  A generalized Petersen graph $GP(n,k)$ is a regular cubic graph on $2n$
vertices (the parameter $k$ is used to define some of the edges). It was
previously shown (Ball et al., 2015) that the cop number of $GP(n,k)$ is at
most $4$, for all permissible values of $n$ and $k$. In this paper we prove
that the cop number of "most" generalized Petersen graphs is exactly $4$. More
precisely, we show that unless $n$ and $k$ fall into certain specified
categories, then the cop number of $GP(n,k)$ is $4$. The graphs to which our
result applies all have girth $8$.
  In fact, our argument is slightly more general: we show that in any cubic
graph of girth at least $8$, unless there exist two cycles of length $8$ whose
intersection is a path of length $2$, then the cop number of the graph is at
least $4$. Even more generally, in a graph of girth at least $9$ and minimum
valency $\delta$, the cop number is at least $\delta+1$.
