The largest component of faulty star graphs

Abstract

The minimum size of the largest component of a network with faults is a useful parameter to make a full evaluation of this network and it is helpful for estimating some other parameters in graph theory. In this paper, we focus on discussing the minimum size of the largest component of star graph Sn which contains faulty edges. Specifically, we show that there exists at most one, two, three, four, five vertices beyond the largest component of Sn−F with F⊆E(Sn) and |F|≤2n−5,3n−8,4n−11,5n−15,6n−19 respectively. As applications, we study the g-extra edge connectivity of Sn for 1≤g≤5, and study the strong Menger edge connectivity of Sn−F with F∈E(Sn).