Path and cycle fault tolerance of bubble-sort graph networks

The bubble-sort graph Bn is one of attractive underlying topologies for distributed systems. Let H be a certain connected subgraph of Bn. The H-structure connectivity of Bn, denoted κ(Bn;H), is the cardinality of a minimal set of subgraphs F={H1,H2,…,Hm} in Bn, such that every Hi∈F is isomorphic to H, and F's removal will disconnect Bn. The H-substructure connectivity of Bn, denoted κs(Bn;H), is the cardinality of a minimal set of subgraphs F={J1,J2,…,Jm}, such that every Ji∈F is isomorphic to a connected subgraph of H, and F's removal will disconnect Bn. The two kinds of connectivity are both generalizations of the classic connectivity. In this paper, we prove that κ(Bn;Pk)=κs(Bn;Pk)=⌈2(n−1)k+1⌉ for n≥6 and odd k≤2n−3, κ(Bn;Pk)=κs(Bn;Pk)=⌈2(n−1)k⌉ for n≥6 and even k≤2n−2, and κ(Bn;C2k)=κs(Bn;C2k)=⌈n−1k⌉ for 6≤2k≤n−1.