Overprescribed Cops and Robbers Anthony Bonato Ryerson University GRASCan 2016

C8C8 Overprescribed Cops and Robbers C C cops win in two steps

C8C8 Overprescribed Cops and Robbers C C cops win in one step C

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 Overprescribed Cops and Robbers

Capture time of Cartesian grids Overprescribed Cops and Robbers

Capture time of hypercubes Theorem (B,Gordinowicz,Kinnersley,Prałat,13) The capture time of Q n is Θ(nlog n). Overprescribed Cops and Robbers

Lower bound Theorem (BGKP,13) For b > 0 a constant, a robber can escape n b 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 capt k (G), where c(G) ≤ k ≤ γ(G) –k-capture time –capt(G) = capt c(G) (G) temporal speed-up: –as c(G) increases to γ(G), capt k (G) monotonically decreases to 1 if k > c(G), we call this Overprescribed Cops and Robbers Overprescribed Cops and Robbers

Trees 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

Bounds Theorem (BGRP,16+) 1.capt k (G) ≥ rad k (G). 2.capt k (G) ≥ (diam(G)-k+1) / 2k Overprescribed Cops and Robbers

Retracts Overprescribed Cops and Robbers

Trees Corollary (BGRP,16+) For a tree T, capt k (G) = rad k (G). Idea: cover by balls (which are retracts) around vertices around metric k-center and use theorem Overprescribed Cops and Robbers

Square grids Overprescribed Cops and Robbers

k-capture time of grids Theorem (BGRP,16+) If k = O(n d ), then capt k (G(d,n)) = Θ(n/k 1/d ). Overprescribed Cops and Robbers

Domination number of hypercubes Overprescribed Cops and Robbers n n= 2 k-1, 2 k 2 n-k

Capture time of hypercubes Overprescribed Cops and Robbers Theorem (BGRP,16+)

Planar graphs (Aigner, Fromme, 84) planar graphs have cop number ≤ 3. (Clarke, 02) outerplanar graphs have cop number ≤ 2. Cops and Robbers25

Capture time of planar graphs Overprescribed Cops and Robbers

capt 3 of planar graphs Theorem (BGRP,16+) If G is a connected planar graph of order n, then capt 3 (G) ≤ (diam(G) +1)n. Overprescribed Cops and Robbers

Questions/directions rcapt k (G): capture time with random initial placement of cops –how far can rcapt k (G) deviate from capt k (G)? capture time of hypercube near domination number bounds on capt 2 (G) if G is outerplanar? Overprescribed Cops and Robbers

Questions/directions Overprescribed Cops and Robbers