Conditional connectivity of recursive interconnection networks respect to embedding restriction

Large-scale multiprocessor systems always take some recursive interconnection networks as underlying topologies. Let Gn be an n-dimensional recursive interconnection network. The m-embedding-restricted connectivity ζm(Gn) (resp. the m-embedding-restricted edge connectivity ηm(Gn)) of Gn is the cardinality of a minimum subset of nodes (resp. edges), if any, whose deletion disconnects Gn and each node of the remaining components lies in an undamaged m-dimensional sub-network Gm. In this paper, we present some relationships between the proposed indices and other conditional connectivity indices in general recursive interconnection networks. We give some bounds on these two indices in k-ary n-cubes and bubble-sort networks. In addition, we determine these two indices in k-ary n-cubes and bubble-sort networks in some cases.