Generalized Fibonacci cubes

Generalized Fibonacci cube Q d ( f ) is introduced as the graph obtained from the d -cube Q d by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γ 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.