Cops and Robbers - PowerPoint Presentation

Slide1

Cops and Robbers

1

Conjectures on Cops and Robbers

Anthony BonatoRyerson University

Joint Mathematics Meetings

AMS

Special SessionSlide2

Cops and Robbers

Cops and Robbers

2

CC

C

RSlide3

Cops and Robbers

Cops and Robbers

3

CC

C

RSlide4

Cops and Robbers

Cops and Robbers

4

CC

C

R

cop

number c(G)

≤

3Slide5
Cops and Robbers

played on a reflexive undirected graph

Gtwo players

Cops C and robber R play at alternate time-steps (cops first) with perfect informationplayers move to vertices along edges; may move 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

5Slide6
Conjectures

conjectures and problems on Cops and Robbers coming from 5 different directions, touch on various aspects of graph theory:

structural, algorithmic, probabilistic, topological…

Cops and Robbers6Slide7

1. How big can the cop number be?

c(n)

= maximum cop number of a connected graph of order nMeyniel Conjecture

: c(n) = O(n1/2).Cops and Robbers

7Slide8

Cops and Robbers

8Slide9

Cops and Robbers

9

Henri Meyniel, courtesy Geňa HahnSlide10

State-of-the-art

(Lu, Peng, 13) proved that

independently proved by (Frieze, Krivelevich, Loh, 11

) and (Scott, Sudakov,11)(Bollobás, Kun, Leader,13): if

p = p(n) ≥ 2.1log n/ n,

then

c(G(n,p)) ≤ 160000n

1/2

log n

(Prałat,Wormald,14+)

: proved

Meyniel’s

conjecture for all

p = p(n

)Cops and Robbers

10Slide11
Graph classes

(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,13): Meyniel’s conjecture holds for diameter 2

graphs, bipartite diameter

3

graphs.

Cops and Robbers

11Slide12
Questions

Soft

Meyniel’s conjecture

: for some ε > 0,c(n) = O(n1-ε

).Meyniel’s conjecture in other graphs classes?bipartite graphsdiameter 3claw-freeCops and Robbers

12Slide13

Cops and Robbers

13

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

Cops and Robbers

14

Fano plane

Heawood graphSlide15

Meyniel extremal families

a family of connected graphs (Gn

: n ≥ 1) is Meyniel extremal if there is a constant d > 0, such that for all n ≥ 1,

c(Gn) ≥ dn1/2IG 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)all other examples of Meyniel extremal families come from combinatorial designs

(B,Burgess,2013)Cops and Robbers

15Slide16
Minimum orders

M

k = minimum order of a

k-cop-win graphM1 = 1, M2

= 4M3 = 10 (Baird, B,12)

see also

(

Beveridge

et al,

14

+)

M

4

=

?

Cops and Robbers16Slide17
Conjectures on

mk

, Mk

Conjecture: Mk monotone increasing.

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

(Baird,

B, 12)

m

k

=

Ω

(k

2

) is equivalent to

Meyniel’s conjecture.Conjecture:

m

k

= M

k

for all

k ≥

4.

Cops and Robbers

17Slide18
2. Complexity

(Berrarducci,

Intrigila

, 93), (Hahn,MacGillivray, 06), (B,Chiniforooshan, 09):

“c(G) ≤ s?” s fixed: in P; running time O(n

2s+3

), n = |V(G)|

(Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08):

if

s

not fixed, then computing the cop number is

NP

-hard

Cops and Robbers

18Slide19
Questions

Goldstein, Reingold Conjecture

: if s

is not fixed, then computing the cop number is EXPTIME-complete.same complexity as say, generalized chesssettled by (Kinnersley,14+)

Conjecture: if s is not fixed, then computing the cop number is not in NP.Cops and Robbers

19Slide20
3. Genus

(Aigner,

Fromme

, 84) planar graphs (genus 0) have cop number ≤ 3.

(Clarke, 02) outerplanar graphs have cop number ≤ 2.

Cops and Robbers

20Slide21
Questions

characterize planar (outer-planar) graphs with cop number

1,2, and

3 (1 and 2)is the dodecahedron the unique smallest order planar

3-cop-win graph?Cops and Robbers

21Slide22
Higher genus

Schroeder’s Conjecture

: If G has genus

k, then c(G) ≤ k +3.true for k = 0

(Schroeder, 01): true for k = 1 (toroidal graphs)

(Quilliot,85):

c(G) ≤

2k

+3

.

(Schroeder,01):

c(G) ≤

floor(3k/2)

+3.

Cops and Robbers

22Slide23

5. VariantsGood guys vs bad guys games in graphs

23

slow

medium

fast

helicopter

slow

traps, tandem-win,

Lazy Cops and Robbers

medium

robot vacuum

Cops and

Robbers

edge searching,

Cops and Fast Robber

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

goodSlide24

Cops and Robbers

24

Distance k Cops and Robber (B,Chiniforooshan,09)

cops can “shoot” robber at some specified distance kplay as in classical game, but capture includes case when robber is distance k from the copsk = 0

is the classical game

C

R

k = 1Slide25

Cops and Robbers

25

Distance k cop number: ck(G)c

k(G) = minimum number of cops needed to capture robber at distance at most kG connected implies

c

k

(G)

≤ diam(G) – 1

for all

k ≥ 1

,

c

k(G) ≤ c

k-1(G)Slide26
When does one cop suffice?

(RJN, Winkler, 83), (

Quilliot,

78)cop-win graphs ↔ cop-win orderingsprovide a structural/ordering characterization of cop-win graphs for:directed graphsdistance

k Cops and Robbersinvisible robber; cops can use traps or alarms/photo radar (Clarke et al,00,01,06…)infinite graphs (Bonato, Hahn, Tardif, 10)

Cops and Robbers

26Slide27
Lazy Cops and Robbers

(

Offner

, Ojakian,14+) only one can move in each roundlazy cop number, cL(G)

(Offner, Ojakian, 14+)

(Bal,B,Kinnsersley,Pralat,14+)

For all

ε

> 0,

.

Cops and Robbers

27Slide28
Questions on lazy cops

Question: Find the asymptotic order of

.

(Bal,B,Kinnsersley,Pralat,14+)

If

G

has genus

g

, then

c

L

(G) =

proved by

using the Gilbert,

Hutchinson,Tarjan

separator theorem

Question: Is

c

L

(G

)

bounded for planar graphs?

Cops and Robbers

28Slide29

29

Cops and Robbers

FirefightingSlide30
A strategy

(MacGillivray, Wang, 03):

If fire breaks out at (

r,c), 1≤r≤c≤n/2, save vertices in following order: (r + 1, c), (r + 1, c + 1), (r + 2, c - 1), (r + 2, c + 2),

(r + 3, c -2),(r + 3, c - 3), ..., (r + c, 1), (r + c, 2c), (r + c, 2c + 1), ..., (r + c, n) strategy saves n(n-r)-(c-1)(n-c) verticesstrategy is optimal assuming fire breaks out in columns (rows)

1,2, n-1, n

Cops and Robbers

30Slide31

¼

-grid conjecture

31

Cops and RobbersSlide32
Infinite hexagonal grid

Conjecture

: one firefighter cannot contain a fire in an infinite hexagonal grid.

Cops and Robbers32Slide33

Cops and Robbers

33Slide34

34

A. Bonato, R.J. Nowakowski, Sketchy

Tweets: Ten Minute Conjectures in Graph Theory, 

The Mathematical Intelligencer 34 (2012)

8-15.

.