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Crucial words and the complexity of some extremal problems for sets of prohibited words

Evdokimov, A. and Kitaev, Sergey (2004) Crucial words and the complexity of some extremal problems for sets of prohibited words. Journal of Combinatorial Theory Series A, 105 (2). pp. 273-289. ISSN 0097-3165

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We introduce the notion of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider three special sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of determining lengths of words that are free from a set of prohibitions is shown to be NP-complete, although the related problem of whether or not a given set of prohibitions is complete is known to be effectively solvable.