Connectivity of Fibonacci cubes, Lucas cubes, and generalized cubes

If f is a binary word and d a positive integer, then the generalized Fibonacci cube Q(d) (f) is the graph obtained from the d-cube Q(d) by removing all the vertices that contain f as a factor, while the generalized Lucas cube Q(d)((f) over bar) is the graph obtained from Q(d) by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Gamma(d) and the Lucas cube Lambda(d) are the graphs Q(d) (11) and Q(d) ((11) over bar), respectively. It is proved that the connectivity and the edge-connectivity of d as well as of Gamma(d) are equal to [d+2/3]. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d <= 9.