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Idea of ReductionA

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P

WALCOM 2012

A

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a d c b h g e f

G /

Idea of Reduction

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Multiple Point-Sets

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G /

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12 points

12 points

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PSE is NP-hard for 3-Connected Graphs

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fg h

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G /

G


Does G / admit a PSE on some point set among

P1, P2, … , Pk ?

February 16, 2012


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Positive Results

A plane 3-tree G

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A construction for G

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a

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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.


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Graphs with Plane 3-Trees as Weak Dual

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Graphs with Plane 3-Trees as Weak Dual

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a b

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fehj

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Convex PSE of Klee Graphs

A klee graph G

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b

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q

op n

mlkj

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ghe

fd r

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dq fo

pn

mr

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geijk

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Does G admits a convex point-set embedding on P ?

P


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Convex PSE of Klee Graphs

A klee graph G

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b

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q

op n

mlkj

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ghe

fd r

Does G admits a convex point-set embedding on P ?

P


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Dynamic Programming

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mlk

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op n

mlk

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d r

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Future Works

Is PSE NP-hard for 4-connected graphs?

Convex PSE algorithms for general klee graphs .

PSE algorithms for klee graphs without convexity constraint.

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Thank You..

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

Documents

s slide

connected graphs g g

s y x g slide

embedding of g

y x g b b b b p slide

s y x slide

d c b e f g g slide

y x g p x y slide