h-extra r-component connectivity of k-ary n-cubes

The [Formula: see text]-ary [Formula: see text]-cube, denoted by [Formula: see text] is one of the important interconnection networks for parallel and distributed computing systems. In this paper, we examine its fault-tolerant properties in the context of [Formula: see text]-extra [Formula: see text]-component connectivity. Let [Formula: see text] be a connected graph. For integers [Formula: see text] and [Formula: see text] an [Formula: see text]-extra [Formula: see text]-component cut is a subset [Formula: see text] of [Formula: see text] such that [Formula: see text] results in a disconnected graph with at least [Formula: see text] components and each component contains more than [Formula: see text] vertices. The [Formula: see text]-extra [Formula: see text]-component connectivity of [Formula: see text] denoted by [Formula: see text] is the minimum size of an [Formula: see text]-extra [Formula: see text]-component cut of [Formula: see text] We investigate the [Formula: see text]-extra [Formula: see text]-component connectivity of [Formula: see text]-ary [Formula: see text]-cubes for [Formula: see text] and [Formula: see text] and determine that [Formula: see text] for [Formula: see text] and [Formula: see text] Furthermore, we derive an upper bound on the [Formula: see text]-extra [Formula: see text]-component connectivity of [Formula: see text]-ary [Formula: see text]-cubes for [Formula: see text] and [Formula: see text] as [Formula: see text]