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 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

C

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 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

C R

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 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

R C

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 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

C R

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 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

R C

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 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

C R

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 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

R C

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 2/19

Strategies

Runs: A run on G is a sequence ρk = (v1, v2, v3, v4, . . . , vk)

◮ v2i+1 is a position of C ◮ v2i is a position of R ◮ vi+2 ∈ N(vi)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 3/19

Strategies

Runs: A run on G is a sequence ρk = (v1, v2, v3, v4, . . . , vk) Strategies:

◮ A strategy sC for C is a map sC : (v1, . . . , v2i−1, v2i) → v2i+1

s.t. v2i+1 ∈ N(v2i−1)

◮ A strategy sR for R is a map sR : (v1, . . . , v2i, v2i+1) → v2i+2

s.t. v2i+2 ∈ N(v2i)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 3/19

Strategies

Runs: A run on G is a sequence ρk = (v1, v2, v3, v4, . . . , vk) Strategies:

◮ A strategy sC for C is a map sC : (v1, . . . , v2i−1, v2i) → v2i+1

s.t. v2i+1 ∈ N(v2i−1)

◮ A strategy sR for R is a map sR : (v1, . . . , v2i, v2i+1) → v2i+2

s.t. v2i+2 ∈ N(v2i) Positional strategies:

◮ A positional strategy sC for C is a map sC : V × V → V s.t.

wC = sC(vC, vR) ∈ N[vC]

◮ A positional strategy sR for R is a map sR : V × V → V s.t.

wR = sR(vR, vC) ∈ N[vR]

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 3/19

Winning strategies

Winning strategy: A (positional) strategy sC is a winning strategy if for any strategy sR, if C follows sC and R follows sR, 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 sC is a winning strategy if for any strategy sR, if C follows sC and R follows sR, 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 sC (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 sequence of moves.

◮ x: the last position of C before it

catches R

◮ y: the position of R when C enters x

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 sequence of moves.

◮ x: the last position of C before it

catches R

◮ y: the position of R when C enters x

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 sequence of moves.

◮ x: the last position of C before it

catches R

◮ y: the position of R when C enters x

R C

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

◮ y: the position of R when C enters x

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 sequence of moves.

◮ x: the last position of C before it

catches R

◮ y: the position of R when C enters x

R C

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

◮ y: the position of R when C enters x

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 sequence of moves.

◮ x: the last position of C before it

catches R

◮ y: the position of R when C enters x

R C y 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

◮ y: the position of R when C enters x

R cannot escape, i.e., N[y] ⊆ N[x]

R C y 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

◮ 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]

R C y 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

G \ {y} is still cop-win

Proposition

G is cop-win = ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win Idea of the proof:

◮ in G′ = G \ {y}, C uses a winning strategy for G ◮ each time it goes to y in G, it goes to x in G′

◮ it is a legal move since N[y] ⊆ N[x] ◮ C “remembers” if its position in G is x or y 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19

G \ {y} is still cop-win

Proposition

G is cop-win = ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win Idea of the proof:

◮ in G′ = G \ {y}, C uses a winning strategy for G ◮ each time it goes to y in G, it goes to x in G′

◮ it is a legal move since N[y] ⊆ N[x] ◮ C “remembers” if its position in G is x or y

◮ Consider a run ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) in G’ where C

follows the strategy in G′

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19

G \ {y} is still cop-win

Proposition

G is cop-win = ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win Idea of the proof:

◮ in G′ = G \ {y}, C uses a winning strategy for G ◮ each time it goes to y in G, it goes to x in G′

◮ it is a legal move since N[y] ⊆ N[x] ◮ C “remembers” if its position in G is x or y

◮ Consider a run ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) in G’ where C

follows the strategy in G′

◮ Consider the run ρ = (v1, v′ 2, v3, v′ 4, . . . , ) in G where C

follows the strategy in G

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19

G \ {y} is still cop-win

Proposition

G is cop-win = ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win Idea of the proof:

◮ in G′ = G \ {y}, C uses a winning strategy for G ◮ each time it goes to y in G, it goes to x in G′

◮ it is a legal move since N[y] ⊆ N[x] ◮ C “remembers” if its position in G is x or y

◮ Consider a run ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) in G’ where C

follows the strategy in G′

◮ Consider the run ρ = (v1, v′ 2, v3, v′ 4, . . . , ) in G where C

follows the strategy in G

◮ if v2i+1 = v′ 2i+1, then v2i+1 = y and v′ 2i+1 = x

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19

G \ {y} is still cop-win

Proposition

G is cop-win = ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win Idea of the proof:

◮ in G′ = G \ {y}, C uses a winning strategy for G ◮ each time it goes to y in G, it goes to x in G′

◮ it is a legal move since N[y] ⊆ N[x] ◮ C “remembers” if its position in G is x or y

◮ Consider a run ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) in G’ where C

follows the strategy in G′

◮ Consider the run ρ = (v1, v′ 2, v3, v′ 4, . . . , ) in G where C

follows the strategy in G

◮ if v2i+1 = v′ 2i+1, then v2i+1 = y and v′ 2i+1 = x ◮ G is cop-win ⇒ ∃i, v2i+1 = v′ 2i

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19

G \ {y} is still cop-win

Proposition

G is cop-win = ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win Idea of the proof:

◮ in G′ = G \ {y}, C uses a winning strategy for G ◮ each time it goes to y in G, it goes to x in G′

◮ it is a legal move since N[y] ⊆ N[x] ◮ C “remembers” if its position in G is x or y

◮ Consider a run ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) in G’ where C

follows the strategy in G′

◮ Consider the run ρ = (v1, v′ 2, v3, v′ 4, . . . , ) in G where C

follows the strategy in G

◮ if v2i+1 = v′ 2i+1, then v2i+1 = y and v′ 2i+1 = x ◮ G is cop-win ⇒ ∃i, v2i+1 = v′ 2i = y

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 6/19

A recursive characterization of cop-win graphs

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 7/19

A recursive characterization of cop-win graphs

Proposition

G is cop-win ⇐ ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win We have a positional winning strategy s′

C : V′ × V′ → V′ for C in

G′ = G \ {y} We construct a positional strategy for C in G:

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 7/19

A recursive characterization of cop-win graphs

Proposition

G is cop-win ⇐ ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win We have a positional winning strategy s′

C : V′ × V′ → V′ for C in

G′ = G \ {y} We construct a positional strategy for C in G:

◮ if C can catch R, it does

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 7/19

A recursive characterization of cop-win graphs

Proposition

G is cop-win ⇐ ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win We have a positional winning strategy s′

C : V′ × V′ → V′ for C in

G′ = G \ {y} We construct a positional strategy for C in G:

◮ if C can catch R, it does ◮ if vR = y, then C plays as in G′:

◮ sC(vC, vR) := s′

C(vC, vR)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 7/19

A recursive characterization of cop-win graphs

Proposition

G is cop-win ⇐ ⇒ ∃x, y ∈ V, N[y] ⊆ N[x] and G \ {y} is cop-win We have a positional winning strategy s′

C : V′ × V′ → V′ for C in

G′ = G \ {y} We construct a positional strategy for C in G:

◮ if C can catch R, it does ◮ if vR = y, then C plays as in G′:

◮ sC(vC, vR) := s′

C(vC, vR)

◮ if vR = y, then C plays as in G′ when R is in x:

◮ sC(vC, vR) := s′

C(vC, x)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 7/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y ◮ let ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) where

◮ v′

2i+1 = v2i+1 ∈ V′

◮ v′

2i = v2i ∈ V′ if v′ 2i = y

◮ v′

2i = x ∈ V′ if v′ 2i = y

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y ◮ let ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) where

◮ v′

2i+1 = v2i+1 ∈ V′

◮ v′

2i = v2i ∈ V′ if v′ 2i = y

◮ v′

2i = x ∈ V′ if v′ 2i = y

◮ ρ′ is a run in G′ where C follows s′ C

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y ◮ let ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) where

◮ v′

2i+1 = v2i+1 ∈ V′

◮ v′

2i = v2i ∈ V′ if v′ 2i = y

◮ v′

2i = x ∈ V′ if v′ 2i = y

◮ ρ′ is a run in G′ where C follows s′ C ◮ s′ C is a winning strategy for G′ ⇒ ∃i, v′ 2i+1 = v′ 2i

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y ◮ let ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) where

◮ v′

2i+1 = v2i+1 ∈ V′

◮ v′

2i = v2i ∈ V′ if v′ 2i = y

◮ v′

2i = x ∈ V′ if v′ 2i = y

◮ ρ′ is a run in G′ where C follows s′ C ◮ s′ C is a winning strategy for G′ ⇒ ∃i, v′ 2i+1 = v′ 2i ◮ if v2i = v′ 2i = v′ 2i+1 = v2i+1, then C wins

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y ◮ let ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) where

◮ v′

2i+1 = v2i+1 ∈ V′

◮ v′

2i = v2i ∈ V′ if v′ 2i = y

◮ v′

2i = x ∈ V′ if v′ 2i = y

◮ ρ′ is a run in G′ where C follows s′ C ◮ s′ C is a winning strategy for G′ ⇒ ∃i, v′ 2i+1 = v′ 2i ◮ if v2i = v′ 2i = v′ 2i+1 = v2i+1, then C wins ◮ if v2i = y, then v2i+1 = v′ 2i+1 = v′ 2i = x

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

sC is a winning strategy in G

Suppose that G is not cop-win and consider a run ρ = (v1, v2, v3, v4, . . . , ) in G where C follows sC

◮ if R wins, then ∀i, v2i+1 = y ◮ let ρ′ = (v′ 1, v′ 2, v′ 3, v′ 4, . . . , ) where

◮ v′

2i+1 = v2i+1 ∈ V′

◮ v′

2i = v2i ∈ V′ if v′ 2i = y

◮ v′

2i = x ∈ V′ if v′ 2i = y

◮ ρ′ is a run in G′ where C follows s′ C ◮ s′ C is a winning strategy for G′ ⇒ ∃i, v′ 2i+1 = v′ 2i ◮ if v2i = v′ 2i = v′ 2i+1 = v2i+1, then C wins ◮ if v2i = y, then v2i+1 = v′ 2i+1 = v′ 2i = x ◮ v2i+2 ∈ N[y] ⊆ N[x] = N[v2i+1] and C can catch R at the

next step

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 8/19

Cop-win graphs are dismantlable graphs

Theorem (Nowakowski and Winkler; Quilliot ’83)

A graph G is cop-win iff there exists a dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, N[vi, Gi] ⊆ N[vj] Gi: graph induced by Xi = {v1, v2, . . . , vi} Examples of cop-win graphs: trees, cliques, chordal graphs

v6 v3 v2 v4 v5 v1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 9/19

Cop-win graphs are dismantlable graphs

Theorem (Nowakowski and Winkler; Quilliot ’83)

A graph G is cop-win iff there exists a dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, N[vi, Gi] ⊆ N[vj] Gi: graph induced by Xi = {v1, v2, . . . , vi} Examples of cop-win graphs: trees, cliques, chordal graphs

v3 v2 v4 v5 v1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 9/19

Cop-win graphs are dismantlable graphs

Theorem (Nowakowski and Winkler; Quilliot ’83)

A graph G is cop-win iff there exists a dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, N[vi, Gi] ⊆ N[vj] Gi: graph induced by Xi = {v1, v2, . . . , vi} Examples of cop-win graphs: trees, cliques, chordal graphs

v3 v2 v4 v1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 9/19

Cop-win graphs are dismantlable graphs

Theorem (Nowakowski and Winkler; Quilliot ’83)

A graph G is cop-win iff there exists a dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, N[vi, Gi] ⊆ N[vj] Gi: graph induced by Xi = {v1, v2, . . . , vi} Examples of cop-win graphs: trees, cliques, chordal graphs

v3 v2 v1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 9/19

Cop-win graphs are dismantlable graphs

Theorem (Nowakowski and Winkler; Quilliot ’83)

A graph G is cop-win iff there exists a dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, N[vi, Gi] ⊆ N[vj] Gi: graph induced by Xi = {v1, v2, . . . , vi} Examples of cop-win graphs: trees, cliques, chordal graphs

v2 v1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 9/19

Cop-win graphs are dismantlable graphs

Theorem (Nowakowski and Winkler; Quilliot ’83)

A graph G is cop-win iff there exists a dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, N[vi, Gi] ⊆ N[vj] Gi: graph induced by Xi = {v1, v2, . . . , vi} Examples of cop-win graphs: trees, cliques, chordal graphs

v1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 9/19

Cop & Robber Game with Speeds

A game between one cop C moving at speed s′ and one robber R moving at speed s Same game as before except that at each step

◮ C traverses at most s′ edges ◮ R traverses at most s edges ◮ R can traverse the position of C

C

◮ C has speed s′ = 1 ◮ R has speed s = 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 10/19

Cop & Robber Game with Speeds

A game between one cop C moving at speed s′ and one robber R moving at speed s Same game as before except that at each step

◮ C traverses at most s′ edges ◮ R traverses at most s edges ◮ R can traverse the position of C

C R

◮ C has speed s′ = 1 ◮ R has speed s = 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 10/19

Cop & Robber Game with Speeds

A game between one cop C moving at speed s′ and one robber R moving at speed s Same game as before except that at each step

◮ C traverses at most s′ edges ◮ R traverses at most s edges ◮ R can traverse the position of C

R C

◮ C has speed s′ = 1 ◮ R has speed s = 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 10/19

Cop & Robber Game with Speeds

A game between one cop C moving at speed s′ and one robber R moving at speed s Same game as before except that at each step

◮ C traverses at most s′ edges ◮ R traverses at most s edges ◮ R can traverse the position of C

C R

◮ C has speed s′ = 1 ◮ R has speed s = 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 10/19

Cop & Robber Game with Speeds

A game between one cop C moving at speed s′ and one robber R moving at speed s Same game as before except that at each step

◮ C traverses at most s′ edges ◮ R traverses at most s edges ◮ R can traverse the position of C

C R

◮ C has speed s′ = 1 ◮ R has speed s = 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 10/19

Cop & Robber Game with Speeds

A game between one cop C moving at speed s′ and one robber R moving at speed s Same game as before except that at each step

◮ C traverses at most s′ edges ◮ R traverses at most s edges ◮ R can traverse the position of C

C R

◮ C has speed s′ = 1 ◮ R has speed s = 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 10/19

(s, s′)-Cop-win Graphs and (s, s′)-dismantlability

A graph G is (s,s′)-cop-win if C (moving at speed s′) can win whatever R (moving at speed s) does

Remark

If s < s′, every graph is (s,s′)-cop-win

Theorem (C., Chepoi, Nisse, Vaxès ’11)

A graph G is (s,s′)-cop-win if and only if there exists a (s,s′)-dismantling order v1, v2, . . . , vn such that ∀i > 1, ∃j < i, Bs(vi, G) ∩ Xi ⊆ Bs′(vj) Xi = {v1, v2, . . . , vi}

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 11/19

Why cannot we “remove” vertices from G y x y x v v w w

◮ B3(y, G) ⊆ B2(x, G) ◮ in G′ = G \ {y}, dG′(v, w) = 4 while dG(v, w) = 2 ◮ one cannot use the strategy for G in G′

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 12/19

δ-hyperbolic graphs

A graph (or a metric space) is δ-hyperbolic if for every four points a, b, c, d, d(a, b)+d(c, d) ≤ max{d(a, c)+d(b, d), d(a, d)+d(b, c)}+2δ The hyperbolicity δ∗ of a graph G is the minimal value of δ such that G is δ-hyperbolic

a b d c ≤ δ

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 13/19

δ-hyperbolic graphs

A graph (or a metric space) is δ-hyperbolic if for every four points a, b, c, d, d(a, b)+d(c, d) ≤ max{d(a, c)+d(b, d), d(a, d)+d(b, c)}+2δ The hyperbolicity δ∗ of a graph G is the minimal value of δ such that G is δ-hyperbolic Examples:

◮ Trees and cliques are 0-hyperbolic ◮ Cycles are n 4-hyperbolic ◮ Square grids are

√ n − 1-hyperbolic

◮ Chordal graphs are 1-hyperbolic

[Brinkmann, Koolen, Moulton ’01]

a d c b

n 4 n 4 n 4 n 4 23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 13/19

δ-hyperbolic graphs

A graph (or a metric space) is δ-hyperbolic if for every four points a, b, c, d, d(a, b)+d(c, d) ≤ max{d(a, c)+d(b, d), d(a, d)+d(b, c)}+2δ The hyperbolicity δ∗ of a graph G is the minimal value of δ such that G is δ-hyperbolic Examples:

◮ Trees and cliques are 0-hyperbolic ◮ Cycles are n 4-hyperbolic ◮ Square grids are

√ n − 1-hyperbolic

◮ Chordal graphs are 1-hyperbolic

[Brinkmann, Koolen, Moulton ’01]

a d c b √n − 1 √n − 1

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 13/19

δ-hyperbolic graphs

A graph (or a metric space) is δ-hyperbolic if for every four points a, b, c, d, d(a, b)+d(c, d) ≤ max{d(a, c)+d(b, d), d(a, d)+d(b, c)}+2δ The hyperbolicity δ∗ of a graph G is the minimal value of δ such that G is δ-hyperbolic

Remark

◮ The hyperbolicity of G measures how G is metrically close

from a tree

◮ There exist many definitions of δ-hyperbolicity; they are

equivalent up to a multiplicative factor

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 13/19

Why is hyperbolicity an interesting parameter ?

A notion from Geometric Group Theory [Gromov ’87]

◮ for δ-hyperbolic group, the word problem is solvable in

linear time (it is undecidable for general groups) Some large scale graphs are known to be of small hyperbolicity

◮ the Internet topology can be embedded into a hyperbolic

space [Boguna et al. ’10]

◮ the map of the AS of the Internet has small hyperbolicity

[Cohen et al. ’ 13] Efficient algorithms exist for graphs of small hyperbolicity

◮ Greedy routing algorithms can be expected to perform very

well [Papadopoulos et al. ’09]

◮ Routing labeling schemes with small labels and small

additive error [Chepoi et al ’12]

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 14/19

Cop, robber and hyperbolicity

◮ Characterization of hyperbolicity via cop and robber games

◮ δ-hyperbolic graphs are (2s, s + 2δ)-cop-win for any s ◮ (s, s′)-cop-win graphs are O(s2)-hyperbolic when s′ < s

◮ An efficient algorithm to approximate the hyperbolicity of a

graph

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 15/19

δ-hyperbolic graphs are (2s, s + 2δ)-cop-win

Proposition (from Chepoi, Estellon ’07)

Any δ-hyperbolic graph is (2s, s + 2δ)-dismantlable, and thus (2s, s + 2δ)-cop-win

◮ Consider any BFS ordering of V(G)

from a vertex u

◮ For all v, let v′ be a vertex on a

shortest path from v to u such that d(v, v′) = s

u v ′ v w s ≤ 2s ≤ d(u, v)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 16/19

δ-hyperbolic graphs are (2s, s + 2δ)-cop-win

Proposition (from Chepoi, Estellon ’07)

Any δ-hyperbolic graph is (2s, s + 2δ)-dismantlable, and thus (2s, s + 2δ)-cop-win Let w ∈ B2s(v) ∩ Xv d(u, v′) + d(v, w) ≤ d(u, v′) + 2s ≤ d(u, v) + s d(v, v′) + d(u, w) ≤ s + d(u, v)

u v ′ v w s ≤ 2s ≤ d(u, v)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 16/19

δ-hyperbolic graphs are (2s, s + 2δ)-cop-win

Proposition (from Chepoi, Estellon ’07)

Any δ-hyperbolic graph is (2s, s + 2δ)-dismantlable, and thus (2s, s + 2δ)-cop-win Let w ∈ B2s(v) ∩ Xv d(u, v′) + d(v, w) ≤ d(u, v′) + 2s ≤ d(u, v) + s d(v, v′) + d(u, w) ≤ s + d(u, v) Consequently, d(v′, w) + d(u, v) ≤ s + d(u, v) + 2δ d(v′, w) ≤ s + 2δ

u v ′ v w s ≤ 2s ≤ d(u, v)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 16/19

δ-hyperbolic graphs are (2s, s + 2δ)-cop-win

Proposition (from Chepoi, Estellon ’07)

Any δ-hyperbolic graph is (2s, s + 2δ)-dismantlable, and thus (2s, s + 2δ)-cop-win

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 16/19

δ-hyperbolic graphs are (2s, s + 2δ)-cop-win

Proposition (from Chepoi, Estellon ’07)

Any δ-hyperbolic graph is (2s, s + 2δ)-dismantlable, and thus (2s, s + 2δ)-cop-win In the other direction (not today)

Theorem (C., Chepoi, Papasoglu, Pecatte ’14)

G is (s, s′)-cop-win = ⇒ G is δ-hyperbolic with δ = 16(s+s′)2

s−s′

+ 1

2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 16/19

Computing the hyperbolicity

Assume the distance-matrix of G has been computed Computing the hyperbolicity δ∗(G)

◮ 4 points condition: O(n4)

Computing an approximation of δ∗(G)

◮ fixing one point: a 2-approx. in O(n3)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 17/19

Computing the hyperbolicity

Assume the distance-matrix of G has been computed Computing the hyperbolicity δ∗(G)

◮ 4 points condition: O(n4) ◮ Using (max, min)-matrix product: O(n3.69)

[Fournier, Ismail, Vigneron ’12] Computing an approximation of δ∗(G)

◮ fixing one point: a 2-approx. in O(n3) ◮ Using (max, min)-matrix product: a 2-approx. in O(n2.69)

[Fournier, Ismail, Vigneron ’12]

◮ a (2 + ǫ)-approx. in O( 1 ǫ n2.38)

[Duan ’14]

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 17/19

Computing the hyperbolicity

Assume the distance-matrix of G has been computed Computing the hyperbolicity δ∗(G)

◮ 4 points condition: O(n4) ◮ Using (max, min)-matrix product: O(n3.69)

[Fournier, Ismail, Vigneron ’12] Computing an approximation of δ∗(G)

◮ fixing one point: a 2-approx. in O(n3) ◮ Using (max, min)-matrix product: a 2-approx. in O(n2.69)

[Fournier, Ismail, Vigneron ’12]

◮ a (2 + ǫ)-approx. in O( 1 ǫ n2.38)

[Duan ’14]

Theorem (C., Chepoi, Papasoglu, Pecatte ’14)

From the distance-matrix of G, one can compute a constant approximation of δ∗(G) in O(n2)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 17/19

Approximation Algorithm for δ∗

Approx-δ∗(G,α) Consider a BFS ordering ≺ of V(G) from any vertex u ; For all v, let fα(v) be on a shortest path from v to u such that d(v, fα(v)) = 2α ; for all v ∈ V do if B4α(v, G) ∩ Xv ⊆ B3α(fα(v), G) then return NO return YES;

u fα(v) v 2α Xv

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 18/19

Approximation Algorithm for δ∗

Approx-δ∗(G,α) Consider a BFS ordering ≺ of V(G) from any vertex u ; For all v, let fα(v) be on a shortest path from v to u such that d(v, fα(v)) = 2α ; for all v ∈ V do if B4α(v, G) ∩ Xv ⊆ B3α(fα(v), G) then return NO return YES;

u fα(v) v 2α Xv

NO ≺ is not (2(2α), 2α + α)-dismantling = ⇒ δ∗ > α

2

YES G is (4α, 3α)-dismantlable = ⇒ δ∗ ≤ 16(7α)2

α

+ 1

2 = 784α + 1 2

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 18/19

Approximation Algorithm for δ∗

Approx-δ∗(G,α) Consider a BFS ordering ≺ of V(G) from any vertex u ; For all v, let fα(v) be on a shortest path from v to u such that d(v, fα(v)) = 2α ; for all v ∈ V do if B4α(v, G) ∩ Xv ⊆ B3α(fα(v), G) then return NO return YES;

u fα(v) v 2α Xv

NO ≺ is not (2(2α), 2α + α)-dismantling = ⇒ δ∗ > α

2

YES G is (4α, 3α)-dismantlable = ⇒ δ∗ ≤ 16(7α)2

α

+ 1

2 = 784α + 1 2

We can find α∗ α∗/2 ≤ δ∗ ≤ 784α∗ + 1

2

1569-approx. of δ∗(G)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 18/19

Approximation Algorithm for δ∗

Approx-δ∗(G,α) Consider a BFS ordering ≺ of V(G) from any vertex u ; For all v, let fα(v) be on a shortest path from v to u such that d(v, fα(v)) = 2α ; for all v ∈ V do if B4α(v, G) ∩ Xv ⊆ B3α(fα(v), G) then return NO return YES;

u fα(v) v 2α Xv

Complexity: Approx-δ∗(G,α) runs in time O(n2)

Proposition

One can compute a 1569-approximation of δ∗ in time O(n2 log δ∗)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 18/19

Approximation Algorithm for δ∗

Approx-δ∗(G,α) Consider a BFS ordering ≺ of V(G) from any vertex u ; For all v, let fα(v) be on a shortest path from v to u such that d(v, fα(v)) = 2α ; for all v ∈ V do if B4α(v, G) ∩ Xv ⊆ B3α(fα(v), G) then return NO return YES;

u fα(v) v 2α Xv

We can avoid to recompute everything when we increase α

Theorem

One can compute a 1569-approximation of δ∗ in time O(n2)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 18/19

Conclusion

◮ Characterization of hyperbolicity via a cop and robber

game Different notions that are qualitatively equivalent

◮ (s, s′)-cop-win graphs ◮ (s, s′)-dismantlability ◮ bounded hyperbolicity

◮ Links between (s, s′)-dismantlability and hyperbolicity hold

for infinite graphs

◮ A constant-factor approximation of the hyperbolicity in

O(n2) (starting from the distance-matrix)

23/11/2017 Jouer au gendarme et au voleur pour approximer l’hyperbolicité 19/19