All Good (Bad) Words Consisting of Four Blocks

Abstract

The generalized Fibonacci cube $Q_d(f)$ is the graph obtained from the hypercube $Q_d$ by removing all vertices that contain a given binary word $f$. A binary word $f$ is called good if $Q_d(f)$ is an isometric subgraph of $Q_d$ for all $d\ge 1$, and bad otherwise. A non-extendable sequence of contiguous equal digits in a word $f$ is called a block of $f$. The question to determine the good (bad) words consisting of at most three blocks was solved by Ilié, Klavžar, and Rho. This question is further studied in the present paper. All the good (bad) words consisting of four blocks are determined completely, and all bad $2$-isometric words among consisting of at most four blocks words are found to be $1100$ and $0011$.