[PPT] - Recoloring bounded treewidth graphs Marthe Bonamy, Nicolas Bousquet PowerPoint Presentation

SLIDE 1

Recoloring bounded treewidth graphs

Marthe Bonamy, Nicolas Bousquet LIRMM, Montpellier, France

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 1/13

SLIDE 2

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 3

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 4

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 5

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 6

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 7

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 8

Recoloring graphs

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 9

Recoloring graphs ⇒ Reconfiguration graphs

Solutions // Vertices. Adjacent solutions // Neighbors.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 2/13

SLIDE 10

Reconfiguration graph

More formally

k-Reconfiguration graph of G

◮ Vertices: Proper k-colorings of G ◮ Edges between any two k-colorings which differ on exactly one

vertex.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 3/13

SLIDE 11

Reconfiguration graph

More formally

k-Reconfiguration graph of G

◮ Vertices: Proper k-colorings of G ◮ Edges between any two k-colorings which differ on exactly one

vertex.

Remark

Two colorings equivalent up to color permutation are distinct.

=

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 3/13

SLIDE 12

Interesting questions

◮ Two solutions:

◮ Are in the same connected component? ◮ What distance between them? Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 4/13

SLIDE 13

Interesting questions

◮ Two solutions:

◮ Are in the same connected component? ◮ What distance between them?

◮ Reconfiguration graphs:

◮ Connex? ◮ What diameter? Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 4/13

SLIDE 14

k-mixing graphs

k-mixing

A graph is k-mixing if its k-reconfiguration graph is connected.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 5/13

SLIDE 15

k-mixing graphs

k-mixing

A graph is k-mixing if its k-reconfiguration graph is connected. 1 1 2 2 3 3 n n

Gap

No function f on the chromatic number ensures that G is k-mixing if k ≥ f (χ).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 5/13

SLIDE 16

State of the art

Theorem (Cereceda, van den Heuvel, Johnson ’07)

Determining if a bipartite graph is 3-mixing is co-NP hard.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 6/13

SLIDE 17

State of the art

Theorem (Cereceda, van den Heuvel, Johnson ’07)

Determining if a bipartite graph is 3-mixing is co-NP hard.

Recoloring diameter

Given a k-mixing graph, the recoloring diameter is in O(A(n)) if the diameter of the k-reconfiguration graph is bounded by C × A(n). (n is the number of vertices)

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 6/13

SLIDE 18

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 19

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 20

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 21

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 22

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 23

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 24

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 25

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 26

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 27

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 28

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 29

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 30

Upper bounds on recoloring

Theorem (Cereceda)

As long as k ≥ n + 1, the clique Kn is k-mixing in O(n).

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

Trees are 3-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 7/13

SLIDE 31

Chordal graphs

◮ No induced cycle of length at least 4. ◮ The graph admits a clique tree.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 8/13

SLIDE 32

Chordal graphs

◮ No induced cycle of length at least 4. ◮ The graph admits a clique tree.

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

The chordal graphs are (k + 1)-mixing in O(n2) for every k ≥ χ + 1.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 8/13

SLIDE 33

Chordal graphs

◮ No induced cycle of length at least 4. ◮ The graph admits a clique tree.

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

The chordal graphs are (k + 1)-mixing in O(n2) for every k ≥ χ + 1.

◮ Compute a clique-tree. Find a vertex which only appears in

the bag of a leaf.

◮ Identify it with a vertex of the bag of its parent.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 8/13

SLIDE 34

Chordal graphs

◮ No induced cycle of length at least 4. ◮ The graph admits a clique tree.

Theorem (Bonamy, Johnson, Lignos, Paulusma, Patel ’12)

The chordal graphs are (k + 1)-mixing in O(n2) for every k ≥ χ + 1.

◮ Compute a clique-tree. Find a vertex which only appears in

the bag of a leaf.

◮ Identify it with a vertex of the bag of its parent.

Questions

◮ Does the same hold for bounded treewidth graphs? ◮ And for perfect graphs?

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 8/13

SLIDE 35

Perfect graphs: a counter-example

1 1 2 2 3 3 n n

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 9/13

SLIDE 36

Bounded Treewidth graphs

Definition

◮ tw(G) =minH{χ(H) − 1|G ⊆ H, H chordal}.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 10/13

SLIDE 37

Bounded Treewidth graphs

Definition

◮ tw(G) =minH{χ(H) − 1|G ⊆ H, H chordal}. ◮ G admits a tree decomposition where each bag has size at

most tw(G) + 1 and each edge appears in at least one bag.

1 2 3 4 5 6 7 8 9

1 3 4 5 6 1 2 2 4 5 7 4 9 4 9 5 9 8 9

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 10/13

SLIDE 38

Bounded Treewidth graphs

Definition

◮ tw(G) =minH{χ(H) − 1|G ⊆ H, H chordal}. ◮ G admits a tree decomposition where each bag has size at

most tw(G) + 1 and each edge appears in at least one bag.

1 2 3 4 5 6 7 8 9

1 3 4 5 6 1 2 2 4 5 7 4 9 4 9 5 9 8 9

Theorem (Cereceda et al.)

Every k-degenerate graph is (k + 2)-mixing in 2n.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 10/13

SLIDE 39

Our main result

Theorem (Bonamy, B.)

Every graph G is (tw(G) + 2)-mixing in O(n2).

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 11/13

SLIDE 40

Our main result

Theorem (Bonamy, B.)

Every graph G is (tw(G) + 2)-mixing in O(n2).

Optimal

◮ tw(Kn) = n − 1, so tw(G) + 2 colors are necessary. ◮ Since 3-colorings of paths have a quadratic recoloring

diameter, the quadratic bound is necessary.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 11/13

SLIDE 41

Sketch of the proof

1 2 3 4 5 6 7 8 9

1 3 4 5 6 1 2 2 4 5 7 4 9 4 9 5 9 8 9

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 12/13

SLIDE 42

Sketch of the proof

1 2 3 4 5 6 7 8 9

1 3 4 5 6 1 2 2 4 5 7 4 9 4 9 5 9 8 9

◮ There exists a tree decomposition such that every bag has size

tw(G) + 2.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 12/13

SLIDE 43

Sketch of the proof

1 2 3 4 5 6 7 8 9

1 3 4 5 6 1 2 2 4 5 7 4 9 4 9 5 9 8 9

◮ There exists a tree decomposition such that every bag has size

tw(G) + 2.

◮ Objective: identify vertices with their parents.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 12/13

SLIDE 44

Sketch of the proof

1 2 3 4 5 6 7 8 9

1 3 4 5 6 1 2 2 4 5 7 4 9 4 9 5 9 8 9

◮ There exists a tree decomposition such that every bag has size

tw(G) + 2.

◮ Objective: identify vertices with their parents. ◮ Waiting for the identification with a not too costful operation.

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 12/13

SLIDE 45

Conclusion

Question

◮ Are k-degenerate graphs (k + 2)-mixing in O(n2)?

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 13/13

SLIDE 46

Conclusion

Question

◮ Are k-degenerate graphs (k + 2)-mixing in O(n2)? ◮ And planar graphs?

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 13/13

SLIDE 47

Conclusion

Question

◮ Are k-degenerate graphs (k + 2)-mixing in O(n2)? ◮ And planar graphs? ◮ Other classes of graphs? (cographs and distance hereditary

graphs are (χ + 1)-mixing (Bonamy, B. ’13+))

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 13/13

SLIDE 48

Conclusion

Question

◮ Are k-degenerate graphs (k + 2)-mixing in O(n2)? ◮ And planar graphs? ◮ Other classes of graphs? (cographs and distance hereditary

graphs are (χ + 1)-mixing (Bonamy, B. ’13+))

◮ How does the diameter decrease when there are more colors?

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 13/13

SLIDE 49

Conclusion

Question

◮ Are k-degenerate graphs (k + 2)-mixing in O(n2)? ◮ And planar graphs? ◮ Other classes of graphs? (cographs and distance hereditary

graphs are (χ + 1)-mixing (Bonamy, B. ’13+))

◮ How does the diameter decrease when there are more colors?

Thanks for your attention !

Marthe Bonamy, Nicolas Bousquet Recoloring bounded treewidth graphs 13/13