For any tree or cycle on n vertices, the shortest representing word has length 2n-2; the number of such words has a closed formula, and a new two-letter pattern representation works for every graph except one open case.
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Enumeration and Extensions of Word-representants
For any tree or cycle on n vertices, the shortest representing word has length 2n-2; the number of such words has a closed formula, and a new two-letter pattern representation works for every graph except one open case.