The self-concatenation of isometric strings is isometric

The generalized Fibonacci Qd(f) is defined as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a contiguous substring. This idea was introduced by Ilić, Klavžar and Rho. A binary string f is called isometric if Qd(f) is an isometric subgraph of Qd for all d≥1, otherwise it is called non-isometric. In this paper, we prove that a string f is isometric if and only if fn is isometric for any n≥1. This result can help us to construct more isometric strings and significantly increase the number of isometric strings.