  We study transition fronts for one-dimensional reaction-diffusion equations
with compactly perturbed ignition-monostable reactions. We establish an almost
sharp condition on reactions which characterizes the existence and
non-existence of fronts. In particular, we prove that a strong inhomogeneity in
the reaction prevents formation of transition fronts, while a weak
inhomogeneity gives rise to a front. Our work extends results and methods
introduced by J. Nolen, J.M. Roquejoffre, L. Ryzhik, and A. Zlato\v{s}.
