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 κ g ( 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 t g ( 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 B n , k is κ g ( B n , k ) = n + g ( k − 2 ) − 1 for 4 ≤ k ≤ n − 1 and 0 ≤ g ≤ n − k . Afterwards, we show that g -extra diagnosability of B n , 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 t g ( B n , k ) = n + g ( k − 1 ) − 1 , respectively.