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

Graphs on surfaces ? ? S0 S1

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

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

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

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

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

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

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

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

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

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

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

Non-orientable surfaces

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)

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

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 )

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?

Directed graphs C R C sources are bad for cops!

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

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

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

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