Hyper Star Fault Tolerance of Hierarchical Star Networks

Let [Formula: see text] and [Formula: see text] be two connected subgraphs of an interconnection network [Formula: see text]. If the removal of any minimum [Formula: see text]-structure-cut (respectively, minimum [Formula: see text]-substructure-cut) splits [Formula: see text] into exactly two components, one of which is isomorphic to [Formula: see text], then [Formula: see text] is said to be hyper[Formula: see text]-connected (respectively, hyper sub-[Formula: see text]-connected). The hierarchical star network [Formula: see text] is one of alternative interconnection networks for multiprocessor systems. Let [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we prove that (i) both the [Formula: see text]-structure connectivity and the sub-[Formula: see text]-structure connectivity of [Formula: see text] are [Formula: see text]; and (ii) both the [Formula: see text]-structure connectivity and the sub-[Formula: see text]-structure connectivity of [Formula: see text] are [Formula: see text]; and (iii) [Formula: see text] is hyper [Formula: see text]-connected and hyper sub-[Formula: see text]-connected, where [Formula: see text] is the complete graph with one vertex and [Formula: see text] is a star with [Formula: see text] vertices.