On the Grundy number of a graph
On the Grundy number of a graph ´ eric ´ Havet Fred
Mascotte Project
Leonardo Sampaio
INRIA/I3S(CNRS UNS) – 1 / 32
Graph colourings ❖ Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
Graph colourings
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 2 / 32
Graph colourings Graph colourings ❖ Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
● k-colouring: a function c : V (G) →{1, 2, ..., k} or (equivalently)
a partition S1 , S2 , ..., Sk of V (G). ● proper colouring: adjacent vertices receive distinct colours. ● chromatic number : χ(G) = min{k | G admits proper k
colouring}. ● Determining χ(G) is N P -hard.
Brooks: G connected: χ(G) ≤ ∆(G) unless G is either a complete graph or an odd cycle, when χ(G) = ∆(G) + 1.
Mascotte Project
INRIA/I3S(CNRS UNS) – 3 / 32
Graph colourings Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
Greedy colourings
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 4 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 5 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
INRIA/I3S(CNRS UNS) – 6 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
INRIA/I3S(CNRS UNS) – 7 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
INRIA/I3S(CNRS UNS) – 8 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
3
INRIA/I3S(CNRS UNS) – 9 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
3
1
INRIA/I3S(CNRS UNS) – 10 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
3
1
4
INRIA/I3S(CNRS UNS) – 11 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
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1
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With reverse ordering:
1 Mascotte Project
2
INRIA/I3S(CNRS UNS) – 12 / 32
Greedy colourings(cont.) Graph colourings
● A colouring is greedy iff:
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
✦ For every j < i, if v is in Si it has a neighbour in Sj .
(P )
● A colouring satisfying Property (P ) is a greedy colouring
relative to a vertex ordering in which the vertices of Si precede those of Sj , i < j. ● Grundy number :Γ(G) = max {k | G has greedy k-colouring }. ● Obviously, Γ(G) ≤ ∆(G) + 1.
Mascotte Project
INRIA/I3S(CNRS UNS) – 13 / 32
Graphs with large Γ(G) − χ(G) Graph colourings
t=1
t=2
Greedy colourings
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t=4
t=k
❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 14 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
Inexistence of a Brooks type theorem for the Grundy number
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 15 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete. Proof ● Reduction from 3-edge colourability of 3-regular graphs. ● Build the incidence graph of the instance of this problem:
Fixed-parameter complexity of the Grundy number
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Future Works
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Mascotte Project
INRIA/I3S(CNRS UNS) – 16 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 17 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 18 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 19 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 20 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem
Fixed-parameter complexity of the Grundy number
❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 21 / 32
Parameterized Greedy colouring problem Graph colourings
Is the following problem FPT:
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
G REEDY C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: Γ(G) ≥ k ?
● There is an algorithm to solve it in time O(n
2k−1
).
● It just checks the existence of k-atoms in G.
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 22 / 32
t-atoms Graph colourings
1−atom
3−atoms
2−atom
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring
(t−1)−atom
matching
independent set
A typical t−atom
❖ Fixed-parameter complexity of the dual of greedy colouring Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 23 / 32
Dual of greedy colouring Graph colourings
Telle proved that the following is FPT:
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
D UAL OF C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: χ(G) ≤ |V (G)| − k? We have proved that the following analog is FPT: D UAL OF G REEDY C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: Γ(G) ≥ |V (G)| − k ?
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 24 / 32
Fixed-parameter complexity of the dual of greedy colouring Graph colourings
Theorem 1. To determine if Γ(G) ≥ |V (G)| − k is FPT.
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
Proof ● If c is a Grundy colouring that uses at least |V (G)| − k, then
there are at most 2k vertices in the colour classes of size > 1. ● These are actually the colour classes that ’saves’ colours.
G size 1 colour classes
Future Works size >1 colour classes (# smaller than k)
C1
Mascotte Project
C2
C3
Ck‘
INRIA/I3S(CNRS UNS) – 25 / 32
Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number
G
complement of G
size 1 colour classes
Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
size >1 colour classes (# smaller than k)
C1
C2
C3
Ck‘ vertex cover in the complement (at most 2k vertices)
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 26 / 32
Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring
Lemma 2. Let G = (V, E) be a graph and k ≥ 0 an integer. Then, Γ(G) ≥ |V | − k if and only there is a vertex cover C of G such that GhCi admits a greedy colouring (C1 , C2 , . . . , Ck′ ) with the following properties: P1: |C| − k ≤ k′ ≤ k; P2: |Ci | ≥ 2, for every 1 ≤ i ≤ k′ ; P3: For each v ∈ V \C and for every 1 ≤ i ≤ k′ , there is u ∈ Ci such that uv ∈ E.
❖ Fixed-parameter complexity of the dual of greedy colouring Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 27 / 32
Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Graphs with Γ(G) ≥ |V (G)| − k characterized by a vertex
cover in G with some properties. ● There is algorithm to enumerate all minimal vertex covers of
size ≤ 2k in time O(22k |V |): but need to consider the ones that are not minimal.
❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
● Every vertex cover contains a minimal one.
● We need an algorithm to determine if a minimal vertex cover
is contained in a ”good” one.
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 28 / 32
Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
Lemma 3. Let k be an integer, G = (V, E) a graph and Cmin a minimal vertex cover of G of size at most 2k. It’s FPT to decide if Cmin is contained in a suitable vertex cover C. The ideia is: ● Given a minimal vertex cover C, generate all proper
(≤ k)-colourings of G[C]. ● Given such colouring, if it is ”good” we answer yes. ● Else, define ”defective” colour classes, and for the vertices
that are not coloured, such ”defective” colour classes have a set of ”candidates” that we may place at it.
Future Works
● We only need to determine if we can find a matching between
candidates and defective colour classes to decide if the minimal vertex cover is contained in a ”good” one.
Mascotte Project
INRIA/I3S(CNRS UNS) – 29 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
Future Works
INRIA/I3S(CNRS UNS) – 30 / 32
Future Works Graph colourings Greedy colourings
●
Does Dual of Greedy colouring have a polynomial kernel?
●
Is the following FPT:
Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
G REEDY C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: Γ(G) ≥ k ?
INRIA/I3S(CNRS UNS) – 31 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
Thanks!
INRIA/I3S(CNRS UNS) – 32 / 32
Mascotte Project
Leonardo Sampaio
INRIA/I3S(CNRS UNS) – 1 / 32
Graph colourings ❖ Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
Graph colourings
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 2 / 32
Graph colourings Graph colourings ❖ Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
● k-colouring: a function c : V (G) →{1, 2, ..., k} or (equivalently)
a partition S1 , S2 , ..., Sk of V (G). ● proper colouring: adjacent vertices receive distinct colours. ● chromatic number : χ(G) = min{k | G admits proper k
colouring}. ● Determining χ(G) is N P -hard.
Brooks: G connected: χ(G) ≤ ∆(G) unless G is either a complete graph or an odd cycle, when χ(G) = ∆(G) + 1.
Mascotte Project
INRIA/I3S(CNRS UNS) – 3 / 32
Graph colourings Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
Greedy colourings
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 4 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 5 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
INRIA/I3S(CNRS UNS) – 6 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
INRIA/I3S(CNRS UNS) – 7 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
INRIA/I3S(CNRS UNS) – 8 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
3
INRIA/I3S(CNRS UNS) – 9 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
1
2
3
1
INRIA/I3S(CNRS UNS) – 10 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
1
Mascotte Project
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INRIA/I3S(CNRS UNS) – 11 / 32
Greedy colourings Graph colourings
● Greedy colouring: produced by the greedy algorithm.
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Ordering V (G) as σ = v1 < v2 < ... < vn : assigns to vi the
smallest positive integer such that the (partial) colouring remains proper. Example:
Future Works
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With reverse ordering:
1 Mascotte Project
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INRIA/I3S(CNRS UNS) – 12 / 32
Greedy colourings(cont.) Graph colourings
● A colouring is greedy iff:
Greedy colourings ❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
✦ For every j < i, if v is in Si it has a neighbour in Sj .
(P )
● A colouring satisfying Property (P ) is a greedy colouring
relative to a vertex ordering in which the vertices of Si precede those of Sj , i < j. ● Grundy number :Γ(G) = max {k | G has greedy k-colouring }. ● Obviously, Γ(G) ≤ ∆(G) + 1.
Mascotte Project
INRIA/I3S(CNRS UNS) – 13 / 32
Graphs with large Γ(G) − χ(G) Graph colourings
t=1
t=2
Greedy colourings
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❖ Greedy colourings ❖ Graphs with large Γ(G) − χ(G) Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 14 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
Inexistence of a Brooks type theorem for the Grundy number
Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 15 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete. Proof ● Reduction from 3-edge colourability of 3-regular graphs. ● Build the incidence graph of the instance of this problem:
Fixed-parameter complexity of the Grundy number
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Mascotte Project
INRIA/I3S(CNRS UNS) – 16 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
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Future Works v3
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Mascotte Project
INRIA/I3S(CNRS UNS) – 17 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 18 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 19 / 32
Inexistence of a Brooks type theorem for the Grundy number Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number ❖ Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
To decide if a bipartite graph G satisfies Γ(G) = ∆(G) + 1 is NP-complete.
v1 e1 e2 v2 e3 e4
Future Works v3
e5 e6
v4
Mascotte Project
INRIA/I3S(CNRS UNS) – 20 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem
Fixed-parameter complexity of the Grundy number
❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring Future Works
Mascotte Project
INRIA/I3S(CNRS UNS) – 21 / 32
Parameterized Greedy colouring problem Graph colourings
Is the following problem FPT:
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
G REEDY C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: Γ(G) ≥ k ?
● There is an algorithm to solve it in time O(n
2k−1
).
● It just checks the existence of k-atoms in G.
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t-atoms Graph colourings
1−atom
3−atoms
2−atom
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring
(t−1)−atom
matching
independent set
A typical t−atom
❖ Fixed-parameter complexity of the dual of greedy colouring Future Works
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Dual of greedy colouring Graph colourings
Telle proved that the following is FPT:
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
D UAL OF C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: χ(G) ≤ |V (G)| − k? We have proved that the following analog is FPT: D UAL OF G REEDY C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: Γ(G) ≥ |V (G)| − k ?
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Fixed-parameter complexity of the dual of greedy colouring Graph colourings
Theorem 1. To determine if Γ(G) ≥ |V (G)| − k is FPT.
Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
Proof ● If c is a Grundy colouring that uses at least |V (G)| − k, then
there are at most 2k vertices in the colour classes of size > 1. ● These are actually the colour classes that ’saves’ colours.
G size 1 colour classes
Future Works size >1 colour classes (# smaller than k)
C1
Mascotte Project
C2
C3
Ck‘
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Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number
G
complement of G
size 1 colour classes
Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
size >1 colour classes (# smaller than k)
C1
C2
C3
Ck‘ vertex cover in the complement (at most 2k vertices)
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Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring
Lemma 2. Let G = (V, E) be a graph and k ≥ 0 an integer. Then, Γ(G) ≥ |V | − k if and only there is a vertex cover C of G such that GhCi admits a greedy colouring (C1 , C2 , . . . , Ck′ ) with the following properties: P1: |C| − k ≤ k′ ≤ k; P2: |Ci | ≥ 2, for every 1 ≤ i ≤ k′ ; P3: For each v ∈ V \C and for every 1 ≤ i ≤ k′ , there is u ∈ Ci such that uv ∈ E.
❖ Fixed-parameter complexity of the dual of greedy colouring Future Works
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Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number
● Graphs with Γ(G) ≥ |V (G)| − k characterized by a vertex
cover in G with some properties. ● There is algorithm to enumerate all minimal vertex covers of
size ≤ 2k in time O(22k |V |): but need to consider the ones that are not minimal.
❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
● Every vertex cover contains a minimal one.
● We need an algorithm to determine if a minimal vertex cover
is contained in a ”good” one.
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Fixed-parameter complexity of the dual of greedy colouring Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number ❖ Parameterized Greedy colouring problem ❖ t-atoms ❖ Dual of greedy colouring ❖ Fixed-parameter complexity of the dual of greedy colouring
Lemma 3. Let k be an integer, G = (V, E) a graph and Cmin a minimal vertex cover of G of size at most 2k. It’s FPT to decide if Cmin is contained in a suitable vertex cover C. The ideia is: ● Given a minimal vertex cover C, generate all proper
(≤ k)-colourings of G[C]. ● Given such colouring, if it is ”good” we answer yes. ● Else, define ”defective” colour classes, and for the vertices
that are not coloured, such ”defective” colour classes have a set of ”candidates” that we may place at it.
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● We only need to determine if we can find a matching between
candidates and defective colour classes to decide if the minimal vertex cover is contained in a ”good” one.
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Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
Future Works
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Future Works Graph colourings Greedy colourings
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Does Dual of Greedy colouring have a polynomial kernel?
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Is the following FPT:
Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
G REEDY C OLOURING Instance: A graph G and an integer k. Parameter: k. Question: Γ(G) ≥ k ?
INRIA/I3S(CNRS UNS) – 31 / 32
Graph colourings Greedy colourings Inexistence of a Brooks type theorem for the Grundy number Fixed-parameter complexity of the Grundy number Future Works
Mascotte Project
Thanks!
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