Structure connectivity and substructure connectivity of hypercubes

The connectivity of a network - the minimum number of nodes whose removal will disconnect the network - is directly related to its reliability and fault tolerability, hence an important indicator of the network's robustness. In this paper, we extend the notion of connectivity by introducing two new kinds of connectivity, called structure connectivity and substructure connectivity, respectively. Let H be a certain particular connected subgraph of G. The H-structure connectivity of graph G, denoted κ(G;H), is the cardinality of a minimal set of subgraphs F={H1',H2',. . .,Hm'} in G, such that every Hi'∈F is isomorphic to H, and F's removal will disconnect G. The H-substructure connectivity of graph G, denoted κ^s(G;H), is the cardinality of a minimal set of subgraphs F={J₁, J₂, . . ., J_m}, such that every J_i∈F is a connected subgraph of H, and F's removal will disconnect G. In this paper, we will establish both κ(Q_n;H) and κ^s(Q_n;H) for the hypercube Q_n and H∈{K₁, K_1,1, K_1,2, K_1,3, C₄}.