Issue No. 07 - July (1997 vol. 8)

ISSN: 1045-9219

pp: 727-737

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/71.598347

ABSTRACT

<p><b>Abstract</b>—The <it>generalized Fibonacci cubes</it> (abbreviated to GFCs) [<ref rid="bibl07274" type="bib">4</ref>] were recently proposed as a class of interconnection topologies, which cover a spectrum ranging from regular graphs such as the hypercube to semiregular graphs such as the <it>second-order Fibonacci cube</it> in [<ref rid="bibl07271" type="bib">1</ref>]. It has been shown that the <it>k</it>th order GFC of dimension <it>n</it> + <it>k</it> is equivalent to an <it>n</it>-cube for 0 ≤<it>n</it> < <it>k</it>; and it is a proper subgraph of an <it>n</it>-cube for <it>n</it>≥<it>k</it> [<ref rid="bibl07274" type="bib">4</ref>]. Thus, a <it>k</it>th order GFC of dimension <it>n</it> + <it>k</it> can be obtained from the <it>n</it>-cube for all <it>n</it>≥<it>k</it> by removing certain nodes from an <it>n</it>-cube. This problem is very simple when no faulty node exists in an <it>n</it>-cube; but it becomes very complex if some faulty nodes appear in an <it>n</it>-cube. In this paper, we first consider the following open problem: How can a maximal (in terms of the number of nodes) <it>generalized Fibonacci cube</it> be distinguished from a <it>faulty hypercube</it> [<ref rid="bibl07273" type="bib">3</ref>] which can also be considered as a <it>Fault-tolerant embedding</it> in hypercubes. Then, we shall show how to directly embed a GFC into a faulty hypercube and prove that if no more than three faulty nodes exist, then an <tmath>$\left\lfloor {{\textstyle{n \over 2}}} \right\rfloor {\rm th}$</tmath> order GFC of dimension <tmath>$n+\left\lfloor {{\textstyle{n \over 2}}} \right\rfloor $</tmath> can be directly embedded into an <it>n</it>-cube in the worst case, for <it>n</it> = 4 or <it>n</it>≥ 6.</p>

INDEX TERMS

Direct embedding, fault-tolerant, generalized Fibonacci cubes, hypercubes, faulty hypercube, second-order Fibonacci cube.

CITATION

Tzong-Wann Kao, Feng-Shr Jiang, Shi-Jinn Horng, "Embedding of Generalized Fibonacci Cubes in Hypercubes with Faulty Nodes",

*IEEE Transactions on Parallel & Distributed Systems*, vol. 8, no. , pp. 727-737, July 1997, doi:10.1109/71.598347