1. Discrete Mathematics and its Applications Lecture 8 – Cop-win Graphs
Miniconference on the Mathematics of Computation
AM8002
Discrete Mathematics and its Applications Lecture 8 – Cop-win Graphs
Dr. Anthony Bonato
Ryerson University

2. Reminder: 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)|

3. Cop-win graphs
consider the case when one cop has a winning strategy; i.e. c(G) = 1
cop-win graphs
introduced by (Nowakowski, Winkler, 83) and independently by (Quilliot, 78)

5. R.J. Nowakowski, P. Winkler Vertex-to-vertex pursuit in a graph, Discrete Mathematics 43 (1983)
5 pages
> 480 citations (most for either author)

6. Examples
Cliques
Graphs with universal vertices
Trees.
What about…?

7. Retracts let H be an induced subgraph of G
a homomorphism f: G → H is a retraction if f(x) = x for all x in V(H). We say that H is a retract of G.
examples:
H is a single vertex (recall G is reflexive).
Let H be the subgraph induced by {1,2,3,4}:
- the mapping sending 5 to 4
fixing all other vertices is a
retraction;
- what if we map 5 to 2? 2 3 4 1 5

8. Retracts and cop number
Theorem 9.1: If H is a retract of G, then
c(H) ≤ c(G).
proof uses shadow strategy
Corollary: If G is cop-win, then so is H.

9. Retracts, continued
Theorem 9.2: If H is a retract of G, then c(G) ≤ max{c(H),c(G-H)+1}.

10. Exercise
9.4 Prove the previous theorem: Theorem 9.2: If H is a retract of G, then c(G) ≤ max{c(H),c(G-H)+1}.

11. Characterization
node u is a corner if there is a v such that N[v] contains N[u]
v is the parent; u is the child
a graph is dismantlable if we can iteratively delete corners until there is only one vertex

12. Exercise
9.5 What are the corners in trees? 9.6 Which vertices are corners in cliques?

13. A dismantlable graph

14. A simple lemma
Lemma 9.3: If G is cop-win, then G contains at least one corner.

15. Exercise
9.7 Prove Lemma 9.2.

16. Characterization Theorem 9.4 (Nowakowski, Winkler 83; Quilliot,78)
A graph is cop-win if and only if it is dismantlable.
idea: cop-win graphs always have corners; retract corner and play shadow strategy;
- dismantlable graphs are cop-win by induction

17. Cop-win orderings 5 1 4 3 2 a permutation v1, v2, … , vn of V(G) is a
cop-win ordering if there exist vertices w1, w2, …, wn such that for all i, wi is the parent of vi in the subgraph induced V(G) \ {vj : j > i}.
a cop-win ordering dismantlability 5 1 4 3 2

18. Exercise 9.8 Explain why the following graph is cop-win.
9.9 Explain why a hypercube Qn, where n > 1, is never cop-win.