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Conjectures on Cops and Robbers

Joint Mathematics Meetings AMS Special Session. Conjectures on Cops and Robbers. Anthony Bonato Ryerson University. Cops and Robbers. C. C. R. C. Cops and Robbers. C. C. R. C. Cops and Robbers. C. R. C. C. cop number c(G) ≤ 3. Cops and Robbers.

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Conjectures on Cops and Robbers

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  1. Joint Mathematics Meetings AMS Special Session Conjectures on Cops and Robbers Anthony Bonato Ryerson University Cops and Robbers

  2. Cops and Robbers C C R C Cops and Robbers

  3. Cops and Robbers C C R C Cops and Robbers

  4. Cops and Robbers C R C C cop number c(G) ≤ 3 Cops and Robbers

  5. 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 Robbers

  6. Conjectures • conjectures and problems on Cops and Robbers coming from 5 different directions, touch on various aspects of graph theory: • structural, algorithmic, probabilistic, topological… Cops and Robbers

  7. 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

  8. Cops and Robbers

  9. Henri Meyniel, courtesy Geňa Hahn Cops and Robbers

  10. 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)) ≤ 160000n1/2log n • (Prałat,Wormald,14+): proved Meyniel’s conjecture for all p = p(n) Cops and Robbers

  11. 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

  12. 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

  13. 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

  14. Example Fano plane Heawood graph Cops and Robbers

  15. 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 (B,Burgess,2013) Cops and Robbers

  16. Minimum orders • Mk = minimum order of a k-cop-win graph • M1 = 1, M2 = 4 • M3 = 10 (Baird, B,12) • see also (Beveridge et al, 14+) • M4 = ? Cops and Robbers

  17. Conjectures on mk, Mk Conjecture: Mk monotone increasing. • mk = minimum order of a connected G such that c(G) ≥ k • (Baird, B, 12) mk= Ω(k2) is equivalent to Meyniel’s conjecture. Conjecture:mk= Mk for all k ≥ 4. Cops and Robbers

  18. 2. 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

  19. 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,14+) Conjecture: if s is not fixed, then computing the cop number is not in NP. Cops and Robbers

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

  21. 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

  22. 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

  23. 5. VariantsGood guys vs bad guys games in graphs bad good

  24. 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

  25. 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

  26. 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

  27. Lazy Cops and Robbers • (Offner, Ojakian,14+) only one can move in each round • lazy cop number, cL(G) • (Offner, Ojakian, 14+) • (Bal,B,Kinnsersley,Pralat,14+) For all ε > 0, . Cops and Robbers

  28. Questions on lazy cops • Question: Find the asymptotic order of . • (Bal,B,Kinnsersley,Pralat,14+)If G has genus g, then cL(G) = • proved by using the Gilbert, Hutchinson,Tarjan separator theorem • Question: Is cL(G) bounded for planar graphs? Cops and Robbers

  29. Firefighting Cops and Robbers

  30. 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 Robbers

  31. ¼-grid conjecture Cops and Robbers

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

  33. Cops and Robbers

  34. . A. Bonato, R.J. Nowakowski, Sketchy Tweets: Ten Minute Conjectures in Graph Theory, The Mathematical Intelligencer34 (2012) 8-15.

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