Cop Numbers for Grids

• Cop numbers for grids define the minimum number of cops required to guarantee the capture of a robber in various pursuit-evasion games on grid graphs.
• While the classical cop number for an n-dimensional grid is exactly n, variants like toroidal, bridge-burning, or those with faster robbers significantly alter the required number of cops.
• Results on cop numbers for grids rely on combinatorial, geometric, and isoperimetric techniques, revealing sharp transitions in tractability depending on the game rules.

Cop numbers for grids refer to the minimum number of cops required to guarantee the capture (or containment) of a robber in various pursuit-evasion games played on finite or infinite grid graphs. The paper of this parameter has generated a comprehensive body of results encompassing classical, accelerated, fast-robber, burning-edge, helicopter, directed, and other capture paradigms; as well as quantitative bounds for associated capture times and cop densities. These results draw on combinatorial, probabilistic, geometric, and structural considerations, with key theorems establishing tight lower and upper bounds, efficient pursuit strategies, and sharp separations between pursuit variants.

1. The Classical Cop Number on Grids

In the classical cops-and-robber game, both cops and the robber alternate moves of length one to neighboring vertices. The cop number for a grid is the smallest number of cops that guarantees capture regardless of the robber's strategy.

For an $n$-dimensional grid of size $d_0 \times \dotsm \times d_{n-1}$, the cop number is exactly $n$:

$$\boxed{\text{Cop number for an } n\text{-dimensional grid is } n}$$

This is both necessary and sufficient: each cop can control a coordinate axis, and an initial configuration with fewer than $n$ cops allows a robber to indefinitely preserve a coordinate gap (0909.1381). For the $n\times n$ 2D grid, this specializes to $c(\text{2D grid})=2$.

Single Cop Strategies in 2D

On a 2D grid, a single cop can win if and only if the initial Manhattan distance $D_0(0) = |C_{0,0}-R_0| + |C_{0,1}-R_1|$ is even. If the initial parity is odd, the robber can evade forever; if even, the coordinate-minimizing cop strategy ensures capture in $O(d_0+d_1)$ moves, and is asymptotically optimal (0909.1381).

2. Cop Numbers for Variants of Grids

Directed and Toroidal Grids

For grids with toroidal closure (periodic boundary conditions), cop numbers behave differently:

• On a standard $m\times n$ toroidal grid $T_{m,n}$, precisely $3$ cops are necessary and sufficient to guarantee capture (Luccio et al., 2017).
• For a semi-torus (periodic in only one dimension), $2$ suffice as in the classical planar grid.
• In directed ("straight-ahead") quadrangulations of the torus or Klein bottle, the cop number is bounded by a constant independent of grid size; e.g., $c(G) \leq 319$ for any such straight-ahead $4$-regular quadrangulation, and $c(G)\leq 13$ for $k$-regularly oriented toroidal grids (Maza et al., 2019).

Hyperopic, Surrounding, and Bridge-Burning Variants

• Hyperopic cop number: For any planar grid, the hyperopic cop number—where the robber is invisible if adjacent to all cops—is at most $3$ (and at least $2$ for grids) (Bonato et al., 2017).
• Surrounding cop numbers: For vertex-surround and edge-surround variants (where the cops win by blocking all exits rather than occupying the robber's vertex), cop numbers can be much larger—not bounded by a function of the classical cop number and maximum degree. Even in bounded-degree grids, these parameters can be arbitrarily large, showing that containment is fundamentally harder than capture (Jungeblut et al., 2023).
• Bridge-burning cop number: If each edge traversed by the robber is destroyed, then for a $m\times n$ grid,

$$\left\lceil \frac{mn}{121} \right\rceil \leq c_b(G_{m,n}) \leq \text{slightly more than } \frac{mn}{112}$$

The burning model makes grids much harder to police: the cop number is linear in the grid area, as compared to a constant in the classical model (Kinnersley et al., 2018).

3. Fast and Accelerated Robber Variants

Robber Moves Faster Than Cops

If the robber has speed $R$ while cops move at unit speed:

• For sufficiently large $R$ ($R=25$ in (Balister et al., 2016)), the minimum number of cops required on an $n\times n$ grid satisfies

$$f_R(n) \geq \exp\left( \Omega\left(\frac{\log n}{\log\log n}\right) \right)$$

This is a dramatic leap: a modest speed advantage for the robber increases the cop number to superpolynomial in $n$. For unit speed, only two cops are needed (Balister et al., 2016).

• For speed-$R$ robbers, the best known upper bound is

$$f_R(n) \leq n \cdot \left(\frac{R-1}{R+1}\right) + O(1)$$

• For speed-2 robbers, recent results give the first $o(n)$ upper bound for 2D grids:

$c([n]^2, \text{speed 2}) \leq n^{0.999}$

with a lower bound of $\Omega(n^{0.03})$ (Gillott, 19 Apr 2025).

Both Sides Move Faster (Accelerated Model)

In the speed-$(s,s)$ ("accelerated") model, where both sides can move up to $s$ steps per turn:

• The key result is

$$c_{s,s}(T_1 \square \ldots \square T_d) = \begin{cases} 1, & d=1\text{ or }2 \ \left\lceil \frac{d}{2} \right\rceil, & d \geq 3,\, \min_i \operatorname{diam}(T_i) \geq 2s \end{cases}$$

For 2D grids, this means only one cop is needed for any $s\geq 2$ in sufficiently large grids (Kinnersley et al., 25 Jun 2025).

This result formalizes that increasing the speed for both sides allows the cops to reduce their number by at most one compared to the classic model.

4. Covering Games and Infinitely Fast Robbers

Covering Pursuit

In the covering game, after each cop turn, the requirement is that there is a cop on the robber's vertex; the robber moves up to distance $2$. For the $[n]^2$ grid, the main results are:

$$\Omega\left(n^{1.357}\right) \leq c([n]^2) \leq n^{1.999}$$

This resolves the open question of whether $o(n^2)$ cops suffice affirmatively (Gillott, 19 Apr 2025).

The proof techniques use recursive tiling and movement of "nets" of cop teams, and density-based lower bounds.

Infinite-Speed Robbers

In the model where the robber may traverse any cop-free path on each turn (infinite speed), the cop number for the $n\times n$ grid is tightly bounded:

$n-1 \leq c_\infty(P_n \square P_n) \leq n$

For higher dimensions, the cop number grows as the "surface area" of the grid:

$$c_\infty(P_n^{\square d}) = \Theta(n^{d-1}/\sqrt{d})$$

This connects to isoperimetric inequalities and treewidth—the minimal set of cops needed to confine the robber to a region (Kinnersley et al., 2021).

5. Cop Number in the Overprescribed Game: Speed-Ups with More Cops

Adding more cops beyond the minimal value decreases the capture time on grids polynomially:

$\capt_k(G^d_n) = \Theta\left(\frac{n}{k^{1/d}}\right)$

Thus, increasing the number of cops yields predictable polynomial improvements in capture time for grids, unlike the highly nonlinear curves found for hypercubes and some random graphs (Bonato et al., 2016).

6. Cop-Width, Flip-Width, and Strong Colouring Numbers

Cop-width and flip-width generalize classic cop numbers in settings where cops can perform "helicopter moves" or alter the underlying graph structure. For grids:

$$\mathrm{copwidth}_r(\text{grid}) \leq \mathrm{scol}_{4r}(\text{grid}) = O(r)$$

Classes of graphs (including grids) with linear strong colouring numbers also enjoy linear cop-width and flip-width, ensuring efficient capture strategies even against fast robbers in these generalized variants (Hickingbotham, 2023).

7. Summary Table: Cop Numbers in Grid Variants

Game/Variant	2D Grid Cop Number	Asymptotic/Formula	Reference
Classical (speed 1, planar grid)	$2$	$c(G^2_n) = 2$	(0909.1381)
Classical (torus)	$3$	$c(T_{m,n}) = 3$	(Luccio et al., 2017)
Hyperopic (planar grid)	$2$ or $3$	$c(G) \leq CH(G) \leq 3$	(Bonato et al., 2017)
Bridge-burning	$\Theta(n^2)$	$\Omega(mn/121) \leq c_b \leq O(mn)$	(Kinnersley et al., 2018)
Directed (straight-ahead torus)	$\leq 13$	$c(G) \leq 13$ or $319$	(Maza et al., 2019)
Fast robber, $R\gg1$	$\exp(\Omega(\log n / \log\log n))$	Lower bound, superpolynomial	(Balister et al., 2016)
Both sides accelerated, $s\ge2$	$1$ (2D); $\lceil d/2\rceil$ (dD)	$c_{s,s}(G) = \lceil d/2 \rceil$	(Kinnersley et al., 25 Jun 2025)
Covering pursuit	$n^{1.357}$ to $n^{1.999}$	$o(n^2)$ bound proved	(Gillott, 19 Apr 2025)
Infinite speed robber	$n-1 \leq c_\infty \leq n$	Surface-area driven	(Kinnersley et al., 2021)

8. Impact and Open Directions

The paper of cop numbers on grids has established that grid-like graphs tend to be structurally tractable in classical and many algorithmic variants, but can exhibit sharp transitions to intractability (superpolynomial growth) in pursuit settings where the robber has substantial speed advantage or can exploit burning or "covering" mechanics. Bounds frequently rely on combinatorial, geometric, and isoperimetric techniques, and pursuit strategies often generalize via product structure decomposition. Recent advances include explicit polynomial and near-linear upper bounds in several fast-robber models, resolutions of longstanding open threshold questions, and bounding cop-widths via strong colouring numbers.

Ongoing questions include tightening bounds for specific fast-robber (fixed $s>1$) or infinitely-fast settings, explicit computation of containment variants in high-degree grids, and extension of tight strategies to tori, higher dimensions, or strongly directed grid variants.