Fibonacci (p, r)-cubes which are partial cubes

Fibonacci (p, r)-cube is an interconnection topology, which unifies a wide range of connection topologies, such as the hypercube, classical Fibonacci cube, postal network, etc. It is known that classical Fibonacci cubes are partial cubes. In this paper we show that a Fibonacci (p, r)-cube is partial cube if and only if either p = 1, or p >= 2 and r <= p + 1. Furthermore, we show that for Fibonacci (p, r)-cubes, almost-median graphs, semi-median graphs and partial cubes are all equivalent.