  Hairpin completion and its variant called bounded hairpin completion are
operations on formal languages, inspired by a hairpin formation in molecular
biology. Another variant called hairpin lengthening has been recently
introduced and studied on the closure properties and algorithmic problems
concerning several families of languages. In this paper, we introduce a new
operation of this kind, called hairpin incompletion which is not only an
extension of bounded hairpin completion, but also a restricted (bounded)
variant of hairpin lengthening. Further, the hairpin incompletion operation
provides a formal language theoretic framework that models a bio-molecular
technique nowadays known as Whiplash PCR. We study the closure properties of
language families under both the operation and its iterated version. We show
that a family of languages closed under intersection with regular sets,
concatenation with regular sets, and finite union is closed under one-sided
iterated hairpin incompletion, and that a family of languages containing all
linear languages and closed under circular permutation, left derivative and
substitution is also closed under iterated hairpin incompletion.
