GRASCan 2016
Overprescribed
Cops and Robbers
Anthony Bonato
Ryerson University
C8
C
C
• cops win in
two steps
Overprescribed Cops and
Robbers
C8
C
C
• cops win in
one step
C
Overprescribed Cops and
Robbers
Capture time of a graph
• the length of Cops and Robbers was considered first as
capture time
• (B,Hahn,Golovach,Kratochvíl,09) capture time of G:
length of game with c(G) cops assuming optimal play,
written capt(G)
– if G is cop-win, then capt(G) ≤ n - 4 if n ≥ 7 (see also
(Gavanciak,10))
– capt(G) ≤ n/2 for many families of cop-win graphs including
trees, chordal graphs
– examples of planar graphs with capt(G) = n - 4
Overprescribed Cops and
Robbers
Capture time of trees
Lemma (B, Perez-Gimenez,Reineger,Prałat,16+):
For a tree T, we have that
capt(T) = rad(T).
Proof sketch:
• for capt(T) ≤ rad(T), place C on a central vertex and use
the zombie strategy
• for rad(T) ≤ capt(T), notice that any other initial
placement of C results in R choosing a vertex distance
> rad(T) away
– R stays put
Overprescribed Cops and
Robbers
Cop number of products of trees
Theorem (Maamoun,Meyniel,87): The cop number
d+1
of the Cartesian product of d trees is
.
2
• no reference to the length of the game; i.e
capture time of grids or the hypercube
Overprescribed Cops and
Robbers
Capture time of Cartesian grids
Theorem (Merhabian,10): The capture time of the
Cartesian product of two trees T1 and T2 is
diam(T1) + diam(T2)) / 2 .
In particular, the capture time of the m x n Cartesian
grid is
(m + n)/2−1 .
Overprescribed Cops and
Robbers
Capture time of hypercubes
Theorem (B,Gordinowicz,Kinnersley,Prałat,13)
The capture time of Qn is
Θ(nlog n).
Overprescribed Cops and
Robbers
Lower bound
Theorem (BGKP,13) For b > 0 a constant, a robber
can escape nb cops for at least
(1-o(1))1/2 n log n
rounds.
– probabilistic method: play with a random
robber
– Coupon collector and large deviation bounds
Overprescribed Cops and
Robbers
Add more cops!
Overprescribed Cops and
Robbers
k-capture time
• define captk(G), where c(G) ≤ k ≤ γ(G)
– k-capture time
– capt(G) = captc(G)(G)
• temporal speed-up:
– as c(G) increases to γ(G),
captk(G) monotonically decreases to 1
• if k > c(G), we call this Overprescribed Cops and Robbers
Overprescribed Cops and
Robbers
Trees
• for k > 0, metric k-center is a set S, |S| ≤ k, that
minimizes
max 𝑑(𝑣, 𝑆)
𝑣∈𝑉(𝐺)
• radk(G) is this minimum
– k = 1, then radk(G) = rad(G)
• NP-complete to find metric k-centers (Vazirani,03)
• radk is monotone on retracts
Overprescribed Cops and
Robbers
Example: k = 1
Overprescribed Cops and
Robbers
Example: k = 2
Overprescribed Cops and
Robbers
Example: k = 3
Overprescribed Cops and
Robbers
Example: k = 4
Overprescribed Cops and
Robbers
Example: k = 5 = 𝛾
Overprescribed Cops and
Robbers
Bounds
Theorem (BGRP,16+)
1. captk(G) ≥ radk(G).
2. captk(G) ≥ (diam(G)-k+1) / 2k
Overprescribed Cops and
Robbers
Retracts
• following theorem is key:
Theorem (BGRP,16+)
Suppose V can be decomposed into t-many vertex sets of
retracts Gi of G. Then
captk(G)≤ max captk(Gi)
1≤𝑖≤𝑡
Overprescribed Cops and
Robbers
Trees
Corollary (BGRP,16+)
For a tree T,
captk(G) = radk(G).
• Idea: cover by balls (which are retracts) around vertices
around metric k-center and use theorem
Overprescribed Cops and
Robbers
Square grids
• G(d,n) = d-dimensional Cartesian n-grid
d+1
• (Maamoun,Meyniel,87): c(G(d,n)) =
2
• (Merhabian,10): capt(G(d,n)) =
1
nd
2
• 𝛾(G(d,n)) = Θ(nd)
Overprescribed Cops and
Robbers
log 2 𝑑
k-capture time of grids
Theorem (BGRP,16+)
If k = O(nd), then
captk(G(d,n)) = Θ(n/k1/d).
Overprescribed Cops and
Robbers
Domination number of hypercubes
• 𝛾 𝑄𝑛 is open for general n
• 𝛾 𝑄𝑛 ≤ 2𝑛−3 if n ≥ 7
n
𝛾 𝑄𝑛
3
2
4
4
5
7
6
12
n= 2k-1, 2k
2n-k
Overprescribed Cops and
Robbers
Capture time of hypercubes
Theorem (BGRP,16+)
Overprescribed Cops and
Robbers
Planar graphs
•
(Aigner, Fromme, 84) planar graphs have cop
number ≤ 3.
•
(Clarke, 02) outerplanar graphs have cop
number ≤ 2.
Cops and Robbers
25
Capture time of planar graphs
Theorem (BGRP,16+)
If G is a connected planar graph and k ≥ 12 𝑛, then
captk(G) ≤ 6∙rad(G)log n.
• proof uses Planar Separator Theorem
(Alon,Seymour,Thomas,90)
• works also if robber has infinite speed
• generalizes to higher genus (only k changes, not bound)
Overprescribed Cops and
Robbers
capt3 of planar graphs
Theorem (BGRP,16+)
If G is a connected planar graph of order n,
then
capt3(G) ≤ (diam(G) +1)n.
Overprescribed Cops and
Robbers
Questions/directions
• rcaptk(G): capture time with random initial
placement of cops
– how far can rcaptk(G) deviate from captk(G)?
• capture time of hypercube near domination
number
• bounds on capt2(G) if G is outerplanar?
Overprescribed Cops and
Robbers
Questions/directions
• examples of planar graphs with large 2-, 3- or
even 𝑛-capture times?
– example of planar G with capt2,3(G) = Θ(n2)?
• capt(G) of toroidal Cartesian grid with three
cops?
• higher genus?
– toroidal graph with c = 4???
Overprescribed Cops and
Robbers