Download - Lecture 12 - Variants of Cops and Robbers Dr. Anthony Bonato Ryerson University AM8002 Fall 2014
Lecture 12 - Variants of Cops and Robbers
Dr. Anthony BonatoRyerson University
AM8002Fall 2014
Variants
• many possible variants exist for Cops and Robbers
• power or speed of cops or robber can be changed in many ways:• the robber is faster• the robber is invisible; there maybe traps or
alarms• the cops have further reach, or can teleport• the robber can fight back
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Photo radar number
• play as in Cops and Robbers in a cop-win graph, but robber is invisible
• cops can place photo radar on edges xy: indicates when the robber is on x or y, and which direction he exits the edge
• photo radar number, written pr(G), minimum number of photo radars needed on edges to catch robber
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Photo Radar
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t
a
b
c
d
e
f
Tandem-win graphs
• pair of cops play, but always must be distance at most one apart
• a graph is tandem-win if one pair of cops playing in tandem can capture the robber
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C C
Nearly irreducible vertices
• a vertex u is nearly irreducible if there is a vertex v such that N(u) is contained in N[v]– note that u need not be joined to v (as in the
case of a corner)
Theorem 12.1 (Clarke, 2002) Let u be nearly irreducible. Then G is tandem-win iff G-u is tandem win.
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Example
• a tandem-win graph with no nearly irreducible vertices
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Discussion
• Why is the following graph tandem-win?
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Complementary Cops and Robbers
• cops move on edges, robber moves on non-edges (i.e. on edges of the complement)
• least number of cops needed to capture the robber with these rules is CC(G)
Theorem 12.2 (Hill,08) For a graph G,
γ(G) - 1 ≤ CC(G) ≤ γ(G).
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CC(G) = k
Corollary 12.3 (Hill,08) If CC(G) = k, then G has a set of k+1 vertices, at least two of which are adjacent, which dominate the graph.
• does not give a characterization…
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Distance k Cops and Robber• cops can “shoot” robber at some specified
distance k• play as in classical game, but capture includes
case when robber is distance k from the cops– k = 0 is the classical game
C
R
k = 1
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The distance k cop number
• ck(G) = minimum number of cops needed to capture robber at distance at most k
• G connected implies
ck(G) ≤ diam(G) – 1
• for all k ≥ 1,
ck(G) ≤ ck-1(G)
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Example: k = 1
C
R
c1(G) > 1
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Example
C C
Rc1(G) = 2
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ck(n)
• ck(n) = maximum value of ck(G) over connected G of order n
• Meyniel conjecture:
c0(n) = O(n1/2).
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Random graphs
• for random graphs G(n,p) with p = p(n), the behaviour of distance k cop number is complicated
Theorem 12.4 (Bonato et al,09)
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Zig-zag functions
• for x in (0,1), define
fk(x) = log E(ck(G(n,nx-1))) / log n
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The robber fights back!
• robber can attack neighbouring cop
• one more cop needed in this graph (check)
CC
C
R
cc number
• let cc(G) be the minimum number of cops needed with these rules
Lemma 12.5 For a graph G,
c(G) ≤ cc(G) ≤ 2c(G).
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20
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Firefighter
• G simple, undirected, connected graph• fire spreads from a vertex over discrete time-
steps or rounds• vertices are on fire, protected, or clear• fire can spread to all available adjacent vertices• firefighter can protect one vertex in each round
• (Hartnell, 95) introduced Firefighter– simplified model for the spread of a fire/disease/virus
in a network
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Saving vertices
• one-player game• firefighter aims to maximize the number of
clear or protected (ie saved) vertices
• sn(G,v) = maximum number of saved vertices in G if a fire starts at v
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Examples
• sn(Pn,v) = n-1, if v is an end-vertex
= n-2, else
• sn(Kn,v) = 1
• Theorem (MacGillivray, P. Wang, 03): sn(Qn,v) = n
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Surviving rate
• (Cai, W. Wang, 09) surviving rate of G,
ρ(G) = expected percentage of vertices saved if fire starts at a random
vertex
)(
2),(
1)(
GVv
vGsnn
G
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Example: path
Lemma 12.6:
2
2
2
221
)1(2)2)(2(1
),(1
)(
nn
nnnn
vPsnn
PVv
nn
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Results on ρ(G)
• (Cai, W. Wang, 10): ρ(G) ≥ 1 – Θ(log n /n) if G is outerplanar
• (Finbow, P. Wang, W. Wang, 10):
if G has size at most (4/3 – ε)n, then ρ(G) ≥ 6/5ε, where 0 < ε < 5/27
• (Prałat, 10):
if G has size at most (15/11 – ε)n, then ρ(G) ≥ 1/60ε, where 0 < ε < 1/2 (15/11 best possible)
Open problem:Infinite hexagonal grid
• can one cop contain the fire?
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Aside: Minimum orders
• Mk = minimum order of a k-cop-win graph
• M1 = 1, M2 = 4
• M3 = 10 (Baird, Bonato,13)
– see also (Beveridge et al, 2014+)
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Questions
• M4 = ?
• are the Mk monotone increasing?– for example, can it happen that M344 < M343?
• mk = minimum order of a connected G such that c(G) ≥ k
• (Baird, Bonato, 13) mk = Ω(k2) is equivalent to Meyniel’s conjecture.
• mk = Mk for all k ≥ 4?29
Good guys vs bad guys games in graphs
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slow medium fast helicopter
slow traps, tandem-win
medium robot vacuum Cops and Robbers edge searching eternal security
fast cleaning distance k Cops and Robbers
Cops and Robbers on disjoint edge sets
The Angel and Devil
helicopter seepage Helicopter Cops and Robbers, Marshals, The Angel and Devil,Firefighter
Hex
badgood