Characterization of test-sets for overlap-free morphisms

We give a characterization of all the sets X such that any morphism h on {a,b} is overlap-free if and only if for each x in X, h(x) is overlap-free. As a consequence, we observe the particular case X={bbabaa} which improves the previous characterization of Berstel–Séébold (Discrete Appl. Math. 46 (1993) 275–281).