A negative answer to a problem on generalized Fibonacci cubes

Generalized Fibonacci cube Qn(f)Qn(f) is the graph obtained from the nn-cube QnQn by removing all vertices that contain a given binary string ff as a consecutive substring. A binary string ff is called bad if Qn(f)Qn(f) is not an isometric subgraph of QnQn for some nn, and the smallest such integer nn, denoted by B(f)B(f), is called the index of ff. Ilić, Klavžar and Rho posed a problem that if Qn(f)Qn(f) is not an isometric subgraph of QnQn, is there a dimension n′n′ such that Qn(f)Qn(f) can be isometrically embedded into Qn′Qn′? We give a negative answer to this problem by showing that if ff is bad, then for any n≥B(f)n≥B(f), Qn(f)Qn(f) cannot be isometrically embedded to any hypercube.