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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,
2012
Point-Set Embeddings
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A plane graph G An embedding of G on P
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2WALCOM 2012February 16,
2012
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
3WALCOM 2012February 16,
2012
WALCOM 2012February 16,
2012
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
11
S
WALCOM 2012February 16,
2012
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)
5
yx
c1c2
c|S|
910
11
S
yx
WALCOM 2012February 16,
2012
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)
6
yx
c1c2
c|S|
910
11
S
yx
G
WALCOM 2012February 16,
2012
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)
7
yx
G
B B B B
P
WALCOM 2012February 16,
2012
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)
8
yx
G P
x
y
WALCOM 2012February 16,
2012
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)
9
yx
G P
x
y
WALCOM 2012February 16,
2012
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
WALCOM 2012February 16,
2012
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
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
WALCOM 2012February 16,
2012
Sketch of Kaufmann and Wiese’s Proof
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Hamiltonian Cycle 1-Bend P.S.E. (3-connected graphs)
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
e
How to get rid off bends?
WALCOM 2012February 16,
2012
Our Result
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Point-Set Embeddability is NP-hard for 3-Connected Graphs
G /G
WALCOM 2012February 16,
2012
Lets Try...
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v1 v2
v3
v4
v5
v6
G
Does G contain a Hamiltonian Cycle?
Does G / admits a PSE on P ?
P
G /
???
WALCOM 2012February 16,
2012
Problems…
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P
G /
P
?
WALCOM 2012February 16,
2012
Tricks…
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WALCOM 2012
Idea of Reduction
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G /
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WALCOM 2012 18
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G /
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Idea of ReductionA
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WALCOM 2012
A
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G /
Idea of Reduction
WALCOM 2012
A
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Multiple Point-Sets
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G /
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12 points
12 points
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WALCOM 2012
PSE is NP-hard for 3-Connected Graphs
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G /
G
Does G contain a Hamiltonian Cycle?
Does G / admit a PSE on some point set among
P1, P2, … , Pk ?
February 16, 2012
WALCOM 2012February 16,
2012
Positive Results
A plane 3-tree G
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A construction for G
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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.
WALCOM 2012February 16,
2012
Graphs with Plane 3-Trees as Weak Dual
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WALCOM 2012February 16,
2012
Graphs with Plane 3-Trees as Weak Dual
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WALCOM 2012February 16,
2012
Convex PSE of Klee Graphs
A klee graph G
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fd r
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Does G admits a convex point-set embedding on P ?
P
WALCOM 2012February 16,
2012
Convex PSE of Klee Graphs
A klee graph G
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b
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op n
mlkj
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ghe
fd r
Does G admits a convex point-set embedding on P ?
P
WALCOM 2012February 16,
2012
Dynamic Programming
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WALCOM 2012February 16,
2012
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..