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GRASCan 2012 . What is left to do on Cops and Robbers?. Anthony Bonato Ryerson University. Where to next?. we focus on 6 research directions on the topic of Cops and Robbers games by no means exhaustive. 1. How big can the cop number be?. c(n) = maximum cop number of a connected
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GRASCan 2012 What is left to do on Cops and Robbers? Anthony Bonato Ryerson University Cops and Robbers
Where to next? • we focus on 6 research directions on the topic of Cops and Robbers games • by no means exhaustive Cops and Robbers
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
Henri Meyniel, courtesy Geňa Hahn Cops and Robbers
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
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
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
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
Example Fano plane Heawood graph Cops and Robbers
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
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
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
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
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
5. Planar graphs • (Aigner, Fromme, 84) planar graphs have cop number ≤ 3. • (Clarke, 02) outerplanar graphs have cop number ≤ 2. Cops and Robbers
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
6. VariantsGood guys vs bad guys games in graphs bad good Cops and Robbers
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
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
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
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
Infinite hexagonal grid • can one cop contain the fire? Fighting Intelligent Fires Anthony Bonato
Fill in the blanks… bad good Cops and Robbers