Computing generalized de Bruijn sequences

De Bruijn sequences of order n represent the set An of all words of length n over a given alphabet A in the sense that they contain occurrences of each of these words (they actually contain exactly one occurrence of each of these words). Recently, the computational problem of representing subsets of An by partial words, which are sequences that may have holes or don't-care symbols that match each letter of A, was considered and shown to be in NP. However, membership in P remained open. In this paper, we show that deciding if a subset S of An is representable by a partial word can be done in polynomial time with respect to the size n|S| of the input. We also describe a polynomial-time algorithm that determines all integers h for representation by partial words with exactly h holes. Moreover, our algorithms construct representation words. Our approach is graph theoretical.