Reliability Analysis of (n, k)-Bubble-Sort Networks Based on Extra Conditional Fault

Given a graph G=(V(G),V(E)), a non-negative integer g and a set of faulty vertices F⊆V(G), the g-extra connectivity of G, denoted by κg(G), is the smallest cardinality of F, whose value of deletion, if exists, will disconnect G and give each remaining component at least g+1 vertices. The g-extra diagnosability of the graph G, denoted by tg(G), is the maximum cardinality of the set F of fault vertices that the graph can guarantee to identify under the condition that each fault-free component has more than g vertices. In this paper, we determine that g-extra connectivity of (n,k)-bubble-sort network Bn,k is κg(Bn,k)=n+g(k−2)−1 for 4≤k≤n−1 and 0≤g≤n−k. Afterwards, we show that g-extra diagnosability of Bn,k under the PMC model (4≤k≤n−1 and 0≤g≤n−k) and MM* model (4≤k≤n−1 and 0≤g≤min{n−k−1,k−2}) is tg(Bn,k)=n+g(k−1)−1, respectively.