Fault-tolerant embedding of linear arrays and rings in the star graph

Methods are presented to embed Hamiltonian paths (H-paths) and Hamiltonian cycles (H-cycles) in a star graph Sn in a faulty environment. The models considered include single-processor failure, double-process failure, and multiple-processor failures. All three models are applied to an H-path/cycle, which is formed by visiting all the (n!4!)S4s in an Sn in a particular order. An optimal embedding is obtained in the case of single-processor failure, wherein the length of the H-path/cycle is shown to be (n! − 2). The multiple-processor failure model is developed based on single and double processor failure models. In this case the length of the H-cycle that can be embedded is shown to be (n! − 2f), where f ≤ n − 2 is the number of faults. Another case of multiple-failure scenario is investigated by assuming that all faults are contained in a single Sm, m < n. The network in this case, is shown to reduce to a cluster-star graph. It is proven that it is always possible to formulate an H-cycle of length (n! − m!) in such a network.