  Large-scale Monte Carlo simulations are used to explore the effect of
quenched disorder on one dimensional, non-equilibrium kinetic Ising models with
locally broken spin symmetry, at zero temperature (the symmetry is broken
through spin-flip rates that differ for '+' and '-' spins). The model is found
to exhibit a continuous phase transition to an absorbing state. The associated
critical behavior is studied at zero branching rate of kinks, through analysis
spreading of '+' and '-' spins and, of the kink density. Impurities exert a
strong effect on the critical behavior only for a particular choice of
parameters, corresponding to the strongly spin-anisotropic kinetic Ising model
introduced by Majumdar et al. Typically, disorder effects become evident for
impurity strengths such that diffusion is nearly blocked. In this regime, the
critical behavior is similar to that arising, for example, in the
one-dimensional diluted contact process, with Griffiths-like behavior for the
kink density. We find variable cluster exponents, which obey a hyperscaling
relation, and are similar to those reported by Cafiero et al. We also show that
the isotropic two-component AB -> 0 model is insensitive to reaction-disorder,
and that only logarithmic corrections arise, induced by strong disorder in the
diffusion rate.
