Generalized Fibonacci cubes

Generalized Fibonacci cube Q(d)(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 Gamma(d) is Q(d)(11). The question whether Q(d)(f) is an isometric subgraph of Q(d) is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Q(d)(10) and Q(d)(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. (C) 2011 Elsevier B.V. All rights reserved.