Cop and Robber Game and Hyperbolicity J. Chalopin 1 V. Chepoi 1 . - PowerPoint PPT Presentation

Cop and Robber Game and Hyperbolicity J. Chalopin 1 V. Chepoi 1 . Papasoglu 2 T. Pecatte 3 P 1 LIF , CNRS & Aix-Marseille Université 2 Mathematical Institute, University of Oxford 3 ÉNS de Lyon GRASTA, 31/03/2014 GRASTA’14 Cop and Robber Game and Hyperbolicity 1/15

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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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; 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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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: ◮ C traverses at most 1 edge; R ◮ 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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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 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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

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 GRASTA’14 Cop and Robber Game and Hyperbolicity 2/15

Cop-win graphs are dismantlable graphs A graph G is cop-win if C can win v 6 whatever R does Theorem (Nowakowski and Winkler; Quilliot ’83) v 5 v 3 A graph G is cop-win iff there exists a dismantling order v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , N [ v i , G i ] ⊆ N [ v j ] v 4 v 1 v 2 G i : graph induced by X i = { v 1 , v 2 , . . . , v i } Examples of cop-win graphs: trees, cliques, chordal graphs, bridged graphs GRASTA’14 Cop and Robber Game and Hyperbolicity 3/15

Cop-win graphs are dismantlable graphs A graph G is cop-win if C can win whatever R does Theorem (Nowakowski and Winkler; Quilliot ’83) v 5 v 3 A graph G is cop-win iff there exists a dismantling order v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , N [ v i , G i ] ⊆ N [ v j ] v 4 v 1 v 2 G i : graph induced by X i = { v 1 , v 2 , . . . , v i } Examples of cop-win graphs: trees, cliques, chordal graphs, bridged graphs GRASTA’14 Cop and Robber Game and Hyperbolicity 3/15

Cop-win graphs are dismantlable graphs A graph G is cop-win if C can win whatever R does Theorem (Nowakowski and Winkler; Quilliot ’83) v 3 A graph G is cop-win iff there exists a dismantling order v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , N [ v i , G i ] ⊆ N [ v j ] v 4 v 1 v 2 G i : graph induced by X i = { v 1 , v 2 , . . . , v i } Examples of cop-win graphs: trees, cliques, chordal graphs, bridged graphs GRASTA’14 Cop and Robber Game and Hyperbolicity 3/15

Cop-win graphs are dismantlable graphs A graph G is cop-win if C can win whatever R does Theorem (Nowakowski and Winkler; Quilliot ’83) v 3 A graph G is cop-win iff there exists a dismantling order v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , N [ v i , G i ] ⊆ N [ v j ] v 1 v 2 G i : graph induced by X i = { v 1 , v 2 , . . . , v i } Examples of cop-win graphs: trees, cliques, chordal graphs, bridged graphs GRASTA’14 Cop and Robber Game and Hyperbolicity 3/15

Cop-win graphs are dismantlable graphs A graph G is cop-win if C can win whatever R does Theorem (Nowakowski and Winkler; Quilliot ’83) A graph G is cop-win iff there exists a dismantling order v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , N [ v i , G i ] ⊆ N [ v j ] v 1 v 2 G i : graph induced by X i = { v 1 , v 2 , . . . , v i } Examples of cop-win graphs: trees, cliques, chordal graphs, bridged graphs GRASTA’14 Cop and Robber Game and Hyperbolicity 3/15

Cop-win graphs are dismantlable graphs A graph G is cop-win if C can win whatever R does Theorem (Nowakowski and Winkler; Quilliot ’83) A graph G is cop-win iff there exists a dismantling order v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , N [ v i , G i ] ⊆ N [ v j ] v 1 G i : graph induced by X i = { v 1 , v 2 , . . . , v i } Examples of cop-win graphs: trees, cliques, chordal graphs, bridged graphs GRASTA’14 Cop and Robber Game and Hyperbolicity 3/15

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 ◮ C traverses at most s ′ edge; ◮ R traverses at most s edge. ◮ C has speed s ′ = 1 ◮ R has speed s = 2 GRASTA’14 Cop and Robber Game and Hyperbolicity 4/15

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 ◮ C traverses at most s ′ edge; ◮ R traverses at most s edge. R ◮ C has speed s ′ = 1 ◮ R has speed s = 2 GRASTA’14 Cop and Robber Game and Hyperbolicity 4/15

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 ′ edge; ◮ R traverses at most s edge. R C ◮ C has speed s ′ = 1 ◮ R has speed s = 2 GRASTA’14 Cop and Robber Game and Hyperbolicity 4/15

Cop & Robber Game with Speeds A game between one cop C moving at R speed s ′ and one robber R moving at speed s Same game as before except that at each step ◮ C traverses at most s ′ edge; ◮ R traverses at most s edge. C ◮ C has speed s ′ = 1 ◮ R has speed s = 2 GRASTA’14 Cop and Robber Game and Hyperbolicity 4/15

Cop & Robber Game with Speeds A game between one cop C moving at R speed s ′ and one robber R moving at speed s Same game as before except that at each step C ◮ C traverses at most s ′ edge; ◮ R traverses at most s edge. ◮ C has speed s ′ = 1 ◮ R has speed s = 2 GRASTA’14 Cop and Robber Game and Hyperbolicity 4/15

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 ◮ C traverses at most s ′ edge; ◮ R traverses at most s edge. R ◮ C has speed s ′ = 1 ◮ R has speed s = 2 GRASTA’14 Cop and Robber Game and Hyperbolicity 4/15

( 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 v 1 , v 2 , . . . , v n such that ∀ i > 1 , ∃ j < i , B s ( v i , G \ v j ) ∩ X i ⊆ B s ′ ( v j ) X i = { v 1 , v 2 , . . . , v i } GRASTA’14 Cop and Robber Game and Hyperbolicity 5/15

Two kinds of ( s , s ′ ) -dismantlability An ordering v 1 , v 2 , . . . , v n of the vertices of V ( G ) is ◮ ( s , s ′ ) -dismantling if ∀ i > 1 , ∃ j < i , B s ( v i , G \ v j ) ∩ X i ⊆ B s ′ ( v j ) ◮ ( s , s ′ ) ∗ -dismantling if ∀ i > 1 , ∃ j < i , B s ( v i , G ) ∩ X i ⊆ B s ′ ( v j ) Remarks ⇒ ( s , s − 1 ) -dismantling if s ′ < s ◮ ( s , s ′ ) -dismantling = ◮ ( s , s ′ ) ∗ -dismantling = ⇒ ( s , s ′ ) -dismantling ◮ ( s , s − 1 ) -dismantling = ⇒ ( s , s − 1 ) ∗ -dismantling ◮ G is ( s , s ) ∗ -dismantlable iff G s is dismantlable GRASTA’14 Cop and Robber Game and Hyperbolicity 6/15