Fibonacci \((p, r)\)-Cubes which are Partial Cubes

Abstract

The Fibonacci \((p, r)\)-cube is an interconnection topology that unifies various connection topologies, including the hypercube, classical Fibonacci cube, and postal network. While classical Fibonacci cubes are known to be partial cubes, we demonstrate that a Fibonacci \((p, r)\)-cube is a partial cube if and only if either \(p = 1\) or \(p \geq 2\) and \(r \leq p + 1\). Furthermore, we establish that for Fibonacci \((p, r)\)-cubes, the properties of being almost-median graphs, semi-median graphs, and partial cubes are equivalent.