Stephane Durocher
Debajyoti Mondal
Department of Computer Science
University of Manitoba
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February 16, 2012
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A point set P
A plane graph G
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An embedding of G on P
A plane graph G
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Reference
Graph Class
Time complexity
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
3-Connected graphs
NP-complete
Klee Graphs (Graphs having
plane 3-trees as dual)
O(n8) under
convexity
This Presentation
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3)
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3-Partition  Point-Set Embeddability (2-connected graphs)
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } ,
S1={10, 10, 12} ,
S2={ 9, 11, 12} ,
S3={ 9, 9,14} ,
c1
x
y
B = 32
x
S4={ 10, 11,11}
c1
c2
S
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c|S|
11
y
A chain
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3-Partition  Point-Set Embeddability (2-connected graphs)
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } ,
S1={10, 10, 12} ,
x
S2={ 9, 11, 12} ,
S3={ 9, 9,14} ,
S4={ 10, 11,11}
c1
c2
S
9
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c|S|
11
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x
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B = 32
y
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3-Partition  Point-Set Embeddability (2-connected graphs)
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } ,
S1={10, 10, 12} ,
x
S2={ 9, 11, 12} ,
B = 32
S3={ 9, 9,14} ,
S4={ 10, 11,11}
c1
c2
S
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c|S|
11
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G
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3-Partition  Point-Set Embeddability (2-connected graphs)
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } ,
S1={10, 10, 12} ,
S2={ 9, 11, 12} ,
B = 32
S3={ 9, 9,14} ,
B
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S4={ 10, 11,11}
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3-Partition  Point-Set Embeddability (2-connected graphs)
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } ,
S1={10, 10, 12} ,
S2={ 9, 11, 12} ,
B = 32
S3={ 9, 9,14} ,
S4={ 10, 11,11}
y
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x
G
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3-Partition  Point-Set Embeddability (2-connected graphs)
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } ,
S1={10, 10, 12} ,
S2={ 9, 11, 12} ,
B = 32
S3={ 9, 9,14} ,
S4={ 10, 11,11}
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G
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Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs)
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Does G contain a
Hamiltonian Cycle?
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Does G admits a
1-bend PSE on P ?
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Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs)
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If G contains a Hamiltonian Cycle, then G admits a 1-bend PSE on P
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Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs)
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How to get rid off
bends?
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P
If G admits a 1-bend PSE on P, then G contains a Hamiltonian Cycle
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Point-Set Embeddability is NP-hard for 3-Connected Graphs
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Does G contain a
Hamiltonian Cycle?
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Does G / admits a
PSE on P ?
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Does G contain a
Hamiltonian Cycle?
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Does G / admit a PSE on
some point set among
P1, P2, … , Pk ?
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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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A plane 3-tree G
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A construction for G
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A klee graph G
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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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