  We examine holographic complexity in time-dependent Vaidya spacetimes with
both the complexity$=$volume (CV) and complexity$=$action (CA) proposals. We
focus on the evolution of the holographic complexity for a thin shell of null
fluid, which collapses into empty AdS space and forms a (one-sided) black hole.
In order to apply the CA approach, we introduce an action principle for the
null fluid which sources the Vaidya geometries, and we carefully examine the
contribution of the null shell to the action. Further, we find that adding a
particular counterterm on the null boundaries of the Wheeler-DeWitt patch is
essential if the gravitational action is to properly describe the complexity of
the boundary state. For both the CV proposal and the CA proposal (with the
extra boundary counterterm), the late time limit of the growth rate of the
holographic complexity for the one-sided black hole is precisely the same as
that found for an eternal black hole.
