  A set $V$ is said to be separated by subsets $V_1,\ldots,V_k$ if, for every
pair of distinct elements of $V$, there is a set $V_i$ that contains exactly
one of them. Imposing structural constraints on the separating subsets is often
necessary for practical purposes and leads to a number of fascinating (and, in
some cases, already classical) graph-theoretic problems.
  In this work, we are interested in separating the vertices of a random graph
by path-connected vertex sets $V_1,\ldots,V_k$, jointly forming a separating
system. First, we determine the size of the smallest separating system of
$G(n,p)$ when $np\to \infty$ up to lower order terms, and exhibit a threshold
phenomenon around the sharp threshold for connectivity. Second, we show that
random regular graphs of sufficiently high degree can typically be optimally
separated by $\lceil \log_2 n\rceil$ sets. Moreover, we provide bounds for the
minimum degree threshold for optimal separation of general graphs.
