The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] 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 [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined 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 [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.
Menger's theorem for countable graphs
