1. Almost all cop-win graphs contain a universal vertex
Anthony Bonato
Ryerson University
Random cop-win graphs Anthony Bonato

2. Random cop-win graphs Anthony Bonato
Cops and Robbers C C R C
Random cop-win graphs Anthony Bonato

3. Random cop-win graphs Anthony Bonato
Cops and Robbers C C R C
Random cop-win graphs Anthony Bonato

4. Random cop-win graphs Anthony Bonato
Cops and Robbers C R C C
cop number c(G) ≤ 3
Random cop-win graphs Anthony Bonato

5. Random cop-win graphs Anthony Bonato
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) ≤ δ(G)
Random cop-win graphs Anthony Bonato

6. Fast facts about cop number
(Aigner, Fromme, 84) introduced parameter
G planar, then c(G) ≤ 3
(Berrarducci, Intrigila, 93), (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
Random cop-win graphs Anthony Bonato

7. 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

8. 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+)
(Bollobás, Kun, Leader, 08+): if
p = p(n) ≥ 2.1log n/ n, then
c(G(n,p)) ≤ n1/2log n
(Prałat,Wormald,11+): removed log factor
Random cop-win graphs Anthony Bonato

9. 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

10. 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
and play shadow strategy;
- dismantlable graphs are cop-win by induction
Random cop-win graphs Anthony Bonato

11. Dismantlable graphs
8/25/2019
Anthony Bonato

12. Dismantlable graphs unique corner!
part of an infinite family that maximizes capture time (Bonato, Hahn, Golovach, Kratochvíl,09)
8/25/2019
Anthony Bonato

13. Random cop-win graphs Anthony Bonato
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 5 1 4 3 2
Random cop-win graphs Anthony Bonato

14. 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

15. 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

16. 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

17. 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

18. 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

19. Random cop-win graphs Anthony Bonato
Corollaries
Corollary (BKP,11+) The number of labeled cop-win graphs is
Random cop-win graphs Anthony Bonato

20. 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

21. 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

22. 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

23. 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

24. Random cop-win graphs Anthony Bonatox v
Random cop-win graphs Anthony Bonato

25. 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 B is cop-win is at most 2-n+2+i+o(n)
Random cop-win graphs Anthony Bonato

26. Random cop-win graphs Anthony Bonato
Problems
higher cop number?
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 > (Clarke, MacGillivray,09+)
characterizing cop-win planar or outer-planar graphs
Random cop-win graphs Anthony Bonato

27. preprints, reprints, contact: Google: “Anthony Bonato”
Random cop-win graphs Anthony Bonato

28. Random cop-win graphs Anthony Bonato

29. Geometric model for OSNs
journal relaunch
new editors
accepting theoretical and empirical papers on complex networks, OSNs, biological networks
Geometric model for OSNs