Jouer au gendarme et au voleur pour approximer lhyperbolicit Jrmie - PowerPoint PPT Presentation

Jouer au gendarme et au voleur pour approximer l’hyperbolicité Jérémie Chalopin LIF , CNRS & Aix-Marseille Université 23 novembre 2017 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 1/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex ◮ R chooses a vertex Step-by-step: ◮ C traverses at most 1 edge ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex ◮ R chooses a vertex Step-by-step: C ◮ C traverses at most 1 edge ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex R ◮ R chooses a vertex Step-by-step: C ◮ C traverses at most 1 edge ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex R C ◮ R chooses a vertex Step-by-step: ◮ C traverses at most 1 edge ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex C ◮ R chooses a vertex Step-by-step: R ◮ C traverses at most 1 edge ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex ◮ R chooses a vertex Step-by-step: R ◮ C traverses at most 1 edge C ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex R ◮ R chooses a vertex Step-by-step: ◮ C traverses at most 1 edge C ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Cop & Robber Game A game between one cop C and one robber R on a graph G Initialization: ◮ C chooses a vertex R C ◮ R chooses a vertex Step-by-step: ◮ C traverses at most 1 edge ◮ R traverses at most 1 edge Winning Condition: ◮ C wins if it is on the same vertex as R ◮ R wins if it can avoid C forever 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Strategies Runs: A run on G is a sequence ρ k = ( v 1 , v 2 , v 3 , v 4 , . . . , v k ) ◮ v 2 i + 1 is a position of C ◮ v 2 i is a position of R ◮ v i + 2 ∈ N ( v i ) 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 3/19

Strategies Runs: A run on G is a sequence ρ k = ( v 1 , v 2 , v 3 , v 4 , . . . , v k ) Strategies: ◮ A strategy s C for C is a map s C : ( v 1 , . . . , v 2 i − 1 , v 2 i ) �→ v 2 i + 1 s.t. v 2 i + 1 ∈ N ( v 2 i − 1 ) ◮ A strategy s R for R is a map s R : ( v 1 , . . . , v 2 i , v 2 i + 1 ) �→ v 2 i + 2 s.t. v 2 i + 2 ∈ N ( v 2 i ) 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 3/19

Strategies Runs: A run on G is a sequence ρ k = ( v 1 , v 2 , v 3 , v 4 , . . . , v k ) Strategies: ◮ A strategy s C for C is a map s C : ( v 1 , . . . , v 2 i − 1 , v 2 i ) �→ v 2 i + 1 s.t. v 2 i + 1 ∈ N ( v 2 i − 1 ) ◮ A strategy s R for R is a map s R : ( v 1 , . . . , v 2 i , v 2 i + 1 ) �→ v 2 i + 2 s.t. v 2 i + 2 ∈ N ( v 2 i ) Positional strategies: ◮ A positional strategy s C for C is a map s C : V × V → V s.t. w C = s C ( v C , v R ) ∈ N [ v C ] ◮ A positional strategy s R for R is a map s R : V × V → V s.t. w R = s R ( v R , v C ) ∈ N [ v R ] 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 3/19

Winning strategies Winning strategy: A (positional) strategy s C is a winning strategy if for any strategy s R , if C follows s C and R follows s R , then C wins 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 4/19

Winning strategies Winning strategy: A (positional) strategy s C is a winning strategy if for any strategy s R , if C follows s C and R follows s R , then C wins Proposition C has a winning strategy in G ⇐ ⇒ C has a positional winning strategy A graph G is cop-win if C has a winning strategy s C (i.e., C can win whatever R does) 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 4/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible C R 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a C sequence of moves. ◮ x : the last position of C before it catches R R ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a C sequence of moves. ◮ x : the last position of C before it catches R R ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a sequence of moves. ◮ x : the last position of C before it catches R R C ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a R sequence of moves. ◮ x : the last position of C before it catches R C ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a R C sequence of moves. ◮ x : the last position of C before it catches R ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a R sequence of moves. ◮ x : the last position of C before it catches R C ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a sequence of moves. ◮ x : the last position of C before it catches R x y R C ◮ y : the position of R when C enters x 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a sequence of moves. ◮ x : the last position of C before it catches R x y R C ◮ y : the position of R when C enters x R cannot escape, i.e., N [ y ] ⊆ N [ x ] 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

What are the cop-win graphs? We assume that ◮ C catches R as soon as possible ◮ R escapes for as long as possible Consider a cop-win graph G and a sequence of moves. ◮ x : the last position of C before it catches R x y R C ◮ y : the position of R when C enters x R cannot escape, i.e., N [ y ] ⊆ N [ x ] Proposition G is cop-win = ⇒ ∃ x , y ∈ V , N [ y ] ⊆ N [ x ] 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 5/19

G \ { y } is still cop-win Proposition G is cop-win = ⇒ ∃ x , y ∈ V , N [ y ] ⊆ N [ x ] and G \ { y } is cop-win 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19