Cube-complements of generalized Fibonacci cubes

Abstract

The generalized Fibonacci cube Qh(f) is the graph obtained from the h-cube Qh by removing all vertices that contain a given binary string f as a substring. If G is an induced subgraph of Qh, then the cube-complement of G is the graph induced by the vertices of Qh which are not in G. In particular, the cube-complement of a generalized Fibonacci cube Qh(f) is the subgraph of Qh induced by the set of all vertices that contain f as a substring. The questions whether a cube-complement of a generalized Fibonacci cube is (i) connected, (ii) an isometric subgraph of a hypercube or (iii) a median graph are studied. Questions (ii) and (iii) are completely solved, i.e. the sets of binary strings that allow a graph of this class to be an isometric subgraph of a hypercube or a median graph are given. The cube-complement of a daisy cube is also studied.