  \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.
