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.

Conditional diagnosability of alternating group networks

“…Yang et al [19] proved that the hypercube of size n with a set F of at most 2n − 3 failed processors has a connected component of size 2 n − |F | − 1, and that it has a connected component of size 2 n − |F | − 2 if |F | 3n − 6. Similar result on a star graph S n was also obtained by Yang et al [20]. Cheng and Lipman [21] then gave a more detailed result as follows:…”

Conditional Fault Diagnosability of Dual-Cubes

“…Yang et al [44][45][46] proved that the hypercube Q n with f faulty processors has a component of size at least 2 n − f − 1 if f ≤ 2n − 3, and of size at least 2 n − f − 2 if f ≤ 3n − 6. Yang et al [47] also obtained a similar result for the star graph S n . Cheng et al [11,14] gave a more detailed result for S n .…”

Fault diagnosability of arrangement graphs

“…Yang et al [33,34] also obtained that a similar result for the star graph S n . Cheng et al [7,10] gave a more detail result for S n .…”