on graphs & winning strategy - kentdragan/st-spring2016/cops vs robbers.pdf · 2016-05-09 ·...
TRANSCRIPT
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ON GRAPHS & WINNING STRATEGY
Heather M. Michaud
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Outline• Simplified cops vs. robbers in real life • Cop-win graphs and strategy
• examples of cops winning/losing • corners • retracts • dismantling • strategy and solution
• Variations • Bounds on cop-number
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Problem: Cops vs. Robbers• Given a map, a cop, and a robber - can the cop
catch the robber, and if so, how?
• The cop and robber always see each other
• The cop and robber take turns moving (but can pass), starting with the cop
• The cop and robber move at the same speed
• If the cop catches the robber, the cop wins
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Similar problems• More applications in other pursuit/evasion
scenarios
• search and rescue
• modeling network security problems
• surveillance and tracking
• artificial intelligence in games
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Map and Graph Construction
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• Cops and robber each occupy a node on a graph, and take turns moving to adjacent nodes via the edges.
• Graph is reflexive • Cops win if they can occupy
the same space as the robber • Robber wins if he never gets
caught (indefinitely evades)
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Graph Problem
Is the constructed graph a cop-win graph?
If yes, what strategy should the cop use to win?
Cops and Robbers on graphs introduced independently by Nowakowski and Winkler (1983)
and Quilliot (1978)
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Preliminaries
• The cop-number of a graph, c(G), is the minimum number of cops to catch a robber in G.
• If c(G)=k, then G is k-cop-win.
• If c(G)=1, then G is cop-win.
• Examples…
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cop-win graph
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cop-win graph
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robber-win graph
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robber-win graph
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Corners• Definition: A vertex u is a corner (or a trap, pitfall, or
irreducible) if there is some vertex v such that N[u]⊆N[v].
• Lemma 1: If G is cop-win, then it has at least one corner.
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Retracts• Definition: Let H be an induced subgraph of G formed by
deleting one vertex. We say that H is a retract of G if there is a homomorphism f from G onto H so that f(x)=x for x∈V(H)
• The subgraph formed by deleting a corner u is a retract, given by the mapping
• Theorem 2: If H is a retract of G, then c(H) ≤ c(G). • Corollary 3: If G is cop-win, then so is each retract of G.
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uv v
G H
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Dismantling• Definition: A graph is dismantlable if some sequence
of deleting corners results in the graph K1.
• Theorem 4: A graph G is cop-win if and only if it is dismantlable.
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3 2
14
3 2
1
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Cop-win Ordering• Definition: A cop-win ordering is a sequence of
positive integers [n] such that for each i<n, the vertex i is a corner in the subgraph induced by {i, i+1, …, n}.
• A cop-win ordering for the above graph is {1,2,3,4,5}40
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3 2
1
cop-win ordering: i, i+1, ……, j, j+1, …., n
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Cop-win Strategy (Preliminaries)
Assume [n] is a cop-win ordering of G • The winning strategy involves the cop “shadowing”
the movements of the robber in increasingly larger induced subgraphs of G
• for 1≤i≤n define Gi = G↾{n, n-1, …, i} (the increasingly smaller subgraphs of G)
• for each 1 ≤ i ≤ n-1, let fi : Gi→Gi+1 (the retractions which remove a corner to make a smaller subgraph)
• Fi = fi-1 o …. o f2 o f1 (a composition of retractions - this will be the “shadow” that the cop will follow)
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Cop-win Strategy (Preliminaries)
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54
3 2
1G1
1 2 3 4 5
f2
2 3 4 5
f3
3 4 5
f4
4 5
f1 1 2 3 4 5
f5
55
4
3 2
1G2
54
3 2
1G3
54
3 2
1G4
54
3 2
1G5
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Cop-win Strategy (Preliminaries)
43
54
3 2
1G1
1 2 3 4 5
f2
2 3 4 5
f3
3 4 5
f4
4 5
f1 1 2 3 4 5
f5
55
4
3 2
1G2
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3 2
1G3
54
3 2
1G4
54
3 2
1G5
Example: F2(5)=5 (the shadow of 5 in G2)
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Cop-win Strategy (Preliminaries)
44
54
3 2
1G1
1 2 3 4 5
f2
2 3 4 5
f3
3 4 5
f4
4 5
f1 1 2 3 4 5
f5
55
4
3 2
1G2
54
3 2
1G3
54
3 2
1G4
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3 2
1G5
Example: F4(3)=5 (the shadow of 3 in G4)
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Cop-win Strategy (Preliminaries)
45
54
3 2
1G1
1 2 3 4 5
f2
2 3 4 5
f3
3 4 5
f4
4 5
f1 1 2 3 4 5
f5
55
4
3 2
1G2
54
3 2
1G3
54
3 2
1G4
54
3 2
1G5
Example: F5(1)=5 (the shadow of 1 in G5)
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
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R
54
3 2
1C
Turn 0 C:5 R:1
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
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R
54
3 2
1
Turn 1… (in progress)
C
Turn 0 C:5 R:1
Cop shadows robber. F4(1)=5
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
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54
3 2
1
Turn 1… (in progress)
C
Turn 0 C:5 R:1
Cop shadows robber. F4(1)=5
Robber runs away.
R
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
49
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
50
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2… (in progress)
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
51
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Cop shadows robber. F3(2)=3
Turn 2… (in progress)
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
52
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Cop shadows robber. F3(2)=3
Turn 2… (in progress)
Robber runs away.
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
53
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2 C:3 R:1
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
54
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2 C:3 R:1Turn 3… (in progress)
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
55
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2 C:3 R:1Turn 3… (in progress)Cop shadows robber.
F2(1)=2
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
56
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2 C:3 R:1Turn 3… (in progress)Cop shadows robber.
F2(1)=2Robber runs away.
![Page 57: ON GRAPHS & WINNING STRATEGY - Kentdragan/ST-Spring2016/cops vs robbers.pdf · 2016-05-09 · • If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec8ee0601d5005ba84fb171/html5/thumbnails/57.jpg)
Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
57
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2 C:3 R:1Turn 3 C:2 R:5Turn 4… (in progress)
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Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
58
54
3 2
1
Turn 1 C:5 R:2
C
Turn 0 C:5 R:1
R
Turn 2 C:3 R:1Turn 3 C:2 R:5Turn 4… (in progress)
Cop shadows robber. F1(5)=5
![Page 59: ON GRAPHS & WINNING STRATEGY - Kentdragan/ST-Spring2016/cops vs robbers.pdf · 2016-05-09 · • If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec8ee0601d5005ba84fb171/html5/thumbnails/59.jpg)
Cop-win Strategy• Cop starts at Gn, (the shadow of
all other positions, including the robber’s, under Fn)
• Suppose the robber is on u and the cop occupies the shadow of the robber, Fi(u), in Gi
• If the robber moves to v, then the cop moves onto the image Fi-1(v) in the larger graph Gi-1.
59
54
3 2
1
Turn 1 C:5 R:2Turn 0 C:5 R:1
R
Turn 2 C:3 R:1Turn 3 C:2 R:5Turn 4 C:5
C
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Cop-win Strategy
• May not be the fastest strategy, but guarantees the robber will be captured in at most n moves.
• Also not a unique for a graph because it relies on the cop-win ordering, which is not unique.
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Hardness• Checking if c(G)=k for some fixed k can be done in
polynomial time.
• Determining k in which c(G)=k, when k is not fixed is NP-hard
• Determining the cop-number of a general graph is NP-hard, but for cop-win graphs, the cop-number is exactly 1.
• We can check in polynomial time if a graph is cop-win
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Problem Solution• Given a map, a cop, and a robber - (1) can the cop catch
the robber, and if so, (2) how? (1) In our example, yes! Determine the graph’s cop-win ordering in polynomial time
(2) Given the cop-win ordering, find a guaranteed win strategy for the cop using this ordering
• (we make some assumptions about where the robber will go to finish the example, but the strategy will work regardless of where robber picks to go next)
• Cop follows the shadow of the robber in maps with increasingly more corners, until he finally reaches him
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Variations• Imperfect information (limited visibility): alarm indicates
general location of robber, blind-spots of security system • Cops set traps to catch or impede robbers • Tandem-win: two cops patrol together within close
proximity of each other • A cop only needs to “see” a robber within distance k to
catch/shoot him • Minimize capture time (steps for cops to win) • Varying speeds of cops and robbers • Minimize number of cops to win
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robber-win graph
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2-cops
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2-cops
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2-cops
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2-cops
2-cop-win graph
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How many cops are needed, at most?
• Every graph can catch a robber with n cops. • Upper bound
• on unconnected graphs: c(G) = O(n) • on connected graphs [Lu and Peng, 2011]:
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How many cops are needed, at least?
• Lower bound • Meyniel’s Conjecture: c(n)=O(√n)
• Depends on the graph • For n≥4 0, c(Pn) = c(Wn) = c(Kn) = 1. • For Cn n≥4, c(Cn) = 2. • [Aigner and Fromme, 1984] Planar graphs are 3-cop-win • [Seymour and Thomas, 1993] A k-cop-win graph has tree width k-1. • [Chepoi, 1997] Bridged graphs are cop-win graphs.
• Other interesting bounds based on girth (length of minimum order cycle), minimum degree, maximum degree
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References(1) Anthony Bonato and Richard Nowakowski, The Game of Cops
and Robbers on Graphs, Student Mathematical Library, Vol. 61, 2011.
(2) Richard Nowakowski and Peter Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math 43, 235-239, 1983.
(3) A. Quilliot, “Homomorhismes, points fixes, rétractions et jeux de poursuite dans les graphes, les ensembles ordonnés et les espaces métriques,” Thése d’État, Université de Paris VI, 1983.
(4) A game of cops & robbers. http://math.ucsd.edu/~fan/152/arch/coprob/. Mostafa Azizi and Fan Chung Graham, 2002.
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Questions?
78