1 Conjectures on Cops and Robbers 2.0 Anthony Bonato Ryerson University Toronto, Canada Ryerson

100 The sum of the first nine primes is = is the sum of the first four cube numbers: = 100 There are 4950 edges in K 100 The crossing number of K 11 is 100 The cop number of Q 199 is 100 Cops and Robbers2

3 C C R

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 Robbers4

Observations 5Cops and Robbers

Applications of Cops and Robbers robotics –mobile computing –missile-defense –gaming counter-terrorism –intercepting messages or agents Cops and Robbers6

Conjectures conjectures and problems on Cops and Robbers coming from five different directions, touch on various aspects of graph theory: –structural, algorithmic, probabilistic, topological… Cops and Robbers7

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(n 1/2 ). Cops and Robbers8

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10 Henri Meyniel, courtesy Geňa Hahn

Sketch of Frankl’s proof Cops and Robbers11

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

For random graphs (Bollobás, Kun, Leader,13): if p = p(n) ≥ 2.1log n/ n, then a.a.s. c(G(n,p)) ≤ n 1/2 log 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 Robbers13

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 Robbers14

Questions Soft Meyniel’s conjecture: for some ε > 0, c(n) = O(n 1-ε ). Meyniel’s conjecture in other graphs classes? –bipartite graphs –diameter 3 –claw-free Cops and Robbers15

Cops and Robbers16 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 Robbers17 Fano plane Heawood graph

Meyniel extremal families a family of connected graphs (G n : n ≥ 1) is Meyniel extremal 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) other examples of Meyniel extremal families come from combinatorial designs and finite geometries (B,Burgess,2013) Cops and Robbers18

(BB,13) New ME families Cops and Robbers19

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) 20Cops and Robbers

Properties of polarity graphs order q 2 +q+1 (q,q+1)-regular C 4 -free diameter 2 21Cops and Robbers

Meyniel Extremal Theorem (Bonato,Burgess,13) Let q be a prime power. If G q is a polarity graph of PG(2, q), then q/2 ≤ c(G q ) ≤ q Cops and Robbers

Lower bounds Theorem (Bonato, Burgess,13) If G is connected and K 2,t -free, then c(G) ≥ δ(G) / t. –applies to polarity graphs: t = 2 23Cops and Robbers

Minimum orders M k = minimum order of a k-cop-win graph M 1 = 1, M 2 = 4 M 3 = 10 (Baird, B,12) –see also (Baird,Beveridge,B, et al, 14) M 4 = ? Cops and Robbers24

2. Complexity (Berrarducci, Intrigila, 93), (Hahn,MacGillivray, 06), (B,Chiniforooshan, 09): “c(G) ≤ s?” s fixed: in P; running time O(n 2s+3 ), n = |V(G)| (Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): if s not fixed, then computing the cop number is NP-hard Cops and Robbers25

Questions 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 Robbers26

3. Genus (Aigner, Fromme, 84) planar graphs (genus 0) have cop number ≤ 3. (Clarke, 02) outerplanar graphs have cop number ≤ 2. Cops and Robbers27

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 Robbers28

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 Robbers29

5. Variants Good guys vs bad guys games in graphs 30 slowmediumfasthelicopter slowtraps, tandem-win, Lazy Cops and Robbers mediumrobot vacuumCops and Robbersedge searching, Cops and Fast Robber eternal 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 Robbers31 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 R k = 1

Cops and Robbers32 Distance k cop number: c k (G) c k (G) = minimum number of cops needed to capture robber at distance at most k G connected implies c k (G) ≤ diam(G) – 1 for all k ≥ 1, c k (G) ≤ c k-1 (G)

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 Robbers33

Lazy Cops and Robbers Cops and Robbers34

Questions on lazy cops Cops and Robbers35

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Zombie horde Cops and Robbers38

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 Robbers39

(B,Mitsche,Perez-Gimenez,Pralat,16) s k (G): probability survivor wins if k zombies play, assuming optimal play s k+1 (G) ≤ s k (G) for all k, and s k (G) → 0 as k → ∞ zombie number of G is z(G) = min{k ≥ c(G): s k (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 Robbers40

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 Robbers41

Cartesian grids (Tosic, 87) c(G H) ≤ c(G) + c(H) Theorem (BMPGP,16) For n ≥ 2, z(P n P n ) = 2, so Z(P n P n ) =1. Cops and Robbers42

Toroidal grids Cops and Robbers43

survivor fights back? (Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Pralat, Stacho, 13): The robber strikes back In: Proceedings of ICC3, Cops and Robbers44

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46 A. Bonato, R.J. Nowakowski, Sketchy Tweets: Ten Minute Conjectures in Graph Theory, The Mathematical Intelligencer 34 (2012) Cops and Robbers