Almost all k-cop-win graphs contain a dominating set of cardinality k

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∈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)−knk2n2/2−(1/2−log2(1−2−k))n.