Cluster connectivity of hypercube-based networks under the super fault-tolerance condition

The connectivity of a graph G, kappa(G), is the minimum cardinality over all vertex-cuts in G, and the value of kappa(G) can be determined using Menger's theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. A graph G is super connected if its minimum vertex-cut is always composed of a vertex's neighborhood. In this article we define the super H-connectivity kappa'(G vertical bar H) and the super H*-connectivity kappa'(G vertical bar H*) as new measures to evaluate the connectedness of G, for which H denotes a connected graph that represents the structure of the clustered faults, and H* denotes the union of the set of all connected subgraphs of H and the set of the trivial graph. Then we establish both kappa'(Q(n)vertical bar H) and kappa'(Q(n)&VERBARH*) for H is an element of{k(1)(,m) &VERBAR m = 1, 2, 3, 4} boolean OR {P-4, C-4 }, where Q(n) denotes the n-dimensional hypercube, K-1,K-m denotes the m-star structure for m >= 1, P-4 denotes a path of order four and C-4 is a cycle of order four. (C) 2021 Elsevier B.V. All rights reserved.