Capture Time of Cop-Win Graphs

• Capture Time of Cop-Win Graphs is a key parameter in Cops and Robbers games quantifying the minimum rounds for one cop to guarantee capture on specific graph structures known as cop-win graphs.
• Studies establish extremal bounds on capture time, showing it is at most $n-4$ for classical $n$-vertex cop-win graphs and exploring variations like speed-$s$ models and directed graphs.
• Analyzing capture time provides insights into structural graph properties like dismantling orders, informs algorithmic approaches for optimal strategies, and reveals trade-offs in resource allocation for pursuit-evasion scenarios.

The capture time of cop-win graphs is a central parameter in the theory of Cops and Robbers games on graphs, measuring the minimum number of rounds required for a single cop to guarantee capture of a robber under optimal play. The paper of capture time not only elucidates the strategic complexity of pursuit-evasion dynamics in graphs but also connects combinatorial structure to quantitative temporal bounds across a range of graph families and variations of the game.

1. Structural Characterizations and Dismantlings

Cop-win graphs are precisely those that admit certain elimination orders, commonly formalized as dismantling schemes. In the classical game (unit speed, full visibility), a graph $G$ is cop-win if and only if there exists an order $v_1, \ldots, v_n$ such that for each $i < n$ there is $j > i$ with

$N_1(v_i) \cap X_i \subseteq N_1(v_j)$

where $X_i = \{v_i, v_{i+1}, \ldots, v_n\}$ and $N_1(v)$ is the closed neighborhood. This structure ensures that at each step, a "corner" vertex is dominated and can be folded away, progressively shrinking the regions where the robber can hide (Chalopin et al., 2010).

Capture time in this context is intimately linked to the length and nature of this vertex elimination order. Each "dismantling" move can reduce the number of cop moves needed to restrict the robber's movement, with the worst-case total being the length of the order.

2. Extremal Results and Explicit Bounds

The tightest possible bound for capture time in cop-win graphs of $n$ vertices was established by Bonato et al. (2009) and Gavenčiak (2010), who proved

$\operatorname{capt}(G) \leq n-4$

for $n \ge 7$, and this is achieved by specific extremal constructions (Offner et al., 2017, Kinnersley, 2017). The extremal cop-win graphs with maximal capture time are characterized by a unique "corner rank profile": their structure ensures each step in the dismantling can only eliminate one vertex, forcing the cop to methodically traverse the graph.

The main formula relating corner rank (a stratification of the graph's vertices by elimination step) to capture time is

$\operatorname{capt}(G) = a - r$

where $a$ is the maximum corner rank and $r$ indicates whether the highest-rank vertex dominates the previous level ($r = 1$) or not ($r = 0$) (Offner et al., 2016, Offner et al., 2017).

3. Variations: Speed-Ups, Graph Powers, and Acceleration

A natural extension is to allow both the cop and the robber to move up to $s$ edges per turn (the “accelerated” or speed-$s$ model). For cop-win graphs in this variant, capture time is determined by the capture time in the $s$-th power of the graph, $G^s$: $$\operatorname{capt}_s(G) = \operatorname{capt}_1(G^s)$$ This reduction enables the transfer of structural arguments from the classical setting to the accelerated one (Kinnersley et al., 25 Jun 2025).

For $s=2$, it is proven that there exist cop-win graphs on $n$ vertices with

$\operatorname{capt}_2(G) = n-7$

showing that the maximal capture time decreases only by a small additive constant compared to the $s=1$ case ($n-4$). This is achieved by constructing families where each addition of a unique corner increases capture time by one, recursively ensuring near-linear capture time even at higher speeds.

4. Algorithmic and Probabilistic Approaches

Beyond adversarial robber models, capture time has been studied under stochastic dynamics. For instance, when the robber moves as a simple random walk, the expected capture time is shown to be at most $n + o(n)$, and this remains tight for certain lollipop graphs (Komarov et al., 2013). In the cop vs. gambler variation, where the opponent “hops” between vertices using a fixed distribution, the expected capture time is exactly $n$ for the known gambler, independent of the cop-win structure (Komarov et al., 2013).

Algorithmically, optimal strategies to minimize capture time can be explicitly constructed using rank-based techniques or value-iteration and dynamic programming (notably in stochastic or concurrent variants) (Konstantinidis et al., 2015, Slettnes et al., 2017). The existence of efficient algorithms for cop-win recognition extends to computation of capture time using corner rank or other dismantling-based decompositions.

5. Trade-offs, Throttling, and Resource Optimization

The cop-throttling number $$\operatorname{th}_c(G) = \min_k (k + \operatorname{capt}_k(G))$$ quantifies the trade-off between appointments of additional cops and reduction of capture time. In many cop-win families (trees, chordal graphs, grids), capture time with more than one cop (“overprescribed” gameplay) can be substantially smaller, and the sum $k + \operatorname{capt}_k(G)$ admits $O(\sqrt n)$ bounds (Breen et al., 2017, Bonato et al., 2016, Bonato et al., 2019). For trees,

$$\operatorname{capt}_k(T) = \min_{|S|=k} \max_{v \in V(T)} \operatorname{dist}(v, S)$$

and

$$\operatorname{th}_c(T) = \min_k (k + \operatorname{capt}_k(T))$$

attained by optimal central placements.

A related “product throttling” parameter (minimizing the person-hours, $k \cdot (1+\operatorname{capt}_k)$) provides further operational insights for resource management (Bonato et al., 2019).

6. Special Graph Classes and Variants

In Cartesian products, such as grids and hypercubes, capture time exhibits explicit dependencies on dimension, graph size, and the number of cops. For instance, in $m \times n$ grids,

$$\operatorname{capt}_2(G) = \left\lfloor \frac{m+n-2}{2} \right\rfloor$$

whereas for hypercubes $Q_n$, $\operatorname{capt}(Q_n) = \Theta(n \log n)$ when the minimum number of cops are used (Mehrabian, 2010, Bonato et al., 2013).

In the burning bridges model, where the robber destroys edges upon use, capture time for certain cop-win graphs can become as large as $\Theta(n^2)$ or $O(n^3)$ (Kinnersley et al., 2018).

In the game of Cops and Robbers on directed graphs, the capture time of cop-win digraphs can be quadratic ($\Theta(n^2)$), in stark contrast to the linear bound for undirected cop-win graphs (Khatri et al., 2018, Kinnersley, 2017).

7. Structural Insights and Open Problems

The extremal paper of capture time reveals that the graphs maximizing capture time for given order are tightly constrained by their rank configuration (corner-rank or equivalent stratification). These "hard" structures provide a blueprint for constructing worst-case instances and for understanding the interplay between graph geometry and pursuit complexity (Offner et al., 2017).

Open questions remain regarding full classification of extremal families for each possible capture time in cop-win graphs, the exact effect of increased cop or robber speed ($s > 2$), and fine-grained thresholds for cop numbers on high-dimensional products as acceleration grows (Kinnersley et al., 25 Jun 2025). Connections to Gromov hyperbolicity and structural decomposition further tie capture time to broader graph geometric properties (Chalopin et al., 2010).

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Model/Class	Max Capture Time (1 cop, $n$-vertex)	Method/Formula
Classical cop-win	$n-4$	Corner rank, dismantling order
Accelerated ($s=2$)	$n-7$	Capture time in $G^2$
Trees (radius $r$)	$r$	Central placement
2D grid ($m \times n$)	$\left\lfloor \frac{m+n-2}{2} \right\rfloor$	Central placement
Hypercube $Q_n$	$\Theta(n \log n)$	Phased coordinate strategy
Claw, even-hole-free ($k=2$)	$\leq 2n$	Layered BFS strategy
Burning bridges (special)	$\Omega(n^2)$–$O(n^3)$	Path avoidance, Eulerian walks

Capture time thus unifies combinatorial structure, graph invariants, and algorithmic strategies across pursuit-evasion paradigms, demonstrating both universal phenomena (e.g., dismantling-based bounds) and fine distinctions depending on game variants, graph class, and resources.