The h-Extra r-Component Connectivity of Half Hypercube

To support the design of more reliable parallel and distributed systems, this work delves into the fault tolerance of interconnection networks and investigates the [Formula: see text]-extra [Formula: see text]-component connectivity of networks. The [Formula: see text]-extra [Formula: see text]-component connectivity of a connected network [Formula: see text], denoted by [Formula: see text], is defined as the minimum cardinality of a vertex set whose removal disconnects [Formula: see text] into at least [Formula: see text] components, each having at least [Formula: see text] vertices. This paper focuses on the [Formula: see text]-extra [Formula: see text]-component connectivity of [Formula: see text]-dimensional half hypercube networks [Formula: see text] and derives formulas for [Formula: see text]-extra [Formula: see text]-component connectivity and [Formula: see text]-extra [Formula: see text]-component connectivity of [Formula: see text]: [Formula: see text] ([Formula: see text]) and [Formula: see text] ([Formula: see text]). Furthermore, we prove that the upper bound [Formula: see text] ([Formula: see text]) is tight for [Formula: see text] and [Formula: see text], providing more accurate and valuable insights into the [Formula: see text]-extra [Formula: see text]-component connectivity of [Formula: see text].