A characterization of non-isometric binary words

Abstract In this paper, first, we introduce two classification criteria of non-isometric words, and characterize every type of non-isometric words. Then, according to those characterizations, a family of words that 2-isometric but not 3-isometric is found, and the words with maximum and minimum index are characterized. Next, an O ( | f | 2 ) algorithm is designed, which is presented to compute the index of a non-isometric word and determine what class that a word belongs to. Finally, several problems are listed for further investigation.

[1]  Sandi Klavzar,et al.  Structure of Fibonacci cubes: a survey , 2013, J. Comb. Optim..

[2]  Guangfu Wang,et al.  The self-concatenation of isometric strings is isometric , 2017, Discret. Math..

[3]  W. Imrich,et al.  Product Graphs: Structure and Recognition , 2000 .

[4]  Wen-Jing Hsu,et al.  Generalized fibonacci cubes are mostly hamiltonian , 1994, J. Graph Theory.

[5]  Sandi Klavzar,et al.  Connectivity of Fibonacci cubes, Lucas cubes, and generalized cubes , 2015, Discret. Math. Theor. Comput. Sci..

[6]  Sandi Klavzar,et al.  Parity Index of Binary Words and Powers of Prime Words , 2012, Electron. J. Comb..

[7]  Sandi Klavzar,et al.  Generalized Fibonacci cubes , 2012, Discret. Math..

[8]  Sandi Klavar.  On median nature and enumerative properties of Fibonacci-like cubes , 2005 .

[9]  Sandi Klavzar,et al.  On the Wiener index of generalized Fibonacci cubes and Lucas cubes , 2015, Discret. Appl. Math..

[10]  Sandi Klavzar,et al.  Asymptotic number of isometric generalized Fibonacci cubes , 2012, Eur. J. Comb..

[11]  Wen-Jing Hsu,et al.  Fibonacci Cubes-A New Interconnection Technology , 1993, IEEE Trans. Parallel Distributed Syst..

[12]  Sandi Klavzar,et al.  On isomorphism classes of generalized Fibonacci cubes , 2016, Eur. J. Comb..

[13]  Heping Zhang,et al.  Proofs of two conjectures on generalized Fibonacci cubes , 2016, Eur. J. Comb..

[14]  Jianxin Wei.  The structures of bad words , 2017, Eur. J. Comb..

[15]  Jianxin Wei.  All good (bad) words consisting of four blocks , 2017, Ars Comb..

[16]  Sandi Klavzar,et al.  The index of a binary word , 2012, Theor. Comput. Sci..

[17]  Heping Zhang,et al.  A negative answer to a problem on generalized Fibonacci cubes , 2017, Discret. Math..

[18]  Heping Zhang,et al.  Solution to a conjecture on words that are bad and 2-isometric , 2015, Theor. Comput. Sci..