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

1/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cop and robber games when the robber can hide and ride

J´ er´ emie Chalopin1 Victor Chepoi1 Nicolas Nisse2 Yann Vax` es1

1 Lab. Informatique Fondamentale, Univ. Aix-Marseille, CNRS, Marseille, France 2 MASCOTTE, INRIA, I3S, CNRS, UNS, Sophia Antipolis, France

Seminario, Universidade Federal do Cear´ a, April 9th, 2010

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

2/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

3/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber/pursuit-evasion/graph searching

Capture an intruder in a network a team of cops/searchers vs. a robber/intruder/fugitive Combinatorial Problem: search/cop number: minimum number of searchers to capture the fugitive whatever he does in a network. Algorithmic Problem: strategy (sequence of moves) of the searchers allowing them to capture the fugitive

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

4/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Taxonomy of graph searching games

Capture = Surround and/or Occupy the same node as the fugitive

robber’s characteristics bounded speed arbitrary fast visible invisible visible invisible Cops turn by turn & X X ? Occupy Robber graph searching simultaneous ? X treewidth pathwidth

• Occ. & Sur.

Capture = Surround the fugitive

simultaneous ? ? ? routing Surround reconfig.

? = No studies (as far as I know) X = Very few studies

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

5/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

6/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Focus on Cops & Robber games

Capture an intruder in a network (= a graph) C plays with a team of cops R plays with one robber Cops’ goal: C: Capture the robber Capture = a Cop occupies the same vertex as the Robber Robber’s goal: R: Perpetually evade

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

7/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cops & robber games [Nowakowski and Winkler; Quilliot, 83]

Initialization:

1 C places the cops; 2 R places the robber.

Step-by-step: each cop traverses at most 1 edge; the robber traverses at most 1 edge. Robber apprehended: A cop occupies the same vertex as the robber.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

8/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cop number

Easy cases for the Cops n cops in n-node graphs k cops in a graph with dominating set ≤ k → Robber’s dead An easy case for the Robber in a C4 against a single cop

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

8/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cop number

Easy cases for the Cops n cops in n-node graphs k cops in a graph with dominating set ≤ k → Robber’s dead An easy case for the Robber in a C4 against a single cop Minimize the number of Cops Capture the robber using as few cops as possible Given a graph G: the minimum called cop-number, cn(G).

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

9/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Simple Examples

Cop number in cliques and trees

cn (Kn) = 1 cn (T) = 1

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

What is the cop-number of a n ∗ m grid?

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

10/32 Intro Def. Cop-Win Run Hide Shoot Concl.

What about grids?

cn(grid)=2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

11/32 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(nk) 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.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

12/32 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) ≥ dt, where d + 1 = minimum degree, girth ≥ 8t − 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]

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

13/32 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) ≤ 3/2g + 3 [Schr¨

• der, 01]

Minor free graph G excluding a minor H: cn(G) ≤ |E(H \ {x})|, where x is any non-isolated vertex

• f H [Andreae, 86]

General upper bound For any connected graph G, cn(G) ≤ O(n/log(n)) [Chiniforooshan, 08]

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

13/32 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) ≤ 3/2g + 3 [Schr¨

• der, 01]

Minor free graph G excluding a minor H: cn(G) ≤ |E(H \ {x})|, where x is any non-isolated vertex

• f 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).

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

14/32 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: speedR ≥ speedC = 1 cns(G) min number of cops to capture a robber with speed s in G Computational hardness Computing cns 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, cns(G) ≤ function(s) ⇒ algorithm in time O(nfunction(s)) Cop-number is unbounded in planar graphs ∀s > 1, ∀n: then cns(Gridn) = Ω(√log n). ∀H planar with an induced subgraph GridΩ(2k2), cn(H) ≥ k.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

15/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

16/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Three variants we consider

When cops and robber can ride s = speedR ≥ speedC = 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

16/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Three variants we consider

When cops and robber can ride s = speedR ≥ speedC = 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

17/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

18/32 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) = {v1, · · · , vn} and for any i < n, there is j > i s.t. N[vi] ⊆ N[vj] in the subgraph induced by vi, · · · , vn Trees, chordal graphs, bridged graphs (...) are cop win.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

19/32 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) = {v1, · · · , vn}, ∀i < n, ∃j > i, s.t. N1(vi, G) ∩ Xi ⊆ N1(vj, Gi) with Xi = {vi, · · · , vn} Theorem Characterization of CWFR(s, s′) G ∈ CWFR(s, s′) iff V (G) = {v1, · · · , vn}, ∀i < n, ∃j > i, s.t. Ns(vi, G \ {vj}) ∩ Xi ⊆ Ns′(vj) with Xi = {vi, · · · , vn}

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

20/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Cop-win graphs and hyperbolicity

Theorem Characterization of CWFR(s, s′) G ∈ CWFR(s, s′) iff V (G) = {v1, · · · , vn}, ∀i < n, ∃j > i, s.t. Ns(vi, G \ {vj}) ∩ Xi ⊆ Ns′(vj) with Xi = {vi, · · · , vn}

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 > 2δ ≥ 0, and G a δ-hyperbolic graph, G ∈ CWFR(2r, r + δ)

Theorem Cop-win ”leads” to hyperbolicity

If s ≥ 2s′, then any G ∈ CWFR(s, s′) is (s − 1)-hyperbolic. Question: ∀s > s′, any G ∈ CWFR(s, s′) is f (s)-hyperbolic?

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

21/32 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) = {v1, · · · , vn}, ∀i < n, ∃j > i, s.t. Ns(vi, G \ {vj}) ∩ Xi ⊆ N1(vj) with Xi = {vi, · · · , vn} 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 its parent-block.

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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 = {v1, · · · , vn}, N3(v1, G \ {y}) ⊆ N1(y) and G \ {v1} ∈ CWFR(3)

induction on the number of blocks

• if G is 2-connected

induction on |V (G)| We prove G is dominated by y v1 b y either G\{v1} is 2−connected dominated by b

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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 = {v1, · · · , vn}, N3(v1, G \ {y}) ⊆ N1(y) and G \ {v1} ∈ CWFR(3)

induction on the number of blocks

• if G is 2-connected

induction on |V (G)| We prove G is dominated by y

w y v1 y v1 y is the dominating articulation point

• r G\{v1} is not 2−connected and
• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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 = {v1, · · · , vn}, N3(v1, G \ {y}) ⊆ N1(y) and G \ {v1} ∈ 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

B

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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 = {v1, · · · , vn}, N3(v1, G \ {y}) ⊆ N1(y) and G \ {v1} ∈ 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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 = {v1, · · · , vn}, N3(v1, G \ {y}) ⊆ N1(y) and G \ {v1} ∈ 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

22/32 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 = {v1, · · · , vn}, N3(v1, G \ {y}) ⊆ N1(y) and G \ {v1} ∈ 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

23/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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 ??

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

24/32 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) 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

25/32 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

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

25/32 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)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

25/32 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(k2) without degree-1 vertex then, ∃v ∈ V , xy ∈ E, Nk(v, G \ xy) ⊆ N1(y)

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

26/32 Intro Def. Cop-Win Run Hide Shoot Concl.

CWW(2)

Lemma Necessity

If G ∈ CWW(2) then V = {v1, · · · , vn}, and ∀i, ∃xy ∈ E(Gi+1) (possibly x = y) N2(vi, G \ xy) ∩ Gi ⊆ N1(y)

Lemma Sufficiency

If V = {v1, · · · , vn}, and ∀i, ∃xy ∈ E(Gi+1) (possibly x = y) N2(vi, G \ xy) ∩ Gi ⊆ N1(y) and, if x = y, then N2(vi, G \ y) ∩ Gi ⊆ N2(x, G \ y) then G ∈ CWW(2)

Not sufficient

d e h i f a b c g

Not necessary

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

27/32 Intro Def. Cop-Win Run Hide Shoot Concl.

CWW(k) when k odd

Lemma Sufficiency for any k odd

If V = {v1, · · · , vn}, and ∀i, ∃xy ∈ E(Gi+1) (possibly x = y) Nk(vi, G \ xy) ∩ Gi ⊆ N1(y) then G ∈ CWW(k)

But it is not necessary...

1 k−2 1 k−2

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

28/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

29/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Finally, let’s advantage the cop :(

CWRC(k) = {G| C wins when capturing at dist. ≤ k} Theorem bipartite graphs in CWRC(1) A bipartite graph G is in CWRC(1) iff V = {v1, · · · , vn} s.t. {vn−1, vn} ∈ E and ∀i, ∃j > i, {vj, vi} / ∈ E and N(vi, Gi) ⊆ N1(vj) Characterization of CWRC(k) seems harder, even for k = 1...

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

30/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Outline

1

General ProblemS

2

Cops & Robber: Definitions, Examples, State of the art

3

Three new variants, Our Problem

4

Cop-win graphs when the Robber/Cops can run

5

The witness version: when the Robber can hide

6

Radius of Capture: when the Cop can shoot

7

Conclusions

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

31/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Perspectives

In case speedR = speedC = 1 G of genus g ⇒ cn(G) ≤ 3

2g + 3. [Schr¨

• der, 01]

Conjecture: G of genus g ⇒ cn(G) ≤ g + 3. General upper bound for cn ? for any connected graph G, cn(G) ≤ O(n/logn). [Chiniforooshan, 08] Conjecture: cn(G) ≤ O(√n). In case speedR > speedC Ω(

• log(n)) ≤ cn(Squaren) ≤ O(n). Exact value?

What about other graphs’classes ? Full characterization of cop-win graphs when witness? Characterization of cop-win graphs when the cop can ”shoot”?

• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride

32/32 Intro Def. Cop-Win Run Hide Shoot Concl.

Muito obrigado

Quest˜

• es?
• J. Chalopin, V. Chepoi, N. Nisse, Y. Vax`

es Cop and robber games when the robber can hide and ride