Reconstruction from subwords ∗

In the paper two variants of a combinatorial problem for the set F n q of sequences of length n over the alphabet Fq = {0, 1, .., q − 1} are considered, with some applications. The original problem was the following: for a given word w ∈ F n q , what is the smallest integer k such that we can reconstruct w if we know all of its subwords of length k. This problem was solved by Lothaire [8] . We consider the following variant of this problem: the n-letter word w = w1...wn (which is called a DNA-word) is composed over an alphabet consisting of q complement pairs:{i, ī : i = 0, .., q − 1}; and denote by w∗ its reverse complement, i.e. w∗ = w̄n...w̄1. A DNA-word u is called a subword of w if it is a subword of either w or w∗. (Another formulation is that we identify w and w∗.) We want to reconstruct w from its subwords of length k. We give a simple proof for k = n − 1, and apply this result for determining the automorphism group of the poset of DNA-words of length at most n, partially ordered by the above subword relation.