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Generalized Fibonacci cubes

Generalized Fibonacci cubes,Aleksandar Ilic,Sandi Klavÿzar,Yoomi Rho

Generalized Fibonacci cubes   (Citations: 1)
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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 d is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non- embeddable 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 cases Qd(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.
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