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Strict quasi-diagonal dominance of the system matrix is known to be sufficient for a linear autonomous system with arbitrary time delays in off-diagonal interactions to be stable. A small perturbation of the matrix yields a perturbed system with the same dominance, and, hence, stability properties. In this paper, it is shown that quasi-diagonal dominance is also necessary for stability with respect to small perturbations and arbitrary off-diagonal time delays. Weaker necessary conditions are given for systems which are themselves stable for all time delays, but which have perturbations that are unstable for certain delays.