SLIDE 1
3-coloring graphs without induced paths on seven vertices
Oliver Schaudt
Universit¨ at zu K¨
- ln, Institut f¨
3-coloring graphs without induced paths on seven vertices Oliver - - PowerPoint PPT Presentation
3-coloring graphs without induced paths on seven vertices Oliver - - PowerPoint PPT Presentation
3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K oln, Institut f ur Informatik with Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Maya Stein, and Mingxian Zhong Graph coloring Graph coloring
SLIDE 2
SLIDE 3
Graph coloring
◮ a proper k-coloring is an assignment of numbers
{1, 2, . . . , k} to the vertices such that any two adjacent vertices receive distinct numbers
SLIDE 4
Graph coloring
◮ a proper k-coloring is an assignment of numbers
{1, 2, . . . , k} to the vertices such that any two adjacent vertices receive distinct numbers
SLIDE 5
Graph coloring
◮ a proper k-coloring is an assignment of numbers
{1, 2, . . . , k} to the vertices such that any two adjacent vertices receive distinct numbers 1 1 1 1 2 2 2 3 3 4 4
SLIDE 6
Graph coloring
◮ a proper k-coloring is an assignment of numbers
{1, 2, . . . , k} to the vertices such that any two adjacent vertices receive distinct numbers 1 1 1 1 2 2 2 3 3 4 4
◮ the related decision problem is called k-colorability
SLIDE 7
Graph coloring
◮ a proper k-coloring is an assignment of numbers
{1, 2, . . . , k} to the vertices such that any two adjacent vertices receive distinct numbers 1 1 1 1 2 2 2 3 3 4 4
◮ the related decision problem is called k-colorability ◮ it is NP-complete for every k ≥ 3
SLIDE 8
k-colorability in H-free graphs
SLIDE 9
k-colorability in H-free graphs
◮ fix some graph H
SLIDE 10
k-colorability in H-free graphs
◮ fix some graph H ◮ a graph is H-free if it does not contain H as an induced
subgraph
SLIDE 11
k-colorability in H-free graphs
◮ fix some graph H ◮ a graph is H-free if it does not contain H as an induced
subgraph
Theorem (Lozin and Kaminski, 2007)
Let H be any graph that is not the disjoint union of paths. Then k-colorability is NP-complete in the class of H-free graphs, for all k ≥ 3.
SLIDE 12
k-colorability in Pt-free graphs
SLIDE 13
k-colorability in Pt-free graphs
◮ Pt is the path on t vertices
SLIDE 14
k-colorability in Pt-free graphs
◮ Pt is the path on t vertices ◮ complexity of k-colorability in the class of Pt-free graphs:
SLIDE 15
k-colorability in Pt-free graphs
◮ Pt is the path on t vertices ◮ complexity of k-colorability in the class of Pt-free graphs:
k/t 4 5 6 7 8 9 . . . 3 O(m) O(nα) O(mnα) P∗ ? ? . . . 4 O(m) P ? NPC NPC NPC . . . 5 O(m) P NPC NPC NPC NPC . . . 6 O(m) P NPC NPC NPC NPC . . . . . . . . . . . . . . . . . . . . . . . . ...
SLIDE 16
k-colorability in Pt-free graphs
◮ Pt is the path on t vertices ◮ complexity of k-colorability in the class of Pt-free graphs:
k/t 4 5 6 7 8 9 . . . 3 O(m) O(nα) O(mnα) P∗ ? ? . . . 4 O(m) P ? NPC NPC NPC . . . 5 O(m) P NPC NPC NPC NPC . . . 6 O(m) P NPC NPC NPC NPC . . . . . . . . . . . . . . . . . . . . . . . . ...
◮ *) our contribution
SLIDE 17
3-colorability in P7-free graphs
SLIDE 18
3-colorability in P7-free graphs
◮ algorithm is split into two cases
SLIDE 19
3-colorability in P7-free graphs
◮ algorithm is split into two cases ◮ input graphs with or without a triangle
SLIDE 20
3-colorability in P7-free graphs
◮ algorithm is split into two cases ◮ input graphs with or without a triangle
SLIDE 21
3-colorability in P7-free graphs
◮ algorithm is split into two cases ◮ input graphs with or without a triangle
SLIDE 22
3-colorability in P7-free graphs
◮ algorithm is split into two cases ◮ input graphs with or without a triangle ◮ triangle-free case is on arxiv (since September)
SLIDE 23
3-colorability in P7-free graphs
◮ algorithm is split into two cases ◮ input graphs with or without a triangle ◮ triangle-free case is on arxiv (since September) ◮ case of graphs contaning a triangle is in preparation
SLIDE 24
Triangle-free case
SLIDE 25
Triangle-free case
Theorem (BCMSSZ, 2014)
Given a (P7,triangle)-free graph G, it can be decided in O(|V (G)|7) time whether G admits a 3-coloring.
SLIDE 26
Triangle-free case
Theorem (BCMSSZ, 2014)
Given a (P7,triangle)-free graph G, it can be decided in O(|V (G)|7) time whether G admits a 3-coloring.
◮ algorithm based on a mild structural analysis
SLIDE 27
Triangle-free case
Theorem (BCMSSZ, 2014)
Given a (P7,triangle)-free graph G, it can be decided in O(|V (G)|7) time whether G admits a 3-coloring.
◮ algorithm based on a mild structural analysis
& smart enumeration
SLIDE 28
Triangle-free case
Theorem (BCMSSZ, 2014)
Given a (P7,triangle)-free graph G, it can be decided in O(|V (G)|7) time whether G admits a 3-coloring.
◮ algorithm based on a mild structural analysis
& smart enumeration & reduction to 2-SAT
SLIDE 29
Triangle-free case
Theorem (BCMSSZ, 2014)
Given a (P7,triangle)-free graph G, it can be decided in O(|V (G)|7) time whether G admits a 3-coloring.
◮ algorithm based on a mild structural analysis
& smart enumeration & reduction to 2-SAT
◮ it actually solves the list 3-colorability problem
SLIDE 30
Triangle-free case
Theorem (BCMSSZ, 2014)
Given a (P7,triangle)-free graph G, it can be decided in O(|V (G)|7) time whether G admits a 3-coloring.
◮ algorithm based on a mild structural analysis
& smart enumeration & reduction to 2-SAT
◮ it actually solves the list 3-colorability problem ◮ every vertex is equipped with a subset of {1, 2, 3} of
admissible colors
SLIDE 31
SLIDE 32
◮ first we determine the core structure of the input graph:
SLIDE 33
◮ first we determine the core structure of the input graph:
SLIDE 34
SLIDE 35
SLIDE 36
SLIDE 37
SLIDE 38
SLIDE 39
◮ outside its core structure, the graph is bipartite
SLIDE 40
SLIDE 41
◮ the non-trival components outside the core structure are
well-behaved
SLIDE 42
◮ the non-trival components outside the core structure are
well-behaved
C1 C2
SLIDE 43
◮ the non-trival components outside the core structure are
well-behaved
C1 C2 Ti
SLIDE 44
◮ the non-trival components outside the core structure are
well-behaved
C1 C2 Ti
N(C1)
SLIDE 45
◮ the non-trival components outside the core structure are
well-behaved
C1 C2 Ti
N(C1) N(C2)
SLIDE 46
◮ the non-trival components outside the core structure are
well-behaved
C1 C2 Ti
N(C1) N(C2)
. . .
SLIDE 47
Triangle-free case
SLIDE 48
Triangle-free case
◮ first we enumerate some partial colorings of the sets Ti
SLIDE 49
Triangle-free case
◮ first we enumerate some partial colorings of the sets Ti
◮ determined by the nested neighborhoods of the non-trivial
components
SLIDE 50
Triangle-free case
◮ first we enumerate some partial colorings of the sets Ti
◮ determined by the nested neighborhoods of the non-trivial
components
◮ leaves a 2-SAT problem for the coloring of the non-trivial
components outside the core structure
SLIDE 51
Triangle-free case
◮ first we enumerate some partial colorings of the sets Ti
◮ determined by the nested neighborhoods of the non-trivial
components
◮ leaves a 2-SAT problem for the coloring of the non-trivial
components outside the core structure
◮ then we enumerate some partial colorings of the sets Dj
SLIDE 52
Triangle-free case
◮ first we enumerate some partial colorings of the sets Ti
◮ determined by the nested neighborhoods of the non-trivial
components
◮ leaves a 2-SAT problem for the coloring of the non-trivial
components outside the core structure
◮ then we enumerate some partial colorings of the sets Dj
◮ leaves a 2-SAT problem for the trivial components
SLIDE 53
Triangle-free case
◮ first we enumerate some partial colorings of the sets Ti
◮ determined by the nested neighborhoods of the non-trivial
components
◮ leaves a 2-SAT problem for the coloring of the non-trivial
components outside the core structure
◮ then we enumerate some partial colorings of the sets Dj
◮ leaves a 2-SAT problem for the trivial components
◮ then we color the rest of the core structure
SLIDE 54
Open problems
SLIDE 55
Open problems
◮ Is there a t such that 3-colorability is NP-complete in Pt-free
graphs?
◮ Is k-colorability FPT in the class of P5-free graphs? ◮ Are there finitely many 4-vertex-critical P6-free graphs?
SLIDE 56