On shuffled-square-free words

A word u is a shuffle of words v and w, which we denote by u∈v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the left-to-right ordering of the letters from v and w. In case u∈v⧢v for some word v, the word u is called a shuffled-square. A word u is shuffled-square...

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Bibliographic Details
Published in:	Theoretical computer science Vol. 941; pp. 91 - 103
Main Authors:	Bulteau, Laurent, Jugé, Vincent, Vialette, Stéphane
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 04-01-2023 Elsevier
Subjects:	Combinatorics Complexity Computer Science Shuffle Shuffle Combinatorics Complexity
Online Access:	Get full text
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Summary:	A word u is a shuffle of words v and w, which we denote by u∈v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the left-to-right ordering of the letters from v and w. In case u∈v⧢v for some word v, the word u is called a shuffled-square. A word u is shuffled-square-free if it does not have a non-empty factor (i.e., non-empty sequence of adjacent letters) that is a shuffled-square. Our contribution in this context is two-fold. First, we prove that there exist arbitrarily long shuffled-square-free words in any alphabet with six letters or more, thereby improving on a previous result of Guégan and Ochem. Furthermore, we show that recognizing shuffled-square-free words on arbitrary alphabets is NP-complete.
ISSN:	0304-3975 1879-2294
DOI:	10.1016/j.tcs.2022.10.028