cops and robbers: directions and generalizations
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DESCRIPTIONGRASTA 2012 . Cops and Robbers: Directions and Generalizations. Anthony Bonato Ryerson University. Happy 60 th Birthday RJN May your searching never end. 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. - PowerPoint PPT Presentation
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Cops and Robbers1Cops and Robbers: Directions and GeneralizationsAnthony BonatoRyerson UniversityGRASTA 2012 Happy 60th Birthday RJN
May your searching never end.Cops and Robbers2Cops and Robbers
Cops and Robbers3CCCRCops and Robbers
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Cops and Robbers5CCCRcop number c(G) 3Cops and Robbersplayed on reflexive undirected graphs Gtwo players Cops C and robber R play at alternate time-steps (cops first) with perfect informationplayers move to vertices along edges; allowed to moved to neighbors or pass cops try to capture (i.e. land on) the robber, while robber tries to evade captureminimum number of cops needed to capture the robber is the cop number c(G)well-defined as c(G) |V(G)|Cops and Robbers6Basic facts on the cop numberc(G) (G) (the domination number of G)far from sharp: paths
trees have cop number 1one cop chases the robber to an end-vertex
cop number can vary drastically with subgraphsadd a universal vertex
Cops and Robbers7How big can the cop number be?c(n) = maximum cop number of a connected graph of order n
Meyniels Conjecture: c(n) = O(n1/2).
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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 Robbers11
Graph classes(Aigner, Fromme,84): Planar (outerplanar) graphs have cop number at most 3 (2).(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 Robbers1212Cops and Robbers13How 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 Robbers14
Fano planeHeawood graphMeyniel extremal families a family of connected graphs (Gn: n 1) is Meyniel extremal (ME) 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)(Praat,10) cop number = q+1Cops and Robbers15Diameter 2(Lu, Peng, 12): If G has diameter 2, then c(G) 2n1/2 - 1.
diameter 2 graphs satisfy Meyniels conjecture
proof uses the probabilistic method
Question: are there explicit Meyniel extremal families whose members are diameter two?Cops and Robbers16Polarity graphssuppose PG(2,q) has points P and lines L. A polarity is a function : P L such that for all points p,q, p (q) iff q (p).
eg of orthogonal polarity: point mapped to its orthogonal complement polarity graph: vertices are points, x and y adjacent if x (y)Cops and Robbers17
Properties of polarity graphsorder q2+q+1(q,q+1)-regularC4-free(Erds, Rnyi, Ss,66), (Brown,66) orthogonal polarity graphs C4-free extremaldiameter 2(Godsil, Newman, 2008) have unbounded chromatic number as q Cops and Robbers18Meyniel ExtremalTheorem (B,Burgess,12+) Let q be a prime power. If Gq is a polarity graph of PG(2, q), then q/2 c(Gq) q + 1.
lower bound: lemmaupper bound: direct analysisCops and Robbers19ME method (BB,12+)
Cops and Robbers20Lower boundsLemma (Aigner,Fromme, 1984) If G is a connected graph of girth at least 5, then c(G) (G).
Lemma (BB,12+) If G is connected and K2,t-free, then c(G) (G) / t.applies to polarity graphs: t = 2Cops and Robbers21Sketch of proof: Lower boundCops and Robbers22RN(R)C< t neighbours attackedSketch of proof: Upper boundCops and Robbers23RCN2(u)uSketch of proof: Upper boundCops and Robbers24RN2(u)C q cops move to N(u)ut-orbit graphs(Fredi,1996) described a family of K2,t+1-free extremal graphs of order (q2 -1)/t and which are (q,q+1)-regular for prime powers q.
gives rise to a new family of ME graphs which are K2,t+1-freeCops and Robbers25(BB,12+) New ME families
Cops and Robbers26BIBDsa BIBD(v, k, ) is a pair (V, B), where V is a set of v points, and B is a set of k-subsets of V, called blocks, such that each pair of points is contained in exactly blocks.
Theorem (BB,12+) The cop number of the IG of a BIBD(v, k, ) is between k and r, the replication number.Cops and Robbers27Sketch of prooflower bound: girth 6; apply AF lemma and Fishers inequality
upper bound: Cops and Robbers28
CR28Block Intersection graphsgiven a block design (V,B), its block intersection graph has vertices equalling blocks, with blocks adjacent if they intersectCops and Robbers29
BIG cop numberTheorem (BB,12+) If G is the block intersection graph of a BIBD(v, k, 1), then c(G) k. If v > k(k-1)2 + 1, then c(G) = k.
gives families with unbounded cop number; not ME
also considered point graphsCops and Robbers30QuestionsSoft Meyniels conjecture: for some > 0,c(n) = O(n1-).
Meyniels conjecture in other graphs classes?bounded chromatic numberbipartite graphsdiameter 3claw-free
ME families from something other than designs?extremal graphs?Cops and Robbers31
32R.J. Nowakowski, P. Winkler Vertex-to-vertex pursuit in a graph, Discrete Mathematics 43 (1983) 235-239.
5 pages> 200 citations (most for either author)Cops and Robbers33The NW relation(Nowakowski,Winkler,83) introduced a sequence of relations characterizing cop-win graphs
u 0 v if u = vu i v if for all x in N[u], there is a y in N[v] such that x j y for some j < i.Cops and Robbers34ExampleCops and Robbers35uvwyzu 1 v u 2 w Characterizationthe relations are i monotone increasing; thus, there is an integer k such that k = k+1 write:k = Theorem (Nowakowski, Winkler, 83)A cop has a winning strategy iff is V(G) x V(G).Cops and Robbers36k copsmay define an analogous relation but in V(G) x V(Gk) (categorical product)
(Clarke,MacGillivray,12) k cops have a winning strategy iff the relation is V(G) x V(Gk).Cops and Robbers37Axioms for pursuit gamesa pursuit game G is a discrete-time process satisfying the following:Two players, Left L and Right R.Perfect-information.There is a set of allowed positions PL for L; similarly for Right.For each state of the game and each player, there is a non-empty set of allowed moves. Each allowed move leaves the position of the other player unchanged.There is a set of allowed start positions I a subset of PL x PR.The game begins with L choosing some position pL and R choosing qR such that (pL, qR) is in I.After each side has chosen its initial position, the sides move alternately with L moving first. Each side, on its turn, must choose an allowed move from its current position.There is a subset of final positions, F. Left wins if at any time, the current position belongs to F. Right wins the current position never belongs to F.Cops and Robbers38Examples of pursuit gamesCops and Robbersplay on graphs, digraphs, orders, hypergraphs, etc.play at different speeds, or on different edge setsCops and Robbers with trapsDistance k Cops and RobbersTandem-win Cops and RobbersRestricted ChessHelicopter Cops and RobbersMaker-Breaker GamesSeepageScared Robber
Cops and Robbers39Relational characterizationgiven a pursuit game G, we may define relations on PL x PR as follows:
pL 0 qR if (pL, qR) in F.pL i qR if Right is on qR and for every xR in PR such that if Right has an allowed move from (pL, qR) to (pL, xR), there exists yL in PL such that xR j yL for some j < i and Left has an allowed move from (pL, xR) to (yL, xR).
define analogously as beforeCops and Robbers40CharacterizationTheorem (B, MacGillivray,12) Left has a winning strategy in the a pursuit game G if and only if there exists pL in PL, which is the first component of an ordered pair in I, such that for all qR in PR with (pL, qR) in I there exists wL in the set of allowed moves for Left from pL such that qR wL.
gives rise to a min/max expression for the length of the gameCops and Robbers41Length of gamefor an allowed start position (pL, qR), define
Corollary (BM,12+) If Left has a winning strategy in the a pursuit game G, then assuming optimal play, the length of the game is
where IL is the set positions for Left which are the first component of an ordered pair in I.
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CGT(Berlekamp, Conway, Guy, 82) A combinatorial game satisfies:
There are two players, Left and Right.There is perfect information.There is a set of allowed positions in the game.The rules of the game specify how the game begins and, for each player and each position, which moves to other positions are allowed.The players alternate moves.The game ends when a position is reached where no moves are possible for the player whose turn it is to move. In normal play the last player to move wins.
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Example: NIMCops and Robbers44
Pursuit CGTTheorem (BM,12+)Every pursuit game is a combinatorial game.Not every combinatorial game is a pursuit game.
uses characterization of (Smith, 66) via game digraphs Nim is a counter-example for item (2)Cops and Robbers45Position independencea pursuit game G is position independent if: if the game is not over, the set of availabl