Almost all cop-win graphs contain a universal vertex

We consider cop-win graphs in the binomial random graph G ( n , 1 / 2 ) . We prove that almost all cop-win graphs contain a universal vertex. From this result, we derive that the asymptotic number of labelled cop-win graphs of order n is equal to ( 1 + o ( 1 ) ) n 2 n 2 / 2 − 3 n / 2 + 1 .