Component conditional fault tolerance of hierarchical folded cubic networks

Abstract

For the sake of achieving higher reliability, conditional connectivity has gradually become well-known. Component connectivity, as a kind of conditional connectivity, is an extension of the classical connectivity. Given a simple and undirected graph G and a nonnegative integer r, the (r+1)-component connectivity of graph G, say cκr+1(G), is the minimum number of a vertex cut whose removal causes the surviving graph to have at least r+1 components. Another basic parameter of reliability, diagnosability is often related to the number of components in the surviving graph. The (r+1)-component diagnosability, say ctr+1(G), is the maximum size of fault sets subject to at least r+1 components in the surviving graph, provided that all faulty vertices can be detected. A graph is t/k-diagnosable if all faulty vertices can be isolated into a set in which up to k vertices are fault-free, provided that the number of faulty vertices is at most t. As a two-level interconnection network, the hierarchical folded cubic network HFQ(n) possesses a great deal of nice properties. In this paper, we first show that the n-dimensional hierarchical folded cubic network is tightly super connected. Then, we explore that cκr+1(HFQ(n))=r(n+1)−(r2)+1(n≥4,1≤r≤n−4), and obtain that ctr+1(HFQ(n))=(r+1)n−(r2)+2(n≥4,1≤r≤n−4) under the PMC and MM* models. Last, we show that the hierarchical folded cubic network HFQ(n) is [(k+1)n−(k2)+2]/k-diagnosable, where n≥4 and 1≤k≤n−4.