Embeddings of Linear Arrays, Rings and 2-D Meshes on Extended Lucas Cube Networks

A Fibonacci string is a length n binary string containing no two consecutive 1s. Fibonacci cubes (FC), Extended Fibonacci cubes (ELC) and Lucas cubes (LC) are subgraphs of hypercube defined in terms of Fibonacci strings. All these cubes were introduced in the last ten years as models for interconnection networks and shown that their network topology posseses many interesting properties that are important in parallel processor network design and parallel applications. In this paper, we propose a new family of Fibonacci-like cube, namely Extended Lucas Cube (ELC). We address the following network simulation problem : Given a linear array, a ring or a two-dimensional mesh; how can its nodes be assigned to ELC nodes so as to keep their adjacent nodes near each other in ELC ?. We first show a simple fact that there is a Hamiltonian path and cycle in any ELC. We prove that any linear array and ring network can be embedded into its corresponding optimum ELC (the smallest ELC with at least the number of nodes in the ring) with dilation 1, which is optimum for most cases. Then, we describe dilation 1 embeddings of a class of meshes into their corresponding optimum ELC.