  Systems with non-Hermitian skin effects are very sensitive to the imposed
boundary conditions and lattice size, and thus an important question is whether
non-Hermitian skin effects can survive when deviating from the open boundary
condition. To unveil the origin of boundary sensitivity, we present exact
solutions for one-dimensional non-Hermitian models with generalized boundary
conditions and study rigorously the interplay effect of lattice size and
boundary terms. Besides the open boundary condition, we identify the existence
of non-Hermitian skin effect when one of the boundary hopping terms vanishes.
Apart from this critical line on the boundary parameter space, we find that the
skin effect is fragile under any tiny boundary perturbation in the
thermodynamic limit, although it can survive in a finite size system. Moreover,
we demonstrate that the non-Hermitian Su-Schreieffer-Heeger model exhibits a
new phase diagram in the boundary critical line, which is different from either
open or periodical boundary case.
