Almost all cop-win graphs contain a universal vertex

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Almost all cop-win graphs contain a universal vertex

Anthony BonatoRyerson University

CanaDAM 2011

Cop number of a graph

• the cop number of a graph, written c(G), is an elusive graph parameter– few connections to other graph parameters– hard to compute– hard 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 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) ≤ γ(G)


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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(n2s+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


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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).


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State-of-the-art• (Lu, Peng, 09+) proved that

• independently proved by (Scott, Sudakov,10+), and (Frieze, Krivelevich, Loh, 10+)

• even proving c(n) = O(n1-ε)

for some ε > 0 is open


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no

nOnc2log))1(1(2

)(

Cop-win case

• consider the case when one cop has a winning strategy– cop-win graphs

• introduced by (Nowakowski, Winkler, 83), (Quilliot, 78) – cliques, universal vertices– trees– chordal graphs


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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

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 induction


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Dismantlable graphs


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Dismantlable graphs


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

(Bonato, Hahn, Golovach, Kratochvíl,09)

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

23

4

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Cop-win Strategy (Clarke, Nowakowski, 2001)

• V(G) = [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 copRandom cop-win graphs

Anthony Bonato12


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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

51 2 3 4

Typical cop-win graphs• what is a random cop-win graph?

• G(n,1/2) and condition on being cop-win

• probability of choosing a cop-win graph on the uniform space of labeled graphs of ordered n


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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))log2n.

-matches the domination number


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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!


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Main result

Theorem (B,Kemkes, Prałat,11+)

In G(n,1/2),P(cop-win) = (1+o(1))n2-n+1


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Corollaries

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


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Corollaries

Un = number of labeled graphs with a universal vertexCn = number of labeled cop-win graphs

Corollary (BKP,11+)

That is, almost all cop-win graphs contain a universal vertex.


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.1lim

n

n

n CU

Strategy of proof

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

1. P(cop-win + ∆ ≤ n – 3) ≤ 2-(1+ε)n

2. P(cop-win + ∆ = n – 2) = 2-(3-log23)n+o(n)


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P(cop-win + ∆ ≤ n – 3) ≤ 2-(1+ε)n • consider cases based on number of parents:

a. there is a cop-win ordering whose vertices in their initial segments of length 0.05n have more than 17 parents.

b. 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.

c. 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.

d. there exists a vertex w with co-degree between 2 and n2/3, such that wi = w for i ≤ 0.05n.


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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 Bonato22


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A B

w

v

x

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)


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2

0

2n

i in

Problems• do 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 Bonato25


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• preprints, reprints, contact:Google: “Anthony Bonato”


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Almost all cop-win graphs contain a universal vertex

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