Almost all cop-win graphs contain a universal vertex

Abstract

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 $(1+o\left(1\right))n{2}^{n^2/2-3n/2+1}$.