Lengths of irreducible and delicate words (Q2153402)

The paper studies words which avoid repetitions in a fragile way. Here ``repetition'' may mean square, overlap or cube, while ``fragile way'' means that the word does not avoid the repetition in question as soon as it is slightly modified. Finally, the slight modification may mean 1) inserting any letter from the alphabet into the word (including at the beginning or end), then the word is called \textit{extremal}; 2) deleting an interior letter, then the word is called \textit{irreducible}; or 3) changing any letter to a different one from the alphabet, then the word is called \textit{delicate}. The notion of extremal words was introduced by \textit{J. Grytczuk} et al. [Electron. J. Comb. 27, No. 1, Research Paper P1.48, 9 p. (2020; Zbl 1435.05007)], and further results about them were obtained in [\textit{L. Mol} and \textit{N. Rampersad}, Contrib. Discrete Math. 16, No. 1, 8--19 (2021; Zbl 1467.68149)] and in [\textit{L. Mol} et al., Electron. J. Comb. 27, No. 4, Research Paper P4.42, 15 p. (2020; Zbl 1462.68151)]. The notion of irreducibility was introduced by \textit{T. Harju} in [Theor. Comput. Sci. 862, 155--159 (2021; Zbl 1497.68404)], where lengths for which there exist irreducible square-free ternary words are characterized. The present paper does the same for irreducible overlap-free binary words and irreducible cube-free binary words. The notion of delicate words seems to be introduced in the present paper, although a similar concept was studied in [\textit{S. Brown} et al., Theor. Inform. Appl. 40, No. 3, 473--484 (2006; Zbl 1110.68117)]. The present paper determines the possible lengths of delicate square-free ternary words, overlap-free binary words, and delicate cube-free binary words. An infinite family of overlap-free binary words that are simultaneously extremal, irreducible, and delicate is also exhibited.