\textit{Pursuit-evasion games} have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. \textsc{Cops and Robber} (\CR) is one of the most well-known pursuit-evasion games played on graphs, where multiple \textit{cops} pursue a single \textit{robber}. The aim is to compute the \textit{cop number} of a graph, $k$, which is the minimum number of cops that ensures the \textit{capture} of the robber. From the viewpoint of parameterized complexity, \CR is W[2]-hard parameterized by $k$~[Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the \textit{vertex cover number} ($\mathsf{vcn}$). First, we establish that $k \leq \frac{\mathsf{vcn}}{3}+1$. Second, we prove that \CR parameterized by $\mathsf{vcn}$ is \FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for \CR parameterized by $\mathsf{vcn}$ to admit a polynomial compression. We extend our exponential kernels to the parameters \textit{cluster vertex deletion number} and \textit{deletion to stars number}, and design a linear vertex kernel for \textit{neighborhood diversity}. Additionally, we extend all of our results to several well-studied variations of \CR.

翻译：追逃博弈由于在人工智能、机器人运动规划、数据库理论、分布式计算以及算法理论中的广泛应用，数十年来被深入研究。警察与强盗问题（Cops and Robber, \CR）是图论中最著名的追逃博弈之一，其中多名警察追捕一名强盗。其目标是计算图的警察数 $k$，即确保捕获强盗所需的最少警察数量。从参数化复杂性的角度来看，以 $k$ 为参数时，\CR 是 W[2]-难的 [Fomin 等，TCS, 2010]。因此，我们转而研究输入图的结构参数。首先考察顶点覆盖数（$\mathsf{vcn}$）。我们证明：第一，$k \leq \frac{\mathsf{vcn}}{3}+1$；第二，通过设计指数核，以 $\mathsf{vcn}$ 为参数的 \CR 属于 \FPT。进一步，我们指出以 $\mathsf{vcn}$ 为参数的 \CR 不太可能存在多项式压缩，从而补充了上述结果。我们将指数核推广至参数簇顶点删除数和星图删除数，并为邻域多样性设计了线性顶点核。此外，我们将所有结果推广至 \CR 的多个经典变体。