Reconfiguring Independent Sets in Cographs
Marthe Bonamy
Nicolas Bousquet
July 3, 2014
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Independent Set Reconfiguration
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Independent Set Reconfiguration
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Independent Set Reconfiguration
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Independent Set Reconfiguration
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Independent Set Reconfiguration ⇒ Reconfiguration Graph
Solutions // Vertices. Closest solutions // Neighbors.
TARk (G )
k=1
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Independent Set Reconfiguration ⇒ Reconfiguration Graph
Solutions // Vertices. Closest solutions // Neighbors.
TARk (G )
k=2
Marthe Bonamy, Nicolas Bousquet
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Reconfiguration Graphs
Two solutions:
In the same connected component?
What distance between them?
Marthe Bonamy, Nicolas Bousquet
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Reconfiguration Graphs
Two solutions:
In the same connected component?
What distance between them?
Reconfiguration graph:
Connected?
Maximal diameter of a connected component?
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Reconfiguration Graphs
Two solutions:
In the same connected component?
What distance between them?
Reconfiguration graph:
Connected?
Maximal diameter of a connected component?
Colorings, Dominating sets, Vertex covers...
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Reconfiguration Graphs
Two solutions:
In the same connected component?
What distance between them?
Reconfiguration graph:
Connected?
Maximal diameter of a connected component?
Colorings, Dominating sets, Vertex covers...
Token Addition & Removal, Token Jumping, Token Sliding...
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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State of the Art
Theorem (Hearn, Demaine ’05, Kamiński, Medvedev, Milanič ’12)
G known to be perfect or subcubic planar:
Are α, β in the same connected component of TARk (G )?
PSPACE-complete.
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State of the Art
Theorem (Hearn, Demaine ’05, Kamiński, Medvedev, Milanič ’12)
G known to be perfect or subcubic planar:
Are α, β in the same connected component of TARk (G )?
PSPACE-complete.
Efficient algorithms for:
claw-free graphs,
line graphs,
chordal graphs...
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Cographs
Cographs: P4 -free graphs.
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Cographs
Cographs: P4 -free graphs.
⇒ cograph
Cograph
Cograph
⇒ cograph
Cograph
Cograph
⇒ cograph
Marthe Bonamy, Nicolas Bousquet
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Our Results
Theorem (Bonsma ’14)
G cograph, α, β ∈ TARk (G ) ⇒ Decide in O(n2 ) whether α and β
in the same connected component.
Question (Bonsma ’14)
?
G cograph ⇒ Decide in Poly (n) whether TARk (G ) is connected.
Marthe Bonamy, Nicolas Bousquet
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Our Results
Theorem (Bonsma ’14)
G cograph, α, β ∈ TARk (G ) ⇒ Decide in O(n2 ) whether α and β
in the same connected component.
Question (Bonsma ’14)
?
G cograph ⇒ Decide in Poly (n) whether TARk (G ) is connected.
Theorem (B., Bousquet ’14+)
G cograph ⇒ Decide in O(n3 ) whether TARk (G ) is connected.
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Our Results
Theorem (Bonsma ’14)
G cograph, α, β ∈ TARk (G ) ⇒ Decide in O(n2 ) whether α and β
in the same connected component.
Question (Bonsma ’14)
?
G cograph ⇒ Decide in Poly (n) whether TARk (G ) is connected.
Theorem (B., Bousquet ’14+)
G cograph, α, β ∈ TARk (G ) ⇒ Decide in O(n) whether α and β
in the same connected component.
Theorem (B., Bousquet ’14+)
G cograph ⇒ Decide in O(n3 ) whether TARk (G ) is connected.
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Proof
Take your favorite G and k.
Marthe Bonamy, Nicolas Bousquet
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Proof
Take your favorite G and k.
Build the decomposition
tree in O(n).
D
J
o
J
o
J
o
o
o
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Reconfiguring Independent Sets in Cographs
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Proof
Take your favorite G and k.
Build the decomposition
tree in O(n).
Pick good and bad sides.
D
J
o
J
o
J
o
o
o
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Proof
Take your favorite G and k.
Build the decomposition
tree in O(n).
Pick good and bad sides.
D
Maximal stable sets ⇔
”Stable-searches”.
J
o
J
o
J
o
o
o
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Proof
Take your favorite G and k.
Build the decomposition
tree in O(n).
Pick good and bad sides.
D
Maximal stable sets ⇔
”Stable-searches”.
J
o
J
o
J
o
o
Find bad side B with
smallest α(B).
o
Marthe Bonamy, Nicolas Bousquet
Reconfiguring Independent Sets in Cographs
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Proof
Take your favorite G and k.
Build the decomposition
tree in O(n).
Pick good and bad sides.
D
Maximal stable sets ⇔
”Stable-searches”.
J
o
J
o
J
o
o
Marthe Bonamy, Nicolas Bousquet
o
Find bad side B with
smallest α(B).
Maximal stable set in
G \ (B ∪ N(B)) of size
k−α(B) ≤ · ≤ k+α(B)−1?
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Proof
Take your favorite G and k.
Build the decomposition
tree in O(n).
Pick good and bad sides.
D
Maximal stable sets ⇔
”Stable-searches”.
J
J
o
J
o
o
Marthe Bonamy, Nicolas Bousquet
o
Find bad side B with
smallest α(B).
Maximal stable set in
G \ (B ∪ N(B)) of size
k−α(B) ≤ · ≤ k+α(B)−1?
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Conclusion
Question (Bonsma’14)
?
G cograph, α, β ∈ TARk (G ) ⇒ Decide in Poly (n) whether α and
β at distance at most `.
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Conclusion
Question (Bonsma’14)
?
G cograph, α, β ∈ TARk (G ) ⇒ Decide in Poly (n) whether α and
β at distance at most `.
Thanks for your attention!
Marthe Bonamy, Nicolas Bousquet
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