The paper introduces the concept of tilde-isometric words, which generalizes the notion of Hamming-isometric words by considering an edit distance based on swap and mismatch operations, called the tilde-distance.
The key highlights and insights are:
The tilde-distance allows swap operations in addition to replacements, making the situation more complex compared to the Hamming distance case. Swap operations can be equivalent to two replacements but count as one in the tilde-distance.
A word is tilde-isometric if for any pair of equal-length words that avoid the word, there exists a tilde-transformation (sequence of swap and replacement operations) that transforms one word into the other while keeping all intermediate words free of the given word.
The paper provides a complete characterization of tilde-non-isometric words in terms of special configurations in their overlaps, including 1-tilde-error overlaps, 2-tilde-error overlaps with non-adjacent errors, and specific patterns of 2-tilde-error overlaps.
The proof of the characterization theorem is technical, involving careful analysis of the possible tilde-transformations between pairs of tilde-witnesses (words that witness a word is tilde-non-isometric).
The results show that the sets of Hamming-isometric and tilde-isometric words are incomparable, with examples of words that are one but not the other.
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