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Cheng, Note on applications of linearly many faults, The Comput. Chang, A note on super connectivity of the bouwer graph, J. Hsieh, Conditional diagnosability of augmented cubes under the PMC model, IEEE Trans. Horng, Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes, Appl. The parameters can provide more accurate measurements for the reliability and fault-tolerance of the corresponding systems. For 0 ≤ ℓ ≤ n − 1 and n ≥ 1, this paper determines ℓ -embedded edge-connectivity of n -dimensional augmented cube, η ℓ ( A Q n ), and shows exact values η ℓ ( A Q n ) = 2 ℓ ( 2 n − 2 ℓ ) − δ, where δ = 1 if ℓ = 0, and δ = 0 otherwise. It retains many favorable properties of the hypercube and possesses several embedded properties that the hypercube and other variations do not have. The augmented cube, denoted by A Q n, proposed by Choudum and Sunitha, is a momentous variant of the hypercube as an interconnection topology of parallel computing. Under this circumstance, in 2012, Yang and Wang first proposed the conception of ℓ -embedded edge-connectivity η ℓ ( G n ) of G n, which is defined as the minimum number of links whose removal results in several disconnected components, and each processor is contained in an ℓ -dimensional subnetwork G ℓ. Since the presence of faulty links or processors may disconnect the entire network, one hopes that every remaining processor lies in an undamaged lower-dimensional subnetwork. Let G n be a recursive n -dimensional network. In the contemporary world, to meet the increasing need to deal with massive data, the interconnection networks for large-scale parallel and distributed systems need to expand for stronger scalability requirements.
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