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

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Cop and robber games when the robber can hide and ride emie Chalopin 1 Victor Chepoi 1 Nicolas Nisse 2 J er es 1 Yann Vax` 1 Lab. Informatique Fondamentale, Univ. Aix-Marseille, CNRS, Marseille, France 2 MASCOTTE, INRIA, I3S, CNRS, UNS,

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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

4th Workshop on GRAph Searching, Theory and Applications GRASTA, Dagstuhl, February 17th 2011

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

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

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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 captured: 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

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SLIDE 3

2/19

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 captured: 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

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SLIDE 4

2/19

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 captured: 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

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SLIDE 5

2/19

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 captured: 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

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SLIDE 6

2/19

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 captured: 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

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SLIDE 7

2/19

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 captured: 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

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SLIDE 8

2/19

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 captured: 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

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SLIDE 9

2/19

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 captured: 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

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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

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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

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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

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SLIDE 13

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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

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SLIDE 14

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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

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SLIDE 15

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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

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SLIDE 16

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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

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SLIDE 17

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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

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SLIDE 18

4/19

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

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SLIDE 19

4/19

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

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SLIDE 20

4/19

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

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SLIDE 21

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Simple Examples

Cop number in cliques and trees

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

Cliques and trees are Cop-Win

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

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

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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] 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

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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

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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] recently improved [Lu & Peng 09, Scott & Sudakov 10]

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

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

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SLIDE 25

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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] recently improved [Lu & Peng 09, Scott & Sudakov 10] Meyniel’s Conjecture: ∀ 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

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Faster protagonists [Fomin,Golovach,Kratochvil,N.,Suchan]

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

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Three variants we consider

When cops and robber can ride s = speedR ≥ speedC = s′ When the robber can hide (witness) [Clarke 08] The robber is visible only every k steps. When the cops can shoot (radius of capture)[BCP 10] Robber captured when at distance k from a cop.

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

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

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SLIDE 28

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Three variants we consider

When cops and robber can ride s = speedR ≥ speedC = s′ CWFR(s, s′) When the robber can hide (witness) [Clarke 08] The robber is visible only every k steps. CWW(k) When the cops can shoot (radius of capture)[BCP 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

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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

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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) 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} For the proof: more general game with X ⊆ V (G) X-game: C and R occupy only X but can pass through V (G) Induction on |X|

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

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

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Cop-win graphs and hyperbolicity

Any graph G is δ-hyperbolic for some δ ≥ 0 the smaller δ, the closer the metric of G is to the metric of a tree. Theorem Hyperbolicity helps the cop

∀s′ > 2δ ≥ 0, and G a δ-hyperbolic graph, G ∈ CWFR(2s′, s′ + 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

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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} CWFR(∞) ⊆ · · · ⊆ CWFR(3) ⊆ CWFR(2) ⊆ CWFR(1)

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

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

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SLIDE 33

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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} CWFR(∞) = · · · = CWFR(3) ⊂ CWFR(2) ⊂ CWFR(1)

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

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

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SLIDE 34

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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

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The witness version

When the robber can hide (witness) [Clarke DM’08] The robber is visible only every k steps.

CWW(k) = {G| C wins against R visible every k steps }

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

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

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

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

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

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SLIDE 37

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

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

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

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SLIDE 38

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

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

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

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SLIDE 39

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

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

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

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SLIDE 40

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

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

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

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SLIDE 41

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

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

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

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SLIDE 42

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The witness version

Relationship with fast robber CWFR(s) ⊆ CWW(s) Lemma invisibility is weaker than speed ∀s ≥ 2, CWFR(s) ⊂ CWW(s)

diameter 2, no dominating vertex, 2-connected ⇒ G / ∈ CWFR(s), s ≥ 2.

G ∈ CWW(s) importance of edge-separator

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

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

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SLIDE 43

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The witness version

Recall that more speed does not help agains one speed-1 cop ∀s ≥ 3, CWFR(3) = CWFR(s) here: different behaviour 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

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SLIDE 44

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The witness version

Recall that more speed does not help agains one speed-1 cop ∀s ≥ 3, CWFR(3) = CWFR(s) here: different behaviour 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

slide-45
SLIDE 45

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The witness version

Recall that more speed does not help agains one speed-1 cop ∀s ≥ 3, CWFR(3) = CWFR(s) here: different behaviour 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

slide-46
SLIDE 46

14/19

The witness version

Recall that more speed does not help agains one speed-1 cop ∀s ≥ 3, CWFR(3) = CWFR(s) here: different behaviour 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

slide-47
SLIDE 47

14/19

The witness version

Recall that more speed does not help agains one speed-1 cop ∀s ≥ 3, CWFR(3) = CWFR(s) here: different behaviour 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

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SLIDE 48

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The witness version

Recall that more speed does not help agains one speed-1 cop ∀s ≥ 3, CWFR(3) = CWFR(s) here: different behaviour 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

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SLIDE 49

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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

∀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

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SLIDE 50

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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

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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

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Finally, let’s advantage the cop :(

When the cops can shoot (radius of capture)[BCP 10] Robber captured when at distance k from a 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

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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 ? 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