Relationship Between Extra Connectivity And Component Connectivity In Networks

Abstract

Abstract Connectivity is a classic measure for reliability of a multiprocessor system in the case of processor failures. Extra connectivity and component connectivity are two important indicators of the reliability of a multiprocessor system in presence of failing processors. The $h$-extra connectivity $\kappa _{h}(G)$ of a graph $G$ is the minimum number of nodes whose removal will disconnect $G$, and every remaining component has at least $h+1$ nodes. Moreover, the $h$-component connectivity $c\kappa _{h}(G)$ of $G$ is the minimum number of nodes whose deletion results in a graph with at least $h$ components. However, the extra connectivity and component connectivity of many well-known networks have been independently investigated. In this paper, we determine the relationship between extra connectivity and component connectivity of general networks. As applications, the extra connectivity and component connectivity are explored for some well-known networks, including complete cubic networks, hierarchical cubic networks, generalized exchanged hypercubes, dual-cube-like networks, Cayley graphs generated by transposition trees and hierarchical hypercubes as well.