The index of a string consisting of 4 blocks

TL;DR

This paper determines the index of strings with four blocks in generalized Fibonacci cubes, extending previous work on strings with up to three blocks, and explores conditions for strings to be considered good.

Contribution

It extends the determination of the index to strings with four blocks and analyzes the necessity of the 'good' string condition for even-block strings.

Findings

The index of a string with four blocks is explicitly determined.

Conditions for a string to be 'good' are established for even-block strings.

The study generalizes previous results on strings with fewer blocks.

Abstract

Generalized Fibonacci cube $Q_{d}(f)$, introduced by Ili\'{c}, Klav\v{z}ar and Rho, is the graph obtained from the $d$-hypercube $Q_{d}$ by removing all vertices that contain $f$ as a substring. The smallest integer $d$ such that $Q_{d}(f)$ is not an isometric subgraph of $Q_{d}$ is called the index of $f$. A non-extendable sequence of contiguous equal digits in a string $\mu$ is called a block of $\mu$. The question that determine the index of a string consisting of at most 3 blocks is solved by Ili\'{c}, Klav\v{z}ar and Rho. This question is further studied and the index of a string consisting of 4 blocks is determined, and the necessity of a string being good is also given for the strings with even blocks.