On the Conditional $P_k$-connectivity of Hypercube-Based Architectures

Abstract

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 $P_k = \langle v_1,v_2,\cdots ,v_k\rangle$, 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 $k \ge 1$, 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 $\kappa ^2(G \vert P_k)$, is the cardinality of the minimum 2-restricted $P_k$-cut in G. On the other hand, a $P_k$-cut F in G is a 2-extra $P_k$-cut if the smallest component of $G-F$ contains at least three nodes. Then the 2-extra $P_k$-connectivity of G, denoted by $\kappa _2(G \vert P_k)$, 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 $\kappa ^2(Q_n \vert P_k)$ and $\kappa _2(Q_n \vert P_k)$ for $k=2,3,4$.