Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

The Complexity of Grundy Coloring and its Variants

Joint work with Florent Foucaud, Eun Jung Kim, and Florian Sikora

Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Hungary (MTA SZTAKI)

FPT seminar of October 7, 2015 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

I Order the vertices v1, v2,..., vn to #colors used by the coloring.

I : vi gets the first color c(vi ) that does not appear in its neighborhood.

I Connected version: ∀i, G[v1 ∪ ... ∪ vi ] is connected. I Weak version: vi can be colored with any color 6 c(vi ). AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Grundy number Γ(G), connected/weak Grundy number

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Grundy coloring

The worst way of reasonably coloring a graph. AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

I Connected version: ∀i, G[v1 ∪ ... ∪ vi ] is connected. I Weak version: vi can be colored with any color 6 c(vi ). AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Grundy number Γ(G), connected/weak Grundy number

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Grundy coloring

The worst way of reasonably coloring a graph.

I Order the vertices v1, v2,..., vn to maximize #colors used by the greedy coloring.

I Greedy coloring: vi gets the first color c(vi ) that does not appear in its neighborhood. AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

I Connected version: ∀i, G[v1 ∪ ... ∪ vi ] is connected. I Weak version: vi can be colored with any color 6 c(vi ). AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Grundy number Γ(G), connected/weak Grundy number

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Grundy coloring

The worst way of reasonably coloring a graph.

I Order the vertices v1, v2,..., vn to maximize #colors used by the greedy coloring.

I Greedy coloring: vi gets the first color c(vi ) that does not appear in its neighborhood. AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Grundy number Γ(G), connected/weak Grundy number

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Grundy coloring

The worst way of reasonably coloring a graph.

I Order the vertices v1, v2,..., vn to maximize #colors used by the greedy coloring.

I Greedy coloring: vi gets the first color c(vi ) that does not appear in its neighborhood.

I Connected version: ∀i, G[v1 ∪ ... ∪ vi ] is connected. I Weak version: vi can be colored with any color 6 c(vi ). AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Grundy coloring

The worst way of reasonably coloring a graph.

I Order the vertices v1, v2,..., vn to maximize #colors used by the greedy coloring.

I Greedy coloring: vi gets the first color c(vi ) that does not appear in its neighborhood.

I Connected version: ∀i, G[v1 ∪ ... ∪ vi ] is connected. I Weak version: vi can be colored with any color 6 c(vi ).

Grundy number Γ(G), connected/weak Grundy number 1 6 1 3 2 4

1 2 5 4

1 3 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Do you see a Grundy coloring reaching color 6? 6 1 3 2 4

1 2 5 4

1 3 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

Do you see a Grundy coloring reaching color 6? 6 1 3 4

1 2 5 4

1 3 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2

Do you see a Grundy coloring reaching color 6? 6 1 3 4

2 5 4

1 3 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2

1

Do you see a Grundy coloring reaching color 6? 6 1 3 4

2 5 4

3 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2

1

1

Do you see a Grundy coloring reaching color 6? 6 1 3 4

2 5 4

3 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2

1

2 1

Do you see a Grundy coloring reaching color 6? 6 1 3 4

2 5 4

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2

1

1 3 2

Do you see a Grundy coloring reaching color 6? 6 1 3 4

2 5

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2

1 4

1 3 2

Do you see a Grundy coloring reaching color 6? 6 1 3

2 5

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2 4 1 4

1 3 2

Do you see a Grundy coloring reaching color 6? 6 1 3

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

2 4 1 5 4

1 3 2

Do you see a Grundy coloring reaching color 6? 6

3

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 1 2 4 1 5 4

1 3 2

Do you see a Grundy coloring reaching color 6? 6

3

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 1 2 4

1 2 5 4

1 3 2

Do you see a Grundy coloring reaching color 6? 6

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 1 3 2 4

1 2 5 4

1 3 2

Do you see a Grundy coloring reaching color 6? A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 6 1 3 2 4

1 2 5 4

1 3 2

Do you see a Grundy coloring reaching color 6? A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 6 1 3 2 4

1 2 5 4

1 3 2

Was it a weak Grundy coloring? A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 6 1 3 2 4

1 2 5 4

1 3 2

Was it a connected Grundy coloring? A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 6 1 3 2 4

1 2 5 4

1 3 2

(Minimal) witness = (minimal) having the same X Grundy number, where X ∈ { −1, weak, connected}. I 2011: Weak Grundy defined by Kierstead and Saoub.

I 2014: Connected Grundy defined by Benevides et al.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

A brief History of Grundy colorings

I 1939: Studied in directed acyclic graphs by Grundy.

I 1979: Formally defined by Kristen and Selkow.

I 1983: Ochromatic number defined by Simmons.

I 1987: Erd¨os et al. proved that ochromatic number = Grundy number. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

A brief History of Grundy colorings

I 1939: Studied in directed acyclic graphs by Grundy.

I 1979: Formally defined by Kristen and Selkow.

I 1983: Ochromatic number defined by Simmons.

I 1987: Erd¨os et al. proved that ochromatic number = Grundy number.

I 2011: Weak Grundy defined by Kierstead and Saoub.

I 2014: Connected Grundy defined by Benevides et al. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Algorithmic motivations

I Γ(G) upper bounds the number of colors used by any greedy heuristic for Min Coloring. I Γ(G) 6 Cχ(G) on some classes of graphs gives a C-approximation for Min Coloring. I Online coloring.

I see Sampaio’s PhD thesis for further motivations. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

More questionable motivations

I (Weak) Grundy Coloring is to (Independent) Dominating Set what Coloring is to Independent Set.

I Is Sudoku more interesting than Grundy Coloring? i I Idea for commercialization: rename color i to 2 and set the goal to 2048. A

parameter XP FPT k n2k−1 ? w nO(w 2) ? k + w - 2O(kw)n

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Complexity of computing the Grundy number

k = Γ(G) and w denotes the of the graph XP algorithm: nf (κ); FPT algorithm: f (κ)nO(1).

I NP-hard on (co-)bipartite, chordal, line, claw-free graphs. 2k−1 I Solvable in n [Z ’06]. O(kw) 1 O(w 2) I Solvable in 2 n and in n [TP ’97].

1 one can show that k 6 1 + w log n [TP ’97] A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Complexity of computing the Grundy number

k = Γ(G) and w denotes the treewidth of the graph XP algorithm: nf (κ); FPT algorithm: f (κ)nO(1).

I NP-hard on (co-)bipartite, chordal, line, claw-free graphs. 2k−1 I Solvable in n [Z ’06]. O(kw) 1 O(w 2) I Solvable in 2 n and in n [TP ’97].

parameter XP FPT k n2k−1 ? w nO(w 2) ? k + w - 2O(kw)n

1 one can show that k 6 1 + w log n [TP ’97] A 4

1 2 3

1 1 2

1

A minimal witness is of size at most 2k−1. (and its vertices are at distance 6 k of the vertex colored by k) Theorem (Zaker ’06) The Grundy number can be computed in O(f (k)n2k−1 ).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

How many vertices (at most) do we need to achieve color k? A

1 2 3

1 1 2

1

A minimal witness is of size at most 2k−1. (and its vertices are at distance 6 k of the vertex colored by k) Theorem (Zaker ’06) The Grundy number can be computed in O(f (k)n2k−1 ).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

How many vertices (at most) do we need to achieve color k?

4 A

1 1 2

1

A minimal witness is of size at most 2k−1. (and its vertices are at distance 6 k of the vertex colored by k) Theorem (Zaker ’06) The Grundy number can be computed in O(f (k)n2k−1 ).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

How many vertices (at most) do we need to achieve color k?

4

1 2 3 A

1

A minimal witness is of size at most 2k−1. (and its vertices are at distance 6 k of the vertex colored by k) Theorem (Zaker ’06) The Grundy number can be computed in O(f (k)n2k−1 ).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

How many vertices (at most) do we need to achieve color k?

4

1 2 3

1 1 2 A

A minimal witness is of size at most 2k−1. (and its vertices are at distance 6 k of the vertex colored by k) Theorem (Zaker ’06) The Grundy number can be computed in O(f (k)n2k−1 ).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

How many vertices (at most) do we need to achieve color k?

4

1 2 3

1 1 2

1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

How many vertices (at most) do we need to achieve color k?

4

1 2 3

1 1 2

1

A minimal witness is of size at most 2k−1. (and its vertices are at distance 6 k of the vertex colored by k) Theorem (Zaker ’06) The Grundy number can be computed in O(f (k)n2k−1 ). Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Complexity of computing the Grundy number

Outline [BFKS ’15]: ∗ n I Solvable in time O (2.443 ).

I FPT in various classes such as H-minor free graphs, chordal graphs, claw-free graphs. ∗ o(w log w) I An O (2 ) algorithm would contradict the ETH. A Outline [BKFS ’15]: ∗ 2O(k) I Weak Grundy Coloring is solvable in O (2 ) but not in 22o(k) 2o(n+m) under the ETH.

I Connected Grundy Coloring is NP-complete for k = 7.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Complexity of the variants

I Weak Grundy Coloring is NP-complete [GV ’97]. I Connected Grundy Coloring is NP-complete on chordal graphs, co-bipartite graphs [B+ ’14]. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Complexity of the variants

I Weak Grundy Coloring is NP-complete [GV ’97]. I Connected Grundy Coloring is NP-complete on chordal graphs, co-bipartite graphs [B+ ’14].

Outline [BKFS ’15]: ∗ 2O(k) I Weak Grundy Coloring is solvable in O (2 ) but not in 22o(k) 2o(n+m) under the ETH.

I Connected Grundy Coloring is NP-complete for k = 7. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Grundy Coloring A Can we improve on this trivial algorithm?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Exact exponential algorithm

Try all possible orderings and run the corresponding greedy coloring: O∗(n!). A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Exact exponential algorithm

Try all possible orderings and run the corresponding greedy coloring: O∗(n!).

Can we improve on this trivial algorithm? I Γ(S) = max{Γ(S \ X), X ind. dom. set in G[S]} + 1.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Solving Grundy Coloring

In a minimal witness:

C1 C2 C3

1 2 3

1 2 3

1 2 3

1 2

S I C is an independent dominating set in G[ C ]. i i6j6k j Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Solving Grundy Coloring

In a minimal witness:

C1 C2 C3

1 2 3

1 2 3

1 2 3

1 2

S I C is an independent dominating set in G[ C ]. i i6j6k j I Γ(S) = max{Γ(S \ X), X ind. dom. set in G[S]} + 1. A

A

The same scheme solves Weak Grundy Coloring in time 2.716n.

O∗(2n) aglorithms? Polynomial space? Connected Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Solving Grundy Coloring

I Enumerating the independent dominating sets takes time ∗ n n O (3 3 ) = O(1.443 ). i I So, filling a cell of the table takes 1.443 for a subset of size i.

I Hence, the total running time is Pn n i n n i=0 i 1.443 = (1 + 1.443) = 2.443 . A

A

O∗(2n) aglorithms? Polynomial space? Connected Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Solving Grundy Coloring

I Enumerating the independent dominating sets takes time ∗ n n O (3 3 ) = O(1.443 ). i I So, filling a cell of the table takes 1.443 for a subset of size i.

I Hence, the total running time is Pn n i n n i=0 i 1.443 = (1 + 1.443) = 2.443 .

The same scheme solves Weak Grundy Coloring in time 2.716n. A

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Solving Grundy Coloring

I Enumerating the independent dominating sets takes time ∗ n n O (3 3 ) = O(1.443 ). i I So, filling a cell of the table takes 1.443 for a subset of size i.

I Hence, the total running time is Pn n i n n i=0 i 1.443 = (1 + 1.443) = 2.443 .

The same scheme solves Weak Grundy Coloring in time 2.716n.

O∗(2n) aglorithms? Polynomial space? Connected Grundy? A

A

A Why does it imply an FPT algorithm for Grundy Coloring?

Minimal witnesses have size at most 2k−1. So, there are less than k222k graphs B to try.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On H-minor free graphs

Induced Subgraph Isomorphism: B induced subgraph of A?

Theorem (FG ’01) ISI is FPT in |V (B)| on H-minor-free graphs. A

A

A

Minimal witnesses have size at most 2k−1. So, there are less than k222k graphs B to try.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On H-minor free graphs

Induced Subgraph Isomorphism: B induced subgraph of A?

Theorem (FG ’01) ISI is FPT in |V (B)| on H-minor-free graphs.

Why does it imply an FPT algorithm for Grundy Coloring? A

A

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On H-minor free graphs

Induced Subgraph Isomorphism: B induced subgraph of A?

Theorem (FG ’01) ISI is FPT in |V (B)| on H-minor-free graphs.

Why does it imply an FPT algorithm for Grundy Coloring?

Minimal witnesses have size at most 2k−1. So, there are less than k222k graphs B to try. A

Besides, ω(G) 6 Γ(G). Therefore, tw(G) 6 Γ(G) − 1 O(tw(G)Γ(G)) O(Γ(G)2) run FPT algorithm in 2 = 2 .

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On chordal graphs

Fact: for any G, tw(G) = ω(G) − 1. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On chordal graphs

Fact: for any chordal graph G, tw(G) = ω(G) − 1.

Besides, ω(G) 6 Γ(G). Therefore, tw(G) 6 Γ(G) − 1 O(tw(G)Γ(G)) O(Γ(G)2) run FPT algorithm in 2 = 2 . A

A

A

Guess a vertex v colored by k := Γ(G) in a minimal witness. This minimal witness is in Nk [v] of size roughly ∆k . Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. Observe that k 6 ∆ + 1.

Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT.

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)). A

A

A

This minimal witness is in Nk [v] of size roughly ∆k . Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. Observe that k 6 ∆ + 1.

Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT.

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)).

Guess a vertex v colored by k := Γ(G) in a minimal witness. A

A

A

Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. Observe that k 6 ∆ + 1.

Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT.

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)).

Guess a vertex v colored by k := Γ(G) in a minimal witness. This minimal witness is in Nk [v] of size roughly ∆k . A

A

A

Observe that k 6 ∆ + 1.

Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT.

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)).

Guess a vertex v colored by k := Γ(G) in a minimal witness. This minimal witness is in Nk [v] of size roughly ∆k . Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. A

A

A Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT.

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)).

Guess a vertex v colored by k := Γ(G) in a minimal witness. This minimal witness is in Nk [v] of size roughly ∆k . Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. Observe that k 6 ∆ + 1. A

A

A

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)).

Guess a vertex v colored by k := Γ(G) in a minimal witness. This minimal witness is in Nk [v] of size roughly ∆k . Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. Observe that k 6 ∆ + 1.

Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT. A

A

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

On claw-free graphs

Observation Grundy Coloring is solvable in time O∗(∆∆O(∆)).

Guess a vertex v colored by k := Γ(G) in a minimal witness. This minimal witness is in Nk [v] of size roughly ∆k . Check if one of the k∆k k-coloring of Nk [v] is a Grundy Coloring. Observe that k 6 ∆ + 1.

Observation For any class such that ∆(G) 6 f (Γ(G)), Grundy Coloring is FPT.

∆(G) In a claw-free graph, if d(v) = ∆(G), then χ(G[N(v)]) > 2 . Besides, Γ(G) > χ(G) > χ(G[N(v)]) holds in any graph. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Weak Grundy Coloring 1 I The probability that a witness is well colored is at least . k2k−1 I Solving the instance is easier with this extra information. 2k−1 I Repeat 100k tries to have a small probability of failure.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Color Coding

Theorem k−1 Weak Grundy Coloring is solvable in O∗(k2 ).

I Color each vertex uniformly at random between 1 and k. I Solving the instance is easier with this extra information. 2k−1 I Repeat 100k tries to have a small probability of failure.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Color Coding

Theorem k−1 Weak Grundy Coloring is solvable in O∗(k2 ).

I Color each vertex uniformly at random between 1 and k. 1 I The probability that a witness is well colored is at least . k2k−1 2k−1 I Repeat 100k tries to have a small probability of failure.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Color Coding

Theorem k−1 Weak Grundy Coloring is solvable in O∗(k2 ).

I Color each vertex uniformly at random between 1 and k. 1 I The probability that a witness is well colored is at least . k2k−1 I Solving the instance is easier with this extra information. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Color Coding

Theorem k−1 Weak Grundy Coloring is solvable in O∗(k2 ).

I Color each vertex uniformly at random between 1 and k. 1 I The probability that a witness is well colored is at least . k2k−1 I Solving the instance is easier with this extra information. 2k−1 I Repeat 100k tries to have a small probability of failure. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #1

6 4 1 1 5 3 4 4 6 3 3 2

2 2 1 6 4

5 3

6 3

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #1

1 1

4 4 3 2

2 2 1 6 4

5 3 4 4 6 3

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #1

1 1

3 2

2 2 1 Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #2

1 2 3 1 5 2 4 1 3 6 4 3

2 1 1 2

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #2

1 3 1 5 2 4 1 3 6 4 3

2 1 1 2

5

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #2

1 3 1 2 4 1 3 6 4 3

2 1 1 2

5

6

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Guess #2

1 3 1 2 4 1 3 4 3

2 1 1 Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

... O(k2k ) unsuccessful guesses later . . .

1 4 6 1 3 2 4 1 1 2 5 4

1 3 2 4

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

... O(k2k ) unsuccessful guesses later . . .

1 6 1 3 2 4 1 1 2 5 4

1 3 2 A A unique optimal (weak) Grundy Coloring. Dominant subtree: largest among its siblings.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to binomial trees Tk = v(T1, T2,..., Tk−1)

4

1 2 3

1 1 2

1 A

Dominant subtree: largest among its siblings.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to binomial trees Tk = v(T1, T2,..., Tk−1)

4

1 2 3

1 1 2

1

A unique optimal (weak) Grundy Coloring. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to binomial trees Tk = v(T1, T2,..., Tk−1)

4

1 2 3

1 1 2

1

A unique optimal (weak) Grundy Coloring. Dominant subtree: largest among its siblings. A User guide: show that only v can get color 4, with degree-based considerations.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

The binomial with missing dominant subtrees

v 4 G − T 3 T

2 2

1 1 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

The binomial tree with missing dominant subtrees

v 4 G − T 3 T

2 2

1 1 1

User guide: show that only v can get color 4, with degree-based considerations. 2 1

1

2 1

1

A dlog me How many dominant T3 in Tdlog me+5? 2 > m A

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH. 2 1

1

2 1

1

A dlog me 2 > m A

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

How many dominant T3 in Tdlog me+5? 2 1

1

2 1

1

A

A

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

How do you want to complete the construction for Weak Grundy? 2 1

1

A

A

2 1

How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

1

How do you want to complete the construction for Weak Grundy? 2 1

1

A

A

1

How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

2 x1 x2 x3 x4 x5 x6

1

How do you want to complete the construction for Weak Grundy? 2 1

1

A

A

How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

2 x1 1 x2 x3 x4 x5 x6

1

How do you want to complete the construction for Weak Grundy? 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

How do you want to complete the construction for Grundy? A

A

2 1

1 1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

How do you want to complete the construction for Grundy? A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

1

How do you want to complete the construction for Grundy? A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

2 x1 x2 x3 x4 x5 x6

1

How do you want to complete the construction for Grundy? A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

2 x1 1 x2 x3 x4 x5 x6

1

How do you want to complete the construction for Grundy? 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

Degree-based considerations . . . 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

Number of vertices and edges? 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

Number of vertices and edges? O(n + m) 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

Value of the parameter? 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

Value of the parameter? dlog me + 5 2 1

1

A

A

2 1

1 How do you want to complete the construction for Weak Grundy?

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Fact: solving Monotone Nae-3-Sat in 2o(n+m) would disprove the ETH.

dlog me How many dominant T3 in Tdlog me+5? 2 > m

3 c1 c2 c3 c4 c5 c6 c7 c8

x1 x2 x3 x4 x5 x6

Solving (Weak) Grundy in 22o(k) 2o(n+m) would disprove the ETH. I Grouping technique: partition the variables into k sets of size n/k you can add one gadget per group assignment. n n n k I ”Compression by permutation” trick: (3 k / log k )! > 2 encode a group of n/k variables with a clique of size only n n 3 k / log k . A Might become tight if the k + w algorithm is improved to O∗(kw ) = O∗(w w (log n)w ) = O∗(w w w 2w ) = O∗(w O(w)).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to Grundy Coloring parameterized by treewidth

Binomial tree gadgetry + tricks for fine-grained lower bounds: Theorem Solving Grundy Coloring in O∗(2o(w log w)) would disprove the ETH. n n n k I ”Compression by permutation” trick: (3 k / log k )! > 2 encode a group of n/k variables with a clique of size only n n 3 k / log k . A Might become tight if the k + w algorithm is improved to O∗(kw ) = O∗(w w (log n)w ) = O∗(w w w 2w ) = O∗(w O(w)).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to Grundy Coloring parameterized by treewidth

Binomial tree gadgetry + tricks for fine-grained lower bounds: Theorem Solving Grundy Coloring in O∗(2o(w log w)) would disprove the ETH.

I Grouping technique: partition the variables into k sets of size n/k you can add one gadget per group assignment. A Might become tight if the k + w algorithm is improved to O∗(kw ) = O∗(w w (log n)w ) = O∗(w w w 2w ) = O∗(w O(w)).

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to Grundy Coloring parameterized by treewidth

Binomial tree gadgetry + tricks for fine-grained lower bounds: Theorem Solving Grundy Coloring in O∗(2o(w log w)) would disprove the ETH.

I Grouping technique: partition the variables into k sets of size n/k you can add one gadget per group assignment. n n n k I ”Compression by permutation” trick: (3 k / log k )! > 2 encode a group of n/k variables with a clique of size only n n 3 k / log k . A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Back to Grundy Coloring parameterized by treewidth

Binomial tree gadgetry + tricks for fine-grained lower bounds: Theorem Solving Grundy Coloring in O∗(2o(w log w)) would disprove the ETH.

I Grouping technique: partition the variables into k sets of size n/k you can add one gadget per group assignment. n n n k I ”Compression by permutation” trick: (3 k / log k )! > 2 encode a group of n/k variables with a clique of size only n n 3 k / log k .

Might become tight if the k + w algorithm is improved to O∗(kw ) = O∗(w w (log n)w ) = O∗(w w w 2w ) = O∗(w O(w)). Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Connected Grundy Coloring 3 1 2

2 1

1 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Connected Grundy number = 3, unbounded witness 3 1 2

2

1 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

Connected Grundy number = 3, unbounded witness 3 1

2

1 2 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2

1

Connected Grundy number = 3, unbounded witness 3 1

2

1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2

1

2

Connected Grundy number = 3, unbounded witness 3 1

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2

1

1 2

Connected Grundy number = 3, unbounded witness 3 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2

2 1

1 2

Connected Grundy number = 3, unbounded witness 3

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 2

2 1

1 2

Connected Grundy number = 3, unbounded witness A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3 1 2

2 1

1 2

Connected Grundy number = 3, unbounded witness I We move along a ”path” P1 of literal vertices: coloring such a vertex by 3 ≡ setting the literal to true.

I We then move along a ”path” P2 of clause vertices cj s: coloring such a vertex by 4 ≡ satisfying the clause.

I To achieve color 7, three special neighbors of the cj s should be colored by 1, 2 and 3 respectively.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

paraNP-hardness of Connected Grundy Coloring

I Reduction from 3Sat-3Occ. I We then move along a ”path” P2 of clause vertices cj s: coloring such a vertex by 4 ≡ satisfying the clause.

I To achieve color 7, three special neighbors of the cj s should be colored by 1, 2 and 3 respectively.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

paraNP-hardness of Connected Grundy Coloring

I Reduction from 3Sat-3Occ.

I We move along a ”path” P1 of literal vertices: coloring such a vertex by 3 ≡ setting the literal to true. I To achieve color 7, three special neighbors of the cj s should be colored by 1, 2 and 3 respectively.

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

paraNP-hardness of Connected Grundy Coloring

I Reduction from 3Sat-3Occ.

I We move along a ”path” P1 of literal vertices: coloring such a vertex by 3 ≡ setting the literal to true.

I We then move along a ”path” P2 of clause vertices cj s: coloring such a vertex by 4 ≡ satisfying the clause. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

paraNP-hardness of Connected Grundy Coloring

I Reduction from 3Sat-3Occ.

I We move along a ”path” P1 of literal vertices: coloring such a vertex by 3 ≡ setting the literal to true.

I We then move along a ”path” P2 of clause vertices cj s: coloring such a vertex by 4 ≡ satisfying the clause.

I To achieve color 7, three special neighbors of the cj s should be colored by 1, 2 and 3 respectively. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

v P i2 1 i1 a4 v1 v1 v2 v2 v3 v3 v4 v4 a6

P2 a9 c1 c2 c3 c4 a11

P1 and P2 for the instance {x1 ∨ ¬x2 ∨ x3}, {x1 ∨ x2 ∨ ¬x4}, {¬x1 ∨ x3 ∨ x4}, {x2 ∨ ¬x3 ∨ x4}. 2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

A connected Grundy coloring setting all the cj s to 4. 2 1 2 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

A connected Grundy coloring setting all the cj s to 4. 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1

A connected Grundy coloring setting all the cj s to 4. 2

3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 2 1

A connected Grundy coloring setting all the cj s to 4. 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1

A connected Grundy coloring setting all the cj s to 4. 3 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 1

A connected Grundy coloring setting all the cj s to 4. 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1

A connected Grundy coloring setting all the cj s to 4. 3 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 1

A connected Grundy coloring setting all the cj s to 4. 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1

A connected Grundy coloring setting all the cj s to 4. 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1

A connected Grundy coloring setting all the cj s to 4. 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3

A connected Grundy coloring setting all the cj s to 4. 3 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 1

A connected Grundy coloring setting all the cj s to 4. 1

1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1

A connected Grundy coloring setting all the cj s to 4. 1 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

A connected Grundy coloring setting all the cj s to 4. 2 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1

A connected Grundy coloring setting all the cj s to 4. 4 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2

A connected Grundy coloring setting all the cj s to 4. 1 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4

A connected Grundy coloring setting all the cj s to 4. 2 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1

A connected Grundy coloring setting all the cj s to 4. 4 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2

A connected Grundy coloring setting all the cj s to 4. 1 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4

A connected Grundy coloring setting all the cj s to 4. 2 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1

A connected Grundy coloring setting all the cj s to 4. 4 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2

A connected Grundy coloring setting all the cj s to 4. 1 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4

A connected Grundy coloring setting all the cj s to 4. 2 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1

A connected Grundy coloring setting all the cj s to 4. 4 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2

A connected Grundy coloring setting all the cj s to 4. 1 A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4

A connected Grundy coloring setting all the cj s to 4. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2 1 2 1 3 1 3 1 1 3 3 1 1

1 2 4 1 2 4 1 2 4 1 2 4 1

A connected Grundy coloring setting all the cj s to 4. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

a30 a31

a29 a32

a28 a33

a27

a15 a23 a22 a26 a25

a7 a12 a13 a14 a21 a20

a3 a5 a6 a10 a11 a16 a17 a19 a24

a1 a2 a4 a8 a9 a18

The doubly circled vertices are linked to all the clause vertices cj s. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

2 1 2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1 2

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

2 1 2 1 1 2 2 2

1 2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3

1 2

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

2 1 2 1 1 2 2 2

2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3

1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

1 2 1 1 2 2 2

2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3 2

1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

2 1 1 2 2 2

2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3 2 1

1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

3 1 2 3 1

2 1 1 2 2 2

2 1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4

3 2 1

1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

3 1 2 3 1

2 1 1 2 2 2

1 1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4

3 2 1

1 2 1 2

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

3 1 2 3 1

2 1 1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4

3 2 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

3 1 2 3 1

1 1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4

3 2 1 2

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

3 1 2 3 1

1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4

3 2 1 2 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

1 2 3 1

1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4 3

3 2 1 2 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

2 3 1

1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4 3 1

3 2 1 2 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

5 4 2 1 3

3 1

1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

4 3 1 2

3 2 1 2 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

3 1

1 2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2

3 2 1 2 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

3 1

2 2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2

3 2 1 2 1 1

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

3 1

2 2

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2

3 2 1 2 1 1 2

1 2 1 2 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

3 1

2 2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2

3 2 1 2 1 1 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

3 1

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2

3 2 1 2 1 1 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

3

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2 1

3 2 1 2 1 1 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 2 1 3

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5

4 3 1 2 3 1

3 2 1 2 1 1 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

4 1 3

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5 2

4 3 1 2 3 1

3 2 1 2 1 1 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

1 3

2

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5 4 2

4 3 1 2 3 1

3 2 1 2 1 1 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

1 3

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5 4 2

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

1

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5 4 2 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

6

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

1 7

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

2 5

7

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

1

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 3 4

5

7

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

2

1

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 4

5

7

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3

2

1

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 5

7

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3 4

2

1

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. 7

A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3 4

2 5

1

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. A

Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

3 4

2 5

1 7

6

5 4 2 1 3

4 3 1 2 3 1

3 2 1 2 1 1 2 2 2

1 2 1 2 1 1

A connected Grundy coloring achieving color 7. Introduction Grundy Coloring Weak Grundy Coloring Connected Grundy Coloring

Open Questions

I Is Grundy Coloring FPT in the highest color k?

I Is Grundy Coloring FPT in the treewidth w? ∗ n I Is (Weak) Grundy Coloring solvable in O (2 )? ∗ n I Is Connected Grundy Coloring solvable in O (c )?