Octal Games on Graphs Laurent Beaudou 1 , Pierre Coupechoux 2 , - PowerPoint PPT Presentation

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 5/19

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 5/19

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 ◮ Study of cycles, wheels, random graphs, . . . 5/19

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 ◮ Study of cycles, wheels, random graphs, . . . ◮ Scoring version of 0.6 (Duchêne et al. , 2017+) 5/19

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 ◮ Study of cycles, wheels, random graphs, . . . ◮ Scoring version of 0.6 (Duchêne et al. , 2017+) ◮ node-kayles is not an octal game 5/19

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 ◮ Study of cycles, wheels, random graphs, . . . ◮ Scoring version of 0.6 (Duchêne et al. , 2017+) ◮ node-kayles is not an octal game 5/19

Octal games on graphs ◮ arc-kayles (Schaeffer, 1978) is 0.07 ◮ FPT when parameterized by the number of rounds played (Lampis & Mitsou, 2014) ◮ Study of cycles and wheels, some sort of periodicity on specific stars (Huggan & Stevens, 2016) ◮ grim (Adams et al. , 2016) is 0.6 ◮ Study of cycles, wheels, random graphs, . . . ◮ Scoring version of 0.6 (Duchêne et al. , 2017+) ◮ 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

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 ( P n ) = 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 ( P n ) = n mod 3. Corollary A path can be reduced to its length modulo 3 without changing its Grundy value. 6/19

The game 0.33 on subdivided stars Subdivided stars A subdivided star S ℓ 1 ,...,ℓ k is a graph composed of a central vertex connected to k paths of length ℓ 1 , ..., ℓ k . S 1 , 1 , 2 S 1 , 2 , 3 , 6 S 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 S 4 7/19

The game 0.33 on subdivided stars Subdivided stars A subdivided star S ℓ 1 ,...,ℓ k is a graph composed of a central vertex connected to k paths of length ℓ 1 , ..., ℓ k . S 1 , 1 , 2 S 1 , 2 , 3 , 6 S 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 S 4 Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ 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. S 1 , 2 , 3 , 6 7/19

The game 0.33 on subdivided stars Subdivided stars A subdivided star S ℓ 1 ,...,ℓ k is a graph composed of a central vertex connected to k paths of length ℓ 1 , ..., ℓ k . S 1 , 1 , 2 S 1 , 2 , 3 , 6 S 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 S 4 Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ 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. ≡ S 1 , 2 = P 4 S 1 , 2 , 3 , 6 7/19

The game 0.33 on subdivided stars Subdivided stars A subdivided star S ℓ 1 ,...,ℓ k is a graph composed of a central vertex connected to k paths of length ℓ 1 , ..., ℓ k . S 1 , 1 , 2 S 1 , 2 , 3 , 6 S 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 S 4 Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ 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. ≡ ≡ S 1 , 2 = P 4 S 1 , 2 , 3 , 6 P 1 7/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + + 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + + + + 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + + + + + 8/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + 9/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + 9/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + + P ℓ + S 1 , 1 ,ℓ 9/19

The game 0.33 on subdivided stars Lemma For all ℓ , we have G ( S 1 , 1 ,ℓ ) = ℓ mod 3. 9/19

The game 0.33 on subdivided stars Lemma For all ℓ , we have G ( S 1 , 1 ,ℓ ) = ℓ mod 3. Proof We use induction on ℓ . ) = 0 ) = 1 G ( G ( 9/19

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

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

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

The game 0.33 on subdivided stars Lemma For all ℓ , we have G ( S 1 , 1 ,ℓ ) = ℓ mod 3. Proof We use induction on ℓ . G = ℓ + 2 mod 3 9/19

The game 0.33 on subdivided stars Lemma For all ℓ , we have G ( S 1 , 1 ,ℓ ) = ℓ mod 3. Proof We use induction on ℓ . G = ℓ + 2 mod 3 G = ℓ − 1 mod 3 G = ℓ − 2 mod 3 9/19

The game 0.33 on subdivided stars Lemma For all ℓ , we have G ( S 1 , 1 ,ℓ ) = ℓ mod 3. Proof We use induction on ℓ . G = ℓ + 2 mod 3 G = ℓ mod 3 G = ℓ − 1 mod 3 G = ℓ − 2 mod 3 9/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + + P ℓ + S 1 , 1 ,ℓ G ( P ℓ + S 1 , 1 ,ℓ ) = 0 9/19

The game 0.33 on subdivided stars Theorem For all ℓ 1 , . . . , ℓ k , we have G ( S ℓ 1 ,...,ℓ k ) = G ( S ℓ 1 mod 3 ,...,ℓ k mod 3 ) . Proof We prove by induction that G ( S ℓ 1 ,...,ℓ i ,...,ℓ k ) = G ( S ℓ 1 ,...,ℓ i + 3 ,...,ℓ k ) . + + + P ℓ + S 1 , 1 ,ℓ G ( P ℓ + S 1 , 1 ,ℓ ) = 0 ⇒ We only need to study stars with paths of length 1 and 2 9/19

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 2 p 2 p + 1 ∅ 0 Number of paths in the subdivided star 1 2 3 4 5 . . . 2 p 2 p + 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 2 p 2 p + 1 ∅ 0 Number of paths in the subdivided star 1 2 3 4 5 . . . 2 p 2 p + 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 2 p 2 p + 1 0 ∅ 0 1 Number of paths in the subdivided star 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 . . . 2 p 2 p + 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 2 p 2 p + 1 0 ∅ 0 1 Number of paths in the subdivided star 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 . . . 2 p ( 03 ) ∗ 0 3 1 2 0 3 0 2 p + 1 ( 12 ) ∗ 1 2 0 3 1 2 1 2 10/19

The game 0.33 on subdivided bistars Subdivided bistars m S 2 is the graph constructed by The subdivided bistar S 1 joining the central vertices of two subdivided stars S 1 and S 2 by a path of m edges. 1 S 1 , 1 S 1 , 2 2 ∅ 3 S 2 , 4 S 1 , 1 S 1 , 2 , 3 11/19

The game 0.33 on subdivided bistars Subdivided bistars m S 2 is the graph constructed by The subdivided bistar S 1 joining the central vertices of two subdivided stars S 1 and S 2 by a path of m edges. 1 S 1 , 1 S 1 , 2 2 ∅ 3 S 2 , 4 S 1 , 1 S 1 , 2 , 3 Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar. 3 S 2 , 4 S 1 , 2 , 3 11/19

The game 0.33 on subdivided bistars Subdivided bistars m S 2 is the graph constructed by The subdivided bistar S 1 joining the central vertices of two subdivided stars S 1 and S 2 by a path of m edges. 1 S 1 , 1 S 1 , 2 2 ∅ 3 S 2 , 4 S 1 , 1 S 1 , 2 , 3 Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar. ≡ 3 S 2 , 4 3 S 1 , 2 S 1 , 2 , 3 S 1 , 2 11/19

The game 0.33 on subdivided bistars Subdivided bistars m S 2 is the graph constructed by The subdivided bistar S 1 joining the central vertices of two subdivided stars S 1 and S 2 by a path of m edges. 1 S 1 , 1 S 1 , 2 2 ∅ 3 S 2 , 4 S 1 , 1 S 1 , 2 , 3 Theorem Each path of a subdivided bistar can be reduced to its length modulo 3 without changing the Grundy value of the bistar. ≡ ≡ 3 S 2 , 4 3 S 1 , 2 S 1 , 1 , 2 , 2 S 1 , 2 , 3 S 1 , 2 11/19

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. 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 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 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! 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! ) = 0 + ) = 0 G ( G ( 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! ) = 0 + ) = 0 G ( G ( ) = 1 + ) = 0 G ( G ( 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! ) = 0 + ) = 0 G ( G ( ) = 1 + ) = 0 G ( G ( ⇒ Refinement of ≡ 12/19

Refinement of ≡ for subdivided bistars Reminder - Equivalence of games ⇒ ∀ X , J 1 + X and J 2 + X have the same outcome. J 1 ≡ J 2 ⇐ 13/19

Refinement of ≡ for subdivided bistars Reminder - Equivalence of games ⇒ ∀ X , J 1 + X and J 2 + X have the same outcome. J 1 ≡ J 2 ⇐ Refinement of ≡ 1 X and S ′ 1 X are equivalent. S ∼ 1 S ′ ⇐ ⇒ ∀ X , S 13/19

Refinement of ≡ for subdivided bistars Reminder - Equivalence of games ⇒ ∀ X , J 1 + X and J 2 + X have the same outcome. J 1 ≡ J 2 ⇐ Refinement of ≡ 1 X and S ′ 1 X are equivalent. S ∼ 1 S ′ ⇐ ⇒ ∀ X , S ≡ �∼ 1 13/19

Refinement of ≡ for subdivided bistars Reminder - Equivalence of games ⇒ ∀ X , J 1 + X and J 2 + X have the same outcome. J 1 ≡ J 2 ⇐ Refinement of ≡ 1 X and S ′ 1 X are equivalent. S ∼ 1 S ′ ⇐ ⇒ ∀ X , S ≡ �∼ 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 2 p 2 p + 1 0 ∅ 0 1 ∗ Number of paths in the subdivided star 1 2 ∗ 0 2 0 1 ∗ 2 ∗ 3 1 0 2 � 1 ∗ 4 0 1 0 3 � 2 � 5 1 0 1 2 2 � 3 � . . . 2 p ( 03 ) ∗ 0 3 � 1 2 � 0 3 0 2 p + 1 ( 12 ) ∗ 1 2 � 0 3 � 1 2 1 2 14/19

Grundy values of subdivided bistars for the game 0.33 1 S 2 depending on the classes of S 1 and The Grundy value of S 1 S 2 is given by: 15/19

Grundy values of subdivided bistars for the game 0.33 1 S 2 depending on the classes of S 1 and The Grundy value of S 1 S 2 is given by: 2 � 3 � 0 1 1 ∗ 2 2 ∗ 3 0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1 ∗ ⊕ ⊕ 2 ⊕ 0 ⊕ ⊕ ⊕ 2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2 ∗ ⊕ ⊕ 0 ⊕ 1 1 ⊕ 0 2 � 1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 3 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 3 � 0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ where ⊕ is the Nim-sum. 15/19