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

Cops and Robbers C R

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

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

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.

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

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

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

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.

How long do you want to live? R C

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

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

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

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

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

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).

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