Generalized Fibonacci cubes

Generalized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and nonembeddable infinite series are given. The question is completely solved for strings f of length at most five. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial casesQd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f , the f -dimension of a graph is introduced. Several problems and conjectures are also listed.