Cops and Robbers: Directions and Generalizations

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

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

3. Cops and Robbers C C R C Cops and Robbers

4. Cops and Robbers C C R C Cops and Robbers

5. Cops and Robbers C R C C cop number c(G) ≤ 3 Cops and Robbers

6. 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 Robbers

7. 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 Robbers

8. 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). Cops and Robbers

9. Cops and Robbers

10. Henri Meyniel, courtesy Geňa Hahn Cops and Robbers

11. 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 Cops and Robbers

12. 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 Robbers

13. 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 Cops and Robbers

14. Example Fano plane Heawood graph Cops and Robbers

15. 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 Cops and Robbers

16. 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? Cops and Robbers

17. Polarity graphs • suppose PG(2,q) has points P and lines L. A polarity is a functionπ: P→ L such that for all points p,q, p ϵπ(q) iff q ϵπ(p). • eg of orthogonal polarity: point mapped to its orthogonal complement • polarity graph: vertices are points, x and y adjacent if xϵπ(y) Cops and Robbers

18. 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→ ∞ Cops and Robbers

19. Meyniel Extremal • Theorem (B,Burgess,12+) Letqbe 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 Cops and Robbers

20. ME method (BB,12+) Cops and Robbers

21. 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 Cops and Robbers

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

23. Sketch of proof: Upper bound u R N2(u) C Cops and Robbers

24. Sketch of proof: Upper bound u • q cops move to N(u) C R N2(u) Cops and Robbers

25. 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 Cops and Robbers

26. (BB,12+) New ME families Cops and Robbers

27. BIBDs • a BIBD(v, k, λ) is a pair (V, B), where V is a set of vpoints, 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 Robbers

28. Sketch of proof • lower bound: girth 6; apply AF lemma and Fisher’s inequality • upper bound: C R Cops and Robbers

29. 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 Robbers

30. 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 Robbers

31. 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? Cops and Robbers

33. R.J. Nowakowski, P. Winkler Vertex-to-vertex pursuit in a graph, Discrete Mathematics43 (1983) 235-239. • 5 pages • > 200 citations (most for either author) Cops and Robbers

34. 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 Robbers

35. Example w y z v u u ≤1 v u ≤2 w Cops and Robbers

36. Characterization • the relations are ≤imonotone 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 Robbers

37. 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). Cops and Robbers

38. Axioms for pursuit games • a pursuit game G is a discrete-time process satisfying the following: • Two players, Left Land Right R. • Perfect-information. • There is a set of allowed positions PL for L; similarly for Right. • 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. • There is a set of allowed start positions I a subset of PL x PR. • The game begins with L choosing some position pL and R choosing qR such that (pL, qR) is in I. • 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. • 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 Robbers

39. Examples of pursuit games • Cops and Robbers • play on graphs, digraphs, orders, hypergraphs, etc. • play at different speeds, or on different edge sets • Cops and Robbers with traps • Distance k Cops and Robbers • Tandem-win Cops and Robbers • Restricted Chess • Helicopter Cops and Robbers • Maker-Breaker Games • Seepage • Scared Robber Cops and Robbers

40. 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 before Cops and Robbers

41. 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 Cops and Robbers

42. 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. Cops and Robbers

43. CGT (Berlekamp, Conway, Guy, 82) A combinatorial game satisfies: • There are two players, Left and Right. • There is perfect information. • There is a set of allowed positions in the game. • The rules of the game specify how the game begins and, for each player and each position, which moves to other positions are allowed. • The players alternate moves. • 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 Robbers

44. Example: NIM Cops and Robbers

45. Pursuit → CGT Theorem (BM,12+) • Every pursuit game is a combinatorial game. • 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 Robbers

46. 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 Robbers

47. State digraph • Ga position independent pursuit game • GL= (PL, ML) andGR= (PR, MR) are the position digraphs ofG • SG = GL x GRstate digraph ofG • not all edges make sense • ignore these Cops and Robbers

48. 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 Cops and Robbers

49. 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. Cops and Robbers

50. 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) Cops and Robbers