Generalized Lucas cubes

Let $f$ be a binary string and $dge 1.$ Then the generalized Lucas cube $Q_d(overleftharpoon{f})$ is introduced as the graph obtained from the $d$-cube $Q_d$ by removing all vertices that have a circulation containing $f$ as a substring. The question for which $f$ and $d,$ the generalized Lucas cube $Q_d({overleftharpoon f})$ is an isometric subgraph of the $d$-cube $Q_d$ is solved for all binary strings of length at most five. Several isometrically embeddable and non-embeddable infinite series where $f$ is of arbitrary length are given. Some structural properties of generalized Lucas cubes are also presented.