Lecture 12 - Variants of Cops and Robbers

Dr. Anthony BonatoRyerson University

AM8002Fall 2014

Variants

• many possible variants exist for Cops and Robbers

• power or speed of cops or robber can be changed in many ways:• the robber is faster• the robber is invisible; there maybe traps or

alarms• the cops have further reach, or can teleport• the robber can fight back

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Photo radar number

• play as in Cops and Robbers in a cop-win graph, but robber is invisible

• cops can place photo radar on edges xy: indicates when the robber is on x or y, and which direction he exits the edge

• photo radar number, written pr(G), minimum number of photo radars needed on edges to catch robber

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Photo Radar

4

t

a

b

c

d

e

f

Tandem-win graphs

• pair of cops play, but always must be distance at most one apart

• a graph is tandem-win if one pair of cops playing in tandem can capture the robber

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C C

Nearly irreducible vertices

• a vertex u is nearly irreducible if there is a vertex v such that N(u) is contained in N[v]– note that u need not be joined to v (as in the

case of a corner)

Theorem 12.1 (Clarke, 2002) Let u be nearly irreducible. Then G is tandem-win iff G-u is tandem win.

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Example

• a tandem-win graph with no nearly irreducible vertices

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Discussion

• Why is the following graph tandem-win?

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Complementary Cops and Robbers

• cops move on edges, robber moves on non-edges (i.e. on edges of the complement)

• least number of cops needed to capture the robber with these rules is CC(G)

Theorem 12.2 (Hill,08) For a graph G,

γ(G) - 1 ≤ CC(G) ≤ γ(G).

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CC(G) = k

Corollary 12.3 (Hill,08) If CC(G) = k, then G has a set of k+1 vertices, at least two of which are adjacent, which dominate the graph.

• does not give a characterization…

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11

Distance k Cops and Robber• cops can “shoot” robber at some specified

distance k• play as in classical game, but capture includes

case when robber is distance k from the cops– k = 0 is the classical game

C

R

k = 1

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The distance k cop number

• ck(G) = minimum number of cops needed to capture robber at distance at most k

• G connected implies

ck(G) ≤ diam(G) – 1

• for all k ≥ 1,

ck(G) ≤ ck-1(G)

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Example: k = 1

C

R

c1(G) > 1

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Example

C C

Rc1(G) = 2

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ck(n)

• ck(n) = maximum value of ck(G) over connected G of order n

• Meyniel conjecture:

c0(n) = O(n1/2).

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Random graphs

• for random graphs G(n,p) with p = p(n), the behaviour of distance k cop number is complicated

Theorem 12.4 (Bonato et al,09)

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Zig-zag functions

• for x in (0,1), define

fk(x) = log E(ck(G(n,nx-1))) / log n

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The robber fights back!

• robber can attack neighbouring cop

• one more cop needed in this graph (check)

CC

C

R

cc number

• let cc(G) be the minimum number of cops needed with these rules

Lemma 12.5 For a graph G,

c(G) ≤ cc(G) ≤ 2c(G).

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20

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Firefighter

• G simple, undirected, connected graph• fire spreads from a vertex over discrete time-

steps or rounds• vertices are on fire, protected, or clear• fire can spread to all available adjacent vertices• firefighter can protect one vertex in each round

• (Hartnell, 95) introduced Firefighter– simplified model for the spread of a fire/disease/virus

in a network

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Saving vertices

• one-player game• firefighter aims to maximize the number of

clear or protected (ie saved) vertices

• sn(G,v) = maximum number of saved vertices in G if a fire starts at v

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Examples

• sn(Pn,v) = n-1, if v is an end-vertex

= n-2, else

• sn(Kn,v) = 1

• Theorem (MacGillivray, P. Wang, 03): sn(Qn,v) = n

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Surviving rate

• (Cai, W. Wang, 09) surviving rate of G,

ρ(G) = expected percentage of vertices saved if fire starts at a random

vertex

)(

2),(

1)(

GVv

vGsnn

G

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Example: path

Lemma 12.6:

2

2

2

221

)1(2)2)(2(1

),(1

)(

nn

nnnn

vPsnn

PVv

nn

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Results on ρ(G)

• (Cai, W. Wang, 10): ρ(G) ≥ 1 – Θ(log n /n) if G is outerplanar

• (Finbow, P. Wang, W. Wang, 10):

if G has size at most (4/3 – ε)n, then ρ(G) ≥ 6/5ε, where 0 < ε < 5/27

• (Prałat, 10):

if G has size at most (15/11 – ε)n, then ρ(G) ≥ 1/60ε, where 0 < ε < 1/2 (15/11 best possible)

Open problem:Infinite hexagonal grid

• can one cop contain the fire?

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Aside: Minimum orders

• Mk = minimum order of a k-cop-win graph

• M1 = 1, M2 = 4

• M3 = 10 (Baird, Bonato,13)

– see also (Beveridge et al, 2014+)

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Questions

• M4 = ?

• are the Mk monotone increasing?– for example, can it happen that M344 < M343?

• mk = minimum order of a connected G such that c(G) ≥ k

• (Baird, Bonato, 13) mk = Ω(k2) is equivalent to Meyniel’s conjecture.

• mk = Mk for all k ≥ 4?29

Good guys vs bad guys games in graphs

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slow medium fast helicopter

slow traps, tandem-win

medium robot vacuum Cops and Robbers edge searching eternal security

fast cleaning distance k Cops and Robbers

Cops and Robbers on disjoint edge sets

The Angel and Devil

helicopter seepage Helicopter Cops and Robbers, Marshals, The Angel and Devil,Firefighter

Hex

badgood