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

Cops and Robbers C C R Cops and Robbers

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

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

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

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

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

Cops and Robbers

Henri Meyniel, courtesy Geňa Hahn Cops and Robbers

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

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

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

For random graphs (Bollobás, Kun, Leader,13): if p = p(n) ≥ 2.1log n/ n, then a.a.s. c(G(n,p)) ≤ 160000n1/2log 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

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

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

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

Example Fano plane Heawood graph Cops and Robbers

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

(BB,13) New ME families Cops and Robbers

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) Cops and Robbers

Properties of polarity graphs order q2+q+1 (q,q+1)-regular C4-free diameter 2 Cops and Robbers

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. Cops and Robbers

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 Cops and Robbers

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

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

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

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

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

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

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

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

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

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

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 3 2 ). (Bal,B,Kinnsersley,Pralat,15) For all ε > 0, Ω( 2 𝑛 / 𝑛 5 2 +𝜀 ) ≤ 𝑐 𝐿 ( 𝑄 𝑛 ). Cops and Robbers

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

Miniconference on the Mathematics of Computation Cops and Robbers

Miniconference on the Mathematics of Computation Cops and Robbers

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

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

(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

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

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

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). Cops and Robbers

Toroidal grids, continued despite the lower bound, no known upper bound is known for the zombie number of toroidal graphs! Cops and Robbers

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

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… Cops and Robbers

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? Cops and Robbers

Cops and Robbers

. 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