Octal Games on Graphs

Laurent Beaudou1, Pierre Coupechoux2, Antoine Dailly3, Sylvain Gravier4, Julien Moncel2, Aline Parreau3, Éric Sopena5

1LIMOS, Clermont-Ferrand 2LAAS, Toulouse 3LIRIS, Lyon 4Institut Fourier, Grenoble 5LaBRI, Bordeaux

This work is part of the ANR GAG (Graphs and Games).

Games And Graphs

CGTC 2017

1/19 I kayles is 0.137

I cram on a single row is 0.07

I The James Bond Game is 0.007

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

Octal games

Deﬁnition Octal games are:

2/19 I kayles is 0.137

I cram on a single row is 0.07

I The James Bond Game is 0.007

Examples

I nim is 0.3333. . .

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

2/19 I kayles is 0.137

I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

2/19 I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

2/19 I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

2/19 I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

2/19 I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

2/19 I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

2/19 I cram on a single row is 0.07

I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

2/19 I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

I cram on a single row is 0.07

2/19 I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

I cram on a single row is 0.07

2/19 I The James Bond Game is 0.007

Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

I cram on a single row is 0.07

2/19 Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

I cram on a single row is 0.07

I The James Bond Game is 0.007

2/19 Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

I cram on a single row is 0.07

I The James Bond Game is 0.007

2/19 Octal games

Deﬁnition Octal games are:

I impartial games;

I played on heaps of counters;

I whose rules are deﬁned by an octal code.

Examples

I nim is 0.3333. . .

I kayles is 0.137

I cram on a single row is 0.07

I The James Bond Game is 0.007

2/19 I kayles: 0,1,2,3,1,4,3,2,. . . after a pre-period 72 it becomes periodic with period 12;

I cram on a single row: 0,1,1,2,0,3,1,1,. . . after a pre-period 53 it becomes periodic with period 34

I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. . . still open,2 28 values computed!

Examples

I nim: 0,1,2,3,4,5,. . .

Conjecture (Guy) All ﬁnite octal games have ultimately periodic Grundy sequences.

Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0, 1, 2,...

3/19 I kayles: 0,1,2,3,1,4,3,2,. . . after a pre-period 72 it becomes periodic with period 12;

I cram on a single row: 0,1,1,2,0,3,1,1,. . . after a pre-period 53 it becomes periodic with period 34

I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. . . still open,2 28 values computed!

Conjecture (Guy) All ﬁnite octal games have ultimately periodic Grundy sequences.

Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0, 1, 2,... Examples

I nim: 0,1,2,3,4,5,. . .

3/19 I cram on a single row: 0,1,1,2,0,3,1,1,. . . after a pre-period 53 it becomes periodic with period 34

I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. . . still open,2 28 values computed!

Conjecture (Guy) All ﬁnite octal games have ultimately periodic Grundy sequences.

Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0, 1, 2,... Examples

I nim: 0,1,2,3,4,5,. . .

I kayles: 0,1,2,3,1,4,3,2,. . . after a pre-period 72 it becomes periodic with period 12;

3/19 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. . . still open,2 28 values computed!

Conjecture (Guy) All ﬁnite octal games have ultimately periodic Grundy sequences.

Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0, 1, 2,... Examples

I nim: 0,1,2,3,4,5,. . .

I kayles: 0,1,2,3,1,4,3,2,. . . after a pre-period 72 it becomes periodic with period 12;

I cram on a single row: 0,1,1,2,0,3,1,1,. . . after a pre-period 53 it becomes periodic with period 34

3/19 Conjecture (Guy) All ﬁnite octal games have ultimately periodic Grundy sequences.

Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0, 1, 2,... Examples

I nim: 0,1,2,3,4,5,. . .

I kayles: 0,1,2,3,1,4,3,2,. . . after a pre-period 72 it becomes periodic with period 12;

I cram on a single row: 0,1,1,2,0,3,1,1,. . . after a pre-period 53 it becomes periodic with period 34

I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. . . still open,2 28 values computed!

3/19 Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0, 1, 2,... Examples

I nim: 0,1,2,3,4,5,. . .

I kayles: 0,1,2,3,1,4,3,2,. . . after a pre-period 72 it becomes periodic with period 12;

I cram on a single row: 0,1,1,2,0,3,1,1,. . . after a pre-period 53 it becomes periodic with period 34

I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. . . still open,2 28 values computed!

Conjecture (Guy) All ﬁnite octal games have ultimately periodic Grundy sequences. 3/19 Removing connected vertices Removing counters from a heap from a graph

Splitting a heap Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs

4/19 Removing connected vertices from a graph

Splitting a heap Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing counters from a heap

4/19 Removing connected vertices from a graph

Splitting a heap Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing counters from a heap

4/19 Splitting a heap Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

4/19 Splitting a heap Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

4/19 Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap

4/19 Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap

4/19 Disconnecting a graph

Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap

4/19 Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap Disconnecting a graph

4/19 Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap Disconnecting a graph

4/19 Playing on a heap ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap Disconnecting a graph

4/19 ≡ Playing on a path

Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap Disconnecting a graph

Playing on a heap

4/19 Octal games on graphs

Natural generalization of the deﬁnition:

Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph

Splitting a heap Disconnecting a graph

Playing on a heap ≡ Playing on a path

4/19 I Study of cycles and wheels, some sort of periodicity on speciﬁc stars (Huggan & Stevens, 2016)

I arc-kayles (Schaeﬀer, 1978) is 0.07

I FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014)

I grim (Adams et al., 2016) is 0.6

I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

5/19 I grim (Adams et al., 2016) is 0.6

I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) I Study of cycles and wheels, some sort of periodicity on speciﬁc stars (Huggan & Stevens, 2016)

5/19 I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) I Study of cycles and wheels, some sort of periodicity on speciﬁc stars (Huggan & Stevens, 2016) I grim (Adams et al., 2016) is 0.6

5/19 I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

5/19 I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

5/19 I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

5/19 I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

5/19 I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I Study of cycles, wheels, random graphs, . . .

5/19 I node-kayles is not an octal game

Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

5/19 Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

5/19 Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

5/19 Octal games on graphs

I arc-kayles (Schaeﬀer, 1978) is 0.07

I Study of cycles, wheels, random graphs, . . .

I Scoring version of 0.6 (Duchêne et al., 2017+)

I node-kayles is not an octal game

5/19 ∅

The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph.

6/19 ∅

The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph.

6/19 ∅

The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph.

6/19 ∅

The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph.

6/19 The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph.

∅

6/19 Corollary A path can be reduced to its length modulo 3 without changing its Grundy value.

The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph. Remark For every integer n, we have G(Pn) = n mod 3.

6/19 The game 0.33 on graphs Rules In the game 0.33, both players alternate removing one or two adjacent vertices without disconnecting the graph. Remark For every integer n, we have G(Pn) = n mod 3. Corollary A path can be reduced to its length modulo 3 without changing its Grundy value.

6/19 ≡ ≡

S1,2,3,6 S1,2 = P4 P1

Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). In other words, each path of a subdivided star can be reduced to its length modulo 3 without changing the Grundy value of the star.

The game 0.33 on subdivided stars Subdivided stars

A subdivided star S`1,...,`k is a graph composed of a central vertex connected to k paths of length `1, ..., `k .

S1,1,2 S4 S1,2,3,6 S1,1,1,1,1,1,1,1

7/19 ≡ ≡

S1,2 = P4 P1

The game 0.33 on subdivided stars Subdivided stars

A subdivided star S`1,...,`k is a graph composed of a central vertex connected to k paths of length `1, ..., `k .

S1,1,2 S4 S1,2,3,6 S1,1,1,1,1,1,1,1

Theorem

S1,2,3,6

7/19 ≡

The game 0.33 on subdivided stars Subdivided stars

A subdivided star S`1,...,`k is a graph composed of a central vertex connected to k paths of length `1, ..., `k .

S1,1,2 S4 S1,2,3,6 S1,1,1,1,1,1,1,1

Theorem

≡

S1,2,3,6 S1,2 = P4

7/19 The game 0.33 on subdivided stars Subdivided stars

A subdivided star S`1,...,`k is a graph composed of a central vertex connected to k paths of length `1, ..., `k .

S1,1,2 S4 S1,2,3,6 S1,1,1,1,1,1,1,1

Theorem

≡ ≡

S1,2,3,6 S1,2 = P4 P1

7/19 +

+ +

Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3).

8/19 +

+ +

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

8/19 +

+ +

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

8/19 +

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

8/19 +

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ +

8/19 +

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ +

8/19 The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ +

8/19 + +

P` + S1,1,`

G(P` + S1,1,`) = 0

⇒ We only need to study stars with paths of length 1 and 2

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

9/19 +

P` + S1,1,`

G(P` + S1,1,`) = 0

⇒ We only need to study stars with paths of length 1 and 2

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ +

9/19 G(P` + S1,1,`) = 0

⇒ We only need to study stars with paths of length 1 and 2

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ + +

P` + S1,1,`

9/19 G = ` + 2 mod 3

G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

Proof We use induction on `.

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3.

9/19 G = ` + 2 mod 3

G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

G( ) = 0 G( ) = 1

9/19 G = ` + 2 mod 3

G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

9/19 G = ` + 2 mod 3

G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

9/19 G = ` + 2 mod 3

G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

9/19 G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

G = ` + 2 mod 3

9/19 G = ` mod 3

The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

G = ` + 2 mod 3

G = ` − 1 mod 3

G = ` − 2 mod 3

9/19 The game 0.33 on subdivided stars Lemma For all `, we have G(S1,1,`) = ` mod 3. Proof We use induction on `.

G = ` + 2 mod 3

G = ` mod 3 G = ` − 1 mod 3

G = ` − 2 mod 3

9/19 ⇒ We only need to study stars with paths of length 1 and 2

The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ + +

P` + S1,1,`

G(P` + S1,1,`) = 0

9/19 The game 0.33 on subdivided stars Theorem

For all `1, . . . , `k , we have G(S`1,...,`k ) = G(S`1 mod 3,...,`k mod 3). Proof

We prove by induction that G(S`1,...,`i ,...,`k ) = G(S`1,...,`i +3,...,`k ).

+ + +

P` + S1,1,`

G(P` + S1,1,`) = 0

⇒ We only need to study stars with paths of length 1 and 2

9/19 0

2 0

0 1 2

1 2 0 1

0 3 1 2 0

1 2 0 3 1 2

0 3 1 2 0 3 (03)∗ 0

1 2 0 3 1 2 (12)∗ 1 2

Grundy values of subdivided stars for the game 0.33

Number of paths of length 2 in the subdivided star 0 1 2 3 4 5 ... 2p 2p + 1

∅

... Number of paths in the subdivided star 2p

2p + 1

10/19 0

2 0

0 1 2

1 2 0 1

0 3 1 2 0

1 2 0 3 1 2

0 3 1 2 0 3 (03)∗ 0

1 2 0 3 1 2 (12)∗ 1 2

Grundy values of subdivided stars for the game 0.33

Number of paths of length 2 in the subdivided star 0 1 2 3 4 5 ... 2p 2p + 1

∅

... Number of paths in the subdivided star 2p

2p + 1

10/19 0 3 1 2 0 3 (03)∗ 0

1 2 0 3 1 2 (12)∗ 1 2

Grundy values of subdivided stars for the game 0.33

Number of paths of length 2 in the subdivided star 0 1 2 3 4 5 ... 2p 2p + 1

∅ 0

0 1

1 2 0

2 0 1 2

3 1 2 0 1

4 0 3 1 2 0

5 1 2 0 3 1 2

... Number of paths in the subdivided star 2p

2p + 1

10/19 Grundy values of subdivided stars for the game 0.33

Number of paths of length 2 in the subdivided star 0 1 2 3 4 5 ... 2p 2p + 1

∅ 0

0 1

1 2 0

2 0 1 2

3 1 2 0 1

4 0 3 1 2 0

5 1 2 0 3 1 2

... Number of paths in the subdivided star 2p 0 3 1 2 0 3 (03)∗ 0

2p + 1 1 2 0 3 1 2 (12)∗ 1 2

10/19 ≡ ≡

3 3 S1,2,3 S2,4 S1,2 S1,2 S1,1,2,2

Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar.

The game 0.33 on subdivided bistars Subdivided bistars m The subdivided bistar S1 S2 is the graph constructed by joining the central vertices of two subdivided stars S1 and S2 by a path of m edges.

1 2 3 S1,1 S1,1 S1,2 ∅ S1,2,3 S2,4

11/19 ≡ ≡

3 S1,2 S1,2 S1,1,2,2

The game 0.33 on subdivided bistars Subdivided bistars m The subdivided bistar S1 S2 is the graph constructed by joining the central vertices of two subdivided stars S1 and S2 by a path of m edges.

1 2 3 S1,1 S1,1 S1,2 ∅ S1,2,3 S2,4

Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar.

3 S1,2,3 S2,4

11/19 ≡

S1,1,2,2

The game 0.33 on subdivided bistars Subdivided bistars m The subdivided bistar S1 S2 is the graph constructed by joining the central vertices of two subdivided stars S1 and S2 by a path of m edges.

1 2 3 S1,1 S1,1 S1,2 ∅ S1,2,3 S2,4

Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar.

≡

3 3 S1,2,3 S2,4 S1,2 S1,2

11/19 The game 0.33 on subdivided bistars Subdivided bistars m The subdivided bistar S1 S2 is the graph constructed by joining the central vertices of two subdivided stars S1 and S2 by a path of m edges.

1 2 3 S1,1 S1,1 S1,2 ∅ S1,2,3 S2,4

Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar.

≡ ≡

3 3 S1,2,3 S2,4 S1,2 S1,2 S1,1,2,2

11/19 + Playing on a subdivided Playing independently on bistar the two subdivided stars . . . except at the end!

G( ) = 0 G( + ) = 0

G( ) = 1 G( + ) = 0

⇒ Reﬁnement of ≡

The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

12/19 . . . except at the end!

G( ) = 0 G( + ) = 0

G( ) = 1 G( + ) = 0

⇒ Reﬁnement of ≡

The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

+ Playing on a subdivided Playing independently on bistar the two subdivided stars

12/19 . . . except at the end!

G( ) = 0 G( + ) = 0

G( ) = 1 G( + ) = 0

⇒ Reﬁnement of ≡

The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

+ Playing on a subdivided Playing independently on ∼? bistar the two subdivided stars

12/19 G( ) = 0 G( + ) = 0

G( ) = 1 G( + ) = 0

⇒ Reﬁnement of ≡

The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

+ Playing on a subdivided Playing independently on ? bistar the two subdivided stars . . . except at the end!

12/19 G( ) = 1 G( + ) = 0

⇒ Reﬁnement of ≡

The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

+ Playing on a subdivided Playing independently on ? bistar the two subdivided stars . . . except at the end!

G( ) = 0 G( + ) = 0

12/19 ⇒ Reﬁnement of ≡

The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

+ Playing on a subdivided Playing independently on ? bistar the two subdivided stars . . . except at the end!

G( ) = 0 G( + ) = 0

G( ) = 1 G( + ) = 0

12/19 The game 0.33 on subdivided bistars

We want to directly compute the Grundy value of a subdivided bistar by using the Grundy values of its stars.

+ Playing on a subdivided Playing independently on ? bistar the two subdivided stars . . . except at the end!

G( ) = 0 G( + ) = 0

G( ) = 1 G( + ) = 0

⇒ Reﬁnement of ≡

12/19 Reﬁnement of ≡ 0 1 0 1 S ∼1 S ⇐⇒ ∀X , S X and S X are equivalent.

≡

6∼1

The Grundy classes will be split into several classes for ∼1.

Reﬁnement of ≡ for subdivided bistars Reminder - Equivalence of games

J1 ≡ J2 ⇐⇒ ∀X , J1 + X and J2 + X have the same outcome.

13/19 ≡

6∼1

The Grundy classes will be split into several classes for ∼1.

Reﬁnement of ≡ for subdivided bistars Reminder - Equivalence of games

J1 ≡ J2 ⇐⇒ ∀X , J1 + X and J2 + X have the same outcome. Reﬁnement of ≡ 0 1 0 1 S ∼1 S ⇐⇒ ∀X , S X and S X are equivalent.

13/19 The Grundy classes will be split into several classes for ∼1.

Reﬁnement of ≡ for subdivided bistars Reminder - Equivalence of games

J1 ≡ J2 ⇐⇒ ∀X , J1 + X and J2 + X have the same outcome. Reﬁnement of ≡ 0 1 0 1 S ∼1 S ⇐⇒ ∀X , S X and S X are equivalent.

≡

6∼1

13/19 Reﬁnement of ≡ for subdivided bistars Reminder - Equivalence of games

J1 ≡ J2 ⇐⇒ ∀X , J1 + X and J2 + X have the same outcome. Reﬁnement of ≡ 0 1 0 1 S ∼1 S ⇐⇒ ∀X , S X and S X are equivalent.

≡

6∼1

The Grundy classes will be split into several classes for ∼1.

13/19 Equivalence classes of ∼1 for the game 0.33

Number of paths of length 2 in the subdivided star 0 1 2 3 4 5 ... 2p 2p + 1

∅ 0

0 1∗

1 2∗ 0

2 0 1∗ 2∗

3 1 2 0 1∗

4 0 3 1 2 0

5 1 2 0 3 1 2

... Number of paths in the subdivided star ∗ 2p 0 3 1 2 0 3 (03) 0

∗ 2p + 1 1 2 0 3 1 2 (12) 1 2

14/19 0 1 1∗ 2 2∗ 2 3 3 0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1∗ ⊕ ⊕ 2 ⊕ 0 ⊕ ⊕ ⊕ 2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2∗ ⊕ ⊕ 0 ⊕ 1 1 ⊕ 0 2 ⊕ ⊕ ⊕ ⊕ 1 ⊕ ⊕ ⊕ 3 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 3 ⊕ ⊕ ⊕ ⊕ 0 ⊕ ⊕ ⊕ where ⊕ is the Nim-sum.

⇒ The values are still in the range 0; 3 J K

Grundy values of subdivided bistars for the game 0.33

1 The Grundy value of S1 S2 depending on the classes of S1 and S2 is given by:

15/19 ⇒ The values are still in the range 0; 3 J K

Grundy values of subdivided bistars for the game 0.33

1 The Grundy value of S1 S2 depending on the classes of S1 and S2 is given by:

0 1 1∗ 2 2∗ 2 3 3 0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1∗ ⊕ ⊕ 2 ⊕ 0 ⊕ ⊕ ⊕ 2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2∗ ⊕ ⊕ 0 ⊕ 1 1 ⊕ 0 2 ⊕ ⊕ ⊕ ⊕ 1 ⊕ ⊕ ⊕ 3 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 3 ⊕ ⊕ ⊕ ⊕ 0 ⊕ ⊕ ⊕ where ⊕ is the Nim-sum.

15/19 Grundy values of subdivided bistars for the game 0.33

1 The Grundy value of S1 S2 depending on the classes of S1 and S2 is given by:

⇒ The values are still in the range 0; 3 J K

15/19 Equivalence classes of ∼2 for the game 0.33

Number of paths of length 2 in the subdivided star 0 1 2 3 4 5 ... 2p 2p + 1

0 1∗

1 2∗ 0

2 0 1∗ 2∗

3 1 2 0 1∗

4 0 3 1 2 0

5 1 2 0∗ 3 1 2

... Number of paths in the subdivided star ∗ 2p 0 3 1 2 0 3 (03) 0

∗ 2p + 1 1 2 0 3 1 2 (12) 1 2

16/19 ⇒ The values are still in the range 0; 3 J K

Grundy values of subdivided bistars for the game 0.33

2 The Grundy value of S1 S2 depending on the classes of S1 and S2 is given by:

0 0∗ 1 1∗ 1 2 2∗ 2 3 3 0 ⊕ ⊕1 ⊕ 2 ⊕1 ⊕ 0 ⊕1 ⊕ ⊕1 ∗ 0 ⊕1 ⊕1 ⊕1 2 ⊕1 ⊕1 0 ⊕1 ⊕1 ⊕1 1 ⊕ ⊕1 ⊕ 3 ⊕1 ⊕ 1 ⊕1 ⊕ ⊕1 1∗ 2 2 3 0 3 0 1 1 1 0 1 ⊕1 ⊕1 ⊕1 3 ⊕1 ⊕1 1 ⊕1 ⊕1 ⊕1 2 ⊕ ⊕1 ⊕ 0 ⊕1 ⊕ 2 ⊕1 ⊕ ⊕1 2∗ 0 0 1 1 1 2 2 2 3 3 2 ⊕1 ⊕1 ⊕1 1 ⊕1 ⊕1 2 0 ⊕1 1 3 ⊕ ⊕1 ⊕ 1 ⊕1 ⊕ 3 ⊕1 ⊕ ⊕1 3 ⊕1 ⊕1 ⊕1 0 ⊕1 ⊕1 3 1 ⊕1 0

where ⊕ is the Nim-sum and x⊕1y stands for x ⊕ y ⊕ 1.

17/19 Grundy values of subdivided bistars for the game 0.33

2 The Grundy value of S1 S2 depending on the classes of S1 and S2 is given by:

where ⊕ is the Nim-sum and x⊕1y stands for x ⊕ y ⊕ 1.

⇒ The values are still in the range 0; 3 17/19 J K 6≡

Proposition The reduction of paths to their length modulo 3 does not work on trees:

Conjecture For all n ≥ 4, there exists a tree T such that G(T ) = n.

G( ) = 10

The game 0.33 on trees

18/19 6≡

Conjecture For all n ≥ 4, there exists a tree T such that G(T ) = n.

G( ) = 10

The game 0.33 on trees

Proposition The reduction of paths to their length modulo 3 does not work on trees:

18/19 Conjecture For all n ≥ 4, there exists a tree T such that G(T ) = n.

G( ) = 10

The game 0.33 on trees

Proposition The reduction of paths to their length modulo 3 does not work on trees:

6≡

18/19 The game 0.33 on trees

Proposition The reduction of paths to their length modulo 3 does not work on trees:

6≡

Conjecture For all n ≥ 4, there exists a tree T such that G(T ) = n.

G( ) = 10

18/19 I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

Conclusion Summary

19/19 I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

Conclusion Summary

I Natural generalization of octal games on graphs;

19/19 I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

19/19 I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

19/19 Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

19/19 I Studying other graph classes;

I Generalize some results on other octal games.

Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

19/19 I Generalize some results on other octal games.

Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

19/19 Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

19/19 Conclusion Summary

I Natural generalization of octal games on graphs;

I Complete resolution of 0.33 on subdivided stars and bistars: every path can be reduced to its length modulo 3;

I Expression of the Grundy value of a subdivided bistar as a pseudo-sum of its two stars’ Grundy values;

I The result does not hold for trees.

Perspectives

I Prove that trees can have arbitrarily large Grundy values;

I Studying other graph classes;

I Generalize some results on other octal games.

19/19