1. Cops and Robbers1 What is left to do on Cops and Robbers? Anthony Bonato Ryerson University GRASCan 2012

2. Where to next? we focus on 6 research directions on the topic of Cops and Robbers games –by no means exhaustive Cops and Robbers2

3. 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 Robbers3

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5. 5 Henri Meyniel, courtesy Geňa Hahn

6. 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)) ≤ 160000n 1/2 log n (Prałat,Wormald,12+): removed log factor Cops and Robbers 6

7. Graph classes (Aigner, Fromme,84): Planar graphs have cop number at most 3. (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 Robbers7

8. Questions Soft Meyniel’s conjecture: for some ε > 0, c(n) = O(n 1-ε ). Meyniel’s conjecture in other graphs classes? –bounded chromatic number –bipartite graphs –diameter 3 –claw-free Cops and Robbers8

9. 9 2. 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

10. Example Cops and Robbers10 Fano plane Heawood graph

11. 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 (see Andrea Burgess’ talk) Cops and Robbers11

12. 3. Minimum orders M k = minimum order of a k-cop-win graph M 1 = 1, M 2 = 4 M 3 = 10 (Baird, Bonato,12+) –see also (Beveridge et al, 2012+) Cops and Robbers12

13. Questions M 4 = ? are the M k monotone increasing? –for example, can it happen that M 344 < M 343 ? m k = minimum order of a connected G such that c(G) ≥ k (Baird, Bonato, 12+) m k = Ω(k 2 ) is equivalent to Meyniel’s conjecture. m k = M k for all k ≥ 4? Cops and Robbers13

14. 4. 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 Robbers14

15. Questions Goldstein, Reingold Conjecture: if s is not fixed, then computing the cop number is EXPTIME-complete. –same complexity as say, generalized chess Conjecture: if s is not fixed, then computing the cop number is not in NP. speed ups? –can we recognize 2-cop-win graphs in o(n 7 )? –how fast can we recognize cop-win graphs? Cops and Robbers15

16. 5. Planar graphs (Aigner, Fromme, 84) planar graphs have cop number ≤ 3. (Clarke, 02) outerplanar graphs have cop number ≤ 2. Cops and Robbers16

17. 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? edge contraction/subdivision and cop number? –see (Clarke, Fitzpatrick, Hill, RJN, 10) Cops and Robbers17

18. 6. Variants Good guys vs bad guys games in graphs 18 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal 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

19. 19 Distance k Cops and Robber (Bonato,Chiniforooshan,09) (Bonato,Chiniforooshan,Prałat,10) 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

20. Cops and Robbers20 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)

21. When does one cop suffice? cop-win graphs ↔ cop-win orderings (RJN, Winkler, 83), (Quilliot, 78) 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…) –line graphs (RJN,12+) –infinite graphs (Bonato, Hahn, Tardif, 10) Cops and Robbers21

22. The robber fights back! (Haidar,12) robber can attack neighbouring cop one more cop needed in this graph (check) at most min{2c(G),γ(G)} cops needed, in general are c(G)+1 many cops needed? Cops and Robbers22 C C C R

23. Infinite hexagonal grid can one cop contain the fire? Fighting Intelligent Fires Anthony Bonato 23

24. Fill in the blanks… Cops and Robbers24 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal 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

25. Cops and Robbers25