Cops and Robbers1 Cops and Robbers: Directions and Generalizations Anthony Bonato Ryerson University GRASTA 2012

Happy 60 th Birthday RJN May your searching never end. Cops and Robbers2

3 C C C R

4 C C C R

5 C C C R 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)| Cops and Robbers6

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 Cops and Robbers7

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(n 1/2 ). Cops and Robbers8

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10 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)) ≤ n 1/2 log n (Prałat,Wormald,12+): removed log factor Cops and Robbers 11

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. Cops and Robbers12

Cops and Robbers13 How close to n 1/2 ? 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 q 2 +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 Robbers14 Fano plane Heawood graph

Meyniel extremal families a family of connected graphs (G n : n ≥ 1) is Meyniel extremal (ME) if there is a constant d > 0, such that for all n ≥ 1, c(G n ) ≥ dn 1/2 IG of projective planes: girth 6, (q+1)-regular, so have cop number ≥ q+1 –order 2(q 2 +q+1) –Meyniel extremal (must fill in non-prime orders) –(Prałat,10) cop number = q+1 Cops and Robbers15

Diameter 2 (Lu, Peng, 12): If G has diameter 2, then c(G) ≤ 2n 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? Cops and Robbers16

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 π(q) iff q π(p). eg of orthogonal polarity: point mapped to its orthogonal complement polarity graph: vertices are points, x and y adjacent if x π(y) Cops and Robbers17

Properties of polarity graphs order q 2 +q+1 (q,q+1)-regular C 4 -free (Erdős, Rényi, Sós,66), (Brown,66) orthogonal polarity graphs C 4 -free extremal diameter 2 (Godsil, Newman, 2008) have unbounded chromatic number as q→ ∞ Cops and Robbers18

Meyniel Extremal Theorem (B,Burgess,12+) Let q be a prime power. If G q is a polarity graph of PG(2, q), then q/2 ≤ c(G q ) ≤ q + 1. lower bound: lemma upper bound: direct analysis Cops and Robbers19

ME method (BB,12+) Cops and Robbers20

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 K 2,t -free, then c(G) ≥ δ(G) / t. –applies to polarity graphs: t = 2 Cops and Robbers21

Sketch of proof: Lower bound Cops and Robbers22 R N(R) C < t neighbours attacked

Sketch of proof: Upper bound Cops and Robbers23 R C N 2 (u) u

Sketch of proof: Upper bound Cops and Robbers24 R N 2 (u) C q cops move to N(u) u

t-orbit graphs (Füredi,1996) described a family of K 2,t+1 -free extremal graphs of order (q 2 -1)/t and which are (q,q+1)-regular for prime powers q. gives rise to a new family of ME graphs which are K 2,t+1 -free Cops and Robbers25

(BB,12+) New ME families Cops and Robbers26

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. Cops and Robbers27

Sketch of proof lower bound: girth 6; apply AF lemma and Fisher’s inequality upper bound: Cops and Robbers28 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 Cops and Robbers29

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 Cops and Robbers30

Questions Soft Meyniel’s conjecture: for some ε > 0, c(n) = O(n 1-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? Cops and Robbers31

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R.J. Nowakowski, P. Winkler Vertex-to- vertex pursuit in a graph, Discrete Mathematics 43 (1983) pages > 200 citations (most for either author) Cops and Robbers33

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. Cops and Robbers34

Example Cops and Robbers35 u v w yz 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). Cops and Robbers36

k cops may define an analogous relation but in V(G) x V(G k ) (categorical product) (Clarke,MacGillivray,12) k cops have a winning strategy iff the relation ≤ is V(G) x V(G k ). Cops and Robbers37

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 P L 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 P L x P R. 6.The game begins with L choosing some position p L and R choosing q R such that (p L, q R ) 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. Cops and Robbers38

Examples of pursuit games 1.Cops and Robbers –play on graphs, digraphs, orders, hypergraphs, etc. –play at different speeds, or on different edge sets 2.Cops and Robbers with traps 3.Distance k Cops and Robbers 4.Tandem-win Cops and Robbers 5.Restricted Chess 6.Helicopter Cops and Robbers 7.Maker-Breaker Games 8.Seepage 9.Scared Robber Cops and Robbers39

Relational characterization given a pursuit game G, we may define relations on P L x P R as follows: p L ≤ 0 q R if (p L, q R ) in F. p L ≤ i q R if Right is on q R and for every x R in P R such that if Right has an allowed move from (p L, q R ) to (p L, x R ), there exists y L in P L such that x R ≤ j y L for some j < i and Left has an allowed move from (p L, x R ) to (y L, x R ). define ≤ analogously as before Cops and Robbers40

Characterization Theorem (B, MacGillivray,12) Left has a winning strategy in the a pursuit game G if and only if there exists p L in P L, which is the first component of an ordered pair in I, such that for all q R in P R with (p L, q R ) in I there exists w L in the set of allowed moves for Left from p L such that q R ≤ w L. gives rise to a min/max expression for the length of the game Cops and Robbers41

Length of game for an allowed start position (p L, q R ), 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 I L is the set positions for Left which are the first component of an ordered pair in I. Cops and Robbers42

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. Cops and Robbers43

Example: NIM Cops and Robbers44

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) Cops and Robbers45

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, … Cops and Robbers46

State digraph G a position independent pursuit game G L = (P L, M L ) and G R = (P R, M R ) are the position digraphs of G S G = G L x G R state digraph of G –not all edges make sense –ignore these Cops and Robbers47

Relational characterization Corollary (BM,12+) L et G be a position independent pursuit game. If G L is strongly connected and there exists X in P L such that S = X x P R, then Left has a winning strategy in G if and only if ≤ = V (D G ) = P L x P R. generalizes results of NW and CM Cops and Robbers48

Algorithm Theorem (BM,12+) Let G be a position independent pursuit game. Given the graphs G L and G R, if N + G L (P L ) and N + G R (P R ) can be obtained in time O(f(|P L |)) and O(g(|P R )|), respectively, then there is a O(|P L ||P R |f(|P L |)g(|P R |)). algorithm to determine if Left has a winning strategy. Cops and Robbers49

Eg: Cops and Robbers gives an O(n 2k+2 ) algorithm to determine if k cops have a winning strategy matches best known algorithm (Clarke, MacGillivray,12) Cops and Robbers50

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 (v 1,v 2,…,v n ) of V(G) such that for all i 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 Cops and Robbers51

Cop-win ordering: dismantling Cops and Robbers52

Vertex elimination ordering cop-win orderings generalize to pursuit games: Idea: –order vertices of state digraph S G –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 S G admits a removable vertex ordering. Cops and Robbers53

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 Cops and Robbers54

Cops and Robbers55 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 Cops and Robbers56

Another direction: Minimum orders M k = minimum order of a k-cop-win graph M 1 = 1, M 2 = 4 M 3 = 10 (Baird, B, et al, 12+) Cops and Robbers57

Questions M 4 = ? (the (4,5)-cage?) are the M k monotone increasing? –for example, can it happen that M 344 < M 343 ? m k = minimum order of a connected G such that c(G) ≥ k (Baird, B, et al, 12+) m k = Ω(k 2 ) is equivalent to Meyniel’s conjecture. m k = M k for all k ≥ 4? Cops and Robbers58

Good guys vs bad guys games in graphs 59 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal security fastcleaningdistance k Cops and Robbers Cops and Robbers on disjoint edge sets The Angel and Devil helicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil, Firefighter Hex bad good Cops and Robbers

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