Structure connectivity and substructure connectivity of bubble-sort star graph networks

The bubble-sort star graph, denoted BSn, is an interconnection network model for multiprocessor systems, which has attracted considerable interest since its first proposal in 1996 [5]. In this paper, we study the problem of structure/substructure connectivity in bubble-sort star networks. Two basic but important structures, namely path Pi and cycle Ci, are studied. Let T be a connected subgraph of graph G. The T-structure connectivity κ(G; T) of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to T. The T-substructure connectivity κs(G; T) of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to a connected subgraph of T. Both T-structure connectivity and T-substructure connectivity are a generalization of the classic notion of node-connectivity. We will prove that for P2k+1, a path on odd nodes (resp. P2k, a path on even nodes), κ(BSn;P2k+1)=κs(BSn;P2k+1)=⌈2n−3k+1⌉ for n ≥ 4 and k+1≤2n−3 (resp. κ(BSn;P2k)=κs(BSn;P2k)=⌈2n−3k⌉ for n ≥ 5 and k≤2n−3). For a cycle on 2k nodes C2k (there are only cycles on even nodes in BSn), κ(BSn;C2k)=κs(BSn;C2k)=⌈2n−3k⌉ for n ≥ 5 and 2≤k≤n−1.