1. Cops and Robbers Games Played on Graphs
University of Iceland
Mathematics Seminar
Conjectures on
Cops and Robbers Games Played on Graphs
Anthony Bonato
Ryerson University
Toronto, Canada
Cops and Robbers

2. Cops and Robbers C C R
Cops and Robbers

3. Cops and Robbers played on an 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

4. Observations c(T) = 1 if T is a tree c(G) ≤ γ(G)
If G has no 3- or 4-cycles, then δ G ≤ c G .
Cops and Robbers

5. Applications of Cops and Robbers
robotics
mobile computing
missile-defense
gaming
counter-terrorism
intercepting messages
or agents
Cops and Robbers

6. Conjectures
conjectures and problems on Cops and Robbers coming from different directions, touch on various aspects of graph theory:
structural, algorithmic, probabilistic, topological…
Cops and Robbers

7. 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

8. Cops and Robbers

9. Henri Meyniel, courtesy Geňa Hahn
Cops and Robbers

10. Frankl’s bound Theorem (Frankl, 1987) 𝑐 𝑛 =𝑂 𝑛 log log 𝑛 log 𝑛
Cops and Robbers

11. Sketch of Frankl’s proof
Moore bound: 𝑛≤1+∆ 𝑖=0 𝐷−1 (∆−1) 𝑖
there is either an isometric path or closed neighbour set of order
log 𝑛 log log 𝑛
either subgraph can be 1-guarded (via a retraction)
guarding the subgraph costs one cop
Induction.
Cops and Robbers

12. State-of-the-art (Lu, Peng, 13) proved that
independently proved by (Frieze, Krivelevich, Loh, 11) and (Scott, Sudakov,11)
Cops and Robbers

13. For random graphs (Bollobás, Kun, Leader,13): if
p = p(n) ≥ 2.1log n/ n, then a.a.s.
c(G(n,p)) ≤ n1/2log n
(Prałat,Wormald,16): proved Meyniel’s conjecture a.a.s. for all p = p(n)
(Prałat,Wormald,16+): holds a.a.s. for random d-regular graphs, for d ≥ 3
Cops and Robbers

14. 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

15. 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

16. 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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17. Example
Fano plane
Heawood graph
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18. 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)
other examples of Meyniel extremal families come from combinatorial designs and finite geometries (B,Burgess,2013)
Cops and Robbers

19. (BB,13) New ME families
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20. 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
xϵ π(y)
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21. Properties of polarity graphs
order q2+q+1
(q,q+1)-regular
C4-free
diameter 2
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22. Meyniel Extremal
Theorem (Bonato,Burgess,13) Let q be a prime power. If Gq is a polarity graph of PG(2, q), then
q/2 ≤ c(Gq) ≤ q + 1.
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23. Lower bounds Theorem (Bonato, Burgess,13)
If G is connected and K2,t-free, then
c(G) ≥ δ(G) / t.
applies to polarity graphs: t = 2
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24. Minimum orders Mk = minimum order of a k-cop-win graph M1 = 1, M2 = 4
M3 = 10 (Baird, B,12)
see also
(Baird,Beveridge,B, et al, 14)
M4 = ?
Cops and Robbers

25. 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

26. Questions settled by (Kinnersley,15) JCTB
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,15) JCTB
Conjecture: if s is not fixed, then computing the cop number is not in NP.
Cops and Robbers

27. 3. Genus
(Aigner, Fromme, 84) planar graphs (genus 0) have cop number ≤ 3.
(Clarke, 02) outerplanar graphs have cop number ≤ 2.
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28. 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

29. 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

30. 4. Variants Good guys vs bad guys games in graphs
slow
medium
fast
helicopter
traps, tandem-win,
Lazy Cops and Robbers
robot vacuum
Cops and Robbers
edge searching, Cops and Fast Robber
eternal security
cleaning
distance k Cops and Robbers
Cops and Robbers on disjoint edge sets
The Angel and Devil
seepage
Helicopter Cops and Robbers, Marshals, The Angel and Devil,
Firefighter
Hex
bad
good

31. 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
k = 1 R
Cops and Robbers

32. 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

33. 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

34. Lazy Cops and Robbers
(Offner, Ojakian,14+) only one can move in each round
lazy cop number, cL(G)
(Offner, Ojakian, 14)
2 𝑛 /20 ≤ 𝑐 𝐿 ( 𝑄 𝑛 )≤𝑂( 2 𝑛 log 𝑛/ 𝑛 ).
(Bal,B,Kinnsersley,Pralat,15) For all ε > 0,
Ω( 2 𝑛 / 𝑛 𝜀 ) ≤ 𝑐 𝐿 ( 𝑄 𝑛 ).
Cops and Robbers

35. Questions on lazy cops
Question: Find the asymptotic order of 𝑐 𝐿 ( 𝑄 𝑛 ).
(Bal,B,Kinnsersley,Pralat,15) If G has genus g, then cL(G) = 𝑂( 𝑔𝑛 )
proved by using the Gilbert, Hutchinson,Tarjan separator theorem
Question: Is cL(G) bounded for planar graphs?
Cops and Robbers

36. Miniconference on the Mathematics of Computation
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37. Miniconference on the Mathematics of Computation
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38. Miniconference on the Mathematics of Computation
Zombie horde
up to 𝑛/2 - 2 zombies on an induced path will never capture the survivor
Cops and Robbers

39. Zombies and Survivors set of zombies, one survivor
players move at alternate ticks of the clock, from vertex to vertex along edges
zombies choose their initial locations u.a.r.
at each step the zombies move along a shortest path connected to the survivor
if more than one such path, then they choose one u.a.r.
zombies win if one or more can eat the survivor
land on the survivor’s vertex
otherwise, survivor wins
NB: zombies have no strategy!
Cops and Robbers

40. (B,Mitsche,Perez-Gimenez,Pralat,16)
sk(G): probability survivor wins if k zombies play, assuming optimal play
sk+1 (G) ≤ sk (G) for all k, and sk(G) → 0 as k → ∞
zombie number of G is
z(G) = min{k ≥ c(G): sk(G) ≤ ½}
well-defined
z(G) represents the minimum number of zombies such that the probability that they eat the survivor is > ½
note that c(G) ≤ z(G)
Z(G) = z(G) / c(G): cost of being undead
Cops and Robbers

41. Deterministic Zombies
deterministic version of the game:
(Fitzpatrick, Howell, Messinger,Pike,16+) at the SIAM DM 2014
conference in Minneapolis
in the deterministic version, the zombies choose their location, and can choose their geodesics if more than one
example: where c(G) = 2 < z(G) = 3
Cops and Robbers

42. Cartesian grids (Tosic, 87) c(G H) ≤ c(G) + c(H) Theorem (BMPGP,16)
For n ≥ 2, z(Pn Pn) = 2, so Z(Pn Pn) =1.
Cops and Robbers

43. Toroidal grids Tn = Cn Cn (Neufeld, 90): c(Tn) = 3 Theorem (BMPGP,16+)
Let ω = ω(n) be a function tending to infinity with n.
Then a.a.s.
𝑧 𝑇 𝑛 ≥ 𝑛 / (ω log n).
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44. Toroidal grids, continued
despite the lower bound, no known upper bound is known for the zombie number of toroidal graphs!
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45. Necromancers, Zombies, and Survivors
set of j necromancers (i.e. cops, who can access all strategies) and k zombies play vs survivor
(j,k)-necro-win: bad guys win with probability > 1/2
Theorem (BMPGPR,16+)
For large m and n, toroidal grids Cm Cn are (2,1)-necro-win.
If gcd(m,n) ≤ 3, then Cm Cn is (1,2)-necro-win.
Cops and Robbers

46. Problems zombie number of toroidal grid upper bound?
other grids? graph products?
necromancer + k zombies: how large must k be to ensure win on square toroidal grids?
random structures?
binomial random graph, G(n,r), random regular…
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47. Deterministic case (FHMP,16+)
characterize zombie-win graphs
conjecture: 𝑧 Q 𝑛 = 2𝑛 3
lower bound tight; best upper bound: n-1
how large can z(G)/c(C) be?
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48. Cops and Robbers

49. .
A. Bonato, R.J. Nowakowski, Sketchy Tweets: Ten Minute Conjectures in Graph Theory, The Mathematical Intelligencer 34 (2012) 8-15
A. Bonato, Conjectures on Cops and Robbers, invited chapter in the book Graph Theory - Favorite Conjectures and Open Problems, edited by Ralucca Gera, Stephen Hedetniemi, and Craig Larson, 2014.
Cops and Robbers