1. Topological directions in Cops and Robbers
GRASCan’17
Grenfell Campus
Topological directions in Cops and Robbers
Anthony Bonato
Ryerson University

2. Graphs on surfaces ? ? S0 S1

3. Genus of a graph
a graph that can be embedded in an (orientable!) surface with g holes (and no fewer) has genus g
planar: genus 0
toroidal: genus 1 Sg

4. Planar graphs (Aigner, Fromme, 84) planar graphs have cop number ≤ 3
Cops and Robbers

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

6. Proof techniques grow cop territory by induction outerplanar planar
Case 1: no cut vertices
position two cops on a single vertex and fan out
Case 2: block decomposition, retracts, and apply Case 1
planar
isometric path lemma
three types of cop territory
apply induction to end up in one of the types

7. Planar 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?
Cops and Robbers

8. Probabilistic graph searching
Capture time of a graph
the length of Cops and Robbers was considered first as capture time
(B,Hahn,Golovach,Kratochvíl,09) capture time of G: length of game with c(G) cops assuming optimal play, written capt(G)
if G is cop-win, then capt(G) ≤ n - 4; see also (Gavanciak,10)
examples of planar graphs with capt(G) = n – 4
define captk(G), where c(G) ≤ k ≤ γ(G)
k-capture time
capt(G) = captc(G)(G)
Probabilistic graph searching

9. capt3 of planar graphs Theorem (BGRP,17+)
If G is a connected planar graph of order n, then
capt3(G) ≤ (diam(G) +1)n = O(n2).
(Pisantechakool,Tan,16) if G is 3-cop-win, then capt3(G) ≤ 2n
Overprescribed Cops and Robbers

10. Planar graphs with many cops
Theorem (BGRP,17+)
If G is a connected planar graph and k ≥ 12 n , then
captk(G) ≤ 6∙rad(G)log n.
proof uses Planar Separator Theorem (Alon,Seymour,Thomas,90)
works also if robber has infinite speed
generalizes to higher genus (only k changes, not bound)
Overprescribed Cops and Robbers

11. Planar temporal questions
bounds on capt2(G) if G is outerplanar?
examples of connected planar graphs with large 2-, 3- capture times?
does there exist G that is 3-cop-win connected planar,
capt3(G) < 2n?
Cops and Robbers

13. Higher genus
Schroeder’s Conjecture: If G has genus g, then c(G) ≤ g + 3.
true for g = 0
(Schroeder, 01): true for g = 1 (toroidal graphs)
(Quilliot,85): c(G) ≤ 2g + 3.
(Schroeder,01): c(G) ≤ 3g/2 + 3
use 1 or 2 cops to reduce genus of robber territory, then use induction
Cops and Robbers

14. Toroidal graphs
Andreae-Schroeder conjecture: The cop number of a toroidal graph is at most 3.

15. Non-orientable surfaces

16. Cop number on non-orientable surfaces
c(g) = sup of the cop numbers of graphs embedded on an orientable surface of genus g
c’(g) = sup of the cop numbers of graphs embedded on a non- orientable surface of genus g
(Andreae,86): c’(g) = O(g)
(Nowakowski,Schroeder,97): c’(g) ≤ 2g+1
(Clarke,Fiorini,Joret,Theis,12): c( g/2 ) ≤ c’(g) ≤ c(g-1)

17. Cop number for non-orientable surfaces
Clarke-Fiorini-Joret-Theis conjecture: c’(g) = c( g/2 ).

18. Lazy Cops and Robbers
game played analogous to Cops and Robbers, except only one cop may move at a time
lazy cop number cL
(Offner,Ojakian,14): bounds on cL for hypercubes
(Bal,B,Kinnersley,Prałat,15): improved bounds on cL for hypercubes using probabilistic method
(BBKP,15): for a genus g graph, cL = O( gn )

19. Lazy cops on planar graphs
is there an absolute bound on the lazy cop number of planar graphs?
constant?
family where cL goes to infinity?

20. Directed graphs C R C
sources are bad for cops!

21. Directed planar graphs
(Frieze,Krivelevich,Loh,12) For an n-vertex, strongly connected digraph
c(G) = O n ( log log n ) 2 log n
(Loh,Oh,15) For an n-vertex, strongly connected planar digraph
c(G) = O( n )
example of a directed planar graph with cop number > 3

22. Cop number of directed planar graphs
does there exist a family of directed planar graphs with unbounded cop number?
all examples have constant cop number

23. Oriented quotients of the integer grid
(Hosseini,Mohar,17+) orientations of toroidal grids
eg horizontal cycles oriented all left to right
have cop number 3 or 4
(Hermosillo de la Maza,17+)
other orientations of toroidal grids
presented at CanaDAM’17

24. Intersection graphs (Gavenčiak,Gordinowicz,Jelínek,
Klavíc,Kratochvíl,17+)