  We study past horizons in the class of type II Robinson-Trautman vacuum
spacetimes with a cosmological constant. These exact radiative solutions of
Einstein's equations exist in the future of any sufficiently smooth initial
data, and they approach the corresponding spherically symmetric
Schwarzschild-(anti-)de Sitter metric. By analytic methods we investigate the
existence, uniqueness, location and character of the past horizons in these
spacetimes. In particular, we generalize the Penrose-Tod equation for
marginally trapped surfaces, which form such white-hole horizons, to the case
of a nonvanishing cosmological constant, we analyze behavior of its solutions
and visualize their evolutions. We also prove that these horizons are explicit
examples of an outer trapping horizon and a dynamical horizon, so that they are
spacelike past outer horizons.
