1. Lecture 12 - Variants of Cops and Robbers Dr. Anthony Bonato Ryerson University AM8002 Fall 2014

2. Variants many possible variants exist for Cops and Robbers power or speed of cops or robber can be changed in many ways: the robber is faster the robber is invisible; there maybe traps or alarms the cops have further reach, or can teleport the robber can fight back 2

3. Photo radar number play as in Cops and Robbers in a cop-win graph, but robber is invisible cops can place photo radar on edges xy: indicates when the robber is on x or y, and which direction he exits the edge photo radar number, written pr(G), minimum number of photo radars needed on edges to catch robber 3

4. Photo Radar 4 t a b c d e f

5. Tandem-win graphs pair of cops play, but always must be distance at most one apart a graph is tandem-win if one pair of cops playing in tandem can capture the robber 5 C C

6. Nearly irreducible vertices a vertex u is nearly irreducible if there is a vertex v such that N(u) is contained in N[v] –note that u need not be joined to v (as in the case of a corner) Theorem 12.1 (Clarke, 2002) Let u be nearly irreducible. Then G is tandem-win iff G-u is tandem win. 6

7. Example a tandem-win graph with no nearly irreducible vertices 7

8. Discussion Why is the following graph tandem-win? 8

9. Complementary Cops and Robbers cops move on edges, robber moves on non-edges (i.e. on edges of the complement) least number of cops needed to capture the robber with these rules is CC(G) Theorem 12.2 (Hill,08) For a graph G, γ(G) - 1 ≤ CC(G) ≤ γ(G). 9

10. CC(G) = k Corollary 12.3 (Hill,08) If CC(G) = k, then G has a set of k+1 vertices, at least two of which are adjacent, which dominate the graph. does not give a characterization… 10

11. 11 Distance k Cops and Robber 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

12. 12 The distance k cop number c k (G) = minimum number of cops needed to capture robber at distance at most k G connected implies c k (G) ≤ diam(G) – 1 for all k ≥ 1, c k (G) ≤ c k-1 (G)

13. 13 Example: k = 1 C R c 1 (G) > 1

14. 14 Example C C R c 1 (G) = 2

15. 15 c k (n) c k (n) = maximum value of c k (G) over connected G of order n Meyniel conjecture: c 0 (n) = O(n 1/2 ).

16. 16 Random graphs for random graphs G(n,p) with p = p(n), the behaviour of distance k cop number is complicated Theorem 12.4 (Bonato et al,09)

17. 17 Zig-zag functions for x in (0,1), define f k (x) = log E(c k (G(n,n x-1 ))) / log n

18. The robber fights back! robber can attack neighbouring cop one more cop needed in this graph (check) 18 C C C R

19. cc number let cc(G) be the minimum number of cops needed with these rules Lemma 12.5 For a graph G, c(G) ≤ cc(G) ≤ 2c(G). 19

20. 20

21. Firefighter G simple, undirected, connected graph fire spreads from a vertex over discrete time- steps or rounds vertices are on fire, protected, or clear fire can spread to all available adjacent vertices firefighter can protect one vertex in each round (Hartnell, 95) introduced Firefighter –simplified model for the spread of a fire/disease/virus in a network 21

22. Saving vertices one-player game firefighter aims to maximize the number of clear or protected (ie saved) vertices sn(G,v) = maximum number of saved vertices in G if a fire starts at v 22

23. Examples sn(P n,v) = n-1, if v is an end-vertex = n-2, else sn(K n,v) = 1 Theorem (MacGillivray, P. Wang, 03): sn(Q n,v) = n 23

24. Surviving rate (Cai, W. Wang, 09) surviving rate of G, ρ(G) = expected percentage of vertices saved if fire starts at a random vertex 24

25. Example: path Lemma 12.6: 25

26. Results on ρ(G) (Cai, W. Wang, 10): ρ(G) ≥ 1 – Θ(log n /n) if G is outerplanar (Finbow, P. Wang, W. Wang, 10): if G has size at most (4/3 – ε)n, then ρ(G) ≥ 6/5ε, where 0 < ε < 5/27 (Prałat, 10): if G has size at most (15/11 – ε)n, then ρ(G) ≥ 1/60ε, where 0 < ε < 1/2 (15/11 best possible) 26

27. Open problem: Infinite hexagonal grid can one cop contain the fire? 27

28. Aside: Minimum orders M k = minimum order of a k-cop-win graph M 1 = 1, M 2 = 4 M 3 = 10 (Baird, Bonato,13) –see also (Beveridge et al, 2014+) 28

29. Questions M 4 = ? are the M k monotone increasing? –for example, can it happen that M 344 < M 343 ? m k = minimum order of a connected G such that c(G) ≥ k (Baird, Bonato, 13) m k = Ω(k 2 ) is equivalent to Meyniel’s conjecture. m k = M k for all k ≥ 4? 29

30. Good guys vs bad guys games in graphs 30 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal security fastcleaningdistance k Cops and Robbers Cops and Robbers on disjoint edge sets The Angel and Devil helicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil, Firefighter Hex bad good