On Recognizing Words That Are Squares for the Shuffle Product

Abstract

The shuffle of two words u and v of A ^* is the language u ш v consisting of all words u ₁ v ₁ u ₂ v ₂ …u _k v _k , where k ≥ 0 and the u _i and the v _i are the words of A ^* such that u = u ₁ u ₂ …u _k and v = v ₁ v ₂ …v _k . A word u ∈ A ^* is a square for the shuffle product if it is the shuffle of two identical words (i.e., u ∈ v ш v for some v ∈ A ^*). Whereas, it can be tested in polynomial-time whether or not u ∈ v ₁ ш v ₂ for given words u, v ₁ and v ₂ [J.-C. Spehner, Le Calcul Rapide des Mélanges de Deux Mots, Theoretical Computer Science, 1986], we show in this paper that it is NP-complete to determine whether or not a word u is a square for the shuffle product. The novelty in our approach lies in representing words as linear graphs, in which deciding whether or not a given word is a square for the shuffle product reduces to computing some inclusion-free perfect matching.