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Cops and Robbers: Directions and Generalizations
Anthony BonatoRyerson University
GRASTA 2012
Happy 60th Birthday RJN
May your searching never end.
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cop number c(G) ≤ 3
Cops and Robbers• played on reflexive undirected graphs G• two players Cops C and robber R play at alternate
time-steps (cops first) with perfect information• players move to vertices along edges; allowed to
moved 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)|
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Basic facts on the cop number• c(G) ≤ γ(G) (the domination number of G)
– far from sharp: paths
• trees have cop number 1– one cop chases the robber to an end-vertex
• cop number can vary drastically with subgraphs– add a universal vertex
How big can the cop number be?
• c(n) = maximum cop number of a connected graph of order n
• Meyniel’s 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
Graph classes• (Aigner, Fromme,84): Planar (outerplanar)
graphs have cop number at most 3 (2).• (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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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 (ME) 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)– (Prałat,10) cop number = q+1
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Diameter 2• (Lu, Peng, 12): If G has diameter 2, then c(G) ≤ 2n1/2 - 1.
– diameter 2 graphs satisfy Meyniel’s conjecture
• proof uses the probabilistic method
• Question: are there explicit Meyniel extremal families whose members are diameter two?
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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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Properties of polarity graphs• order q2+q+1• (q,q+1)-regular• C4-free• (Erdős, Rényi, Sós,66), (Brown,66) orthogonal
polarity graphs C4-free extremal• diameter 2• (Godsil, Newman, 2008) have unbounded
chromatic number as q→ ∞
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Meyniel Extremal
• Theorem (B,Burgess,12+) Let q be a prime power. If Gq is a polarity graph of PG(2, q), then
q/2 ≤ c(Gq) ≤ q + 1.
• lower bound: lemma• upper bound: direct analysis
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ME method (BB,12+)
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Lower bounds
Lemma (Aigner,Fromme, 1984) If G is a connected graph of girth at least
5, then c(G) ≥ δ(G).
Lemma (BB,12+) If G is connected and K2,t-free, then
c(G) ≥ δ(G) / t.– applies to polarity graphs: t = 2
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Sketch of proof: Lower bound
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R
N(R)
C
< t neighbours attacked
Sketch of proof: Upper bound
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N2(u)
u
Sketch of proof: Upper bound
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R
N2(u)
C
• q cops move to N(u)
u
t-orbit graphs
• (Füredi,1996) � described a family of K2,t+1-free extremal graphs of order (q2 -1)/t and which are (q,q+1)-regular for prime powers q.
• gives rise to a new family of ME graphs which are K2,t+1-free
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(BB,12+) New ME families
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BIBDs• a BIBD(v, k, λ) is a pair (V, B), where V is a set of v
points, and B is a set of k-subsets of V, called blocks, such that each pair of points is contained in exactly λ blocks.
• Theorem (BB,12+) The cop number of the IG of a BIBD(v, k, λ) is between k and r, the replication number.
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Sketch of proof
• lower bound: girth 6; apply AF lemma and Fisher’s inequality
• upper bound:
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C
R
Block Intersection graphs
• given a block design (V,B), its block intersection graph has vertices equalling blocks, with blocks adjacent if they intersect
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BIG cop number
Theorem (BB,12+) If G is the block intersection graph of a BIBD(v, k, 1), then c(G) ≤ k. If v > k(k-1)2 + 1, then c(G) = k.
• gives families with unbounded cop number; not ME
• also considered point 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
• ME families from something other than designs?– extremal graphs?
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R.J. Nowakowski, P. Winkler Vertex-to-vertex pursuit in a graph, Discrete Mathematics 43 (1983) 235-239.
• 5 pages• > 200 citations (most for either author)
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The NW relation
• (Nowakowski,Winkler,83) introduced a sequence of relations characterizing cop-win graphs
• u ≤0 v if u = v• u ≤i v if for all x in N[u], there is a y in N[v]
such that x ≤j y for some j < i.
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Example
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u v w y z
u ≤1 v
u ≤2 w
Characterization
• the relations are ≤i monotone increasing; thus, there is an integer k such that
≤k = ≤k+1 – write:
≤k = ≤Theorem (Nowakowski, Winkler, 83)A cop has a winning strategy iff ≤ is V(G) x V(G).
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k cops
• may define an analogous relation but in V(G) x V(Gk) (categorical product)
• (Clarke,MacGillivray,12) k cops have a winning strategy iff the relation ≤ is V(G) x V(Gk).
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Axioms for pursuit games• a pursuit game G is a discrete-time process satisfying the following:1. Two players, Left L and Right R.2. Perfect-information.3. There is a set of allowed positions PL for L; similarly for Right.4. For each state of the game and each player, there is a non-empty set of
allowed moves. Each allowed move leaves the position of the other player unchanged.
5. There is a set of allowed start positions I a subset of PL x PR.6. The game begins with L choosing some position pL and R choosing qR
such that (pL, qR) is in I.7. After each side has chosen its initial position, the sides move alternately
with L moving first. Each side, on its turn, must choose an allowed move from its current position.
8. There is a subset of final positions, F. Left wins if at any time, the current position belongs to F. Right wins the current position never belongs to F.
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Examples of pursuit games1. Cops and Robbers
– play on graphs, digraphs, orders, hypergraphs, etc.– play at different speeds, or on different edge sets
2. Cops and Robbers with traps3. Distance k Cops and Robbers4. Tandem-win Cops and Robbers5. Restricted Chess6. Helicopter Cops and Robbers7. Maker-Breaker Games8. Seepage9. Scared Robber
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Relational characterization• given a pursuit game G, we may define relations on
PL x PR as follows:
• pL ≤0 qR if (pL, qR) in F.• pL ≤i qR if Right is on qR and for every xR in PR such that if
Right has an allowed move from (pL, qR) to (pL, xR), there exists yL in PL such that xR ≤j yL for some j < i and Left has an allowed move from (pL, xR) to (yL, xR).
• define ≤ analogously as beforeCops and Robbers 40
Characterization
Theorem (B, MacGillivray,12) Left has a winning strategy in the a pursuit game G if and only if there exists pL in PL, which is the first component of an ordered pair in I, such that for all qR in PR with (pL, qR) in I there exists wL in the set of allowed moves for Left from pL such that qR ≤ wL.
• gives rise to a min/max expression for the length of the game
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Length of game• for an allowed start position (pL, qR), define
Corollary (BM,12+) If Left has a winning strategy in the a pursuit game G, then assuming optimal play, the length of the game is
where IL is the set positions for Left which are the first component of an ordered pair in I.
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qpRLLL
II
CGT(Berlekamp, Conway, Guy, 82) A combinatorial game satisfies:
1. There are two players, Left and Right.2. There is perfect information.3. There is a set of allowed positions in the game.4. The rules of the game specify how the game begins and, for each
player and each position, which moves to other positions are allowed.
5. The players alternate moves.6. The game ends when a position is reached where no moves are
possible for the player whose turn it is to move. In normal play the last player to move wins.
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Example: NIM
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Pursuit → CGT
Theorem (BM,12+)1. Every pursuit game is a combinatorial
game.2. Not every combinatorial game is a pursuit
game.
• uses characterization of (Smith, 66) via game digraphs • Nim is a counter-example for item (2)
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Position independence
• a pursuit game G is position independent if: if the game is not over, the set of available moves for a side does not depend on the position of the other side.
• examples: Cops and Robbers …• non-examples: Helicopter Cops and
Robbers, Maker Breaker, …
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State digraph
• G a position independent pursuit game• GL= (PL, ML) and GR= (PR, MR) are the
position digraphs of G
• SG = GL x GR state digraph of G– not all edges make sense– ignore these
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Relational characterization
Corollary (BM,12+) Let G be a position independent pursuit game. If GL is strongly connected and there exists X in PL such that S = X x PR, then Left has a winning strategy in G if and only if ≤ = V (DG) = PL x PR .
• generalizes results of NW and CM
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Algorithm
Theorem (BM,12+) Let G be a position independent pursuit game. Given the graphs GL and GR, if N+
GL(PL)
and N+GR
(PR) can be obtained in time O(f(|PL|)) and O(g(|PR)|), respectively, then there is a
O(|PL||PR|f(|PL|)g(|PR|)).
algorithm to determine if Left has a winning strategy.
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Eg: Cops and Robbers
• gives an O(n2k+2)
algorithm to determine if k cops have a winning strategy
• matches best known algorithm (Clarke, MacGillivray,12)
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Cop-win graphs• node u is a corner if there is a v such that N[v] contains
N[u]• a cop-win ordering of G is an enumeration (v1,v2,…,vn) of
V(G) such that for all i < n, there is a j > i such that
Theorem (Nowakowski, Winkler 83; Quilliot, 78)A graph is cop-win if and only if it has a cop-win ordering.idea: cop-win graphs always have corners; retract corner and play shadow strategy;
- graphs with cop-win orderings are cop-win by induction
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Cop-win ordering: dismantling
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Vertex elimination ordering• cop-win orderings generalize to pursuit games:
Idea: – order vertices of state digraph SG– removable vertices are those whose out-neighbours
are “dominated” by some by a vertex with higher index in the sequence
Corollary (BM,12+): Left has a winning strategy in the position independent pursuit game G if and only if SG admits a removable vertex ordering.
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Infinite case
• analogous characterization holds• k such that ≤k = ≤k+1 is an ordinal, κ
– CR-ordinal• relational characterization and vertex-
ordering hold (now a transfinite sequence)
• NB: κ can be infinite:– cannot think of κ as length of game
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κ = ω
Questions• c(G)≤k? k fixed:
– optimizing complexity for small k?• c(G)≤k? k not fixed:
– (Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): NP-hard
– Goldstein, Reingold Conjecture: EXPTIME-complete.– Conjecture: not in NP.– PSPACE-complete?
• infinite graphs: what are the CR-ordinals κ for cop-win graphs? – same question, but for more than one cop
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Another direction:Minimum orders
• Mk = minimum order of a k-cop-win graph
• M1 = 1, M2 = 4• M3 = 10 (Baird, B, et al, 12+)
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Questions
• M4 = ? (the (4,5)-cage?)• 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, B, et al, 12+) mk = Ω(k2) is equivalent to Meyniel’s conjecture.
• mk = Mk for all k ≥ 4?
Good guys vs bad guys games in graphs
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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
Cops and Robbers
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