  We consider $k$-cop-win graphs in the binomial random graph $G(n,1/2).$ It is
known that almost all cop-win graphs contain a universal vertex. We generalize
this result and prove that for every $k \in N$, almost all $k$-cop-win graphs
contain a dominating set of cardinality $k$. From this it follows that the
asymptotic number of labelled $k$-cop-win graphs of order $n$ is equal to
$(1+o(1)) (1-2^{-k})^{-k} {n \choose k} 2^{n^2/2 - (1/2-\log_2(1-2^{-k})) n}$.
