Abstract
The failure of some key vertices in a network, due to either attacks or malfunctioning, may break the network into several disconnected components-the vertices within each component are connected, but there are no connections between components. When this happens, two measures should be of concern: (1) the number of components in the remaining network; and (2) the size of each component. When a given number of “break vertices”are removed from a network, we hope to have as few components as possible in the remaining network. On the other hand, it is certainly desirable that the components are as large as possible. Therefore both the number of components and the size of the maximal component after removing certain vertices can be a metric of a network's fault tolerability, an important dimension of its robustness. In this paper, we examine the split-star, denoted S 2 n , in terms of this metric. We prove that when an arbitrary subset of vertices D with |D| ≤ 6n-15 is removed from S 2 n , the remaining network will have at most 3 components, with the largest component having at least n! - |D| -3 vertices. With |D| ≤ 8n -21, the remaining network has at most 4 components, with the largest component's size at least n!-|D|-5. As a theoretical application, the r-component connectivity of S 2 n is estimated by using the obtained maximal component and the minimal neighbor-set of independent set of size r. Moreover, we present a bound of r-component edge connectivity on S 2 n by considering the minimal neighbor edge-set of appropriate substructures in S 2 n .
Highlights
Numerous processors in a multiprocessor system are connected, and communicate with each other, via various interconnection networks
In this paper, we prove that when an arbitrary subset of vertices D with |D| ≤ 6n − 15 is removed from Sn2, the remaining network will have at most 3 components, with the largest component having at least n! − |D| − 3 vertices
We prove that when an arbitrary subset of vertices D with |D| ≤ 6n − 15 is removed from Sn2, the remaining network has at most 3 components, and has a component with at least n! − |D| − 3 vertices
Summary
Numerous processors (or vertices) in a multiprocessor system are connected, and communicate with each other, via various interconnection networks. Yang et al [28] found that, when deleting a subset D ⊆ V (Qn) in n-hypercube Qn with |D| ≤ 3n − 6, there exists a component with size ≥ |V (Qn)| − |D| − 2 in the remaining graph. Yang et al [31] proved that, in an n-star graph network Sn with a subset D ⊆ V (Sn) of size |D| ≤ 2n − 4, the size of big component without D is at least |V (Sn)| − |D| − 2. L. Lin et al.: Component Reliability Evaluation on Split-Stars vertices, and a great number of alternative paths (the larger the number is, the higher the connectivity is). Yang et al [30] addressed the maximal component reliability of faulty n-hypercube Qn. Cheng and Lipták [11] gave the linearly many faults in star graph networks.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
