Cops and Robbers - PowerPoint Presentation

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Cops and Robbers

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Cops and Robbers: Directions and Generalizations

Anthony BonatoRyerson University

GRASTA

2012 Slide2

Happy 60

th Birthday RJNMay your searching never end.

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Cops and Robbers

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CC

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CC

C

R

cop

number c(G)

≤

3Slide6
Cops and Robbers

played on reflexive undirected graphs

Gtwo players Cops C

and robber R play at alternate time-steps (cops first) with perfect informationplayers move to vertices along edges; allowed to moved to neighbors or pass

cops try to capture (i.e. land on) the robber, while robber tries to evade captureminimum number of cops needed to capture the robber is the cop number c(G)well-defined as c(G) ≤ |V(G)|

Cops and Robbers

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Basic facts on the cop number

c(G) ≤

γ(G)

(the domination number of G)far from sharp: paths

trees have cop number 1one cop chases the robber to an end-vertexcop number can vary drastically with subgraphsadd a universal vertex

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How big can the cop number be?

c(n) = maximum cop number of a

connected graph of order n

Meyniel’s Conjecture: c(n) = O(n1/2).

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Cops and Robbers

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Henri Meyniel,

courtesy Geňa HahnSlide11

State-of-the-art

(Lu, Peng, 12) proved that

independently proved by (Scott, Sudakov,11) and

(Frieze, Krivelevich, Loh, 11)(Bollobás, Kun, Leader, 12+): if p = p(n) ≥ 2.1log n/ n,

then

c(G(n,p)) ≤ 160000n

1/2

log n

(Prałat,Wormald,12+)

: removed log factor

Cops and Robbers

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Graph classes

(Aigner, Fromme,84): Planar (outerplanar) graphs have cop number at most

3 (2).(Andreae,86): H-minor free graphs have cop number bounded by a constant.

(Joret et al,10): H-free class graphs have bounded cop number iff each component of H is a tree with at most 3 leaves.

(Lu,Peng,12):

Meyniel’s conjecture holds for diameter

2

graphs, bipartite diameter

3

graphs.

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How close to n1/2?

consider a finite projective plane Ptwo lines meet in a unique pointtwo points determine a unique lineexist

4

points, no line contains more than two of them

q

2

+q+1

points; each line (point) contains (is incident with)

q+1

points (lines)

incidence graph (IG)

of

P:

bipartite graph

G(P)

with

red nodes the points of P and

blue

nodes the lines of

P

a point is joined to a line if it is on that lineSlide14

Example

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Fano plane

Heawood graphSlide15

Meyniel extremal families

a family of connected graphs (Gn

: n ≥ 1) is Meyniel extremal (ME) if there is a constant d > 0

, such that for all n ≥ 1, c(Gn) ≥ dn1/2

IG of projective planes

: girth

6

,

(q+1)-

regular, so have cop number

≥ q+1

order

2(q

2

+q+1)

Meyniel extremal (must fill in non-prime orders)

(Prałat,10)

cop number

= q+1

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Diameter 2

(Lu, Peng, 12)

: If G has diameter

2, then c(G) ≤ 2n1/2 - 1.

diameter 2 graphs satisfy Meyniel’s conjectureproof uses the probabilistic methodQuestion: are there explicit

Meyniel extremal families whose members are diameter two?

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Polarity graphs

suppose

PG(2,q) has points

P and lines L. A polarity is a function π

: P→ L such that for all points p,q, p ϵ π(q) iff

q

ϵ

π

(p).

eg of

orthogonal polarity

: point mapped to its orthogonal complement

polarity graph

: vertices are points,

x

and

y

adjacent if

x

ϵ

π

(y)

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Properties of polarity graphs

order

q2+q+1

(q,q+1)-regularC4-free

(Erdős, Rényi, Sós,66), (Brown,66) orthogonal polarity graphs C

4

-free extremal

diameter

2

(Godsil, Newman, 2008)

have unbounded chromatic number as

q→ ∞

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Meyniel Extremal

Theorem

(B,Burgess,12+) Let

q be a prime power. If Gq is a polarity graph of

PG(2, q), then q/2 ≤ c(Gq) ≤ q + 1.lower bound: lemmaupper bound:

direct analysis

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ME method (BB,12+)

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Lower bounds

Lemma

(Aigner,Fromme, 1984) If G

is a connected graph of girth at least 5, then c(G) ≥ δ(G).

Lemma (BB,12+) If G is connected and K2,t-free, then c(G) ≥

δ

(G) / t

.

applies to polarity graphs:

t = 2

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Sketch of proof: Lower bound

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R

N(R)

C

< t

neighbours attackedSlide23

Sketch of proof: Upper bound

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R

C

N

2

(u)

uSlide24

Sketch of proof: Upper bound

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R

N

2

(u)

C

q

cops move to

N(u)

uSlide25

t-orbit graphs

(Füredi,1996) described a family of

K2,t+1-free extremal graphs of order (q2 -1)/t and which are

(q,q+1)-regular for prime powers q.gives rise to a new family of ME graphs which are K2,t+1-free

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(BB,12+) New ME families

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BIBDs

a

BIBD(v, k, λ)

is a pair (V, B), where V is a set of v

points, and B is a set of k-subsets of V, called blocks, such that each pair of points is contained in exactly

λ

blocks.

Theorem

(BB,12+)

The cop number of the IG of a

BIBD(v, k,

λ

)

is between

k

and

r

, the

replication number

.Cops and Robbers

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Sketch of proof

lower bound: girth

6; apply AF lemma and Fisher’s inequalityupper bound:

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Block Intersection graphs

given a block design

(V,B), its block intersection graph

has vertices equalling blocks, with blocks adjacent if they intersectCops and Robbers

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BIG cop number

Theorem

(BB,12+) If G is the block intersection graph of a

BIBD(v, k, 1), then c(G) ≤ k. If v > k(k-1)

2 + 1, then c(G) = k.gives families with unbounded cop number; not MEalso considered point graphs

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Questions

Soft Meyniel’s conjecture: for some ε

> 0,c(n) = O(n1-ε

).Meyniel’s conjecture in other graphs classes?bounded chromatic numberbipartite graphsdiameter 3claw-free

ME families from something other than designs?

extremal graphs?

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R.J. Nowakowski, P. Winkler Vertex-to-vertex pursuit in a graph,

Discrete Mathematics 43 (1983) 235-239.5 pages

> 200 citations (most for either author)Cops and Robbers

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The NW relation

(Nowakowski,Winkler,83)

introduced a sequence of relations characterizing cop-win graphs

u ≤0 v if u = v

u ≤i v if for all x in N[u], there is a y in

N[v]

such that

x ≤

j

y

for some

j < i

.

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Example

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u

v

w

y

z

u ≤

1

v

u ≤

2

w

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Characterization

the relations are

≤i

monotone increasing; thus, there is an integer k such that ≤k = ≤

k+1 write:≤k = ≤Theorem

(Nowakowski, Winkler, 83)

A cop has a winning strategy iff

≤

is

V(G) x V(G).

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k cops

may define an analogous relation but in V(G) x V(Gk

) (categorical product)(Clarke,MacGillivray,12) k

cops have a winning strategy iff the relation ≤ is V(G) x V(Gk).

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Axioms for pursuit games

a

pursuit game G

is a discrete-time process satisfying the following:Two players, Left L and

Right R.Perfect-information.There is a set of allowed positions P

L

for

L

; similarly for Right.

For each state of the game and each player, there is a non-empty set of

allowed moves

. Each allowed move leaves the position of the other player unchanged.

There is a set of

allowed start positions

I

a subset of

P

L

x P

R.

The game begins with

L

choosing some position

p

L

and

R

choosing

q

R

such that

(p

L

, q

R

)

is in

I

.

After each side has chosen its initial position, the sides move alternately with

L

moving first. Each side, on its turn, must choose an allowed move from its current position.

There is a subset of

final positions

,

F

. Left

wins

if

at any time, the current position belongs to

F

. Right

wins

the current position never belongs to

F

.

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Examples of pursuit games

Cops and Robbers

play on graphs, digraphs, orders, hypergraphs, etc.

play at different speeds, or on different edge setsCops and Robbers with traps

Distance k Cops and RobbersTandem-win Cops and RobbersRestricted ChessHelicopter Cops and Robbers

Maker-Breaker Games

Seepage

Scared Robber

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Relational characterization

given a pursuit game

G, we may define relations on

PL x PR as follows:p

L ≤0 qR if (p

L

, q

R

)

in

F.

p

L

≤

i

q

R

if Right is on

qR and for every x

R

in

P

R

such that if Right has an allowed move from

(

p

L

,

q

R

)

to

(

p

L

, x

R

),

there exists

y

L

in

P

L

such that

x

R

≤

j

y

L

for some

j < i

and Left has an allowed move from

(

p

L

, x

R

)

to

(y

L

, x

R

).

define

≤

analogously as before

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Characterization

Theorem

(B, MacGillivray,12) Left has a winning strategy in the a pursuit game

G if and only if there exists pL in PL

, which is the first component of an ordered pair in I, such that for all qR in PR

with

(

p

L

,

q

R

)

in

I

there exists

w

L

in the set of allowed moves for Left from

pL such that q

R

≤

w

L

.

gives rise to a min/max expression for the length of the game

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Length of game

for an allowed start position

(pL

, qR), define

Corollary (BM,12+) If Left has a winning strategy in the a pursuit game G, then assuming optimal play, the length of the game is

where

I

L

i

s the set positions for Left which are the first component of an ordered pair in

I.

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CGT

(

Berlekamp, Conway,

Guy, 82) A combinatorial game satisfies:There are two players,

Left and Right.There is perfect information.There is a set of allowed positions in the game.

The rules of the game specify how the game begins and, for each player and each position, which moves to other positions are allowed.

The players alternate moves.

The game ends when a position is reached where no moves are possible for the player whose turn it is to move. In

normal play

the last player to move wins.

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Example: NIM

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Pursuit → CGT

Theorem

(BM,12+)Every pursuit game is a combinatorial game.

Not every combinatorial game is a pursuit game.uses characterization of (Smith, 66) via game digraphs

Nim is a counter-example for item (2)Cops and Robbers45Slide46
Position independence

a pursuit game

G is position independent

if: if the game is not over, the set of available moves for a side does not depend on the position of the other side.examples: Cops and Robbers …

non-examples: Helicopter Cops and Robbers, Maker Breaker, …Cops and Robbers

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State digraph

G

a position independent pursuit game

GL= (PL

, ML) and GR= (P

R

, M

R

)

are the

position digraphs

of

G

S

G

=

G

L

x GR

state digraph

of

G

not all edges make sense

ignore these

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Relational characterization

Corollary

(BM,12+) Let

G be a position independent pursuit game. If GL is strongly connected and there exists X in

PL such that S = X x PR, then Left has a winning strategy in

G

if and only if

≤ = V (D

G

) =

P

L

x P

R

.

generalizes results of

NW

and

CMCops and Robbers

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Algorithm

Theorem

(BM,12+) Let G

be a position independent pursuit game. Given the graphs GL and GR

, if N+GL(PL) and

N

+

G

R

(

P

R

) can be obtained in time

O(f(|

P

L

|))

and

O(g(|

PR)|), respectively, then there is a

O(|

P

L

||

P

R

|f(|

P

L

|)g(|

P

R

|)).

algorithm to determine if Left has a winning strategy

.

Cops and Robbers

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Eg: Cops and Robbers

gives an

O(n2k+2

) algorithm to determine if k cops have a winning strategymatches best known algorithm (Clarke, MacGillivray,12)

Cops and Robbers50Slide51
Cop-win graphs

node

u is a corner if there is a

v such that N[v] contains N[u]a cop-win ordering

of G is an enumeration (v1,v2,…,v

n

)

of

V(G)

such that for all

i < n

, there is a

j > i

such that

Theorem

(Nowakowski, Winkler 83; Quilliot, 78)

A graph is cop-win if and only if it has a cop-win ordering.

idea:

cop-win graphs always have corners; retract corner and play shadow strategy

;

- graphs with cop-win orderings are cop-win by induction

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Cop-win ordering:

dismantling

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Vertex elimination ordering

cop-win orderings generalize to pursuit games:

Idea: order vertices of state digraph

SGremovable vertices are those whose out-neighbours are “dominated” by some by a vertex with higher index in the sequenceCorollary

(BM,12+): Left has a winning strategy in the position independent pursuit game G if and only if SG admits a

removable vertex ordering

.

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Infinite case

analogous characterization holds

k such that ≤

k = ≤k+1 is an ordinal, κ

CR-ordinalrelational characterization and vertex-ordering hold (now a transfinite sequence)NB: κ can be infinite:cannot think of

κ

as length of game

Cops and Robbers

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Cops and Robbers

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κ = ωSlide56
Questions

c(G)≤k? k

fixed:optimizing complexity for small

k?c(G)≤k? k not fixed:(Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): NP

-hard Goldstein, Reingold Conjecture: EXPTIME-complete.Conjecture: not in NP.

PSPACE

-complete?

infinite graphs: what are the CR-ordinals

κ

for cop-win graphs?

same question, but for more than one cop

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Another direction:

Minimum orders

Mk

= minimum order of a k-cop-win graphM1

= 1, M2 = 4M3 = 10 (Baird, B, et al, 12+)

Cops and Robbers

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Questions

M

4 =

? (the (4,5)-cage?)are the Mk monotone increasing?

for example, can it happen that M344 < M343?m

k

= minimum order of a connected

G

such that

c(G) ≥ k

(Baird, B, et al, 12+)

m

k

=

Ω

(k

2

)

is equivalent to Meyniel’s conjecture.mk = M

k

for all

k ≥ 4

?

Cops and Robbers

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Good guys vs bad guys games in graphs

59

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

bad

good

Cops and RobbersSlide60

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