On the Conditional Pk-connectivity of Hypercube-Based Architectures

An interconnection network is a programmable system that serves to transport data packets efficiently in a systematic manner. From a worst-case perspective, the smaller diameter a network has, the shorter communication delay it can incur. A network's topology is abstractly modeled by a graph. A path of order k in a graph G is a sequence of k distinct nodes, denoted by P-k = < v(1), v(2), center dot center dot center dot, v(k)>, in which any two consecutive nodes are adjacent. The connectivity is a classic index to assess the level of network reliability and fault tolerance. For k >= 1, a set F of node subsets of G is a P-k-cut if G-F is disconnected, and each element of F happens to induce a P-k-subgraph in G. A P-k-cut F in G is a 2-restricted P-k-cut if the minimum degree of G - F is at least two. Then the 2-restricted P-k-connectivity of G, denoted by k(2)(G|P-k), is the cardinality of the minimum 2-restricted Pk-cut in G. On the other hand, a P-k-cut F in G is a 2-extra Pk-cut if the smallest component of G - F contains at least three nodes. Then the 2-extra Pk-connectivity of G, denoted by k(2)(G|P-k), is the cardinality of the minimum 2-extra P-k-cut in G. The hypercube Q(n) is one of the most popular network architectures for high-performance computing. This article is dedicated to figuring out the exact values of both k(2)(Q(n)|P-k) and k(2)(Q(n)|P-k) for k = 2, 3, 4.