  We revisit the notion of initial sets by Xu and Cayrol, i.e., non-empty
minimal admissible sets in abstract argumentation frameworks. Initial sets are
a simple concept for analysing conflicts in an abstract argumentation framework
and to explain why certain arguments can be accepted. We contribute with new
insights on the structure of initial sets and devise a simple non-deterministic
construction principle for any admissible set, based on iterative selection of
initial sets of the original framework and its induced reducts. In particular,
we characterise many existing admissibility-based semantics via this
construction principle, thus providing a constructive explanation on the
structure of extensions. We also investigate certain problems related to
initial sets with respect to their computational complexity.
