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

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

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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 Henri Meyniel, courtesy Geňa Hahn

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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

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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

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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

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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

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Example Cops and Robbers10 Fano plane Heawood graph

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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

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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

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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

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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

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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

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

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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

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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

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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

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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)

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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

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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

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

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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

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Cops and Robbers25

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