Reliability evaluation of Modified bubble-sort graph networks based on structure fault pattern

Abstract

Let G be a graph and H be a fixed connected subgraph. Let H={H1,H2,⋯,Hk} be a set of connected subgraphs of G. The H-structure connectivity (resp. H-substructure connectivity) κ(G;H) (resp. κs(G;H)) is defined as the least cardinality of H such that Hi is isomorphic to H (resp. a connected subgraph of H) for any 1≤i≤k, and H’s deletion makes G disconnected or trivial. As an extension of the classic connectivity, the H-structure (resp. H-substructure) connectivity can better evaluate the fault-tolerance of an interconnection network. In this paper, we focus on the n-dimensional modified bubble-sort graph MBn. We determine κ(MBn;Pl) (resp. κs(MBn;Pl), where n≥5, 2≤l≤2n, and Pl is a path on l vertices; κ(MBn;C2l) (resp. κs(MBn;C2l)), where 6≤2l≤n, and C2l is a cycle on 2l vertices; κ(MBn;T2l) (resp. κs(MBn;T2l)), where 1≤l≤n−2 and T2l is an l-leaves 2-step star. In addition, we give the upper bound of κ(MBn;K1,l) (resp. κs(MBn;K1,l)), where n≥4 and 2≤l≤n and K1,l is a star on l+1 vertices, and prove that the upper bounds are sharp for l=2,3,4.