Lecture 9 - Cop-win Graphs and Retracts

Dr. Anthony BonatoRyerson University

AM8002Fall 2014

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)|

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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)

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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)

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Examples

1. Cliques

2. Graphs with universal vertices

3. Trees.

4. What about…?

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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?

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1

2 3

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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.

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Retracts, continued

Theorem 9.2: If H is a retract of G, then

c(G) ≤ max{c(H),c(G-H)+1}.

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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}.

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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…

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A dismantlable graph

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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. □

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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

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Cop-win orderings

• 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

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1

234

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Discussion

1. Explain why the following graph is cop-win.

2. Explain why a hypercube Qn, where n > 1, is never cop-win.

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Cop-win Strategy (Clarke, Nowakowski, 2001)

• (1,2,…,n) a cop-win ordering• G1 = G, i > 1, Gi: subgraph induced by deleting 1, …, i-1• fi: Gi → Gi+1 retraction mapping i to a fixed one of its parents

• Fi = fi-1 ○… ○ f2 ○ f1

– a homomorphism

• idea: robber on u, think of Fi(u) shadow of robber– cop moves to capture shadow – works as the Fi are homomorphisms

• results in a capture in at most n moves of cop

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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.

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Example

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u v w y z

u ≤1 v

u ≤2 w

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

k cops

• may define an analogous relation but in V(G) x V(Gk) (categorical product)

• Theorem 9.6 (Clarke,MacGillivray,12) k cops have a winning strategy iff the relation ≤ is V(G) x V(Gk).

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