1 Conjectures on Cops and Robbers Anthony Bonato Ryerson University Toronto, Canada Stella Maris College Chennai

Cops and Robbers 2 C C C R

3 C C C R

4 C C C R cop number c(G) ≤ 3

Cops and Robbers played on a reflexive 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 Robbers5

Basic properties Theorem: 1.c(G) ≤ γ (G) (the domination number). 2.The cop number of a cycle is 2. 3.The cop number of a clique is 1. 4.The cop number of the disjoint union of two graphs is the sum of their cop number. 6Cops and Robbers

Applications of Cops and Robbers robotics –mobile computing –missile-defense –gaming counter-terrorism –intercepting messages or agents Cops and Robbers7

Trees Theorem 1.The cop number of a finite tree is 1. 2.The cop number of an infinite tree is 1 or infinite. It is infinite exactly when the tree has no ray: infinite paths. 8Cops and Robbers

Lower bound Theorem (Aigner, Fromme, 84). If G is a graph with girth at least 5, then c(G) ≥ δ(G). Application: c(Petersen) ≥ 3. 9Cops and Robbers

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 Robbers10

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 Robbers11

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Cops and Robbers13 Henri Meyniel, courtesy Geňa Hahn

State-of-the-art (Lu, Peng, 13) proved that –independently proved by (Frieze, Krivelevich, Loh, 11) and (Scott, Sudakov,11) (Bollobás, Kun, Leader,13): if p = p(n) ≥ 2.1log n/ n, then c(G(n,p)) ≤ n 1/2 log n (Prałat,Wormald,15): proved Meyniel’s conjecture for all p = p(n) Cops and Robbers 14

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 Robbers15

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 Robbers16

Cops and Robbers17 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 Robbers18 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) all other examples of Meyniel extremal families come from combinatorial designs (B,Burgess,2013) Cops and Robbers19

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 Robbers20

Conjectures on m k, M k Conjecture: M k monotone increasing. m k = minimum order of a connected G such that c(G) ≥ k (Baird, B, 12) m k = Ω(k 2 ) is equivalent to Meyniel’s conjecture. Conjecture: m k = M k for all k ≥ 4. Cops and Robbers21

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 Robbers22

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) Conjecture: if s is not fixed, then computing the cop number is not in NP. Cops and Robbers23

3. Genus (Aigner, Fromme, 84) planar graphs (genus 0) have cop number ≤ 3. (Clarke, 02) outerplanar graphs have cop number ≤ 2. Cops and Robbers24

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 Robbers25

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 Robbers26

5. Variants Good guys vs bad guys games in graphs 27 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 Robbers

28 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 Robbers29 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 Robbers30

Lazy Cops and Robbers Cops and Robbers31

Questions on lazy cops Cops and Robbers32

33Cops and Robbers Firefighting

A strategy (MacGillivray, Wang, 03): If fire breaks out at (r,c), 1≤r≤c≤n/2, save vertices in following order: (r + 1, c), (r + 1, c + 1), (r + 2, c - 1), (r + 2, c + 2), (r + 3, c -2),(r + 3, c - 3),..., (r + c, 1), (r + c, 2c), (r + c, 2c + 1),..., (r + c, n) –strategy saves n(n-r)-(c-1)(n-c) vertices –strategy is optimal assuming fire breaks out in columns (rows) 1,2, n-1, n Cops and Robbers34

¼ -grid conjecture 35Cops and Robbers

Infinite hexagonal grid Conjecture: one firefighter cannot contain a fire in an infinite hexagonal grid. Cops and Robbers36

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

Thank you! நன்றி ! Cops and Robbers39