Graph Searching Games and Probabilistic Methods PIMS-University of Manitoba Distinguished Lecture Graph Searching Games and Probabilistic Methods Anthony Bonato Ryerson University

Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Cops and Robbers C C R Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Cops and Robbers C C R C Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Cops and Robbers C C R C Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Cops and Robbers C R C C cop number c(G) = 3 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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)| Probabilistic graph searching - Anthony Bonato

Applications of Cops and Robbers robotics AI & mobile computing gaming Network interdiction intercepting messages or agents Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato power of cops: traps, photo radar, walls, capture from distance, teleportation, tandem-win, lazy cops power of robber: speed: fast or infinite, invisible, decoys, barricades, damage, capture cops vertex pursuit games/processes: Firefighting, watchman’s problem, eternal domination, seepage, robot vacuum, robot crawler, acquaintance time, Angel and Devil, burning, graph cleaning, Revolutionaries and Spies, … Probabilistic graph searching - Anthony Bonato

Randomness in vertex pursuit DETERMINISTIC GAME/PROCESS RANDOM GAME/PROCESS DETERMINISTIC BOARD Classical model Cops and drunk Robber, burning, Zombies and Survivors, … RANDOM BOARD Cops and Robbers, Firefighter, Seepage Robot vacuum, acquisition number, graph cleaning, acquaintance time, toppling number… ? Probabilistic graph searching - Anthony Bonato

Zombies

Miniconference on the Mathematics of Computation Probabilistic graph searching - Anthony Bonato

Miniconference on the Mathematics of Computation Probabilistic graph searching - Anthony Bonato

Miniconference on the Mathematics of Computation Zombie horde up to n/2 - 2 zombies on an induced path will never capture the survivor on a cycle Probabilistic graph searching - Anthony Bonato

The game

Probabilistic graph searching - Anthony Bonato 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! Probabilistic graph searching - Anthony Bonato

(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 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato with probability 1− 5 n k = exp − 5k n +𝑂 k n 2 all zombies begin outside the cycle implies: 𝑧 G ~ log 2 5 n and so 𝑍 G ~ log 2 10 n n-5 leaves G Probabilistic graph searching - Anthony Bonato

Zombie number of cycles Theorem (BMPGP,16) If n ≥ 27, then z(Cn) = 4, so Z(Cn) = 2. Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Cartesian grids (Tosic, 87) c(G H) ≤ c(G) + c(H) Theorem (BMPGP,16) For n ≥ 2, z(Pn Pn) = 2. Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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). Probabilistic graph searching - Anthony Bonato

Toroidal grids, continued despite the lower bound, no known subquadratic upper bound is known for the zombie number of toroidal graphs! Probabilistic graph searching - Anthony Bonato

In the beginning…

G(n,p) random graph model (Erdős, Rényi, 63) Miniconference on the Mathematics of Computation G(n,p) random graph model (Erdős, Rényi, 63) p = p(n) a real number in (0,1), n a positive integer G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p 1 2 3 4 5 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Cop number of G(n,p) in G(n,p), the cop number is a random variable Theorem (Bonato, Hahn, Wang, 07) For 0 < p < 1 a constant, then a.a.s. c(G(n,p)) = Θ(log n). Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Sketch of proof upper bound: uses (Dreyer, 00) bound for domination number of G(n,p) lower bound: G is (1,k)-e.c. if for all x and S with |S| ≤ k, there is a node z such that R x S C z G (1,k)-e.c. implies c(G) ≥ k a.a.s. G(n,p) is (1,k)-e.c., where k = (1-ε)log1/1-pn Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Zig-zag for random graphs G(n,p) with p = p(n), the asymptotic behaviour of the cop number is more complicated (Prałat, Łuczak,10) Probabilistic graph searching - Anthony Bonato

Cop-win

Probabilistic graph searching - Anthony Bonato Cop-win graphs node u is a corner if there is a v such that N[v] contains N[u] v is the parent; u is the child a graph is dismantlable if we can iteratively delete corners until there is only one vertex Theorem (Nowakowski, Winkler 83; Quilliot, 78) A graph is cop-win if and only if it is dismantlable. Idea: cop-win graphs always have corners; retract corner and play shadow strategy Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Dismantlable graphs Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Dismantlable graphs unique corner! part of an infinite family that maximizes capture time (Bonato, Hahn, Golovach, Kratochvíl,09) Probabilistic graph searching - Anthony Bonato

Typical cop-win graphs what is a random cop-win graph? G(n,1/2) and condition on being cop-win probability of choosing a cop-win graph on the uniform space of labeled graphs of ordered n Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Universal vertices P(cop-win) ≥ P(universal) = n2-n+1 – O(n22-2n+3) = (1+o(1))n2-n+1 …this is in fact the correct answer! Probabilistic graph searching - Anthony Bonato

Almost all cop-win graphs Theorem (B,Kemkes, Prałat,12) In G(n,1/2), P(cop-win) = (1+o(1))n2-n+1 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Corollary Un = number of labeled graphs with a universal vertex Cn = number of labeled cop-win graphs Corollary (BKP,12) lim 𝑛→∞ 𝑈 𝑛 𝐶 𝑛 = 1. That is, almost all cop-win graphs contain a universal vertex. Probabilistic graph searching - Anthony Bonato

How big can the cop number be?

Probabilistic graph searching - Anthony Bonato c(n) = maximum cop number of a connected graph of order n Meyniel Conjecture: c(n) = O(n1/2). Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato State-of-the-art (Lu, Peng, 13) proved that independently proved by (Frieze, Krivelevich, Loh, 11) and (Scott, Sudakov,11) all these proofs use the probabilistic method Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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,17+): holds a.a.s. for random d-regular graphs, for d ≥ 3 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Example Fano plane Heawood graph Probabilistic graph searching - Anthony Bonato

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) Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato (BB,13) New ME families Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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) Probabilistic graph searching - Anthony Bonato

Properties of polarity graphs order q2+q+1 (q,q+1)-regular C4-free diameter 2 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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. Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato 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 Probabilistic graph searching - Anthony Bonato

Capture time

Probabilistic graph searching - Anthony Bonato Capture time of a graph the length of Cops and Robbers was considered first as capture time (B,Hahn,Golovach,Kratochvíl,09) capture time of G: length of game with c(G) cops assuming optimal play, written capt(G) if G is cop-win, then capt(G) ≤ n - 3 if n ≥ 7 (see also (Gavanciak,10)) capt(G) ≤ n/2 for many families of cop-win graphs including trees, chordal graphs examples of planar graphs with capt(G) = n - 3 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Trees are cop-win C Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Capture time of trees Lemma (B, Perez-Gimenez,Reiniger,Prałat,17): For a tree T, we have that capt(T) = rad(T). Proof sketch: for capt(T) ≤ rad(T), place C on a central vertex and use the zombie strategy for rad(T) ≤ capt(T), notice that any other initial placement of C results in R choosing a vertex distance > rad(T) away R stays put Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Hypercubes Probabilistic graph searching - Anthony Bonato

Cop number of products of trees Theorem (Maamoun,Meyniel,87): The cop number of the Cartesian product of d trees is d+1 2 . no reference to the length of the game; i.e capture time of grids or the hypercube Probabilistic graph searching - Anthony Bonato

Capture time of Cartesian grids Theorem (Merhabian,10): The capture time of the Cartesian product of two trees T1 and T2 is diam(T1) + diam(T2)) / 2 . In particular, the capture time of the m x n Cartesian grid is (m + n)/2−1 . Probabilistic graph searching - Anthony Bonato

Capture time of hypercubes Theorem (B,Gordinowicz,Kinnersley,Prałat,13) The capture time of Qn is Θ(nlog n). Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Lower bound Theorem (BGKP,13) For b > 0 a constant, a robber can escape nb cops for at least (1-o(1))1/2 n log n rounds. probabilistic method: play with a random/drunk robber Coupon collector and large deviation bounds Probabilistic graph searching - Anthony Bonato

Proof of lower bound (sketch) let T= 1/2(n-1)log n, ε = ln((4d+1) ln n) / ln n = o(1). show that a random robber can play (1- ε)T rounds without being captured can play initial round due to expansion next consider a single cop C playing greedily can show process of C capturing R is equivalent to the coupon collector problem using a deviation bound, the probability single cop captures robber is exp(-(n/2)ε/4); via union bound for all nd cops this is o(1) hence, there is SOME deterministic strategy for the robber to survive (1- ε)T rounds Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Add more cops! Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato k-capture time define captk(G), where c(G) ≤ k ≤ γ(G) k-capture time capt(G) = captc(G)(G) temporal speed-up: as c(G) increases to γ(G), captk(G) monotonically decreases to 1 if k > c(G), we call this Overprescribed Cops and Robbers Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Trees for k > 0, metric k-center is a set S, |S| ≤ k, that minimizes max 𝑣∈𝑉(𝐺) 𝑑(𝑣,𝑆) radk(G) is this minimum k = 1, then radk(G) = rad(G) NP-complete to find metric k-centers (Vazirani,03) radk is monotone on retracts retractions monotone on walk length Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Example: k = 1 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Example: k = 2 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Example: k = 3 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Example: k = 4 Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Example: k = 5 = 𝛾 Probabilistic graph searching - Anthony Bonato

Retracts following theorem is key: Theorem (BGRP,17) Suppose V can be decomposed into t-many vertex sets of retracts Gi of G. Then captk(G)≤ max 1≤𝑖≤𝑡 captk(Gi) Idea: Play shadow strategy in each retract. Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Trees Corollary (BGRP,17) For a tree T, captk(G) = radk(G). Idea: cover by balls (which are retracts) around vertices around metric k-center and use theorem Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato Square grids G(d,n) = d-dimensional Cartesian n-grid (Maamoun,Meyniel,87): c(G(d,n)) = d+1 2 (Merhabian,10): capt(G(d,n)) = 1 2 nd log 2 𝑑 𝛾(G(d,n)) = Θ(nd) use dominating sets of paths order Θ(n) Probabilistic graph searching - Anthony Bonato

k-capture time of grids Theorem (BGRP,17) If k = O(nd), then captk(G(d,n)) = Θ(n/k1/d). Idea: cover by sub-grids (retracts) and use theorem Probabilistic graph searching - Anthony Bonato

Domination number of hypercubes 𝛾 𝑄 𝑛 is open for general n 𝛾 𝑄 𝑛 ≤ 2 𝑛−3 if n ≥ 7 n 𝛾 𝑄 𝑛 3 2 4 5 7 6 12 n= 2k-1, 2k 2n-k Probabilistic graph searching - Anthony Bonato

Temporal speed up in hypercubes Theorem (BGRP,17) Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato place cops randomly in the hypercube cops are sufficiently dense, can occupy some Nd(R) a.a.s. apply Hall’s condition to find a perfect matching between Nd(R) and Nd-1(R) d C d-1 R C C C Probabilistic graph searching - Anthony Bonato

Probabilistic graph searching - Anthony Bonato place cops randomly in the hypercube cops are sufficiently dense, can occupy some Nd(R) a.a.s. apply Hall’s condition to find a perfect matching between Nd(R) and Nd-1(R) cops move along this matching and “close in” on R d C d-1 R C C C Probabilistic graph searching - Anthony Bonato

Where to next with Cops and Robbers? Meyniel’s conjecture Soft Meyniel’s conjecture: for some ε > 0, c(n) = O(n1-ε). topological graph theory Schroeder’s conjecture Lazy Cops and Robbers planar graphs invisible robber 0-visibility, limited visibility, hyperopic cops, localization game Probabilistic graph searching - Anthony Bonato

Contact Web: http://www.math.ryerson.ca/~abonato/ Blog: https://anthonybonato.com/ @Anthony_Bonato https://www.facebook.com/anthony.bonato.5 Zombies and Survivors

Thank you!