1. Lecture 9 - Cop-win Graphs and Retracts Dr. Anthony Bonato Ryerson University AM8002 Fall 2014

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)| 2

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) 3

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5. R.J. Nowakowski, P. Winkler Vertex-to- vertex pursuit in a graph, Discrete Mathematics 43 (1983) 235-239. 5 pages > 300 citations (most for either author) 5

6. Examples 1.Cliques 2.Graphs with universal vertices 3.Trees. 4.What about…? 6

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: 1)H is a single vertex (recall G is reflexive). 2)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? 7 1 2 3 4 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. 8

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

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

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 examples: cliques, trees, the following graph… 11

12. A dismantlable graph 12

13. A simple lemma Lemma 9.3: If G is cop-win, then G contains at least one corner. Proof: Consider the second-to-last move of the cop using a winning strategy. No matter what move the robber makes, he will lose in the next round. Hence, the cop must be joined to the robber’s vertex u, and all of its neighbours. It follows that u is a corner. □ 13

14. 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 14

15. Cop-win orderings a permutation v 1, v 2, …, v n of V(G) is a cop-win ordering if there exist vertices w 1, w 2, …, w n such that for all i, w i is the parent of v i in the subgraph induced V(G) \ {v j : j > i}. –a cop-win ordering dismantlability 15 1 2 3 4 5

16. Discussion 1.Explain why the following graph is cop- win. 2.Explain why a hypercube Q n, where n > 1, is never cop-win. 16

17. Cop-win Strategy (Clarke, Nowakowski, 2001) (1,2,…,n) a cop-win ordering G 1 = G, i > 1, G i : subgraph induced by deleting 1, …, i-1 f i : G i → G i+1 retraction mapping i to a fixed one of its parents F i = f i-1 ○… ○ f 2 ○ f 1 –a homomorphism idea: robber on u, think of F i (u) shadow of robber –cop moves to capture shadow –works as the F i are homomorphisms results in a capture in at most n moves of cop 17

18. The NW relation (Nowakowski,Winkler,83) introduced a sequence of relations characterizing cop- win graphs u ≤ 0 v if u = v u ≤ i v if for all x in N[u], there is a y in N[v] such that x ≤ j y for some j < i. 18

19. Example 19 u v w yz u ≤ 1 v u ≤ 2 w

20. Characterization the relations are ≤ i monotone increasing; thus, there is an integer k such that ≤ k = ≤ k+1 –write: ≤ k = ≤ Theorem 8.5 (Nowakowski, Winkler, 83) A cop has a winning strategy iff ≤ is V(G) x V(G). 20

21. k cops may define an analogous relation but in V(G) x V(G k ) (categorical product) Theorem 9.6 (Clarke,MacGillivray,12) k cops have a winning strategy iff the relation ≤ is V(G) x V(G k ). 21