Largest connected component of a star graph with faulty vertices

Abstract

In order to make a full evaluation of an interconnection network, it is essential to estimate the minimum size of a largest connected component of this network provided the faulty vertices in the network may break its connectedness. Star graphs are recognized as promising candidates for interconnection networks. This article addresses the size of a largest connected component of a faulty star graph. We prove that, in an n-star graph (n≥3) with up to 2n−4 faulty vertices, all fault-free vertices but at most two form a connected component. Moreover, all fault-free vertices but exactly two form a connected component if and only if the set of all faulty vertices is equal to the neighbourhood of a pair of fault-free adjacent vertices. These results show that star graphs exhibit excellent fault-tolerant abilities in the sense that there exists a large functional network in a faulty star graph.

Keywords:

• interconnection network
• fault tolerance
• largest connected component
• star graph

CCS Category :

• B.4.3 (hardware, input/output and data communications, interconnections, topology)

Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions, which have greatly improved the quality of the article. This work was supported by the Natural Science Foundation of China (10771227), the program for New Century Excellent Talent of Educational Ministry of China (NCET-05-0759), Doctorate Foundation of Educational Ministry of China (20050611001), and Natural Science Foundation of Chongqing CSTC (2006BB2231 2005BB2191).