1 Almost all cop-win graphs contain a universal vertex - PowerPoint Presentation

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Almost all cop-win graphs contain a universal vertex

Anthony BonatoRyerson University

CanaDAM

2011Slide2
Cop number of a graph

the

cop number of a graph, written c(G), is an elusive graph parameterfew connections to other graph parametershard to computehard to find bounds

structure of k-cop-win graphs with k > 1 is not well understood Random cop-win graphs Anthony Bonato

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Cops and Robbers

played on

reflexive graphs Gtwo players Cops C

and robber R play at alternate time-steps (cops first) with perfect informationplayers 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) ≤

γ

(G)

Random cop-win graphs Anthony Bonato

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Fast facts about cop number

(Aigner, Fromme, 84) introduced parameter

G planar, then c(G) ≤ 3(Berrarducci, Intrigila, 93), (Hahn, MacGillivray,06), (B, Chiniforooshan,10): “

c(G) ≤ s?” s

fixed: running time

O(n

2s+3

), n = |V(G)|

(Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08):

if s not fixed, then computing the cop number is

NP-hard

(Shroeder,01)

G genus g, then c(G) ≤ ⌊ 3g/2 ⌋+3(Joret, Kamiński, Theis, 09) c(G) ≤ tw(G)/2

Random cop-win graphs Anthony Bonato

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Meyniel’s Conjecture

c(n) = maximum cop number of a connected graph of order n

Meyniel Conjecture: c(n) = O(n1/2).

Random cop-win graphs Anthony Bonato

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State-of-the-art

(Lu, Peng, 09+) proved thatindependently proved by (Scott, Sudakov,10+)

, and (Frieze, Krivelevich, Loh, 10+)

even proving

c(n) = O(n

1-

ε

)

for some

ε > 0 is open

Random cop-win graphs Anthony Bonato

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Cop-win case

consider the case when one cop has a winning strategy

cop-win graphsintroduced by (Nowakowski, Winkler, 83), (Quilliot, 78) cliques, universal verticestrees

chordal graphsRandom cop-win graphs Anthony Bonato7Slide8
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

Theorem

(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 inductionRandom cop-win graphs Anthony Bonato8Slide9
Dismantlable graphs

Random cop-win graphs Anthony Bonato

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Dismantlable graphs

Random cop-win graphs Anthony Bonato

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unique corner! part of an infinite family that maximizes capture time

(Bonato, Hahn, Golovach, Kratochvíl,09)Slide11
Cop-win orderings

a permutation

v1, v2, … , v

n of V(G) is a cop-win ordering if there exist vertices w1

, w

2

, …, w

n

such that for all

i, w

i is the parent of v

i

in the subgraph induced

V(G) \ {vj : j < i}. a cop-win ordering dismantlabilityRandom cop-win graphs Anthony Bonato11

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2

3

4

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Cop-win Strategy (Clarke, Nowakowski, 2001)

V(G) = [n] a cop-win orderingG1

= G, i > 1, Gi: subgraph induced by deleting 1, …, i-1f

i

: G

i

→ G

i+1

retraction mapping

i to a fixed one of its parentsF

i =

f

i-1

○… ○ f2 ○ f1 a homomorphismidea: robber on u, think of Fi(u) shadow of robbercop moves to capture shadow works as the Fi are homomorphismsresults in a capture in at most n moves of copRandom cop-win graphs Anthony Bonato

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Random cop-win graphs Anthony Bonato

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Random graphs G(n,p) (Erdős, Rényi, 63)

n

a positive integer,

p = p(n)

a real number in

(0,1)

G(n,p)

: probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability

p

5

1

2

3

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Typical cop-win graphs

what is a random cop-win graph?

G(n,1/2) and condition on being cop-winprobability of choosing a cop-win graph on the uniform space of labeled graphs of ordered n

Random cop-win graphs Anthony Bonato14Slide15
Cop number of

G(n,1/2)

(B,Hahn, Wang, 07), (B,Prałat, Wang,09)A.a.s.c(G(n,1/2)) = (1+o(1))log

2n.-matches the domination number Random cop-win graphs Anthony Bonato

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Universal vertices

P(cop-win) ≥

P(universal) = n2-n+1

– O(n22-2n+3) = (1+o(1))n2-n+1

…this is in fact the correct answer!

Random cop-win graphs Anthony Bonato

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Main result

Theorem

(B,Kemkes, Prałat,11+)In G(n,1/2),

P(cop-win) = (1+o(1))n2-n+1Random cop-win graphs Anthony Bonato

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Corollaries

Corollary

(BKP,11+)The number of labeled cop-win graphs is

Random cop-win graphs Anthony Bonato18Slide19
Corollaries

U

n = number of labeled graphs with a universal vertexC

n = number of labeled cop-win graphsCorollary (BKP,11+)That is,

almost all cop-win graphs contain a

universal vertex

.

Random cop-win graphs Anthony Bonato

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Strategy of proof

probability of being cop-win and not having a universal vertex is very small

P(cop-win + ∆ ≤ n – 3) ≤ 2-(1+

ε)nP(cop-win + ∆ = n – 2) =

2

-(3-log

2

3)

n+o

(n)

Random cop-win graphs Anthony Bonato

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P

(cop-win + ∆ ≤ n – 3) ≤

2-(1+ε)n

consider cases based on number of parents:there is a cop-win ordering whose vertices in their initial segments of length 0.05n have more than 17

parents.

there is a cop-win ordering whose vertices in their initial segments of length

0.05n

have at most

17

parents, each of which has co-degree more than

n2/3

.there is a cop-win ordering whose initial segments of length

0.05n

have between

2 and 17 parents, and at least one parent has co-degree at most n2/3.there exists a vertex w with co-degree between 2 and n2/3, such that wi = w for i ≤ 0.05n.Random cop-win graphs Anthony Bonato21Slide22
P

(cop-win + ∆ = n – 2)

≤ 2-(3-log23)

n+o(n)Sketch of proof: Using (1), we obtain that there is an ε

> 0

such that

P

(cop-win)

≤

P(cop-win and ∆ ≤ n-3) + P

(∆ ≥ n-2)

≤

2

-(1+ε)n + n22-n+1 ≤ 2-n+o(n) (*)if ∆ = n-2, then G has a vertex w of degree n-2, a unique vertex v not adjacent to w.let A be the vertices not adjacent to

v (and adjacent to w

)

let

B

be the vertices adjacent to

v

(and also to

w

)

Claim:

The subgraph induced by

B

is cop-win.

Random cop-win graphs Anthony Bonato

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Random cop-win graphs Anthony Bonato

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A

B

w

v

xSlide24
Proof continued

n

choices for w; n-1 for v choices for

A if |A| = i, then using (*)

, probability that

B

is cop-win

is at most

2

-n+2+i+o(n)Random cop-win graphs Anthony Bonato

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Problemsdo almost all

k-cop-win graphs contain a dominating set of order k?would imply that the number of labeled k

-cop-win graphs of order n is difficulty: no simple elimination ordering for k > 1 (Clarke, MacGillivray,09+)

characterizing cop-win planar graphs

(Clarke, Fitzpatrick, Hill, Nowakowski,10)

: classify the cop-win graphs which have cop number 2 after a vertex is deleted

Random cop-win graphs Anthony Bonato

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Random cop-win graphs Anthony Bonato

26preprints, reprints, contact:

Google: “Anthony Bonato”Slide27

Random cop-win graphs Anthony Bonato

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