Abstract
A k-ary n-cube is a graph with \(k^n\) vertices, each associated to a word of length n over an alphabet of cardinality k. The subgraph obtained deleting those vertices which contain a given k-ary word f as a factor is here introduced and called the k-ary n-cube avoiding f. When, for any n, such a subgraph is isometric to the cube, the word f is said isometric. In the binary case, isometric words can be equivalently defined, independently from hypercubes. A binary word f is isometric if and only if it is good, i.e., for any pair of f-free words u and v, u can be transformed in v by exchanging one by one the bits on which they differ and generating only f-free words. These two approaches are here considered in the case of a k-ary alphabet, showing that they are still coincident for \(k=3\), but they are not from \(k=4\) on. Bad words are then characterized in terms of their overlaps with errors. Further properties are obtained on non-isometric words and their index, in the case of a quaternary alphabet.
Partially supported by INdAM-GNCS Projects 2020-2021, FARB Project ORSA203187 of University of Salerno and TEAMS Project of University of Catania.
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Anselmo, M., Flores, M., Madonia, M. (2021). Quaternary n-cubes and Isometric Words. In: Lecroq, T., Puzynina, S. (eds) Combinatorics on Words. WORDS 2021. Lecture Notes in Computer Science(), vol 12847. Springer, Cham. https://doi.org/10.1007/978-3-030-85088-3_3
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