We discuss formation of cosmic strings associated with a spontaneously broken U(1) symmetry by performing classical field-theoretical simulations. An original U(1) symmetry is explicitly broken down to its subgroup ZN even before spontaneous breaking takes place. We estimate the ratio of explicit breaking to that of spontaneous breaking for which topological defects for N=1 and N=2 are formed. For N=1, a cosmic string attached to a single domain wall can be formed when the amount of the explicit breaking is 3 orders of magnitude smaller than that of the spontaneous breaking. For N=2, no matter how large the explicit breaking is, domain walls are inevitably formed as long as the temperature of the Universe is high enough to restore Z2 symmetry. In that case, cosmic strings are also inevitably formed as long as the amount of the explicit breaking is smaller than that of the spontaneous breaking.