1. Discrete Mathematics and its Applications Lecture 7 – Cops and Robbers
Miniconference on the Mathematics of Computation
AM8002
Discrete Mathematics and its Applications Lecture 7 – Cops and Robbers
Dr. Anthony Bonato
Ryerson University

2. Cops and Robbers C R

3. Cops and Robbers C R
cop number c(G) = 2 C

4. Cops and Robbers played on reflexive graphs G
two players Cops C and robber R play at alternate time-steps (cops first) with perfect information
players 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 capture
minimum number of cops needed to capture the robber is the cop number c(G)
well-defined as c(G) ≤ |V(G)|
Cops and Robbers

5. Basic properties Theorem 8.1: c(G) ≤ γ(G) (the domination number).
The cop number of a cycle is 2.
The cop number of a clique is 1.
The cop number of the disjoint union of two graphs is the sum of their cop number.

6. Applications: multiple-agent moving-target search
example: in video games, player controls robber, while cops are computer generated agents
octile connected maps
4/12/2019
Anthony Bonato

7. agents must be smart and perform calculations quickly
(Greiner et al, 08):
problem in AI
agents must be smart and perform calculations quickly
other applications:
Missile defense
Counter-terrorism
4/12/2019
Anthony Bonato

8. Trees
Theorem 8.2 The cop number of a finite tree is 1.

9. Exercise
8.2 Describe infinitely many non-isomorphic infinite trees, each possessing a ray (i.e. an infinite one way path). 8.3 Prove that an infinite tree with a ray has infinite cop number.

10. How long do you want to live?R C

11. Exercise (revisited)
8.4 Explain why the Petersen graph has cop number ≤ 3.

12. Lower bound
Theorem 8.3. If G is a graph with girth at least 5, then c(G) ≥ δ(G). Application: c(Petersen) ≥ 3.

13. Isometric paths
an isometric path P in G has the property that: dP(x,y) = dG(x,y)
i.e. there are no “short-cuts”
eg: a b c d f

14. Guarding subgraphs
a set of cops guard an induced subgraph H of G if they can arrange things so that after finitely many moves, if the robber enters H, they are immediately caught
eg: H is a clique, or has a universal vertex
H is k-guardable if k cops can guard it

15. Guarding paths Theorem 8.4 (Aigner, Fromme, 84)
Isometric paths are 1-guardable.
simple proof, but powerful tool
can cover a graph by isometric paths, or beats

16. Upper bound on the cop number
let c(n) be the maximum cop number of a connected graph of order n
Theorem 8.5 (Frankl,87)
c(n) = O(n loglogn / log n) = o(n).

17. Moore bound
Theorem 8.6: Let G be a graph of order n with maximum degree ∆ and diameter d. Then n ≤ 1 + ∆((∆ -1)d – 1)/(∆ -2)). = O(∆)d