Proofs of two conjectures on generalized Fibonacci cubes

Wei, Jianxin; Zhang, Heping

doi:10.1016/j.ejc.2015.07.018

Cited by 21 publications

(3 citation statements)

References 13 publications

(45 reference statements)

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“…However, no additional daisy cubes are obtained via this generalization, no matter whether they are partial cubes or not, cf. [28] for the latter aspect of generalized Fibonacci cubes.…”

Section: Examples and Basic Properties Of Daisy Cubesmentioning

confidence: 99%

Daisy cubes and distance cube polynomial

Klavžar¹,

Mollard²

2017

Preprint

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Let X ⊆ {0, 1} n . Then the daisy cube Q n (X) is introduced as the subgraph of Q n induced by the intersection of the intervals I(x, 0 n ) over all x ∈ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial D G,u (x, y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Q k at a given distance from the vertex u. It is proved that if G is a daisy cube, then D G,0 n (x, y) = C G (x + y − 1), where C G (x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then D G,u (x, −x) = 1 holds for every vertex u in G.

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“…However, no additional daisy cubes are obtained via this generalization, no matter whether they are partial cubes or not, cf. [28] for the latter aspect of generalized Fibonacci cubes.…”

Section: Examples and Basic Properties Of Daisy Cubesmentioning

confidence: 99%

Daisy cubes and distance cube polynomial

Klavžar¹,

Mollard²

2017

Preprint

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“…Fibonacci cubes have been introduced by Hsu in [14] and correspond to the case with f = 11. In [15,16,19,20,22] the structure of non-isometric words for alphabets of size 2 and Hamming distance is completely characterized and related to particular properties on their overlaps. The more general case of alphabets of size greater than 2 and Lee distance is studied in [3,4,5].…”

Section: Introductionmentioning

confidence: 99%

Isometric Words based on Swap and Mismatch Distance

Anselmo¹,

Castiglione²,

Flores³

et al. 2023

Preprint

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An edit distance is a metric between words that quantifies how two words differ by counting the number of edit operations needed to transform one word into the other one. A word f is said isometric with respect to an edit distance if, for any pair of f -free words u and v, there exists a transformation of minimal length from u to v via the related edit operations such that all the intermediate words are also ffree. The adjective "isometric" comes from the fact that, if the Hamming distance is considered (i.e., only mismatches), then isometric words are connected with definitions of isometric subgraphs of hypercubes.We consider the case of edit distance with swap and mismatch. We compare it with the case of mismatch only and prove some properties of isometric words that are related to particular features of their overlaps.

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“…Two nodes are linked if and only if they differ in exactly one position, and the mismatch is given by two symbols a and b that verify a = b ± 1 mod d. In order to obtain some variants of hypercubes for which the number of vertices increases slower than in a hypercube, Hsu [8] intro-duced Fibonacci cubes in which nodes are on a binary alphabet and avoid the factor 11. The notion of dary n-cubes has subsequently be extended to define the generalized Fibonacci cube [9,10,18] ; it is the subgraph Q 2 n (f ) of a 2-ary n-cube whose nodes avoid some factor f . In this framework, a binary word f is said to be Lee-isometric when, for any n ≥ 1, Q 2 n (f ) can be isometrically embedded into Q 2 n , that is, the distance between two words u and v vertices of Q 2 n (f ) is the same in Q 2 n (f ) and in Q 2 n .…”

Section: Introductionmentioning

confidence: 99%

Checking whether a word is Hamming-isometric in linear time

Béal¹,

Crochemore²

2021

Preprint

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A finite word f is Hamming-isometric if for any two word u and v of same length avoiding f , u can be transformed into v by changing one by one all the letters on which u differs from v, in such a way that all of the new words obtained in this process also avoid f . Words which are not Hamming-isometric have been characterized as words having a border with two mismatches. We derive from this characterization a linear-time algorithm to check whether a word is Hamming-isometric. It is based on pattern matching algorithms with k mismatches. Leeisometric words over a four-letter alphabet have been characterized as words having a border with two Leeerrors. We derive from this characterization a lineartime algorithm to check whether a word over an alphabet of size four is Lee-isometric.

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Proofs of two conjectures on generalized Fibonacci cubes

Abstract: A binary string f is a factor of string u if f appears as a sequence of |f | consecu-

Cited by 21 publications

References 13 publications

Daisy cubes and distance cube polynomial

Daisy cubes and distance cube polynomial

Isometric Words based on Swap and Mismatch Distance

Checking whether a word is Hamming-isometric in linear time

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