Structure and Substructure Connectivity of Hypercube-Like Networks.

The connectivity of a graph G, K(G), is the minimum number of vertices whose removal disconnects G, and the value of K(G) can be determined using Menger's theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let H be isomorphic to any connected subgraph of G. The H -structure connectivity of G, denoted by K(G; H), is the cardinality of a minimum set F of connected subgraphs in G such that every element of F is isomorphic to H, and the removal of F disconnects G. The H-substructure connectivity of G, denoted by kappa' (G; H), is the cardinality of a minimum set X of connected subgraphs in G whose removal disconnects G and every element of X is isomorphic to a connected subgraph of H. The family of hypercube-like networks includes many welldefined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the m-star K-1,K-m structure, m >= 1, and the 4-cycle C-4 structure. Moreover, we consider the relationships between these parameters and other concepts.