Almost all cop-win graphs contain a universal vertex

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CanaDAM 2011. Almost all cop-win graphs contain a universal vertex. Anthony Bonato Ryerson University. 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

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Almost all cop-win graphs contain a universal vertex

Anthony Bonato

Ryerson University

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

Cops and Robbers
• played on reflexive graphs G
• two players CopsC 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)

Random cop-win graphs Anthony Bonato

• (Aigner, Fromme, 84) introduced parameter
• G planar, then c(G) ≤ 3
• (Berrarducci, Intrigila, 93), (Hahn, MacGillivray,06), (B, Chiniforooshan,10):

“c(G) ≤ s?” sfixed: 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) Ggenus g, then c(G)≤ ⌊ 3g/2 ⌋+3
• (Joret, Kamiński, Theis, 09) c(G)≤ tw(G)/2

Random cop-win graphs Anthony Bonato

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

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

Random cop-win graphs Anthony Bonato

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

Random cop-win graphs Anthony Bonato

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

- dismantlable graphs are cop-win by induction

Random cop-win graphs Anthony Bonato

Dismantlable graphs

Random cop-win graphs Anthony Bonato

Dismantlable graphs
• unique corner!
• part of an infinite family that maximizes capture time (Bonato, Hahn, Golovach, Kratochvíl,09)

Random cop-win graphs Anthony Bonato

Cop-win orderings
• a permutation v1, v2, … , vnof V(G) is a

cop-win ordering if there exist vertices w1, w2, …, wnsuch that for all i, wi is the parent of vi in the subgraph induced V(G) \ {vj : j < i}.

• a cop-win ordering dismantlability

5

1

4

3

2

Random cop-win graphs Anthony Bonato

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 cop

Random cop-win graphs Anthony Bonato

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

4

1

2

3

5

Random cop-win graphs Anthony Bonato

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

Random cop-win graphs Anthony Bonato

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

Random cop-win graphs Anthony Bonato

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

Main result

Theorem (B,Kemkes, Prałat,11+)

In G(n,1/2),

P(cop-win) = (1+o(1))n2-n+1

Random cop-win graphs Anthony Bonato

Corollaries

Corollary (BKP,11+)

The number of labeled cop-win graphs is

Random cop-win graphs Anthony Bonato

Corollaries

Un= number of labeled graphs with a universal

vertex

Cn= number of labeled cop-win graphs

Corollary (BKP,11+)

That is, almost all cop-win graphs contain a

universal vertex.

Random cop-win graphs Anthony Bonato

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

Random cop-win graphs Anthony Bonato

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 Bonato

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

w

A

B

x

v

Random cop-win graphs Anthony Bonato

Proof continued
• n choices for w; n-1 for v
• choices for A
• if |A| = i, then using (*), probability that Bis cop-win is at most 2-n+2+i+o(n)

Random cop-win graphs Anthony Bonato

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 Bonato

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