Download - What is left to do on Cops and Robbers?
![Page 1: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/1.jpg)
Cops and Robbers 1
What is left to do on Cops and Robbers?
Anthony BonatoRyerson University
GRASCan 2012
![Page 2: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/2.jpg)
Cops and Robbers 2
Where to next?
• we focus on 6 research directions on the topic of Cops and Robbers games–by no means exhaustive
![Page 3: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/3.jpg)
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 3
![Page 4: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/4.jpg)
Cops and Robbers 4
![Page 5: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/5.jpg)
Cops and Robbers 5
Henri Meyniel, courtesy Geňa Hahn
![Page 6: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/6.jpg)
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 6
)1(1log))1(1( 22
)( ono
nnOnc
![Page 7: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/7.jpg)
Cops and Robbers 7
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.
![Page 8: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/8.jpg)
Cops and Robbers 8
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
![Page 9: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/9.jpg)
Cops and Robbers 9
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
![Page 10: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/10.jpg)
Example
Cops and Robbers 10
Fano plane Heawood graph
![Page 11: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/11.jpg)
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 11
![Page 12: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/12.jpg)
Cops and Robbers 12
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+)
![Page 13: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/13.jpg)
Cops and Robbers 13
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?
![Page 14: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/14.jpg)
Cops and Robbers 14
4. 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, Kratochvíl, Nisse, Suchan, 08):
if s not fixed, then computing the cop number is NP-hard
![Page 15: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/15.jpg)
Cops and Robbers 15
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?
![Page 16: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/16.jpg)
Cops and Robbers 16
5. Planar graphs• (Aigner, Fromme, 84) planar graphs have cop
number ≤ 3.
• (Clarke, 02) outerplanar graphs have cop number ≤ 2.
![Page 17: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/17.jpg)
Cops and Robbers 17
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)
![Page 18: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/18.jpg)
Cops and Robbers 18
6. VariantsGood guys vs bad guys games in graphs
slow medium fast helicopter
slow traps, tandem-win
medium robot vacuum Cops and Robbers edge searching eternal security
fast cleaning distance k Cops and Robbers
Cops and Robbers on disjoint edge sets
The Angel and Devil
helicopter seepage Helicopter Cops and Robbers, Marshals, The Angel and Devil,Firefighter
Hex
badgood
![Page 19: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/19.jpg)
Cops and Robbers 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
![Page 20: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/20.jpg)
Cops and Robbers 20
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)
![Page 21: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/21.jpg)
Cops and Robbers 21
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)
![Page 22: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/22.jpg)
Cops and Robbers 22
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?
C
C
C
R
![Page 23: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/23.jpg)
Fighting Intelligent Fires Anthony Bonato
23
Infinite hexagonal grid
• can one cop contain the fire?
![Page 24: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/24.jpg)
Cops and Robbers 24
Fill in the blanks…slow medium fast helicopter
slow traps, tandem-win
medium robot vacuum Cops and Robbers edge searching eternal security
fast cleaning distance k Cops and Robbers
Cops and Robbers on disjoint edge sets
The Angel and Devil
helicopter seepage Helicopter Cops and Robbers, Marshals, The Angel and Devil,Firefighter
Hex
badgood
![Page 25: What is left to do on Cops and Robbers?](https://reader036.vdocuments.us/reader036/viewer/2022062410/5681661d550346895dd96e9c/html5/thumbnails/25.jpg)
Cops and Robbers 25