what is left to do on cops and robbers?

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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 - PowerPoint PPT Presentation

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Models of the web graph

Cops and Robbers1What is left to do on Cops and Robbers?Anthony BonatoRyerson UniversityGRASCan 2012 Where to next?we focus on 6 research directions on the topic of Cops and Robbers gamesby no means exhaustiveCops and Robbers21. 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 Robbers3Cops and Robbers4

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Henri Meyniel, courtesy Gea HahnState-of-the-art(Lu, Peng, 12+) proved that

independently proved by (Scott, Sudakov,11) and (Frieze, Krivelevich, Loh, 11)

(Bollobs, Kun, Leader, 12+): if p = p(n) 2.1log n/ n, thenc(G(n,p)) 160000n1/2log n

(Praat,Wormald,12+): removed log factorCops and Robbers6

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+): Meyniels conjecture holds for diameter 2 graphs, bipartite diameter 3 graphs.Cops and Robbers77QuestionsSoft Meyniels conjecture: for some > 0,c(n) = O(n1-).

Meyniels conjecture in other graphs classes?bounded chromatic numberbipartite graphsdiameter 3claw-freeCops and Robbers8Cops and Robbers92. How close to n1/2?consider a finite projective plane Ptwo lines meet in a unique pointtwo points determine a unique lineexist 4 points, no line contains more than two of themq2+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 Pa point is joined to a line if it is on that line

ExampleCops and Robbers10

Fano planeHeawood graphMeyniel 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+1order 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 Robbers113. Minimum ordersMk = minimum order of a k-cop-win graph

M1 = 1, M2 = 4M3 = 10 (Baird, Bonato,12+)

see also (Beveridge et al, 2012+)

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QuestionsM4 = ?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 = (k2) is equivalent to Meyniels conjecture.mk = Mk for all k 4?

Cops and Robbers134. Complexity(Berrarducci, Intrigila, 93), (Hahn,MacGillivray, 06), (B,Chiniforooshan, 09): c(G) s? s fixed: in P; running time O(n2s+3), n = |V(G)|

(Fomin, Golovach, Kratochvl, Nisse, Suchan, 08): if s not fixed, then computing the cop number is NP-hard

Cops and Robbers14QuestionsGoldstein, Reingold Conjecture: if s is not fixed, then computing the cop number is EXPTIME-complete.same complexity as say, generalized chessConjecture: 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 Robbers155. Planar graphs(Aigner, Fromme, 84) planar graphs have cop number 3.

(Clarke, 02) outerplanar graphs have cop number 2.

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Questionscharacterize 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)

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6. VariantsGood guys vs bad guys games in graphs18slowmediumfasthelicopterslowtraps, tandem-winmediumrobot vacuumCops and Robbersedge searchingeternal securityfastcleaningdistance k Cops and RobbersCops and Robbers on disjoint edge setsThe Angel and DevilhelicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil,FirefighterHexbadgoodCops and RobbersCops and Robbers19Distance k Cops and Robber (Bonato,Chiniforooshan,09)(Bonato,Chiniforooshan,Praat,10)cops can shoot robber at some specified distance kplay as in classical game, but capture includes case when robber is distance k from the copsk = 0 is the classical game

CRk = 1Cops and Robbers20Distance k cop number: ck(G)ck(G) = minimum number of cops needed to capture robber at distance at most kG connected implies ck(G) diam(G) 1for all k 1, ck(G) ck-1(G)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 graphsdistance k Cops and Robbersinvisible 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 Robbers21The robber fights back! (Haidar,12) robber can attack neighbouring cop

one more cop needed in this graph (check)at most min{2c(G),(G)} cops needed, in generalare c(G)+1 many cops needed?Cops and Robbers22

CCCRInfinite hexagonal gridcan one cop contain the fire?Fighting Intelligent Fires Anthony Bonato23

Fill in the blanksCops and Robbers24slowmediumfasthelicopterslowtraps, tandem-winmediumrobot vacuumCops and Robbersedge searchingeternal securityfastcleaningdistance k Cops and RobbersCops and Robbers on disjoint edge setsThe Angel and DevilhelicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil,FirefighterHexbadgoodCops and Robbers25