GRASCan 2012
What is left to do on Cops
and Robbers?
Anthony Bonato
Ryerson University
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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
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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
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State-of-the-art
• (Lu, Peng, 12+) proved that
n

c(n)  O (1o (1))
2
log2 n

  n1o (1)

– 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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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
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Example
Fano plane
Heawood graph
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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. Variants
Good guys vs bad guys games in graphs
bad
good
slow
slow
medium
fast
helicopter
eternal
security
traps, tandem-win
medium
robot vacuum
Cops and Robbers
edge searching
fast
cleaning
distance k Cops
and Robbers
Cops and Robbers The Angel
on disjoint edge
and Devil
sets
seepage
Helicopter Cops
and Robbers,
Marshals, The
Angel and Devil,
Firefighter
helicopter
Cops and Robbers
Hex
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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
k=1
R
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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
C
C
R
C
• 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?
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Infinite hexagonal grid
• can one cop contain the fire?
Fighting Intelligent Fires
Anthony Bonato
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Fill in the blanks…
bad
good
slow
slow
medium
fast
helicopter
eternal
security
traps, tandem-win
medium
robot vacuum
Cops and Robbers
edge searching
fast
cleaning
distance k Cops
and Robbers
Cops and Robbers The Angel
on disjoint edge
and Devil
sets
seepage
Helicopter Cops
and Robbers,
Marshals, The
Angel and Devil,
Firefighter
helicopter
Cops and Robbers
Hex
24
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