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Embedding of Generalized Fibonacci Cubes in Hypercubes with Faulty Nodes
July 1997 (vol. 8 no. 7)
pp. 727-737

Abstract—The generalized Fibonacci cubes (abbreviated to GFCs) [4] 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 second-order Fibonacci cube in [1]. It has been shown that the kth order GFC of dimension n + k is equivalent to an n-cube for 0 ≤n < k; and it is a proper subgraph of an n-cube for nk [4]. Thus, a kth order GFC of dimension n + k can be obtained from the n-cube for all nk by removing certain nodes from an n-cube. This problem is very simple when no faulty node exists in an n-cube; but it becomes very complex if some faulty nodes appear in an n-cube. In this paper, we first consider the following open problem: How can a maximal (in terms of the number of nodes) generalized Fibonacci cube be distinguished from a faulty hypercube [3] which can also be considered as a Fault-tolerant embedding 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 $\left\lfloor {{\textstyle{n \over 2}}} \right\rfloor {\rm th}$ order GFC of dimension $n+\left\lfloor {{\textstyle{n \over 2}}} \right\rfloor $ can be directly embedded into an n-cube in the worst case, for n = 4 or n≥ 6.

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Index Terms:
Direct embedding, fault-tolerant, generalized Fibonacci cubes, hypercubes, faulty hypercube, second-order Fibonacci cube.
Citation:
Feng-Shr Jiang, Shi-Jinn Horng, Tzong-Wann Kao, "Embedding of Generalized Fibonacci Cubes in Hypercubes with Faulty Nodes," IEEE Transactions on Parallel and Distributed Systems, vol. 8, no. 7, pp. 727-737, July 1997, doi:10.1109/71.598347
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