The k-ary n-cubes are a generalization of the hypercubes to alphabets of cardinality k, with k>=2. More precisely, a k-ary n-cube is a graph with k^n vertices associated to the k-ary words of length n. Given a k-ary word f, the k-ary n-cube avoiding f is the subgraph obtained deleting those vertices which contain f as a factor. When such a subgraph is isometric to the cube, for any n>=1, the word f is said isometric. A binary word f is isometric if and only if it is Ham-isometric, i.e., for any pair of f-free binary words u and v, u can be transformed in v by complementing the bits on which they differ and generating only f-free words. The case of a k-ary alphabet, with k>=2, is here investigated. From k>=4, the isometricity in terms of cubes is no longer captured by the Ham-isometricity, but by the Lee-isometricity. Then, Ham-isometric and Lee-isometric k-ary words are characterized in terms of their overlaps with errors. The minimal length of two words which witness the non-isometricity of a word f is called its index. The index of f is bounded in terms of its length and the bounds are shown tight by examples.

On k-ary n-cubes and isometric words

Anselmo M.;Flores M.;
2022

Abstract

The k-ary n-cubes are a generalization of the hypercubes to alphabets of cardinality k, with k>=2. More precisely, a k-ary n-cube is a graph with k^n vertices associated to the k-ary words of length n. Given a k-ary word f, the k-ary n-cube avoiding f is the subgraph obtained deleting those vertices which contain f as a factor. When such a subgraph is isometric to the cube, for any n>=1, the word f is said isometric. A binary word f is isometric if and only if it is Ham-isometric, i.e., for any pair of f-free binary words u and v, u can be transformed in v by complementing the bits on which they differ and generating only f-free words. The case of a k-ary alphabet, with k>=2, is here investigated. From k>=4, the isometricity in terms of cubes is no longer captured by the Ham-isometricity, but by the Lee-isometricity. Then, Ham-isometric and Lee-isometric k-ary words are characterized in terms of their overlaps with errors. The minimal length of two words which witness the non-isometricity of a word f is called its index. The index of f is bounded in terms of its length and the bounds are shown tight by examples.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4808294
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact