conjectures on cops and robbers

Cops and Robbers 1

Conjectures on Cops and Robbers

Anthony BonatoRyerson University

Joint Mathematics Meetings AMS Special Session

Cops and Robbers

Cops and Robbers 2

C

C

C

R

Cops and Robbers

Cops and Robbers 3

C

C

C

R

Cops and Robbers

Cops and Robbers 4

C

C

C

R

cop number c(G) ≤ 3

Cops and Robbers• played on a reflexive undirected graph G• two players Cops C and robber R play at alternate

time-steps (cops first) with perfect information• players 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 capture• minimum number of cops needed to capture the

robber is the cop number c(G)– well-defined as c(G) ≤ |V(G)|

Cops and Robbers 5

Cops and Robbers 6

Conjectures

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

– structural, algorithmic, probabilistic, topological…

1. How big can the cop number be?

• c(n) = maximum cop number of a connected graph of order n

Meyniel Conjecture: c(n) = O(n1/2).

Cops and Robbers 7

Cops and Robbers 8

Cops and Robbers 9

Henri Meyniel, courtesy Geňa Hahn

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)) ≤ 160000n1/2log n

• (Prałat,Wormald,14+): proved Meyniel’s conjecture for all p = p(n)

Cops and Robbers 10

)1(1log))1(1( 22

)( ono

nnOnc

Cops and Robbers 11

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 12

Questions

Soft Meyniel’s conjecture: for some ε > 0,c(n) = O(n1-ε).

• Meyniel’s conjecture in other graphs classes?– bipartite graphs– diameter 3– claw-free

Cops and Robbers 13

How close to n1/2?

• consider a finite projective plane P– two lines meet in a unique point– two points determine a unique line– exist 4 points, no line contains more than two of them

• q2+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 line

Example

Cops and Robbers 14

Fano plane Heawood graph

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) ≥ dn1/2

• IG of projective planes: girth 6, (q+1)-regular, so have cop number ≥ q+1– order 2(q2+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 15

Cops and Robbers 16

Minimum orders

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

• M1 = 1, M2 = 4• M3 = 10 (Baird, B,12)

– see also (Beveridge et al, 14+)• M4 = ?

Cops and Robbers 17

Conjectures on mk, Mk

Conjecture: Mk monotone increasing.

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

• (Baird, B, 12) mk = Ω(k2) is equivalent to Meyniel’s conjecture.

Conjecture: mk = Mk for all k ≥ 4.

Cops and Robbers 18

2. Complexity• (Berrarducci, Intrigila, 93), (Hahn,MacGillivray, 06),

(B,Chiniforooshan, 09):

“c(G) ≤ s?” s fixed: in P; running time O(n2s+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 19

Questions

Goldstein, Reingold Conjecture: if s is not fixed, then computing the cop number is EXPTIME-complete.

– same complexity as say, generalized chess

• settled by (Kinnersley,14+)

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

Cops and Robbers 20

3. Genus• (Aigner, Fromme, 84) planar graphs (genus 0)

have cop number ≤ 3.

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

Cops and Robbers 21

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 22

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.

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

badgood

Cops and Robbers 24

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

• 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

Cops and Robbers 25

Distance k cop number: ck(G)

• 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)

Cops and Robbers 26

When does one cop suffice?

• (RJN, Winkler, 83), (Quilliot, 78)cop-win graphs ↔ cop-win orderings

• provide a structural/ordering characterization of cop-win graphs for:– directed graphs– distance k Cops and Robbers– invisible robber; cops can use traps or alarms/photo

radar (Clarke et al,00,01,06…)– infinite graphs (Bonato, Hahn, Tardif, 10)

Cops and Robbers 27

Lazy Cops and Robbers• (Offner, Ojakian,14+) only one can move in each

round– lazy cop number, cL(G)

• (Offner, Ojakian, 14+)

• (Bal,B,Kinnsersley,Pralat,14+) For all ε > 0,.

Cops and Robbers 28

Questions on lazy cops• Question: Find the asymptotic order of .

• (Bal,B,Kinnsersley,Pralat,14+) If G has genus g, then cL(G) = – proved by using the Gilbert, Hutchinson,Tarjan

separator theorem

• Question: Is cL(G) bounded for planar graphs?

Cops and Robbers 29

Firefighting

Cops and Robbers 30

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) vertices– strategy is optimal assuming fire breaks out in

columns (rows) 1,2, n-1, n

Cops and Robbers 31

¼ -grid conjecture

nPPsn nnnn 4

1)),(, (

n largefor ,0every For

22

Cops and Robbers 32

Infinite hexagonal grid

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

Cops and Robbers 33

34

A. Bonato, R.J. Nowakowski, Sketchy Tweets: Ten Minute Conjectures in Graph Theory, The Mathematical Intelligencer 34 (2012) 8-15.

1..