What is left to do on Cops and Robbers?

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

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

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(n1/2). Cops and Robbers

4. Cops and Robbers

5. Henri Meyniel, courtesy Geňa Hahn Cops and Robbers

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)) ≤ 160000n1/2log n • (Prałat,Wormald,12+): removed log factor Cops and Robbers

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 Robbers

8. Questions • Soft Meyniel’s conjecture: for some ε > 0, c(n) = O(n1-ε). • Meyniel’s conjecture in other graphs classes? • bounded chromatic number • bipartite graphs • diameter 3 • claw-free Cops and Robbers

9. 2. 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

10. Example Fano plane Heawood graph Cops and Robbers

11. 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) • all other examples of Meyniel extremal families come from combinatorial designs (see Andrea Burgess’ talk) Cops and Robbers

12. 3. Minimum orders • Mk = minimum order of a k-cop-win graph • M1 = 1, M2 = 4 • M3 = 10 (Baird, Bonato,12+) • see also (Beveridge et al, 2012+) Cops and Robbers

13. Questions • M4 = ? • are the Mk monotone increasing? • for example, can it happen that M344 < M343? • mk = minimum order of a connected G such that c(G) ≥ k • (Baird, Bonato, 12+) mk= Ω(k2) is equivalent to Meyniel’s conjecture. • mk= Mk for all k ≥ 4? Cops and Robbers

14. 4. Complexity • (Berrarducci, Intrigila, 93), (Hahn,MacGillivray, 06), (B,Chiniforooshan, 09): “c(G) ≤ s?” sfixed: 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

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(n7)? • how fast can we recognize cop-win graphs? Cops and Robbers

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

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 Robbers

18. 6. VariantsGood guys vs bad guys games in graphs bad good Cops and Robbers

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 k = 1 R Cops and Robbers

20. 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

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 Robbers

22. The robber fights back! (Haidar,12) • robber can attackneighbouring 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? C C R C Cops and Robbers

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

24. Fill in the blanks… bad good Cops and Robbers

25. Cops and Robbers