The index of a string consisting of 4 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.