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What is left to do on Cops and Robbers?

Anthony BonatoRyerson University

GRASCan 2012

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Where to next?

• we focus on 6 research directions on the topic of Cops and Robbers games–by no means exhaustive

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).

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Henri Meyniel, courtesy Geňa Hahn

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)) ≤ 160000n1/2log n

• (Prałat,Wormald,12+): removed log factor

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

)( ono

nnOnc

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Graph classes• (Aigner, Fromme,84): Planar graphs have cop

number at most 3.• (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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Questions

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

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

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2. 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

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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 (see Andrea Burgess’ talk)

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3. Minimum orders

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

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

– see also (Beveridge et al, 2012+)

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

• mk = Mk for all k ≥ 4?

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4. 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

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Questions• Goldstein, Reingold Conjecture:

if s is not fixed, then computing the cop number is EXPTIME-complete.– same complexity as say, generalized chess

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

• speed ups? – can we recognize 2-cop-win graphs in o(n7)?– how fast can we recognize cop-win graphs?

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5. Planar graphs• (Aigner, Fromme, 84) planar graphs have cop

number ≤ 3.

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

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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?

• edge contraction/subdivision and cop number?– see (Clarke, Fitzpatrick, Hill, RJN, 10)

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6. VariantsGood guys vs bad guys games in graphs

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

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

(Bonato,Chiniforooshan,Prałat,10)• 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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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)

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When does one cop suffice?

• cop-win graphs ↔ cop-win orderings(RJN, Winkler, 83), (Quilliot, 78)• 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…)– line graphs (RJN,12+)– infinite graphs (Bonato, Hahn, Tardif, 10)

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The robber fights back! (Haidar,12) • robber can attack neighbouring cop

• one more cop needed in this graph (check)• at most min{2c(G),γ(G)} cops needed, in general• are c(G)+1 many cops needed?

C

C

C

R

Fighting Intelligent Fires Anthony Bonato

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Infinite hexagonal grid

• can one cop contain the fire?

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Fill in the blanks…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

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