Cop and robber games when the robber can hide and ride emie Chalopin - PowerPoint PPT Presentation

Intro Def. Cop-Win Run Hide Shoot Concl. What about grids? What is the cop-number of a n ∗ m grid? 10/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. What about grids? cn(grid)=2 10/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. State of art: characterization and complexity Characterization of cop-win graphs { G | cn ( G ) = 1 } . [Nowakowski & Winkler, 83; Quilliot, 83; Chepoi, 97] Algorithms: O ( n k ) to decide if cn ( G ) ≤ k . [Hahn & MacGillivray, 06] cn ( G ) ≤ k iff the configurations’graph with k cops is copwin. Complexity: Computing the cop-number is EXPTIME-complete. [Goldstein & Reingold, 95] in directed graphs; in undirected graphs if initial positions are given. 11/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. State of art: lower bound For any graph G with girth ≥ 5 and min degree ≥ d , cn ( G ) ≥ d . [Aigner & Fromme, 84] cn ( G ) ≥ d t , where d + 1 = minimum degree, girth ≥ 8 t − 3. [Frankl, 87] ( ⇒ there are n -node graphs G with cn ( G ) ≥ Ω( √ n )) For any k , n , it exists a k -regular graph G with cn ( G ) ≥ n [Andreae, 84] 12/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. State of art: upper bound Planar graph G : cn ( G ) ≤ 3. [Aigner & Fromme, 84] Bounded genus graph G with genus g : cn ( G ) ≤ 3 / 2 g + 3 [Schr¨ oder, 01] Minor free graph G excluding a minor H : cn ( G ) ≤ | E ( H \ { x } ) | , where x is any non-isolated vertex of H [Andreae, 86] General upper bound For any connected graph G , cn ( G ) ≤ O ( n / log ( n )) [Chiniforooshan, 08] 13/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. State of art: upper bound Planar graph G : cn ( G ) ≤ 3. [Aigner & Fromme, 84] Bounded genus graph G with genus g : cn ( G ) ≤ 3 / 2 g + 3 [Schr¨ oder, 01] Minor free graph G excluding a minor H : cn ( G ) ≤ | E ( H \ { x } ) | , where x is any non-isolated vertex of H [Andreae, 86] General upper bound For any connected graph G , cn ( G ) ≤ O ( n / log ( n )) [Chiniforooshan, 08] Conjecture: For any connected graph G , cn ( G ) ≤ O ( √ n ). 13/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. Faster protagonists [Fomin,Golovach,Kratochvil,N.,Suchan, TCS 2010] Speed = max number of edges traversed in 1 step: speed R ≥ speed C = 1 cn s ( G ) min number of cops to capture a robber with speed s in G Computational hardness Computing cn s for any s ≥ 1 is NP-hard; the parameterized version is W [2]-hard. For s ≥ 2, it is true already on split graphs. Fast robber in interval graphs robber with speed s ≥ 1, cn s ( G ) ≤ function ( s ) ⇒ algorithm in time O ( n function ( s ) ) Cop-number is unbounded in planar graphs ∀ s > 1 , ∀ n : then cn s ( Grid n ) = Ω( √ log n ). ∀ H planar with an induced subgraph Grid Ω(2 k 2 ) , cn ( H ) ≥ k . 14/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. Outline General ProblemS 1 Cops & Robber: Definitions, Examples, State of the art 2 Three new variants, Our Problem 3 Cop-win graphs when the Robber/Cops can run 4 The witness version: when the Robber can hide 5 Radius of Capture: when the Cop can shoot 6 15/32 Conclusions 7 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. Three variants we consider When cops and robber can ride s = speed R ≥ speed C = s ′ When the robber can hide (witness) [Clarke DM’08] The robber is visible only every k steps. When the cops can shoot (radius of capture)[BCP TCS’10] Robber captured when at distance k from a cop. Problem: Characterization of cop-win graphs (cop-win graph: in which one cop always captures the robber) 16/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. Three variants we consider When cops and robber can ride s = speed R ≥ speed C = s ′ CWFR ( s , s ′ ) When the robber can hide (witness) [Clarke DM’08] The robber is visible only every k steps. CWW ( k ) When the cops can shoot (radius of capture)[BCP TCS’10] Robber captured when at distance k from a cop. CWRC ( k ) Problem: Characterization of cop-win graphs (cop-win graph: in which one cop always captures the robber) 16/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. Let us go back to a slow robber Characterization of cop-win graphs { G | cn ( G ) = 1 } . [Nowakowski & Winkler, 83; Quilliot, 83; Chepoi, 97] Theorem: cn ( G ) = 1 iff V ( G ) = { v 1 , · · · , v n } and for any i < n , there is j > i s.t. N [ v i ] ⊆ N [ v j ] in the subgraph induced by v i , · · · , v n Trees, chordal graphs, bridged graphs (...) are cop win. 18/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. One fast cop vs. a fast robber Theorem [Nowakowski & Winkler, 83; Quilliot, 83] G ∈ CWFR (1 , 1) iff V ( G ) = { v 1 , · · · , v n } , ∀ i < n , ∃ j > i , s.t. N 1 ( v i , G ) ∩ X i ⊆ N 1 ( v j , G i ) with X i = { v i , · · · , v n } Characterization of CWFR ( s , s ′ ) Theorem G ∈ CWFR ( s , s ′ ) iff V ( G ) = { v 1 , · · · , v n } , ∀ i < n , ∃ j > i , s.t. N s ( v i , G \ { v j } ) ∩ X i ⊆ N s ′ ( v j ) with X i = { v i , · · · , v n } 19/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. Cop-win graphs and hyperbolicity Characterization of CWFR ( s , s ′ ) Theorem G ∈ CWFR ( s , s ′ ) iff V ( G ) = { v 1 , · · · , v n } , ∀ i < n , ∃ j > i , s.t. N s ( v i , G \ { v j } ) ∩ X i ⊆ N s ′ ( v j ) with X i = { v i , · · · , v n } Any graph G is δ -hyperbolic for som δ ≥ 0 the smaller δ , the closer the metric of G is to the metric of a tree. Theorem Hyperbolicity helps the cop ∀ r > 2 δ ≥ 0, and G a δ -hyperbolic graph, G ∈ CWFR (2 r , r + δ ) Theorem Cop-win ”leads” to hyperbolicity If s ≥ 2 s ′ , then any G ∈ CWFR ( s , s ′ ) is ( s − 1)-hyperbolic. Question: ∀ s > s ′ , any G ∈ CWFR ( s , s ′ ) is f ( s )-hyperbolic? 20/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. One slow cop vs. a fast robber We consider C with speed one CWFR ( s ) = CWFR ( s , 1) G ∈ CWFR ( s ) iff V ( G ) = { v 1 , · · · , v n } , ∀ i < n , ∃ j > i , s.t. N s ( v i , G \ { v j } ) ∩ X i ⊆ N 1 ( v j ) with X i = { v i , · · · , v n } Characterization of CWFR ( s ) G ∈ CWFR ( s ) iff G is Case s = 1 : dismantable Case s = 2 : dually-chordal Case s ≥ 3 : a ”big brother graph” G is a big brother graph if each block (maximal 2-connected comp.) is dominated by its articulation point with 21/32 its parent-block. J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. More speed does not help R vs. a slow cop ∀ s ≥ 3, G ∈ CWFR ( s ) iff G is a big brother graph. G big brother ⇒ G ∈ CWFR ( ∞ ) ⊆ · · · ⊆ CWFR (3) 22/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. More speed does not help R vs. a slow cop ∀ s ≥ 3, G ∈ CWFR ( s ) iff G is a big brother graph. G big brother ⇒ G ∈ either G\{v1} is 2−connected dominated by b CWFR ( ∞ ) ⊆ · · · ⊆ CWFR (3) b Let G ∈ CWFR (3) v1 V = { v 1 , · · · , v n } , N 3 ( v 1 , G \ { y } ) ⊆ N 1 ( y ) and y G \ { v 1 } ∈ CWFR (3) induction on the number of blocks - if G is 2 -connected induction on | V ( G ) | We prove G is dominated by y 22/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. More speed does not help R vs. a slow cop ∀ s ≥ 3, G ∈ CWFR ( s ) iff G is a big brother graph. G big brother ⇒ G ∈ or G\{v1} is not 2−connected and y is the dominating articulation point CWFR ( ∞ ) ⊆ · · · ⊆ CWFR (3) v1 Let G ∈ CWFR (3) y w V = { v 1 , · · · , v n } , N 3 ( v 1 , G \ { y } ) ⊆ N 1 ( y ) and G \ { v 1 } ∈ CWFR (3) induction on the number of blocks - if G is 2 -connected v1 induction on | V ( G ) | We prove G is dominated by y y 22/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. More speed does not help R vs. a slow cop ∀ s ≥ 3, G ∈ CWFR ( s ) iff G is a big brother graph. G big brother ⇒ G ∈ CWFR ( ∞ ) ⊆ · · · ⊆ CWFR (3) Let G ∈ CWFR (3) V = { v 1 , · · · , v n } , N 3 ( v 1 , G \ { y } ) ⊆ N 1 ( y ) and G \ { v 1 } ∈ CWFR (3) induction on the number of blocks - if G is 2 -connected ⇒ dominated - if G is not 2 -connected If B a leaf-block dominated by articulation point B G \ B ∈ CWFR (3) because retract ⇒ G \ B is big brother ⇒ G is big brother 22/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. More speed does not help R vs. a slow cop ∀ s ≥ 3, G ∈ CWFR ( s ) iff G is a big brother graph. G big brother ⇒ G ∈ CWFR ( ∞ ) ⊆ · · · ⊆ CWFR (3) Let G ∈ CWFR (3) V = { v 1 , · · · , v n } , N 3 ( v 1 , G \ { y } ) ⊆ N 1 ( y ) and G \ { v 1 } ∈ CWFR (3) induction on the number of blocks - if G is 2 -connected ⇒ dominated - if G is not 2 -connected If B a leaf-block dominated by articulation point G \ B ∈ CWFR (3) because retract ⇒ G \ B is big brother ⇒ G is big brother If not, escape strategy for R 22/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The witness version CWW ( k ) = { G | C wins against R visible every k steps } CWFR ( s ) ⊆ CWW ( s ) equality ?? 24/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The witness version CWW ( k ) = { G | C wins against R visible every k steps } CWFR ( s ) ⊆ CWW ( s ) equality ?? NO: G with diameter 2 and no dominating vertex ⇒ G / ∈ CWFR ( s ) for any s ≥ 2, but G ∈ CWW ( s ) for any k ≥ 1 24/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The witness version CWW ( k ) = { G | C wins against R visible every k steps } CWFR ( s ) ⊆ CWW ( s ) equality ?? NO: G with diameter 2 and no dominating vertex ⇒ G / ∈ CWFR ( s ) for any s ≥ 2, but G ∈ CWW ( s ) for any k ≥ 1 importance of edge-separator 24/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The witness version CWW ( k ) = { G | C wins against R visible every k steps } Lemma invisibility is weaker than speed ∀ s ≥ 2, CWFR ( s ) ⊂ CWW ( s ) Lemma less visibility helps R ∀ k ≥ 1, there are graphs in CWW ( k ) \ CWW ( k + 1) 24/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The witness version CWW ( k ) = { G | C wins against R visible every k steps } Lemma invisibility is weaker than speed ∀ s ≥ 2, CWFR ( s ) ⊂ CWW ( s ) Lemma less visibility helps R ∀ k ≥ 1, there are graphs in CWW ( k ) \ CWW ( k + 1) 24/32 Question: CWW ( k + 1) ⊂ CWW ( k )? J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The big two-brother graphs CWW = { G |∀ k , C wins vs. R visible every k steps } = � k CWW ( k ) Theorem CWW is the class of the big two-brother graphs G is a big two-brother graph if ∃ y ∈ V or xy ∈ E s.t. x or y dominated a connected comp. C of G \ { x , y } and G \ C is a big two-brother graph 25/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The big two-brother graphs CWW = { G |∀ k , C wins vs. R visible every k steps } = � k CWW ( k ) Theorem CWW is the class of the big two-brother graphs G is a big two-brother graph if ∃ y ∈ V or xy ∈ E s.t. x or y dominated a connected comp. C of G \ { x , y } and G \ C is a big two-brother graph * big two-brother ⇒ CWW (easy) 25/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. The big two-brother graphs CWW = { G |∀ k , C wins vs. R visible every k steps } = � k CWW ( k ) Theorem CWW is the class of the big two-brother graphs G is a big two-brother graph if ∃ y ∈ V or xy ∈ E s.t. x or y dominated a connected comp. C of G \ { x , y } and G \ C is a big two-brother graph * big two-brother ⇒ CWW (easy) * CWW ⇒ big two-brother ∀ k , G ∈ CWW ( k 2 ) without degree-1 vertex then, ∃ v ∈ V , xy ∈ E , N k ( v , G \ xy ) ⊆ N 1 ( y ) 25/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Intro Def. Cop-Win Run Hide Shoot Concl. CWW (2) Not sufficient Lemma Necessity e h If G ∈ CWW (2) then i V = { v 1 , · · · , v n } , and ∀ i , ∃ xy ∈ E ( G i +1 ) a d (possibly x = y ) N 2 ( v i , G \ xy ) ∩ G i ⊆ N 1 ( y ) f g Lemma Sufficiency b c Not necessary If V = { v 1 , · · · , v n } , and ∀ i , ∃ xy ∈ E ( G i +1 ) (possibly x = y ) N 2 ( v i , G \ xy ) ∩ G i ⊆ N 1 ( y ) and, if x � = y , then N 2 ( v i , G \ y ) ∩ G i ⊆ N 2 ( x , G \ y ) then G ∈ CWW (2) 26/32 J. Chalopin, V. Chepoi, N. Nisse, Y. Vax` es Cop and robber games when the robber can hide and ride

Cop and robber games when the robber can hide and ride emie Chalopin - PowerPoint PPT Presentation

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