  We present a realization of directed Ising (DI) type dynamic absorbing state
phase transitions in the context of one-dimensional interfaces, such as the
relaxation of a step on a vicinal surface. Under the restriction that particle
deposition and evaporation can only take place near existing kinks, the
interface relaxes into one of three steady states: rough, perfectly ordered
flat (OF) without kinks, or disordered flat (DOF) with randomly placed kinks
but in perfect up-down alternating order. A DI type dynamic preroughening
transition takes place between the OF and DOF phases. At this critical point
the asymptotic time evolution is controlled not only by the DI exponents but
also by the initial condition. Information about the correlations in the
initial state persists and changes the critical exponents.
