The hairpin completion is a natural operation on formal languages which has been inspired by molecular phenomena in biology and by DNA-computing. In 2009 we presented a (polynomial time) decision algorithm to decide regularity of the hairpin completion. In this paper we provide four new results:

1.) We show that the decision problem is NL-complete.

2.) There is a polynomial time decision algorithm which runs in time O(n^8), this improves our previous results, which provided O(n^{20}).

3.) For the one-sided case (which is closer to DNA computing) the time is O(n^2), only.

4.) The hairpin completion of a regular language is unambiguous linear context-free. This result allows to compute the growth (generating function) of the hairpin completion and to compare it with the growth of the underlying regular language.